Principal component density estimation for scenario generation using normalizing flows

Abstract

Neural networks-based learning of the distribution of non-dispatchable renewable electricity generation from sources, such as photovoltaics (PV) and wind as well as load demands, has recently gained attention. Normalizing flow density models are particularly well suited for this task due to the training through direct log-likelihood maximization. However, research from the field of image generation has shown that standard normalizing flows can only learn smeared-out versions of manifold distributions. Previous works on normalizing flow-based scenario generation do not address this issue, and the smeared-out distributions result in the sampling of noisy time series. In this paper, we exploit the isometry of the principal component analysis (PCA), which sets up the normalizing flow in a lower-dimensional space while maintaining the direct and computationally efficient likelihood maximization. We train the resulting principal component flow (PCF) on data of PV and wind power generation as well as load demand in Germany in the years 2013–2015. The results of this investigation show that the PCF preserves critical features of the original distributions, such as the probability density and frequency behavior of the time series. The application of the PCF is, however, not limited to renewable power generation but rather extends to any dataset, time series, or otherwise, which can be efficiently reduced using PCA.

Normalizing Flow-based Day-Ahead Wind Power Scenario Generation for Profitable and Reliable Delivery Commitments by Wind Farm Operators

“…In Cramer et al (2022b), we showed that normalizing flows sample uncharacteristically noisy scenarios when applied to sample for the distributions of renewable electricity time series, due to their inherent lower-dimensional manifold structure. To address the issue, we proposed dimensionality reduction based on the principal component analysis (PCA).…”

“…The conditional input vectors y are not affected by the PCA. For more information on the effects of manifolds we refer to Brehmer and Cranmer (2020), Behrmann et al (2021), andCramer et al (2022b).…”

“…Due to numerically singular Jacobians and non-invertible transformations (Behrmann et al, 2021;Cramer et al, 2022b), full-space normalizing flows fail to accurately describe the distribution of daily wind time series trajectories residing on lower-dimensional manifolds (Cramer et al, 2022b). Therefore, we use PCA (Pearson, 1901) to reduce the data dimensionality following our recent contribution (Cramer et al, 2022b). We select the number of principal components based on the explained variance ratio, i.e., the amount of information maintained by the PCA (Pearson, 1901).…”

“…The major advantage of normalizing flows is their training via direct log-likelihood maximization, which leads to interpretable loss functions and stable convergence (Rossi, 2018). In prior works, normalizing flows performed well for scenario generation of residential loads Ge et al, 2020) as well as wind and photovoltaic electricity generation (Dumas et al, 2022;Cramer et al, 2022b).…”

“…Herein, we extend our previous work on normalizing flow-based scenario generation (Cramer et al, 2022b) to perform conditional scenario generation Dumas et al, 2021) of wind power generation with wind speed forecasts as conditional inputs, i.e., we use the wind speed forecast to generate day-ahead wind power generation scenarios that are specifically tailored to the given day. We then apply the generated scenarios in a stochastic day-ahead wind electricity producer bidding problem based on Garcia-Gonzalez et al (2008) and Conejo et al (2010).…”