Band Depth Clustering for Nonstationary Time Series and Wind Speed Behavior

Abstract

We explore the behavior of wind speed over time, using the Eastern Wind Dataset published by the National Renewable Energy Laboratory. This dataset gives wind speeds over three years at hundreds of potential wind farm sites. Wind speed analysis is necessary to the integration of wind energy into the power grid; short-term variability in wind speed affects decisions about usage of other power sources, so that the shape of the wind speed curve becomes as important as the overall level. To assess differences in intra-day time series, we propose a functional distance measure, the band distance, which extends the band depth of Lopez-Pintado and Romo (2009). This measure emphasizes the shape of time series or functional observations relative to other members of a dataset, and allows clustering of observations without reliance on pointwise Euclidean distance. To emphasize short-term variability, we examine the short-time Fourier transform of the nonstationary speed time series; we can also adjust for seasonal effects, and use these standardizations as input for the band distance. We show that these approaches to characterizing the data go beyond mean-dependent standard clustering methods, such as k-means, to provide more shape-influenced cluster representatives useful for power grid decisions.

Testing Separability of Functional Time Series

“…Bosq (2000), Horváth and Kokoszka (2012), Kokoszka and Reimherr (2017), but research on spatial fields or panels of time series of functions is relatively new, e.g. Kokoszka et al (2016), Gromenko et al (2016Gromenko et al ( , 2017, French et al (2016), Tupper et al (2017), Liu et al (2017), and Shang and Hyndman (2017). Testing separability of spatiotemporal functional data of the above form is investigated in Constantinou et al (2017), Aston et al (2017), and Bagchi and Dette (2017) under the assumption that the fields X n (⋅, ⋅), 1 ≤ n ≤ N, are independent.…”

Mixed data and classification of transit stops

“…Combining these types of data leads to its own methodological questions, and points to the potential usefulness of the band distance. Euclidean distance is sensitive to the relative scaling of each dimension, and does not handle skewed data well; as discussed in [18], L p -norm distances also become less informative in high dimensions. The band distance, by contrast, uses only the ordering of the observations on each dimension, and so could accommodate features that are scaled differently, skewed, or even not strictly numeric (such as ordinal estimates of non-quantitative factors like those discussed in [9]).…”

“…An alternative method is to use the band distance introduced in [18]. This measure examines the dissimilarity of the rider curves at each pair of stops relative to the overall body of data, from all stops.…”

A Review on Time Series Aggregation Methods for Energy System Models

“…This clustering algorithm was used by numerous authors [14,19,41,55,57,58,74,78,86,139,141,149,150,159,[163][164][165][166][167], either by using the partitioning around medoids (PAM) introduced by Kaufman et al [168] or by using an MILP formulation introduced by Vinod et al [169] and used in several studies [14,41,55,57,139,159,164]. The MILP can be formulated as follows:…”

“…However, it is unclear how it was guaranteed that different attributes were not compared to each other within the warping window which remains a field of future research. Furthermore, a band distance, which is also a pairwise rather than a pointwise distance measure, was used in a k-medoids algorithm by Tupper et al [167], leading to significantly less loss of load when deriving operational decisions for the next day using a stochastic optimization model.…”