**3.2. Synthetically generated signals: scaling laws for the cross-covariance**

In this case pairs of aggregated consumption series, A and B, were obtained by randomly selecting among pairs of the previously generated groups of signals. Different values of the product*na* ⋅*nb*, where *na* is the number of signals in group A and *nb* the same number in group B, were considered, up to the maximum value*na* ⋅*nb* =500. Each aggregation process was characterized by the cross-correlation value between the single signals in the same group and the cross-correlation value between the single signals of the two native groups. The cross-covariance was computed for the different aggregation levels and the scaling law were derived for each process. The results are summarized in Table 3 with reference to equation 17, considering *Coeff* =*E ρab*,*Δ<sup>T</sup>* ⋅*E σa*,*Δ<sup>T</sup>* ⋅*E σb*,*Δ<sup>T</sup>* and α as the exponent of the product *na* ⋅*nb*.


Equation 17 then becomes *<sup>E</sup> cov AB*,*<sup>T</sup>* (*na*, *nb*) <sup>=</sup>*<sup>n</sup>* <sup>2</sup> <sup>⋅</sup>*<sup>E</sup> <sup>ρ</sup>ab*,∆*<sup>t</sup>* <sup>⋅</sup> *<sup>E</sup> <sup>σ</sup>*∆*<sup>t</sup>*

**Time**

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

cal quadratic scaling for cross-covariance.

is 1 second.

values of the cross-correlation coefficient are described in Table 4. They confirm the theoreti‐

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

> 6-7 0.468 1.994 1.2288 7-8 1.066 6.678 1.114 8-9 0.988 7.401 1.0435 9-10 0.891 6.205 1.0756 10-11 0.735 4.336 1.113 11-12 0.791 4.782 1.089 12-13 0.68 4.452 1.092 13-14 0.807 5.322 1.065 14-15 0.827 5.338 1.0688 15-16 0.704 3.857 1.1311 16-17 0.512 2.266 1.1739 17-18 0.634 3.112 1.1666 18-19 0.667 3.594 1.1412 19-20 0.707 5.445 1.0384 20-21 0.68 3.702 1.1253 21-22 0.635 3.412 1.099 22-23 0.397 1.958 1.0771 mean 0.717 4.344 1.1084 confidence limits 95% 0.082 0.759 0.024

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

**E [μ***1]* **E [σ***<sup>2</sup> 1]* **α***var (L/min) (L /min)2* **-**

<sup>2</sup> .The results for different

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

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Water Demand Uncertainty: The Scaling Laws Approach

**Table 3.** Theoretical and experimental values of the scaling law for the cross-covariance for different values of the cross-correlation coefficients in, ρ*a* and ρ*b*, and between A,B, ρ*ab*.

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


**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.

Equation 17 then becomes *<sup>E</sup> cov AB*,*<sup>T</sup>* (*na*, *nb*) <sup>=</sup>*<sup>n</sup>* <sup>2</sup> <sup>⋅</sup>*<sup>E</sup> <sup>ρ</sup>ab*,∆*<sup>t</sup>* <sup>⋅</sup> *<sup>E</sup> <sup>σ</sup>*∆*<sup>t</sup>* <sup>2</sup> .The results for different values of the cross-correlation coefficient are described in Table 4. They confirm the theoreti‐ cal quadratic scaling for cross-covariance.
