Solar energy is a vital renewable source in the ongoing energy transformation. However, the inherent uncertainty of solar-based energy sources prevents them from stable, large-scale integration into power grids. Photovoltaic energy production, reliably reflected by solar irradiance, is highly dependent on temporal changes and spatial-related, difficult-to-predict short-term meteorological fluctuations, such as cloud movements and coverage. Moreover, with various climate zones and dynamically evolving atmospheric conditions, it is impossible to cover all possible weather phenomena with conventional forecasting approaches based on static and centralized algorithms. To mitigate this issue, we proposed a continual adaptation strategy to adjust to local conditions and atmospheric phenomena. The developed incremental learning method leverages the stream of incoming sensor readings and prior model predictions to determine the relevance of the data based on the absolute magnitude of the prediction error. The computed significance level serves as an indicator to decide whether and which model parameters should be revised. The proposed approach has been verified against different latitudes, seasons, and weather conditions, demonstrating its ability to adapt rapidly. The experiments have shown an improvement in solar irradiance forecasting. Compared to the offline model, the mean absolute error has decreased by more than 60%, and the average mean absolute error has reduced by 52 \(W/m^2\) for a 30-min forecast horizon. Moreover, the successful deployment and efficient pipeline processing on resource-constrained single-board computers showcase the strong applicability of the developed irradiance forecasting strategy.