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

2026-06-10

AI Summary

This study addresses the challenge of probabilistic wind power forecasting at a ten-minute resolution over monthly horizons by proposing a novel framework that integrates the stationarity assumption of the Weibull distribution with stochastic differential equations (SDEs). The approach estimates monthly Weibull parameters from SCADA wind speed data and constructs three forward wind speed SDE models—including OU-Weibull and Fokker-Planck formulations—coupled with a heteroscedastic Kalman filter and a bivariate VAR(1) state-space model, ultimately mapping wind speeds to power outputs via an XGBoost power curve. Innovatively embedding Weibull stationarity into long-term forecasting, this work also presents the first systematic comparison of the three SDE modeling paradigms. Experimental results show wind speed CRPS errors of 1.569–1.575 m/s, a sevenfold computational speedup for diffusion-prioritized models, power-domain Wasserstein distances of 26.1–27.6 kW (<1.4% of rated capacity), a monthly energy bias of approximately −7.3%, and exceedance probability errors under 2.2 percentage points.

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.

Problem

• wind power forecasting
• probabilistic forecasting
• stochastic differential equations
• Weibull distribution
• long-horizon prediction

Innovation

• Weibull-stationary SDE
• conditional probabilistic forecasting
• heteroskedastic Kalman filter
• Fokker-Planck modeling
• wind power forecasting