IEEE Power & Energy Magazine - May/June 2022 - 57

A partitioning method divides the observations into
a number, k, of nonoverlapping groups such that each
group contains at least one observation.
24 h of a day), peak and valley hours, and autodependencies
(how consumption in the previous hours is related to the
consumption at the current hour). These static and dynamic
features can be used to characterize, analyze, and forecast
the main consumers' consumption patterns and to group
those households with similar profiles.
Clustering Households Time Series
Clustering Methods for Time-Series Data
Clustering is one of the most important tools in unsupervised
machine learning for grouping unlabeled data sets.
It is a mathematical process to find groups of observations
(in our case, end-use load profiles) in such a way that the
observations within the same cluster have similar characteristics,
while the observations in different clusters vary from
each other. In that way, we expect households in a group to
have a similar consumption pattern but a very different pattern
from households in other groups. The similarity among
households involves measures of distance between the attributes
or characteristic values that describe the time series of
electrical consumption.
There are two main clustering tools: partitioning methods
and hierarchical methods. A partitioning method divides
the observations into a number, k, of nonoverlapping groups
such that each group contains at least one observation. A
hierarchical method starts with singleton clusters (clusters with
a single observation) and joins the clusters at each step in a hierarchical
way.
For time-series data, there are
three approaches to build the clusters:
features based, model based,
and dependence based. The features
can be statistics that summarize the
raw data or the raw data themselves
(in our case, 8,760 features that
correspond to the hourly consumption
of a household). The following
summary features could be used in
place of the original 8,760 raw ones:
✔ mean or median (total or by the
hour, day, month, and so on)
✔ variance or some representative
percentiles
✔ autocorrelation function and
partial autocorrelation function
(for a given set of lags).
may/june 2022
In a model-based approach, a time-series model is selected
for each series, and the difference (in model parameter) is
computed between each pair of models. A time-series model
is a mathematical formulation that relates the value at time
t with the values at the previous moments, usually involving
a set of parameters that must be estimated. Finally, dependence-based
clustering considers the cross dependency among
the series (cross correlations) and is based on computing a
dissimilarity measure.
In all of these approaches, a dissimilarity (distance)
function is needed. There are convenient dissimilarity functions,
for instance, Euclidean and weighted Euclidean,
Fréchet, or Mahalanobis distances using the extracted features.
Figure 2 illustrates the calculation of the Euclidean
and Fréchet distances. The Fréchet distance is the minimum-maximum
distance between the two sets of points
maintaining an increasing order, while the Euclidean distance
is the sum of the squared distances maintaining a
strict order that is point to point (the dashed lines in the
graph). The weighted Euclidean and Mahalanobis distances
give different weights to the point-to-point distances or differences,
respectively. They are especially useful for giving
more importance to some of the features or for taking
into account the variability and/or covariability between
the extracted features. Those extracted features can be, for
instance, percentile profiles, autocorrelations, periodogram
values, estimated model parameters, and so on, as
raw vectors.
Customer A
Features
Fréchet
Distance
Customer B
Features
figure 2. An illustration of Euclidean and Fréchet distance assuming five features.
The Fréchet distance (double arrow) is the min-max distance between the two sets
of points maintaining an increasing order. The Euclidean distance is the sum of the
squared distances (dashed lines).
ieee power & energy magazine
57
IEEE Power & Energy Magazine - May/June 2022

Table of Contents for the Digital Edition of IEEE Power & Energy Magazine - May/June 2022
