**3.1. Synthetically generated signals: scaling laws for the mean and the variance**

In order to confirm the analytical development reported in the previous paragraph, the scal‐ ing laws were derived for groups of synthetically and simultaneously generated consump‐ tion signals. At this aim the Multivariate Streamflow model [28], with a normal probability distribution, was used. Each group was assumed to contain 300 consumption signals with 3600 realizations each, distinguished by different values of the cross-correlation coefficient between them. The correctness of the procedure used to generate each demand series was tested by checking that the mean, the variance and the cross-correlation coefficient of the generated signals were equal to the input parameters of the model. Only little differences were observed (Table 1), which are explained due to the fact that the generated demand ser‐ ies are realizations of a stochastic process and, consequently, their moments necessarily dif‐ fer from the theoretical ones.

**Cross-correlation**

**coefficient** *E[mT]* **<sup>α</sup>**

**Table 2.** Experimental values of the scaling law for the second order moment for different values of the cross-

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

fectly correlated demands the scaling law is quadratic.

Results confirm the linear scaling for the first order moment and show that the variance in‐ creases with the spatial aggregation level according to an exponent that varies between 1 and 2. In theory, for spatially uncorrelated demands the scaling laws is linear and for per‐

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

correlation.

product *na* ⋅*nb*.

0 3.9808 1.0004 0.001 3.5205 1.0541 0.010 1.9864 1.3079 0.025 1.4498 1.4984 0.050 1.2702 1.6403 0.10 1.2940 1.7686 0.20 1.5621 1.8675 0.30 1.8879 1.9139 0.40 2.1713 1.9379 0.50 2.4485 1.9570 0.60 2.7685 1.9692 0.70 3.0803 1.9804 0.80 3.4051 1.9888 0.90 3.6567 1.9945 0.99 3.9960 1.9985

Water Demand Uncertainty: The Scaling Laws Approach

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

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Once the single consumption signals of each group were generated, they were aggregated randomly selecting one at a time, until a maximum of 100 aggregated consumption signals was reached. The first and second order moments, mean and variance, were calculated for each aggregation level. In order to obtain a result as general as possible, the same procedure has been repeated 50 times, aggregating each time different users [25]. The obtained results are summarized in Table 1 and 2, with reference to equation 5.


**Table 1.** Theoretical and experimental values of the scaling law for the first order moment for different values of the cross-correlation.


**3. Validation of the Analytical expressions**

are summarized in Table 1 and 2, with reference to equation 5.

fer from the theoretical ones.

118 Water Supply System Analysis - Selected Topics

**Cross-correlation coefficient**

cross-correlation.

**3.1. Synthetically generated signals: scaling laws for the mean and the variance**

In order to confirm the analytical development reported in the previous paragraph, the scal‐ ing laws were derived for groups of synthetically and simultaneously generated consump‐ tion signals. At this aim the Multivariate Streamflow model [28], with a normal probability distribution, was used. Each group was assumed to contain 300 consumption signals with 3600 realizations each, distinguished by different values of the cross-correlation coefficient between them. The correctness of the procedure used to generate each demand series was tested by checking that the mean, the variance and the cross-correlation coefficient of the generated signals were equal to the input parameters of the model. Only little differences were observed (Table 1), which are explained due to the fact that the generated demand ser‐ ies are realizations of a stochastic process and, consequently, their moments necessarily dif‐

Once the single consumption signals of each group were generated, they were aggregated randomly selecting one at a time, until a maximum of 100 aggregated consumption signals was reached. The first and second order moments, mean and variance, were calculated for each aggregation level. In order to obtain a result as general as possible, the same procedure has been repeated 50 times, aggregating each time different users [25]. The obtained results

> *E[mT]* **α Theoretical Experimental Theoretical Experimental**

0 0.70 0.7003 1.00 0.9996 0.001 0.70 0.7017 1.00 0.9993 0.010 0.70 0.6971 1.00 1.0001 0.025 0.70 0.7020 1.00 0.9989 0.050 0.70 0.7096 1.00 1.0004 0.10 0.70 0.7063 1.00 1.0009 0.20 0.70 0.7086 1.00 0.9994 0.30 0.70 0.7032 1.00 1.0008 0.40 0.70 0.6923 1.00 1.0003 0.50 0.70 0.6942 1.00 0.9985 0.60 0.70 0.6857 1.00 1.0002 0.70 0.70 0.6874 1.00 1.0011 0.80 0.70 0.6852 1.00 0.9998 0.90 0.70 0.6789 1.00 1.0009 0.99 0.70 0.7050 1.00 0.9997

**Table 1.** Theoretical and experimental values of the scaling law for the first order moment for different values of the

**Table 2.** Experimental values of the scaling law for the second order moment for different values of the crosscorrelation.

Results confirm the linear scaling for the first order moment and show that the variance in‐ creases with the spatial aggregation level according to an exponent that varies between 1 and 2. In theory, for spatially uncorrelated demands the scaling laws is linear and for per‐ fectly correlated demands the scaling law is quadratic.
