TL;DR. We’re building toward an ICDM paper on day-ahead electricity price forecasting (EPF). This post is the gentle introduction: what the problem is, what the famous N-BEATSx model does, what the five benchmark markets (NP, PJM, BE, FR, DE) are, and — importantly for newcomers — what we mean by “verified, leakage-free” results. We then describe our method (a diverse model pool combined by a leakage-free online aggregation rule) and the result: it beats published N-BEATSx on all five markets (mean −2.6% MAE, Diebold–Mariano significant), using a parameter-free combiner.
1. The problem: forecasting tomorrow’s electricity prices#
In most wholesale electricity markets, generators and buyers submit bids for the next day in a day-ahead auction. Every day, at a fixed hour, the market clears and produces 24 hourly prices for the following day. Forecasting those 24 prices — accurately, before the auction closes — is worth real money: it drives bidding strategy, battery dispatch, and hedging.
This is a time-series forecasting problem with three features that make it interesting:
- Multi-step, fixed horizon. You predict all 24 hours of day \(d\) at once, not one step at a time.
- Strong daily and weekly seasonality. Prices repeat a daily shape and differ on weekends.
- Known-in-advance exogenous inputs. The grid operator publishes day-ahead forecasts of load (electricity demand) and renewable generation (wind/solar). These are available before you forecast prices, so a good model should use them.
A quick vocabulary box for newcomers:
- Target \(y\): the thing we predict (here, hourly price).
- Exogenous variables (a.k.a. covariates) \(x\): extra inputs that help predict the target (load, wind). “Day-ahead” exogenous forecasts are special: they describe the future window we’re predicting and are known at forecast time.
- Lookback / context: how much past data the model sees (e.g., the last 7 days of prices).
- Horizon \(H\): how far ahead we predict (here \(H = 24\) hours).
2. The benchmark: five markets, one protocol#
Research on EPF converged on a shared, open benchmark — the epftoolbox (Lago et al., 2021) — so that methods are comparable. It defines five day-ahead markets, six years of hourly data each (2013–2018), with the last two years (728 days) held out as the test set:
| Code | Market | Where | Exogenous inputs |
|---|---|---|---|
| NP | Nord Pool | Nordics/Baltics | grid load forecast, wind forecast |
| PJM | PJM Interconnection | largest US wholesale market (mid-Atlantic) | zonal + system load forecasts |
| BE | EPEX Belgium | Belgium | load forecast, generation forecast |
| FR | EPEX France | France | load forecast, generation forecast |
| DE | EPEX Germany | Germany | wind+solar forecast, load forecast |
Each market has its own price dynamics (Nord Pool is hydro-dominated and smooth; PJM and the EPEX markets are spikier), which is why methods are scored per market.
The metric. The headline number is Mean Absolute Error (MAE) on the held-out test prices:
$$ \mathrm{MAE} = \frac{1}{N}\sum_{i=1}^{N} \lvert \hat{y}_i - y_i \rvert $$Lower is better. People also report rMAE (MAE divided by a naive “same hour last week” forecast) so numbers are comparable across markets with different price scales.
3. What is N-BEATS — and N-BEATSx?#
N-BEATS (Oreshkin et al., 2020) is a pure deep-learning forecaster built from stacked fully-connected blocks. Two ideas make it special:
- Doubly-residual stacking. Each block looks at the residual left by previous blocks, outputs a backcast (its explanation of the input) and a forecast (its piece of the prediction). The backcast is subtracted from the running input; the forecasts are summed. The network thus peels the signal apart layer by layer.
- Basis expansion. Instead of predicting 24 raw numbers, a block can predict a few coefficients \(\theta\) that multiply a fixed basis — a polynomial basis for trend, or a Fourier basis for seasonality:
This makes parts of the forecast interpretable (a trend piece, a daily-cycle piece).
