Optimal Control of Part Load Vortex Rope in Francis Turbines

The mitigation of the precessing vortex core developing in the draft tube of Francis turbines operating under part load conditions is crucial to increase the operation flexibility of these hydraulic machines to balance the massive power production of intermittent energy sources. A systematic approach following the optimal control theory is, therefore, presented to control this vortical flow structure. Modal analysis characterizes the part load vortex rope as a self-sustained instability associated with an unstable eigenmode. Based on this physical characteristic, an objective function targeting a zero value of the unstable eigenvalue growth rate is defined and subsequently minimized using an adjoint-based optimization algorithm. We determine an optimal force distribution that successfully quenches the part load vortex rope and sketches the design of a realistic control appendage.