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 have performed particularly well in 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 and can result in the generation of noisy data. To avoid the generation of time series data with unrealistic noise, we propose a dimensionality-reducing flow layer based on the linear principal component analysis (PCA) that sets up the normalizing flow in a lower-dimensional space. 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 to 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 data set, time series, or otherwise, which can be efficiently reduced using PCA.

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.

Electricity is traded on various markets with different time horizons and regulations. Short-term trading becomes increasingly important due to higher penetration of renewables. In Germany, the intraday electricity price typically fluctuates around the day-ahead price of the EPEX spot markets in a distinct hourly pattern. This work proposes a probabilistic modeling approach that models the intraday price difference to the day-ahead contracts. The model captures the emerging hourly pattern by considering the four 15 min intervals in each day-ahead price interval as a four-dimensional joint distribution. The resulting nontrivial, multivariate price difference distribution is learned using a normalizing flow, i.e., a deep generative model that combines conditional multivariate density estimation and probabilistic regression. The normalizing flow is compared to a selection of historical data, a Gaussian copula, and a Gaussian regression model. Among the different models, the normalizing flow identifies the trends most accurately and has the narrowest prediction intervals. Notably, the normalizing flow is the only approach that identifies rare price peaks. Finally, this work discusses the influence of different external impact factors and finds that, individually, most of these factors have negligible impact. Only the immediate history of the price difference realization and the combination of all input factors lead to notable improvements in the forecasts.

To model manifold data using normalizing flows, we propose to employ the isometric autoencoder to design nonlinear encodings with explicit inverses. The isometry allows us to separate manifold learning and density estimation and train both parts to high accuracy. Applied to the MNIST data set, the combined approach generates high-quality images.