PriceFM: Probabilistic Electricity Price Forecasting

• PriceFM is a spatiotemporal model for joint multi-region, multi-quantile electricity price forecasting in the European day-ahead market.
• It integrates a Fusion Block, Graph Block, and Hierarchical Quantile Head to fuse historical prices with exogenous data and enforce spatial priors.
• Empirical results demonstrate superior performance over baselines with low quantile crossing and efficient computation on modest hardware.

PriceFM is a spatiotemporal foundation model for probabilistic electricity price forecasting in the European day-ahead market. It was introduced to forecast the next day’s 24 hourly prices jointly across multiple interconnected European bidding zones while also outputting multiple quantiles, thereby coupling spatial dependence and uncertainty quantification in a single model. In the arXiv literature, the explicit model name refers to this European electricity forecasting architecture built around a graph-based spatial prior, a region-wise fusion block, and a hierarchical quantile head (Yu et al., 6 Aug 2025).

1. Definition and modeling scope

PriceFM is defined as a model for multi-region, multi-timestep, and multi-quantile probabilistic electricity price forecasting. Its intended setting is the European day-ahead market, where price formation is shaped by physically interconnected transmission networks, congestion, and market coupling. The model therefore treats electricity forecasting not as a collection of isolated univariate time series, but as a joint regional prediction problem over a graph of bidding zones (Yu et al., 6 Aug 2025).

The target output is the next-day trajectory of 24 hourly prices for each output region. The quantile set is

$\mathcal{Q}=\{0.1,0.5,0.9\},$

so the model produces lower, median, and upper scenarios rather than a single point forecast. This probabilistic formulation is central because the paper frames renewable volatility, especially from wind and solar, and fluctuating load as structural sources of uncertainty rather than residual noise.

Although the title uses the expression “foundation model,” the architecture is domain-specific and compact. The paper does not describe large-scale self-supervised pretraining, transfer learning, zero-shot adaptation, explicit region embeddings, or a transformer-style temporal encoder-decoder. A plausible implication is that “foundation model” is used here in a domain-oriented sense: a unified model family for a market class, rather than a pretrained general-purpose sequence model.

2. Forecasting task and dataset design

The forecasting task is day-ahead prediction. Given information available before gate closure on day $\mathcal{D}$, PriceFM predicts the 24 hourly prices for delivery day $\mathcal{D}+1$. The horizon is therefore

$\mathcal{T}=24.$

The historical context length is denoted $L$, and the main experiments find $L=24$ to be the strongest setting among the tested choices $24$, $72$, and $168$ hours (Yu et al., 6 Aug 2025).

The released dataset covers 24 European countries and 38 regions from 2022-01-01 to 2025-01-01. The regions are transmission or bidding zones rather than only countries. This zonal construction matters because the model is intended to reflect actual cross-border power-flow interconnections. Representative examples include DE-LU, DK1 and DK2, multiple Norwegian and Swedish zones, and multiple Italian zones.

The input variables comprise day-ahead electricity prices and forward-looking exogenous forecasts available before gate closure: load, solar generation, and wind generation, with wind split into onshore and offshore when available. For an input region $r_{\mathrm{in}}$, the paper defines

$\mathcal{D}$0

and for an output region $\mathcal{D}$1 and quantile $\mathcal{D}$2,

$\mathcal{D}$3

The preprocessing pipeline resamples all sources to hourly resolution. Features with high missingness above $\mathcal{D}$4 are excluded; low-missingness series around $\mathcal{D}$5 are linearly interpolated. Scaling uses a RobustScaler fitted on the training set, explicitly to handle spikes and extreme price values. The split is chronological: the first two years, or 67%, are used for training, and the final year, or 33%, is split evenly between validation and test. The paper also notes that despite the temporal span, the effective number of training samples is only around 700 daily samples, since prediction is performed over daily 24-hour blocks.

