Applied solution techniques for bi-level problems in hydropower equivalent estimation

Hydropower is able to provide flexibility for the energy system by saving water in a reservoir and, in some stations, also the possibility of pumping. However, modeling hydroelectric systems can be challenging due to the complexity of cascade river systems and data limitations. Energy system models are therefore often using a simplified approach called a "hydropower equivalent." Calculating an equivalent involves bi-level optimization that minimizes the differences between a detailed reference model and the simplified system. Previous research has demonstrated the flexibility and effectiveness of applying Particle Swarm Optimization (PSO) to solve bi-level optimization problems. However, its performance has not been compared to other search algorithms, such as Bayesian optimization and random sampling. This study examines the robustness of PSO solutions by comparing them with those of other methods. The Optuna framework is used in this research because it offers implementations of both random sampling and Bayesian optimization. Additionally, it provides a functional ANOVA (Analysis of Variance) to calculate the importance of individual variables, which enhances understanding of the solutions and highlights the significance of each variable. Furthermore, a new combined approach of Bayesian optimization and PSO is proposed. While PSO tends to be more accurate for complex equivalent systems with many parameters, Optuna’s framework provides significant advantages in parameter analysis. A combined approach employing both methods showed promising results for optimizing more complex models. Overall, Optuna enhances the understanding of each parameter’s impact, offering valuable insights for designing equivalent hydropower systems.