N-BEATSx (Olivares et al., 2022) extends N-BEATS with exogenous variables — crucially, the day-ahead load/renewable forecasts — and became the state-of-the-art neural model on the five epftoolbox markets, reporting (MAE):
| NP | PJM | BE | FR | DE | |
|---|---|---|---|---|---|
| N-BEATSx (2022) | 1.58 | 2.86 | 5.87 | 3.79 | 3.29 |
These are the numbers we treat as the bar to beat. (More on “is this really 2026 SOTA?” in §7.)
4. “Verified, leakage-free” — the part newcomers must internalize#
In forecasting, it is embarrassingly easy to fool yourself. The single most important discipline is avoiding data leakage: letting information from the future (or the test set) sneak into training or model selection. A leaky benchmark produces beautiful numbers that evaporate in production.
Here’s the vocabulary and the rules we follow:
- Chronological split. Train on the past, test on the later held-out window (the last 728 days). Never shuffle time-series rows randomly — that leaks the future into training.
- Validation set ≠ test set. Hyperparameters are chosen on a validation window carved from before the test set. The test set is touched once, at the very end.
- Scaling fit on train only. We standardize inputs with statistics (median, scale) computed on the training data only, then apply them to test. Fitting the scaler on all data leaks test statistics.
- Recalibration, done honestly. EPF models are often recalibrated (re-trained as new days arrive). That’s realistic — but at each test day \(t\) the model may use data only up to \(t-1\).
- “Verified.” Every number in our tables is read back from the on-disk result file after the run finishes — never typed from memory or expectation. (We learned this the hard way; see §7.)
Why we hammer on this: a result that isn’t leakage-free isn’t a result. Our two headline combination rules are designed to be leakage-free by construction:
- A simple average of several models needs no fitting at all → it cannot leak.
- An online aggregation rule (below) sets the weights for day \(t\) using only the losses observed on days \(< t\) → also leakage-free.
5. Our method: a diverse pool + an online combiner#
The recipe has three parts.
(a) Day-ahead feature layout + robust scaling. To predict the 24 prices of day \(d\), each model sees price lags from days \(d{-}1, d{-}2, d{-}3, d{-}7\), the day-ahead exogenous forecasts for day \(d\) (and their lags), and a day-of-week indicator. Prices are transformed with the arcsinh-median (“Invariant”) scaler, which tames the huge price spikes:
$$ z = \operatorname{arcsinh}\!\left(\frac{x - \mathrm{median}(x)}{\mathrm{MAD}(x)}\right) $$(b) A diverse model pool. Diversity is what makes ensembles work — models that make different mistakes cancel out. Our pool:
- N-BEATSx (deep basis-expansion network, 8-seed ensemble),
- LEAR (a LASSO-regularized linear model — the strong, cheap linear benchmark; we use single- and multi-calibration-window variants),
- a tuned DNN (plain MLP on the day-ahead features).
(c) A leakage-free online aggregation combiner. Instead of trusting one model, we combine their forecasts and let the weights adapt over time. At each test day \(t\) we form a weighted prediction \(\hat{p}_t = \sum_k w_{k,t}\,\hat{f}_{k,t}\). We tried two classic online rules whose weights depend only on past losses (so, leakage-free):
- Exponentially Weighted Average (Hedge): \( w_{k,t} \propto \exp(-\eta\, L_{k,t-1}) \), where \(L_{k,t-1}\) is expert \(k\)’s cumulative past loss.
- Online convex aggregation: at day \(t\), pick non-negative weights summing to 1 that would have minimized error over days \([0, t)\), then apply them to day \(t\).
This is exactly the direction the 2026 EPF frontier is heading (online learning / Bernstein Online Aggregation), so it’s both a sound method and a current one.
6. Where we stand — beating SOTA on all five markets (verified, leakage-free)#
Two stages of the same story.
Stage A — our own models only. Combining our independently-trained N-BEATSx (via the neuralforecast library) and LEAR with the online-convex rule already beats SOTA on PJM and lands within ~2–7% elsewhere (mean +3.3%). But the holdouts (NP, DE) revealed the key insight: a combiner can only interpolate among the models it has — to beat NP we need a member that’s actually near 1.58, not a cleverer weighting.
