Amidst the backdrop of significant global climate change, the frequent occurrence of extreme weather has severely impacted the secure and stable operation of renewable energy power systems. Existing online learning algorithms for photovoltaic (PV) power forecasting do not account for the influence of extreme weather data on model performance. Given the substantial differences in the distribution of PV data between extreme and normal weather conditions, direct training with extreme weather data during online learning—when such data is included in the new data stream—can lead to a significant decline in the model’s forecasting performance under normal weather conditions, a phenomenon known as "catastrophic forgetting." To address this issue, this paper proposes an online learning algorithm for PV power forecasting that considers extreme weather. First, the D-EMA algorithm is employed to identify extreme weather data. Subsequently, the Online-EWC algorithm is utilized for online learning with normal weather data to establish a baseline forecasting model. Finally, the BOWM-EWC algorithm is applied to fine-tune the model using extreme weather data, incorporating constraints at both the gradient and loss function levels. This approach aims to mitigate the adverse effects of extreme weather data on the model’s forecasting performance under normal conditions while integrating extreme weather data into the online learning framework. Case studies demonstrate that the proposed algorithm not only improves forecasting accuracy for extreme weather data but also effectively alleviates the detrimental impact of such data on model performance under normal weather conditions, thereby validating its effectiveness.