The large-scale integration of wind and solar power has caused the risk of supply-demand imbalances in power systems to rise sharply. Hydropower plays a crucial role in ensuring the stability of the power system and responding to fluctuations in renewable energy. However

when cascade hydropower stations support the random daily energy balance demand

they face multiple challenges

including the complex nonlinear spatial-temporal coupling of hydraulic and power generation processes

the deep coupling of long-term random energy balance demand

reservoir capacity limitations

and long-term comprehensive utilization goals. These challenges severely restrict the flexibility of hydropower. To address these issues

this paper proposes a hydropower mid-term random energy balance support scheduling model

with the objective of maximizing power generation. Based on the spatial-temporal coupling relationship between hydraulic and power generation processes

the model characterizes the random states of energy balance support

such as flow

output

and reservoir capacity

and establishes the randomness constraints for the energy balance support process. The model is then solved by applying Lagrangian duality theory to deterministically reconstruct the random constraints and convert the original model into a mixed-integer quadratic programming (MIQP) model for efficient solution. Compared to traditional methods

the proposed approach demonstrates significant advantages in terms of the adequacy of energy balance support capacity

plan execution safety

as well as solution accuracy and computational efficiency. Additionally

the proposed method is adaptable to various scheduling requirements and supply- demand imbalance risk scenarios. The results show that the proposed method provides effective theoretical support for hydropower in energy balance scheduling for grid stability.