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arxiv: 2606.12097 · v2 · pith:YJSDGNDYnew · submitted 2026-06-10 · 📊 stat.AP · physics.data-an

Weibull-Stationary Stochastic Differential Equations for Conditional Long-Horizon Wind Power Forecasting

Luca Di Persio , Mehrdad Ghadiri This is my paper

Pith reviewed 2026-06-27 07:51 UTC · model grok-4.3

classification 📊 stat.AP physics.data-an
keywords wind power forecastingstochastic differential equationsWeibull distributionprobabilistic forecastingKalman filterSCADA datapower curvecontinuous ranked probability score
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The pith

Three Weibull-stationary SDE models for wind speed yield equivalent probabilistic accuracy in power forecasts, so the fastest one can be used without loss of fidelity.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper builds a one-month-ahead probabilistic wind-power forecasting system at ten-minute resolution. Monthly Weibull shape and scale parameters are estimated from SCADA data with Godambe covariance correction, then forecasted via a heteroskedastic Kalman filter on a bivariate VAR(1) state-space model. Wind-speed trajectories are generated from three positive SDE formulations conditioned on the minimum-mean-square-error Weibull law, and the resulting ensembles are passed through a calibrated XGBoost power curve. On January 2021 data from a Senvion MM92 turbine, the three SDEs produce statistically indistinguishable accuracy while the diffusion-first variant runs roughly seven times faster than the Ornstein-Uhlenbeck-Weibull version.

Core claim

Conditional on the MMSE forecasted Weibull invariant law, the Ornstein-Uhlenbeck-Weibull transform, the Fokker-Planck drift-first diffusion, and the Fokker-Planck diffusion-first model generate wind-speed ensembles whose power-mapped distributions are statistically indistinguishable, with mean CRPS values between 1.569 and 1.575 m/s; the diffusion-first model is therefore preferred on computational grounds, reducing runtime by about a factor of seven, while Wasserstein distances in the power domain remain 26.1-27.6 kW (below 1.4% of rated capacity) and exceedance-probability errors stay below 1.6 percentage points over the 0-1500 kW range.

What carries the argument

The diffusion-first Fokker-Planck SDE for positive wind speeds conditioned on the forecasted Weibull invariant law, which matches the accuracy of the OU-Weibull and drift-first alternatives at lower computational cost.

If this is right

  • The diffusion-first model can be substituted for the OU-Weibull or drift-first formulations without degrading probabilistic accuracy.
  • Exceedance-probability errors remain below 1.6 percentage points over the 0-1500 kW range and rise to about 2.2 percentage points near rated power.
  • Monthly energy-yield bias stays around -7.3% for the examined month.
  • The resulting probability distributions supply decision-relevant inputs for reserve, storage, market, or fatigue problems rather than solving those problems outright.
  • Full marginalisation over the Kalman predictive law of the Weibull parameters is a direct next step left open by the work.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same conditioning strategy on a forecasted invariant law could be tested on solar irradiance or wave-height series to check whether computational savings appear in other renewable domains.
  • Direct insertion of the SDE ensembles into unit-commitment or storage-sizing optimisers would reveal whether the reported Wasserstein distances translate into measurable operational gains.
  • The Godambe covariance correction for parameter estimation from autocorrelated SCADA data may extend to other short-term renewable forecasting pipelines that rely on monthly distributional fits.
  • Repeating the comparison across multiple turbines and seasons would test whether the observed equivalence of the three SDEs is specific to the January 2021 Kelmarsh data or holds more generally.

Load-bearing premise

The monthly Weibull shape and scale parameters estimated from serially dependent SCADA data and forecasted by the heteroskedastic Kalman filter on a bivariate VAR(1) model are accurate enough that conditioning the SDE models on their MMSE values produces simulated power distributions that match observed data.

What would settle it

A statistically significant difference in mean CRPS larger than 0.01 m/s or a Wasserstein distance above 30 kW between the diffusion-first model and either of the other two models on an independent test month would falsify the claim of statistical indistinguishability.

Figures

Figures reproduced from arXiv: 2606.12097 by Luca Di Persio, Mehrdad Ghadiri.

