**4. Stochastic simulation of a network**

The series were divided into time periods of 1 hour to guarantee the stationarity of the proc‐ ess. In Table 5 the estimated values of the expected values of the mean and the variance of the unit user and the exponent α for the scaling law of the variance are reported. The same exponents for the mean were always trivially equal to 1. In these results the first six hours of the day and the last one were excluded because, during the night hours consumptions are very small and therefore their statistics have a poor significance. It was observed that the mean scales linearly with the number of customers. Differently, the variance shows a slight non-linearity with the number of users. It must be underlined that the average dai‐ ly value of the exponent α is 1.1, showing that there is a very weak correlation between

**3.4. Real consumption data: scaling laws for the cross-covariance andcross-correlation**

law of the consumption signals between 11:00 and 12:00 am is graphically reported.

**Figure 4.** Scaling law for the cross-covariance between 11:00 and 12:00am.

Considering the consumption signals belonging to homogeneous users, equation 23 is valid and a quadratic scaling law for the cross-covariance should be expected. This behaviour was confirmed by the measured data for all the time intervals considered. In Figure 4 the scaling

The obtained cross-correlation coefficient between the single user signals was low, being al‐ ways less than 0.05, but increased noticeably when the number of aggregated users in‐ creased, as expected according to equation 22. For groups of 150 aggregated users the crosscorrelation coefficient reached the values shown in Table 6. These results enhance the importance of evaluating the cross-correlation degree at different levels of spatial aggrega‐ tion. Even if the cross-correlation between single-user demand signals is relatively low and less likely to significantly affect the performance of a network, it can largely increase with

the spatial aggregation of users, becoming not negligible at those larger scales.

the considered users.

122 Water Supply System Analysis - Selected Topics

**coefficient**

To illustrate the effect of the uncertainty of water demands on the performance of a net‐ work, particularly, the effect of the level of correlation between consumptions on the out‐ come pressure heads, a simple network simulation was performed. The water distribution network of Hanoi (Fujiwara and Khang, 1990) was considered for this matter (Figure 5).

**Figure 5.** Water distribution network of Hanoi [31].

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 heads for each node and for each correlation level.

Another aspect that emerges from the simulations is the increase of standard deviation of

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the pressure heads at each node of the network, which is illustrated in Figure 7.

**Figure 7.** Standard deviation of the pressure heads *vs* the cross-correlation coefficient between the demands.

standard deviation of the pressure heads increases more than 4 times.

signing and managing WDS.

**5. Conclusions**

The standard deviation of the pressure head verified at each node increases when the crosscorrelation between demands increases. The average standard deviation of the pressure heads along the network when the cross-correlation between demands is equal to 0.01 is 1.35m, while the average standard deviation of the pressure heads when the cross-correla‐ tion is 0.99, is 5.75m. This means that the cross-correlation increases from 0.01 to 0.99 the

The obtained results clearly show that the level of cross-correlation between demands signif‐ icantly affects the performance of a network and should, therefore, not be ignored when de‐

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‐

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 to pressure failures. Figure 6 illustrates this result.

**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 levels of cross-correlation that were considered.

Another aspect that emerges from the simulations is the increase of standard deviation of the pressure heads at each node of the network, which is illustrated in Figure 7.

**Figure 7.** Standard deviation of the pressure heads *vs* the cross-correlation coefficient between the demands.

The standard deviation of the pressure head verified at each node increases when the crosscorrelation between demands increases. The average standard deviation of the pressure heads along the network when the cross-correlation between demands is equal to 0.01 is 1.35m, while the average standard deviation of the pressure heads when the cross-correla‐ tion is 0.99, is 5.75m. This means that the cross-correlation increases from 0.01 to 0.99 the standard deviation of the pressure heads increases more than 4 times.

The obtained results clearly show that the level of cross-correlation between demands signif‐ icantly affects the performance of a network and should, therefore, not be ignored when de‐ signing and managing WDS.
