## 7 Conclusions  
In this paper, various control strategies, i.e., inertial and proportional (droop) control and pitch compensation control, are proposed for VSWTs to provide frequency regulation. Based on these proposed control loops, the VSWT can release synthetic inertia in a hybrid WDPS of San Cristobal Island and help stabilize the system frequency during windfluctuations and power imbalances. An SPBA-based tuning methodology is adopted to tune the hybrid VSWT WDPS. This SPBA methodology tuned the proposed control loops based on numerous performance indices, i.e., IAE, ISE, and controller quality index, Z. The main benefit of adopting this methodology is that it provides an optimal solution for the proposed control and enhances hybrid system effectiveness and stability. The system's behavior tended to become worse in terms of frequency deviation, increased settling time, and increased oscillations if we increased or decreased the gain parameters from the proposed optimum values. This result shows the distinctiveness and usefulness of the optimum solution generated by the SPBA. With the SPBA, the proposed pitch compensation, which enhanced the effectiveness and smoothness of the BPAC, was further simplified, resulting in a reduction in hybrid system complexity. Furthermore, hybrid WDPS tests (real-world scenarios) were conducted under various sets of perturbations, i.e., step (sudden loss of wind generation), ramp (steady decrease in wind power), and random (wind and/or load variations). These perturbations were applied as wind speed in the VSWT and a power imbalance to the DPP in hybrid WDPSs. In the case of a 0.5 pu increase in load demand, the FD was reduced by 1.8% (1.05 Hz). In addition, the reductions in the other quality indices, IAE, ISE, and Z, were 159.65, 16.75, and 83.80%, respectively. Furthermore, it is concluded that the system is stable until the variation in load demand is 0.6 pu. Furthermore, the proposed control loop result under various real-world scenarios (random wind and/or load variations) justifies the PFC support provided to the hybrid WDPS. In summary, the simulation results illustrate the effectiveness of hybrid WDPS performance for FR.  
Conceptualization, J.A.S.-F.; methodology, J.A.S.-F. and M.A.; software, M.A. and J.A.S.-F; validation, M.A. and J.A.S.-F; formal analysis, M.A. and J.A.S.-F; investigation, M.A. and J.A.S.-F; resources, J.A.S.-F; data curation, M.A. and J.A.S.-F; writing--original draft preparation, M.A. and J.A.S.-F; writing--review and editing, J.A.S.-F; visualization, M.A. and J.A.S.-F; supervision, J.A.S.-F; project administration, J.A.S.-F; funding acquisition, J.A.S.-F. All authors have read and agreed to the published version of the manuscript. This research was funded by Universidad Politecnica de Madrid under grant number RP2304330031. Not applicable. Not applicable. Not applicable. Not applicable. Data Availability Statement: All the data supporting the reported results can be found in this paper and the cited references. Conflicts of Interest: The authors declare no conflicts of interest. The funders had no role in the design of the study; in the collection, analysis, or interpretation of the data; in the writing of the manuscript; or in the decision to publish the results.