Mathematical Modeling for the Accurate Prediction and Classification of Renewable Energy Outputs via Ensemble Learning

Renewable energy systems, such as solar, wind, and hydroelectric systems, face inherent stochasticity due to environmental fluctuations, demanding advanced mathematical models for accurate forecasting and classification to ensure grid reliability and sustainability. This study introduces a novel ensemble learning paradigm that integrates supervised frameworks with rigorous mathematical optimization, pioneering the use of polynomial feature expansions and principal component analysis (PCA) to augment a compact 96-observation multivariate dataset encompassing irradiance, wind speed, and hydrological flux. The hyperparameters for the random forest (RF), support vector regression (SVR), and gradient boosting machine (GBM) methods are fine-tuned via grid search and k-fold cross-validation, yielding ground breaking performance: the GBM attains 94.2% accuracy in multiclass classification of energy types and availability, outstripping RF (82.1%) and SVR (71.8%) while achieving an RMSE of 1.86 MW in output prediction, an 85% error slash over SVR and a 10% edge over RF. Sensitivity analyses and partial dependence plots reveal intricate nonlinear interactions, validating GBM sequential gradient descent for heteroscedastic error correction. This innovation not only advances applied mathematics in energy modeling but also delivers practical tools for real-time grid stabilization, optimizing resource allocation, and informing policy amid global de-carbonization goals. With hydroelectric dominance (130 MW average output) and environmental drivers such as wind speed optima (6–7 m/s), our framework empowers stakeholders to mitigate intermittency risks, accelerate renewable integration, and foster equitable, low-carbon energy transitions