Recognition of the developing vortex rope in Francis turbine draft tube based on PSO-CS2

Condition monitoring and early warning of vortex rope are necessary to prevent unit resonance and other detrimental consequences for the Francis turbine. Previous studies focus on pressure pulsation to determine the vortex rope precession frequency directly. However, non-intrusive vibration monitoring is more suitable for industrial applications. The vibration signal is always accompanied by heavy noise. To eliminate these interferences, this study proposes the particle swarm optimization for second-order cyclostationarity (PSO-CS2) to identify the characteristic frequency bands of vortex rope. Firstly, the evolution process is divided into three states based on its physical behavior and spatial location. Secondly, it is pointed out that the vibration signal induced by vortex rope possesses cyclostationarity, allowing for the establishment of cyclostationary models for different vortex rope stages. Subsequently, the CS2 indicator is designed to evaluate the development state of vortex rope, and the PSO-CS2 algorithm is proposed. Finally, the effectiveness of the proposed PSO-CS2 is validated by simulation signals and experimental data from a scaled turbine model. The results demonstrate that the developing stage is dominated by second-order cyclostationary noise in a relatively high-frequency band of vortex rope and the shaft frequency and vortex rope precession frequency are presented in the final stage.