Vibration characteristics analysis and Cross-Vibration-Zone operation strategy optimization for Francis turbine unit in deep peak regulation process

Abstract Francis turbine units (FTUs) frequently traverse vibration zones during deep peak regulation (DPR), posing significant challenges for safe and stable operation. This study proposes an integrated framework for vibration safety control that systematically links data cleansing, dynamic modeling, quantitative evaluation, and strategy optimization. Grid-based density stratified 3D-DBSCAN is developed to eliminate outliers and ensure data reliability, followed by a coupled modeling approach that integrates hydropower transients with vibration dynamics via the method of characteristics. A composite entropy index is then constructed to quantitatively evaluate vibration safety, incorporating vibration amplitude, duration, and impact on regulation performance. Finally, a Cauchy Mutation-enhanced Rime Optimization Algorithm leverages the coupled model to derive optimal operation strategies under safety constraints. Validation against field data during DPR processes shows that simulated vibration values achieved a symmetric mean absolute percentage error of 6.97% relative to measurements, while the optimized strategy reduced vibration wave entropy (VWE) by 11.6% and improved regulation efficiency. This framework provides a practical, systematic solution for vibration-aware DPR of FTUs and supports safe operation under high renewable energy penetration.