A Hybrid Long Short-Term Memory and Fractional Stochastic Differential Equation Model for Solar–Wind Energy Forecasting

Accurate forecasting of renewable energy generation is essential for reliable power system operation and sustainable energy management. However, the intermittent and highly variable nature of solar and wind resources, together with persistent stochastic fluctuations, limits the effectiveness of conventional deterministic forecasting approaches. This study develops a hybrid forecasting framework that combines a multivariate Long Short-Term Memory neural network with a fractional stochastic differential equation residual model for the joint prediction of solar and wind generation in India. The neural network component captures the dominant nonlinear temporal dynamics, whereas the fractional stochastic component models the long-memory dependence remaining in the forecasting residuals through fractional Brownian motion. Long-range persistence is quantified using Hurst exponent estimation, and a Fractional Renewable Complementarity Index is introduced to characterize the persistence-adjusted interaction between solar and wind resources. Experimental results demonstrate that the proposed framework consistently outperforms the standalone neural network model, reducing the root mean squared error from 4.83 to 3.92 for solar generation and from 54.85 to 46.21 for wind generation, corresponding to improvements of approximately 18.84% and 15.75%, respectively. Estimated Hurst exponents greater than 0.5 confirm the existence of persistent long-memory behaviour, providing empirical support for the fractional stochastic formulation. The proposed methodology offers an interpretable and uncertainty-aware framework for renewable energy forecasting, probabilistic decision support, and intelligent power system operation.