Figure 1
Figure 1. Figure 1: Zoomed-in satellite view showing the Kelmarsh Wind Farm[25] [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Weibull Distribution across January months (2017-2020). [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: State-space VAR(1) one-step-ahead Kalman filter forecast of Weibull shape [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Prior and posterior standard deviations of [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Diagnostic display of the forecasted Weibull distribution for January 2021 at [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Pooled MLE estimate of the mean-reversion rate across January months (2017- [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Wind Speed Simulation Using Fokker–Planck Drift-Based SDE. [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Statistical Performance analysis of Using Fokker–Planck Drift-Based SDE. [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Wind Speed Simulation Using Fokker–Planck Diffusion-Based SDE. [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Statistical Performance analysis of Using Fokker–Planck Diffusion-Based SDE. [PITH_FULL_IMAGE:figures/full_fig_p024_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Wind Speed Simulation Using Ornstein–Uhlenbeck SDE. [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Statistical Performance analysis of Using Ornstein–Uhlenbeck SDE. [PITH_FULL_IMAGE:figures/full_fig_p025_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Probabilistic wind power forecast, January 2021, Fokker-Planck drift-based [PITH_FULL_IMAGE:figures/full_fig_p030_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Power coefficient Cp analysis, January 2021, Fokker-Planck drift-first SDE: (a) Cp time series vs. observed; (b) binned Cp vs. wind speed. E¯ sim = 492 MWh. The near-identical bias across all three formulations confirms that the shortfall is not caused by the choice of SDE formulation. Rather, all three models share the same forecasted invariant law ( ˆk49, λˆ 49) and the same decorrelation rate αˆ; hence… view at source ↗
Figure 15
Figure 15. Figure 15: Probabilistic wind power forecast, January 2021, Fokker-Planck diffusion-based [PITH_FULL_IMAGE:figures/full_fig_p032_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Power coefficient Cp analysis, January 2021, Fokker-Planck diffusion-first SDE: (a) Cp time series vs. observed; (b) binned Cp vs. wind speed. speed distribution assigns insufficient probability to the set {v : v ≥ Vrated}, as evidenced by the shared upper-tail compression across all three SDE for￾mulations; and second, the learned XGBoost approximation may introduce smoothing or bias in the plateau regio… view at source ↗
Figure 17
Figure 17. Figure 17: Probabilistic wind power forecast, January 2021, Ornstein-Uhlenbeck-Weibull [PITH_FULL_IMAGE:figures/full_fig_p034_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Power coefficient Cp analysis, January 2021, Ornstein–Uhlenbeck-Weibull SDE: (a) Cp time series vs. observed; (b) binned Cp vs. wind speed. blade radius R = 46 m, and V is the hub-height wind speed [24]. The power coefficient Cp quantifies the fraction of the kinetic energy flux through the rotor disk that is converted into electrical power, by the Betz limit, it is theoretically bounded above by C max p … view at source ↗
read the original abstract

We present a one-month-ahead conditional probabilistic framework for wind-power forecasting at ten-minute resolution. Monthly Weibull shape and scale parameters are estimated from serially dependent SCADA wind-speed data, corrected through a Godambe covariance, and forecast by a heteroskedastic Kalman filter on a bivariate VAR(1) state-space model. Conditional on the MMSE forecasted Weibull invariant law, we construct and compare three positive wind-speed SDE models: an Ornstein-Uhlenbeck-Weibull transform, a Fokker-Planck drift-first diffusion, and a Fokker-Planck diffusion-first model. The simulated wind-speed ensembles are mapped to power through a calibrated XGBoost power curve. Applied to January 2021 data from a Senvion MM92 turbine at Kelmarsh Wind Farm, the three SDE formulations are statistically indistinguishable in probabilistic accuracy, with mean CRPS values between 1.569 and 1.575 m/s. The diffusion-first model is therefore preferred on computational grounds, reducing runtime by about a factor of seven relative to the OU-Weibull model. In the power domain, the Wasserstein distance between simulated and observed distributions is 26.1-27.6 kW, below $1.4\%$ of rated capacity, while the monthly energy-yield bias is about $-7.3\%$ for the examined month. Exceedance-probability errors remain below 1.6 percentage points over the 0-1500 kW range and about 2.2 percentage points near rated power. These quantities provide decision-relevant probabilistic inputs for downstream operational problems, rather than completed reserve, storage, market, or fatigue-optimization decisions. Full marginalisation over the Kalman predictive law of the Weibull parameters is left as a natural extension.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript presents a one-month-ahead conditional probabilistic framework for wind-power forecasting at ten-minute resolution. Monthly Weibull shape and scale parameters are estimated from serially dependent SCADA data with Godambe covariance correction, then forecasted via a heteroskedastic Kalman filter on a bivariate VAR(1) state-space model. Conditional on the MMSE forecasted Weibull invariant law, three positive wind-speed SDE models are constructed and compared: an Ornstein-Uhlenbeck-Weibull transform, a Fokker-Planck drift-first diffusion, and a Fokker-Planck diffusion-first model. Simulated wind-speed ensembles are mapped to power through a calibrated XGBoost power curve. On January 2021 data from a Senvion MM92 turbine at Kelmarsh Wind Farm, the three SDE formulations yield statistically indistinguishable probabilistic accuracy (mean CRPS 1.569–1.575 m/s). The diffusion-first model is preferred for reducing runtime by a factor of approximately seven. In the power domain, Wasserstein distances are 26.1–27.6 kW (<1.4% of rated capacity), monthly energy bias is about −7.3%, and exceedance-probability errors remain below 2.2 percentage points.

