# Multivariate Probabilistic Forecasting of Intraday Electricity Prices using Normalizing Flows

arXiv: https://arxiv.org/abs/2205.13826 · HTML: https://mlsanity.com/paper/2205.13826/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 76 references that are in the map.
- First posted on arXiv 27 May 2022.

## Builds on (most cited first, 18 of 18)

- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [A Survey on Explainable Artificial Intelligence (XAI): Towards Medical XAI](https://mlsanity.com/paper/1907.07374/) (2019)
- [Learning Likelihoods with Conditional Normalizing Flows](https://mlsanity.com/paper/1912.00042/) (2019)
- [Multivariate Probabilistic Time Series Forecasting via Conditioned Normalizing Flows](https://mlsanity.com/paper/2002.06103/) (2020)
- [Forecasting day-ahead electricity prices: A review of state-of-the-art algorithms, best practices and an open-access benchmark](https://mlsanity.com/paper/2008.08004/) (2020)
- [TensorFlow Distributions](https://mlsanity.com/paper/1711.10604/) (2017)
- [Day-ahead electricity price forecasting with high-dimensional structures: Univariate vs. multivariate modeling frameworks](https://mlsanity.com/paper/1805.06649/) (2018)
- [A deep generative model for probabilistic energy forecasting in power systems: normalizing flows](https://mlsanity.com/paper/2106.09370/) (2021)
- [Ensemble Forecasting for Intraday Electricity Prices: Simulating Trajectories](https://mlsanity.com/paper/2005.01365/) (2020)
- [Electricity Price Forecasting: The Dawn of Machine Learning](https://mlsanity.com/paper/2204.00883/) (2022)
- [Principal Component Density Estimation for Scenario Generation Using Normalizing Flows](https://mlsanity.com/paper/2104.10410/) (2021)
- [Revealing drivers and risks for power grid frequency stability with explainable AI](https://mlsanity.com/paper/2106.04341/) (2021)
- [Changes in electricity demand pattern in Europe due to COVID-19 shutdowns](https://mlsanity.com/paper/2004.14864/) (2020)
- [Predictability of Power Grid Frequency](https://mlsanity.com/paper/2004.09259/) (2020)
- [Normalizing Flow-based Day-Ahead Wind Power Scenario Generation for Profitable and Reliable Delivery Commitments by Wind Farm Operators](https://mlsanity.com/paper/2204.02242/) (2022)
- [Ancillary services in Great Britain during the COVID-19 lockdown: a glimpse of the carbon-free future](https://mlsanity.com/paper/2101.04387/) (2021)

## Built on by (most cited first, 14 of 14)

- [On-line conformalized neural networks ensembles for probabilistic forecasting of day-ahead electricity prices](https://mlsanity.com/paper/2404.02722/) (2024)
- [Bayesian Hierarchical Probabilistic Forecasting of Intraday Electricity Prices](https://mlsanity.com/paper/2403.05441/) (2024)
- [Sample-based Uncertainty Quantification with a Single Deterministic Neural Network](https://mlsanity.com/paper/2209.08418/) (2022)
- [Simulation-based Forecasting for Intraday Power Markets: Modelling Fundamental Drivers for Location, Shape and Scale of the Price Distribution](https://mlsanity.com/paper/2211.13002/) (2022)
- [Multivariate Scenario Generation of Day-Ahead Electricity Prices using Normalizing Flows](https://mlsanity.com/paper/2311.14033/) (2023)
- [OrderFusion: Encoding Orderbook for End-to-End Probabilistic Intraday Electricity Price Forecasting](https://mlsanity.com/paper/2502.06830/) (2025)
- [Probabilistic intraday electricity price forecasting using generative machine learning](https://mlsanity.com/paper/2506.00044/) (2025)
- [A general framework for multi-step ahead adaptive conformal heteroscedastic time series forecasting](https://mlsanity.com/paper/2207.14219/) (2022)
- [Least Squares and Marginal Log-Likelihood Model Predictive Control using Normalizing Flows](https://mlsanity.com/paper/2409.17632/) (2024)
- [Creating Probabilistic Forecasts from Arbitrary Deterministic Forecasts using Conditional Invertible Neural Networks](https://mlsanity.com/paper/2302.01800/) (2023)
- [AutoPQ: Automating Quantile estimation from Point forecasts in the context of sustainability](https://mlsanity.com/paper/2412.00419/) (2024)
- [The Evolution of Probabilistic Price Forecasting Techniques: A Review of the Day-Ahead, Intra-Day, and Balancing Markets](https://mlsanity.com/paper/2511.05523/) (2025)
- [Assessing the Performance-Efficiency Trade-off of Foundation Models in Probabilistic Electricity Price Forecasting](https://mlsanity.com/paper/2604.14739/) (2026)
- [Deep Learning for Cross-Border Electricity Price Forecasting: A Comparative Study](https://mlsanity.com/paper/2608.17091/) (2026)

## Abstract (arXiv)

Electricity is traded on various markets with different time horizons and regulations. Short-term intraday trading becomes increasingly important due to the higher penetration of renewables. In Germany, the intraday electricity price typically fluctuates around the day-ahead price of the European Power EXchange (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 probability 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. Furthermore, this work discusses the influence of different external impact factors based on literature insights and impact analysis using explainable artificial intelligence (XAI). The normalizing flow is compared to an informed selection of historical data and probabilistic forecasts using a Gaussian copula and a Gaussian regression model. Among the different models, the normalizing flow identifies the trends with the highest accuracy and has the narrowest prediction intervals. Both the XAI analysis and the empirical experiments highlight that the immediate history of the price difference realization and the increments of the day-ahead price have the most substantial impact on the price difference.