Day-ahead optimal scheduling for cascade hydropower units to mitigate vibration zone crossing induced by intra-day flexibility response

The flexibility demand of the power system has substantially increased with the grid connection of large-scale variable renewable energy (VRE). Hydropower, as a crucial flexible power source, may cause unit outputs to frequently cross vibration zones (VZs) when responding to power fluctuations, posing significant threats to the safe operation of units. Therefore, in the context of building the new power system dominated by renewables, a significant challenge for hydropower scheduling is how to fully exploit the regulation flexibility without overestimating capability or triggering physical limits. This paper addresses this issue by developing a novel day-ahead scheduling approach that considers the limiting effect of VZs on cascade hydropower systems. Firstly, the Stable Regulation Capacity (SRC) is proposed to accurately evaluate the secure regulation capability of hydropower units when responding to VRE forecast deviations. Subsequently, the day-ahead scheduling model that considers crossing VZ constraints is developed to optimize the regulation capacity of units. By introducing the unit operating state vector and applying the infinity norm of the difference between vectors in adjacent periods, accurate identification of whether the output of units crosses VZs is achieved. Furthermore, the model is extended to incorporate dynamic head-dependent boundaries using linear interpolation. Finally, comprehensive simulation analyses are conducted utilizing cascade hydropower systems in southwest China. The research findings indicate that: (1) The day-ahead scheduling model with the proposed VZ constraints effectively prevents units from operating within or frequently crossing VZs. (2) The proposed scheduling approach can significantly enhance the actual secure regulation capability, achieving a 91.0% increase in usable SRC while maintaining a minimal 9.2% loss in conventional hydropower flexibility. (3) The robustness of the derived scheduling solutions is validated across representative days in both dry and flood seasons, with intra-day simulations demonstrating excellent operational stability and reliability during the dispatch process. (4) The proposed model effectively accommodates head-dependent irregular VZs to mitigate VZ crossings induced by intra-day flexibility response.