Significance. If the reported metrics hold after addressing parameter uncertainty, the framework provides a computationally tractable route to generating decision-relevant probabilistic wind-power ensembles by combining stochastic differential equations, state-space forecasting, and machine-learned power curves. The explicit identification of the diffusion-first model as runtime-efficient and the deferral of full marginalization constitute clear, actionable contributions to applied stochastic modeling in renewables.

major comments (2)
  1. [Abstract] Abstract: The central performance claims (CRPS range 1.569–1.575 m/s, Wasserstein distances 26.1–27.6 kW, energy bias −7.3%) are obtained by simulating the three SDEs conditioned solely on the MMSE point forecast of the Weibull shape/scale pair. Because the heteroskedastic Kalman filter on the VAR(1) state-space model produces a non-degenerate predictive covariance (after Godambe correction for serial dependence), the reported ensembles omit integration over the full predictive law of the Weibull parameters. This omission is load-bearing for the claimed statistical indistinguishability and sub-1.4% capacity error; the abstract correctly flags full marginalization as future work, but the current metrics therefore reflect a narrower conditional law than the one-step-ahead predictive distribution.
  2. [Results] Results section (implied by abstract claims): The statement that the three SDE formulations are “statistically indistinguishable” rests on CRPS values differing by at most 0.006 m/s. No standard errors, bootstrap intervals, or formal pairwise tests on the CRPS differences are referenced, making it impossible to assess whether the observed similarity exceeds sampling variability of the 10-minute ensemble evaluation.
minor comments (2)
  1. [Abstract] Abstract: The phrases “about a factor of seven” and “about −7.3%” would be more precise if replaced by exact reported values or accompanied by uncertainty measures.
  2. [Methods] Notation: The distinction between the three Fokker-Planck formulations (drift-first vs. diffusion-first) would benefit from an explicit equation reference or short table summarizing the drift and diffusion coefficients for each model.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful and constructive review. We address each major comment below, indicating whether revisions will be incorporated.

read point-by-point responses

  1. Referee: [Abstract] The central performance claims are obtained by simulating the three SDEs conditioned solely on the MMSE point forecast of the Weibull shape/scale pair. The reported ensembles omit integration over the full predictive law of the Weibull parameters. This omission is load-bearing for the claimed statistical indistinguishability; the abstract correctly flags full marginalization as future work, but the current metrics reflect a narrower conditional law than the one-step-ahead predictive distribution.

    Authors: We thank the referee for this observation. The manuscript explicitly presents a conditional framework in which ensembles are generated given the MMSE forecast of the monthly Weibull parameters; this conditioning is stated in the abstract, methods, and results. Full marginalization over the Kalman predictive covariance is correctly identified as future work owing to its computational cost. The reported CRPS, Wasserstein, and bias metrics are therefore accurate for the conditional model as implemented, which remains a tractable and decision-relevant contribution. No change to the scope or claims is required. revision: no

  2. Referee: [Results] The statement that the three SDE formulations are “statistically indistinguishable” rests on CRPS values differing by at most 0.006 m/s. No standard errors, bootstrap intervals, or formal pairwise tests on the CRPS differences are referenced, making it impossible to assess whether the observed similarity exceeds sampling variability of the 10-minute ensemble evaluation.

    Authors: We agree that uncertainty quantification on the CRPS differences would strengthen the indistinguishability claim. In the revised manuscript we will add bootstrap standard errors (resampling the 10-minute evaluation periods) for the reported CRPS values of each SDE model and will note whether the observed 0.006 m/s spread lies within these intervals. revision: yes

Circularity Check

0 steps flagged

No significant circularity; forecasting setup is self-contained against out-of-sample data.

full rationale

The derivation estimates monthly Weibull parameters from historical SCADA data (with Godambe correction), fits a VAR(1) Kalman filter to the resulting time series of past monthly estimates, produces an MMSE forecast of the next month's parameters, conditions the three SDE models on that forecasted invariant law, simulates ensembles, and maps them via a separately calibrated XGBoost power curve. Performance (CRPS, Wasserstein) is then measured on the held-out January 2021 observations. Because the target month's Weibull law is forecasted rather than estimated from the evaluation data itself, and because the paper explicitly flags full marginalization over the Kalman predictive distribution as future work, none of the central objects reduce by construction to the evaluation quantities. The indistinguishability of the SDE variants follows directly from their shared stationary law rather than from any self-referential fitting step. This is a standard out-of-sample forecasting pipeline with an acknowledged approximation; no load-bearing claim collapses to a fit on the reported metrics.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The framework rests on several data-fitted quantities and standard stochastic-process assumptions rather than parameter-free derivations. The central claim depends on the accuracy of these fitted objects and on the domain assumption that wind speed admits a Weibull invariant law within each month.

free parameters (3)
  • monthly Weibull shape and scale parameters
    Estimated directly from SCADA wind-speed data for each month and used as the target for Kalman forecasting.
  • VAR(1) state-space parameters
    Coefficients of the bivariate VAR(1) model inside the heteroskedastic Kalman filter, fitted to the time series of Weibull parameters.
  • XGBoost power-curve parameters
    Hyperparameters and tree structure of the XGBoost model calibrated to map simulated wind speeds to power output.
axioms (2)
  • domain assumption Wind-speed process admits a stationary Weibull distribution within each calendar month
    Invoked to define the target invariant law for the SDE models and the Kalman forecast target.
  • standard math Fokker-Planck equation governs the evolution of the probability density for the chosen SDE drift and diffusion terms
    Used to construct the drift-first and diffusion-first SDE formulations.

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