| 研究概要 |
A variability of energy consumption is the fraction of total variance over total mean consumption. Real data shows convergence of aggregated variability ith number of customers. We investigate the mathematical reasons of this phenomenon, as well as the subtleties of convergence rate. We show that the results for convergence on real data are consistent with the prediction of a simple sum of random correlated variables. Data granularity of 15-min consumption data : 96 data points per day were obtained to calculate the variability of aggregate consumption. Variability is unchanged when consumption distributions are normalized. That is, for a random variable (or vector) x with mean μ and variance σ^2, the random variable αx, where α>0, has mean αμ, variance α^2σ^2, and variability σ/μ. We see that the variability falls to zero as inverse square root with the number of the terms in the sum, v_<sum>(k)=(σ_<sum>(k))/(μ_<sum>(k))=1/(√<k>)σ/μ. on the real data, we don't see this behavior, which
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is maybe because of small data set size, where 1/(√<k>)=0.1. However, this convergence to nonzero value of variability can also be due to the fact that consumer are not independent. For example, for 10 copies of 1 consumer we get constant nonzero variability independent of cluster size. Customers living in the same town and the data taken in the same time of the year means a lot of correlations in the data. So this correlation should be found and characterized by covariance matrix before analyzing the cluster variability. c_<ij>=<(x_i-μ_j)(x_j-μ_i)>
We conclude that for each time t, will converge to lie within a narrow range. We also conclude that there is a way to calculate/estimate "convergence rate" of variability and the convergence differs by a multiplicative Factor. In conclusion, we see that variability is indeed a good parameter to encode a lot of properties of the real data, and relatively small dataset size are required to faithfully describe the bigger picture, as it converges to a constant value. 隠す
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