**3.3. Real consumption data: scaling laws for the mean and the variance**

**Cross-correlation coefficient Coeff α**

**ρ<sup>a</sup> ρ<sup>b</sup> ρab theoretical experimental theoretical experimental** 0.10 0.10 0.10 0.3754 0.3747 1.00 0.9998 0.20 0.10 0.10 0.3880 0.3907 1.00 0.9982 0.40 0.10 0.10 0.3676 0.3624 1.00 1.0020 0.40 0.20 0.10 0.3463 0.3440 1.00 1.0003 0.80 0.60 0.10 0.3939 0.3908 1.00 1.0009 0.50 0.50 0.10 0.3545 0.3541 1.00 0.9993 0.50 0.50 0.20 0.7449 0.7401 1.00 1.0007 0.50 0.50 0.30 1.0999 1.0923 1.00 1.0004 0.50 0.50 0.40 1.4643 1.4605 1.00 0.9999 0.50 0.50 0.50 1.8408 1.8504 1.00 0.9986

**Table 3.** Theoretical and experimental values of the scaling law for the cross-covariance for different values of the

and*σ<sup>a</sup>* =*σb*, that is, when all the consumptions are homogeneous, and with*na* =*nb*.

**Cross-correlation Coeff α**

**Table 4.** Theoretical and experimental values of the scaling law for the cross-covariance between homogeneous

groups of consumptions and different values of the cross-correlation coefficient.

Results confirm that *α* is always equal to one. However, in this case the scaling does not con‐ sider the number of aggregated users, but their product, and thus the law is not linear but quadratic. A similar approach was also applied in the particular case in which*ρ<sup>a</sup>* =*ρ<sup>b</sup>* =*ρab*,

> **coefficient theoretical experimental theoretical experimental** 0.10 0.40 0.4000 2.00 1.9998 0.20 0.80 0.7914 2.00 1.9997 0.30 1.20 1.2160 2.00 2.0008 0.40 1.60 1.6009 2.00 1.9985 0.50 2.00 2.0059 2.00 1.9992 0.60 2.40 2.3955 2.00 2.0003 0.70 2.80 2.7934 2.00 2.0000 0.80 3.20 3.2043 2.00 1.9999 0.90 3.60 3.6057 2.00 1.9999 0.99 3.96 3.9408 2.00 2.0003

cross-correlation coefficients in, ρ*a* and ρ*b*, and between A,B, ρ*ab*.

120 Water Supply System Analysis - Selected Topics

The parameters of the scaling laws were also derived for a set of real demand data. The in‐ door water uses demand series of 82 single-family homes, with a total of 177 inhabitants, in a building belonging to the IIACP (Italian Association of Council Houses) in the town of Latina were considered [29, 30]. The apartments are inhabited by single-income families, be‐ longing to the same low socioeconomic class.The daily demand series of four different days (4 consecutive Mondays) of the 82 users were considered [25]. For each user the different days of consumptions can be considered different realizations of the same stochastic proc‐ ess. In this way the number of customers was artificially extended to about 300, preserving at the same time the homogeneity of the sample. The temporal resolution of each time series is 1 second.


**Table 5.** Estimated parameters of the scaling laws for the experimental data set of Latina (see [25]).

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 the considered users.

**Time ρ***150users* **Time ρ***150users* 0-1 0.56 13-14 0.49 1-2 0.6 14-15 0.5 3-4 0.5 15-16 0.42 4-5 0.58 16-17 0.49 5-6 0.36 17-18 0.5 6-7 0.48 18-19 0.39 7-8 0.71 19-20 0.51 8-9 0.61 20-21 0.52 9-10 0.39 21-22 0.39 10-11 0.46 22-23 0.47 11-12 0.57 23-24 0.61 12-13 0.48 - -

Water Demand Uncertainty: The Scaling Laws Approach

http://dx.doi.org/10.5772/51542

123

**Table 6.** Estimated values of the mean cross-correlation coefficients between groups of 150 aggregated user from the

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).

experimental data set of Latina.

**4. Stochastic simulation of a network**

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