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

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 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.

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.


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