Grey models have attracted considerable attention as a time series forecasting tool in recent years. Nevertheless, the linear characteristics of the differential equations on which traditional grey models rely frequently result in inadequate predictive accuracy and applicability when addressing intricate nonlinear systems. This study introduces a conformable fractional order unbiased kernel-regularized nonhomogeneous grey model (CFUKRNGM) based on statistical learning theory to address these limitations. The proposed model initially uses a conformable fractional-order accumulation operator to derive distribution information from historical data. A novel regularization problem is then formulated, thereby eliminating the bias term from the kernel-regularized nonhomogeneous grey model (KRNGM). The parameter estimation of the CFUKRNGM model requires solving a linear equation with a lower order than the KRNGM model, and is automatically calibrated through the Bayesian optimization algorithm. Experimental results show that the CFUKRNGM model achieves superior prediction accuracy and greater generalization performance compared to both the KRNGM and traditional grey models.

Reliable energy forecasting is essential for the planning and dispatch of power and fuel systems; however, energy series are often short and exhibit pronounced nonlinearity. To tackle this small sample setting, we propose a gray random vector functional link (GRVFL) framework and further derive a kernelized variant (KGRVFL). In GRVFL, an RVFL network is integrated into gray system modeling, and the parameters are learned via sparsity-regularized regression, enabling stable and reproducible training without backpropagation or evolutionary optimization. Hyperparameters are tuned using Bayesian optimization driven by a Top-k mean absolute percentage error (Top-k MAPE) criterion to improve robustness. To further promote compactness, we introduce a fractional ratio-type Fr-$\ell 1$ penalty and solve the resulting problem efficiently using a fractional coordinate descent (FCD) algorithm. The proposed methods are assessed on six real-world energy datasets using eight evaluation metrics. Comparisons with nine gray model baselines and six machine learning forecasters demonstrate that the sparse KGRVFL (SKGRVFL) achieves higher predictive accuracy and improved training stability under small sample conditions.