Forecasting with an N-dimensional Langevin Equation and a Neural-Ordinary Differential Equation

TL;DR

The paper tackles forecasting day-ahead electricity prices under concurrent stationary and non-stationary dynamics by decomposing the price series S_t into a stationary component X_t described by an $N$-dimensional Langevin equation and a non-stationary residual Y_t learned with a neural-ODE. This two-stage approach pairs an interpretable stochastic differential equation for stationary behavior with a continuous-time neural model to capture non-stationary effects, validated on the Spanish day-ahead market. Results show that LE captures mean-reverting and hourly structure while the NODE corrects non-stationary deviations, improving short-term forecasts relative to naïve baselines, especially under external drift scenarios. The framework offers a robust, extensible methodology for systems exhibiting both stationary and non-stationary dynamics and has potential applications beyond electricity prices.

Abstract

Accurate prediction of electricity day-ahead prices is essential in competitive electricity markets. Although stationary electricity-price forecasting techniques have received considerable attention, research on non-stationary methods is comparatively scarce, despite the common prevalence of non-stationary features in electricity markets. Specifically, existing non-stationary techniques will often aim to address individual non-stationary features in isolation, leaving aside the exploration of concurrent multiple non-stationary effects. Our overarching objective here is the formulation of a framework to systematically model and forecast non-stationary electricity-price time series, encompassing the broader scope of non-stationary behavior. For this purpose we develop a data-driven model that combines an N-dimensional Langevin equation (LE) with a neural-ordinary differential equation (NODE). The LE captures fine-grained details of the electricity-price behavior in stationary regimes but is inadequate for non-stationary conditions. To overcome this inherent limitation, we adopt a NODE approach to learn, and at the same time predict, the difference between the actual electricity-price time series and the simulated price trajectories generated by the LE. By learning this difference, the NODE reconstructs the non-stationary components of the time series that the LE is not able to capture. We exemplify the effectiveness of our framework using the Spanish electricity day-ahead market as a prototypical case study. Our findings reveal that the NODE nicely complements the LE, providing a comprehensive strategy to tackle both stationary and non-stationary electricity-price behavior. The framework's dependability and robustness is demonstrated through different non-stationary scenarios by comparing it against a range of basic naive methods.

Figures (7)

• Figure 1: Schematic diagram of the framework proposed to forecast time series, $\mathbf{S}_t$. The framework consists of a two-stage process. First, a N-dimensional LE approximates the stationary component, $\mathbf{X}_t$, of $\mathbf{S}_t$. Second, the NODE extends $\mathbf{X}_t$ to account for the non-stationary behavior, $\mathbf{Y}_t$, of $\mathbf{S}_t$ that the LE cannot capture. The framework is applied to the electricity day-ahead market, in which case $\mathbf{S}_t$ represents the electricity day-ahead prices.
• Figure 2: (a) Spanish electricity day-ahead price time series and (b) its associated volatility for the year 2021. The volatility is computed as the standard deviation of the last 3 days.
• Figure 3: Simulation of $\mathbf{X}_t$ obtained from the LE in stationary conditions. Initial date: 3/3/2021. Solid line is the true electricity price, $\mathbf{S}_t$. Dashed line corresponds to the mean price over $10^{3}$ simulated paths of $\mathbf{X}_t$. Shaded areas delimit the following percentile ranges: [25, 75] (dark) and [10, 90] (light). Thus, the lightest area at the bottom of the plot encloses the percentiles [10, 25], while the lightest area at the top encloses the percentiles [75, 90].
• Figure 4: Assessment of $\mathbf{X}_t$ obtained from the LE in non-stationary conditions. Each row corresponds to a different scenario with initial dates: (a) 1/1/2021, (b) 8/5/2021, and (c) 6/9/2021. Left column: comparison between the true electricity price, $\mathbf{S}_t$, (solid line) and $10^3$ simulated price paths of $\mathbf{X}_t$. Right column: time evolution of $\mathbf{S}_t - \mathbf{X}_t$. Dashed lines correspond to the mean of $\mathbf{X}_t$ (left column) and mean of $\mathbf{S}_t - \mathbf{X}_t$ (right column) over all simulated paths. Shaded areas in both columns delimit the same percentile ranges as in Fig. \ref{['fig:stationary-gle']}.
• Figure 5: Hourly training dataset $\mathcal{D}^{p}_{n}, \text{with } p=9, \, n=10^3$, of scenario (a) in Fig. \ref{['fig:gle-validation-scenarios']} computed as the difference between the true electricity price, $\mathbf{S}_t$, and the simulated paths of $\mathbf{X}_t$ obtained from the LE. Dashed lines correspond to the mean of $\mathbf{S}_t - \mathbf{X}_t$. Shaded areas delimit the same percentile ranges as in Fig. \ref{['fig:stationary-gle']}. These trajectories are equivalent to the time evolution depicted in Fig. \ref{['fig:gle-validation-scenarios']} (a.ii) but rearranged into the 24 hourly dimensions considered in the LE. Solid lines correspond to the mean of $\mathbf{S}_t - (\mathbf{X}_t + \mathbf{Y}_t)$, i.e., the error between the true price and the out-of-sample prediction of the LE and the NODE.