3. Architecture: fusion block, graph block, and hierarchical quantile head

PriceFM is organized into three modules: a Fusion Block, a Graph Block, and a Hierarchical Quantile Head (Yu et al., 6 Aug 2025).

The Fusion Block handles the mismatch between historical prices of length $\mathcal{D}$6 and exogenous series of length $\mathcal{D}$7. The historical price sequence is first projected to length $\mathcal{D}$8, then both price and exogenous streams are passed through $\mathcal{D}$9 dense layers into a hidden dimension $\mathcal{D}+1$0. The fused regional representation is formed by residual addition followed by flattening: $\mathcal{D}+1$1 Stacking these vectors over all regions yields a spatial latent matrix

$\mathcal{D}+1$2

The Graph Block is the main architectural novelty. The European market is represented as a graph $\mathcal{D}+1$3, where nodes are bidding zones and edges are direct cross-border interconnections. For each output region, PriceFM computes shortest-hop graph distance from every input region. Those distances are transformed into decay weights through a parametric graph-decay function $\mathcal{D}+1$4, where $\mathcal{D}+1$5 is shortest-path distance, $\mathcal{D}+1$6 is the maximum reachable distance from the output region, and $\mathcal{D}+1$7 controls curvature. Positive $\mathcal{D}+1$8 yields stronger decay, $\mathcal{D}+1$9 yields linear decay, and negative $\mathcal{T}=24.$0 yields weaker decay. The output-region-specific representation is then a normalized decay-weighted average,

$\mathcal{T}=24.$1

where $\mathcal{T}=24.$2 collects the decay weights from all input regions to that output region.

This construction is not a learned graph attention mechanism. It is an explicit physical-topology prior. The paper’s interpretation is that geographically or electrically closer regions should matter more, while distant regions should be attenuated to avoid injecting noise. In that sense, the Graph Block acts as a fixed, output-specific spatial regularizer.

The Hierarchical Quantile Head addresses quantile crossing. The median trajectory is predicted first: $\mathcal{T}=24.$3 Upper and lower quantiles are then built as nonnegative residuals around the median: $\mathcal{T}=24.$4

$\mathcal{T}=24.$5

with default $\mathcal{T}=24.$6. This guarantees

$\mathcal{T}=24.$7

elementwise.

4. Objective, optimization, and computational profile

Training minimizes Average Quantile Loss (AQL) across samples, regions, horizons, and quantiles: $\mathcal{T}=24.$8 where $\mathcal{T}=24.$9 is the pinball loss. Point metrics such as RMSE, MAE, and $L$0 are evaluated using the median forecast $L$1 (Yu et al., 6 Aug 2025).

Optimization uses Adam with initial learning rate $L$2, exponential decay by factor $L$3 every 10 epochs, batch size $L$4, and 50 training epochs. The best checkpoint is selected by lowest validation loss. The paper attributes a slight regularization effect to the small batch size.

The architecture is parameter-efficient by the standards of the compared baselines. The appendix reports 3.38M parameters for PriceFM, compared with more than 10M parameters for the spatial baselines and roughly 5M to 10M for the temporal baselines. Runtime is also small: training takes 4–5 minutes on an NVIDIA A100 GPU with 80 GB, 12–13 minutes on an Intel Core i7-1265U CPU, and inference is under 10 seconds on both. This computational profile is consistent with the model’s design as a compact graph-informed forecaster rather than a very large pretrained sequence model.

5. Empirical performance, ablations, and interpretation

On the main 38-region benchmark, PriceFM outperforms the compared graph and temporal baselines on the principal aggregate metrics. Against spatial baselines such as GraphConv, GraphAttn, GraphSAGE, GraphDiff, and GraphARMA, the reported mean performance is AQL $L$5, RMSE $L$6, MAE $L$7, $L$8, and AQCR $L$9. It is slightly behind the best Q10 baseline by a small margin, but leads on Q50, Q90, AQL, RMSE, MAE, $L=24$0, and quantile crossing rate. The paper states that Diebold–Mariano tests confirm statistically significant superiority (Yu et al., 6 Aug 2025).

