Tuesday Wednesday Thursday Friday Saturday Sunday 1.0 1881 Demand (kWh) Monday 0.5 0.0 0 6 12 18 24 0 6 12 18 24 0 6 12 18 24 0 6 12 18 24 0 6 12 18 24 0 6 12 18 24 0 6 12 18 24 Time of Day Percentile 10 25 50 75 90 figure 3. The demand distribution of the least typical household out of the 500 smart meters included in the analysis. Typical and Anomalous Households to study the whole group of household demand distributions, we will first compute the differences in electricity consumption patterns between pairs of households. statistically speaking, we call these differences distances. note that the distance used here refers to the distance between two probability distributions rather than the physical distance between two houses. one way to measure the distance between two distributions is the Jensen-shannon divergence. We have 336 probability distributions per household, one for each half-hour period of the week, so we have 336 Jensen-shannon distance measures for each pair of households. We can measure the overall distance between the distributions from two households by summing these 336 Jensen-shannon distance measures. in this way, we can find the distance between each pair of households in the data set. from these pairwise distances, we can compute a measure of the typicality of a specific household by seeing how many similar houses are nearby according to the Jensen-shannon divergence. if there are many households with similar probability distributions, the typicality measure will be high. But if there are few similar households, the typicality measure will be low. this gives us a way to find anomalies in the data set, which are the smart meters corresponding to the least typical households. the most anomalous (i.e., least typical) household is shown in figure 3. this is may/june 2018 clearly a very strange demand distribution, with extremely low demand almost all of the time, reflected by almost overlapping percentiles. Visualization via Embedding the pairwise distances between households can also be used to create a plot of all households together. if we compute 99 percentiles for 48 half hours per day and seven days a week, each of the household distributions can be thought of as a vector in K -dimensional space where K = 99 # 48 # 7 = 33, 264. to easily visualize these, we need to project them onto a 2-D space. there are several ways of doing this, such as principal component analysis (pca) and multidimensional scaling. We have used a laplacian eigenmap method to keep the most similar points in K-dimensional space as close as possible in the 2-D space. figure 4 shows a 2-D embedding of the 500 households in this data set. the colors are taken from the measure of typicality, with the most typical 1% of points shown in red and the Laplacian Embedding of Smart-Meter Distributions 2 Component 2 distribution) because the data set contains a large number of zeros, making the distribution a mixture of discrete and continuous components. the high skewness of the data, and the nonnegative nature of demand, makes it problematic to use kernel density estimates. there are several advantages to working with percentiles rather than the data directly. problems with missing observations and the specific timing of household events (e.g., parties) are avoided, and attention is focused on the typical behavior of a household throughout the week. although only five percentiles are shown in figure 2, we actually compute percentiles for probabilities 1, 2, ..., 99%. 0 4 −2 −4 5 3 2 1 −2 HDRs 50 1 99 >99 −1 0 Component 1 1 2 figure 4. A 2-D representation of the data from all 500 households. The most typical points are shown in red, and the most unusual are shown in black. HDR: high density region. ieee power & energy magazine 21

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