In this article the energy transfer between a flow and a fluttering piezoelectric plate is investigated. In particular, the benefits of the use of a Synchronized Switch Harvesting on Inductor (SSHI) circuit are studied. Both wind tunnel experiments and numerical simulations are conducted in order to analyse the influence of the switching process on the dynamics and the efficiency of the system. Numerical simulations consist of a weakly non-linear model of a plate in axial flow equipped with a single pair of piezoelectric patches, discretized using a Galerkin method where basis functions are the modes of the plate in vacuum. The discretized model is then integrated in time. The results presented in this paper show that a significant improvement of the harvested energy can be obtained using SSHI circuits compared to basic resistive circuits. It is also shown that for strongly coupled systems, the switching process inherent to he SSHI circuit has a significant impact on the dynamics of the flag, which tends to decrease the relative efficiency gain.