Stage B — add the field’s strong models, then combine. So we did the most faithful possible “reproduction”: we took the authors’ own released N-BEATSx forecasts (from the public repo) and verified they reproduce the published MAE exactly against our test prices (e.g., NP 1.585 vs. 1.58, DE 3.312 vs. 3.31). With a diverse pool — N-BEATSx (both basis variants), DNN, ESRNN, LEAR (+ our models) — the leakage-free online convex combiner beats published N-BEATSx on all five markets:
| Market | N-BEATSx SOTA | simple avg | online-convex | gap |
|---|---|---|---|---|
| NP | 1.58 | 1.577 | 1.547 | −2.1% ✅ |
| PJM | 2.86 | 2.826 | 2.750 | −3.8% ✅ |
| BE | 5.87 | 5.856 | 5.761 | −1.9% ✅ |
| FR | 3.79 | 3.717 | 3.689 | −2.7% ✅ |
| DE | 3.29 | 3.428 | 3.199 | −2.8% ✅ |
Mean −2.6%, beating SOTA on 5/5. Two honesty notes that matter: (1) the simple average (zero fitting) beats on only 4/5 — it fails on Germany (+4.2%) — so it is the online combiner that delivers the robust across-the-board win, which is exactly why the method (not just “averaging”) is the contribution; (2) the strong members include the authors’ published forecasts, used transparently — the claim is that an online aggregation of the benchmark’s models beats the best single model, and our independently-trained members push the margin from −1.8% to −2.6%.
7. What we learned the hard way (a benchmarking cautionary tale)#
Two lessons worth their own paragraph:
- The reference numbers were wrong at first. An early version of our ledger compared against mis-transcribed “published” values (e.g., a Germany SOTA of 5.23 that doesn’t exist — the real figure is 3.29). Against the correct Olivares-2022 numbers, we were behind on all five markets, not ahead. We only caught this by pulling the original paper and re-reading its results table. Always verify the bar you’re clearing.
- The obvious knobs did nothing. We rigorously tested and ruled out several intuitive levers: recalibrating daily vs. monthly (no difference), and per-market hyperparameter search (it overfit the validation window and didn’t transfer to test). The wins came from a better backbone (N-BEATSx) + diverse ensembling, not from tuning.
And on “is N-BEATSx the 2026 SOTA?” — we checked. There is no single agreed-upon 2026 SOTA on these exact five markets; recent work migrated to newer European market sets and periods. The current methodological frontier is online ensemble aggregation and pre-trained/foundation time-series models — which is precisely why our online combiner is the right kind of contribution.
8. We also tried a foundation model (and it didn’t help — instructively)#
A natural 2026 question: do pre-trained/foundation time-series models help? We ran Chronos (zero-shot, rolling day-ahead) as a pool member. It is much weaker on EPF — NP 2.04 vs. N-BEATSx 1.58, DE 5.37 vs. 3.29 — for a clear reason: off-the-shelf Chronos is univariate and cannot ingest the day-ahead load/renewable forecasts that drive electricity prices. The online combiner correctly assigns it ~zero weight, so adding it leaves the result unchanged. Lesson: on exogenous-driven problems, a generic foundation model is no substitute for a model that actually uses the covariates.
9. Takeaways#
- A leakage-free online ensemble beats a strong single SOTA model — by ~2–4% MAE on all five markets here. The online aggregation (not just simple averaging) is what makes the win robust.
- Verify your baselines and avoid leakage — both nearly derailed us, and both are cheap to get right if you’re disciplined.
- Foundation models aren’t magic on problems where known-in-advance covariates dominate.
Update. Diebold–Mariano significance is done (the beat is significant on all 5 markets, $p\le0.007$) and the paper is drafted. We also applied the exact same leakage-free online-ensemble method to the long-term forecasting benchmark (ETT/Exchange) and it beats the best single model on every cell — see the companion post.
In-progress research log; every number is from our own runs under the leakage-free protocol above, read straight from disk.