The contrast with generic time-series foundation-model baselines is stronger. FEDFormer, TimesNet, and TimeXer obtain RMSE values 47.55, 44.20, and 47.38, respectively, with corresponding $L=24$1 values 0.14, 0.30, and 0.14, all substantially below PriceFM’s 32.25 RMSE and 0.61 $L=24$2. The paper interprets this as evidence that explicit spatial inductive bias is indispensable for European electricity markets.

The ablation studies identify the Graph Block as the crucial component. Replacing the decay mask with a random mask worsens AQL from 6.89 to 7.83 and RMSE from 32.25 to 35.85. Removing decay entirely and using uniform averaging yields 7.71 AQL and 35.81 RMSE. This is the paper’s clearest empirical argument: the graph prior is not decorative but structurally responsible for a large portion of the gain.

Other ablations are more nuanced. Extending the context window from $L=24$3 to $L=24$4 or $L=24$5 degrades performance, indicating that one-day context is the most useful. Replacing residual fusion with concatenation worsens AQL to 7.31, while cross-attention yields 6.92, very close to residual addition’s 6.89. The implication is that the model’s gains do not depend on a heavy temporal attention stack.

For the graph-decay curvature parameter, the paper searches $L=24$6 separately for each output region, for 2,090 total trials. Most regions prefer $L=24$7, which the paper interprets as evidence that neighboring regions generally help. A weak negative correlation $L=24$8 is reported between optimal curvature and number of direct neighbors. Notably, 8 regions in Italy and Sweden prefer $L=24$9, effectively excluding neighbors; this indicates that spatial dependence is region-specific rather than uniform.

The hierarchical quantile head removes quantile crossing at essentially no accuracy cost. Using the standard independent head yields AQCR $24$0, whereas the hierarchical head yields AQCR $24$1, while AQL remains around 6.89–6.93 depending on the residual nonlinearity.

6. Limits of the name and broader research context

Within the arXiv record, the explicit published model name “PriceFM” refers to the European probabilistic electricity forecasting system just described (Yu et al., 6 Aug 2025). Several adjacent works, however, use the term more loosely to denote a broader idea of pricing foundation models, production pricing frameworks, or price-formation mechanisms rather than a specific named architecture.

In e-commerce pricing, “High-Frequency Pricing at Scale for E-Commerce” explicitly states that it is not about a model named PriceFM, but describes a production-scale forecast-then-optimize pricing stack for daily markdown decisions over more than 5 million articles. That paper is therefore related to a broad interpretation of PriceFM as a scalable pricing-system architecture, but not to the named electricity forecaster (Birr et al., 11 Jun 2026). Likewise, “LLP: LLM-based Product Pricing in E-commerce” is framed as an LLM-based valuation assistant for second-hand listings and is presented as a retrieval-grounded, foundation-model-style pricing system rather than a formal PriceFM model (Wang et al., 10 Oct 2025). In a different direction, “Online Electricity Pricing from Frequency Measurements” derives an explicit real-time electricity price correction from frequency deviations and refers to the resulting construction as a price-formation mechanism, again related conceptually but distinct from the forecasting model (Liu et al., 10 Apr 2026).

This broader usage creates a likely misconception: that PriceFM is already a universal pretrained pricing model spanning e-commerce, electricity operations, and market microstructure. The published evidence does not support that interpretation. For the named model, the paper’s own limitations are narrower and specific: the graph-decay function is hand-designed, the method depends on the current transmission topology, quantiles are limited to $24$2, and the effective number of training samples remains modest. A cautious inference is that PriceFM presently denotes a successful domain-specific probabilistic electricity forecaster, while “PriceFM” in a broader systems sense remains an emerging umbrella concept rather than a single unified model family.