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| A Stackelberg Game-Theoretic Approach to Tri-Level Planning-Operation Co-Optimization in Hydro-Wind-PV Systems |
| Wang Yanyue1,2, Liu Chongru1,2 |
1. State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources North China Electric Power University Beijing 102206 China;
2. College of Electrical and Electronic Engineering North China Electric Power University Beijing 102206 China |
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Abstract The integrated planning and operation of hydro-wind-photovoltaic (PV) hybrid systems are crucial for mitigating renewable energy volatility and enhancing grid accommodation. Existing research often treats planning and operation in isolation, neglecting their tight coupling and the need to simultaneously optimize economic viability, power supply reliability, and energy yield. This study aims to address this gap by developing a holistic tri-level optimization framework that coordinates these often-conflicting objectives for multi-energy complementary systems.
A tri-level hierarchical model based on Stackelberg game theory is proposed to formulate the coupled planning-operation problem. The leader level optimizes the selection and sequencing of wind and PV projects for integration, based on the descending order of their equity financial internal rate of return (FIRR), ensuring economic feasibility relative to a benchmark rate. The middle level maximizes the firm output of the entire hybrid system during the dry season, given the project portfolio determined by the leader level. The follower level maximizes the total annual energy generation, subject to the constraint that the aggregate output never falls below the firm output determined by the middle level. This hierarchy ensures that decisions cascade from economic selection to reliability assurance and, finally, to energy maximization. The model incorporates comprehensive constraints, including hydrological balances, hydraulic linkages between cascade reservoirs, unit performance limits, minimum/maximum power outputs, reservoir storage and discharge limits, and transmission capacity. A dynamic iterative algorithm solves the tri-level optimization problem by simulating interactions among decision levels until convergence.
The model was validated using real-world data from Basin B, which features four cascade hydropower stations (totaling 2 664 MW) and numerous potential wind/PV projects (totaling 4 645 MW). The economic optimization selected 5 wind farms (270 MW) and 19 PV plants (2 855 MW) for integration, achieving a total system capacity of 5 789 MW while maintaining FIRR above the benchmark. Increasing wind-PV capacity led to a decline in equity FIRR, demonstrating the principle of diminishing marginal economic returns. The firm output increased with added wind-PV capacity, showing initial linear growth followed by nonlinear growth with diminishing gains as hydropower regulation limits were approached. Total annual energy generation increased with higher wind-PV capacity. However, wind-PV curtailment rates rose non-linearly beyond a capacity threshold due to limited hydropower flexibility and transmission constraints. The coordinated operation significantly smoothed the annual and daily output profiles compared to independent operation.
The proposed tri-level Stackelberg game model effectively integrates planning and operation for hydro- wind-PV systems. The results demonstrate that: (1) An economically-driven project selection sequence is essential for ensuring system viability. (2) The firm output is predominantly determined by hydropower regulation capacity and dry-season water availability. Beyond the hydropower compensation capability, the marginal gain in firm output diminishes. (3) A critical trade-off exists between increasing total energy yield and rising curtailment rates, highlighting the necessity for optimal capacity sizing.
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Received: 10 August 2025
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2026, Vol. 41 
