**5. Conclusions**

This work advances the research work presented in [33]. We present a novel self-tunable dual-frequency piezoelectric energy harvester with optimized performances. The dual frequency feature has been thoroughly investigated and we demonstrated that the resonator magnifies the amplitudes at two close fundamental frequencies, enabling simultaneous energy harvesting from both vibration frequencies. The system integrates permanent magnets, whose magnetostatic forces enable the frequency agility of the harvester. In order to simulate the bidirectional frequency tuning effect, we derived a reduced order model of the resonator and the harvester finite element model. The resonator's reduced order model has been experimentally validated and we demonstrated ±18% of bidirectional tuning. Furthermore, the reduced order modelling has been applied to the harvester.

A control algorithm has been developed to drive the tuning mechanism and thereby ensures the self-adaption of the system. The algorithm is based on maximum-voltage tracking and is able to automatically choose the adequate tuning actuator.

Furthermore, we presented experimental results of the piezoelectric harvester. The characterization demonstrated the dual-frequency feature of the harvester and showed that the harvester supplies sufficient voltage and power levels. Additionally, we investigated the efficiency of two commercially available power managemen<sup>t</sup> systems. Further experiments will be performed to evaluate the harvesting system's performance under realistic applications.

Finally, an optimized version of the harvester design has been proposed. This design presents two modes appearing at two close frequencies and an increased operative bandwidth. Further experiments are planned to verify the characteristics of the optimized version.

**Author Contributions:** S.B. and D.H. conceived the overall system design and built the finite element models. S.B. was additionally in charge of the experimental characterization, the implementation of the tuning mechanism, and the data processing. D.H. and T.B. were both supervising the project and reviewing the methodology. C.Y. contributed the reduced order model. Y.R. composed the system-level model and implemented the tuning control algorithm. F.L. was responsible for the automation of the tuning mechanism. A.S. and S.H. performed the

optimization of the harvester design and derived the power calculations. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** We gratefully acknowledge the German Academic Exchange Service "DAAD" for a partial financial support through a Ph.D. scholarship.

**Conflicts of Interest:** The authors declare no conflict of interest.
