## 8 Conclusions  
To address the challenge presented by the fact that the TVSG system struggles to balance transient oscillations and steady-state deviations resulting from active power and grid frequency disturbances, a VSG control strategy founded on Transient Damping Compensation and virtual inertia adaptation is provided. Through theoretical analysis, Matlab R2024a simulations, and semi-physical experimental validation, the following conclusions are drawn:  
1. A theoretical framework for discussing the dynamic performance of TVSG outputs is provided, which helps to explain the reason for the transient oscillation of power outputs in the TVSG. Building on the previous analysis, the problem of steady-state deviation in TVSG outputs caused by fixed damping is proposed.
2. A VSG control strategy founded on Transient Damping Compensation (TDC) is provided, which differs from existing methods and effectively contributes to suppressing transient oscillations and eliminating steady-state deviations of APL outputs.
3. In addition, the root locus distribution trend for analyzing the dynamics of the TDC-VSG is provided, which helps in determining the optimal control parameter in the TDC-VSG. Based on this analysis, a parameter design scheme utilizing the dominant pole method is proposed.
4. Building upon the TDC-VSG strategy, a virtual inertia adaptive control strategy incorporating the ISRU approach rate is proposed, which significantly mitigates frequency deviations and rates of change. Furthermore, the accuracy of the TDC-VSG control strategy is maintained since virtual inertia compensation is utilized in the APL system. The proposed strategy also additionally improves the system's anti-interference capabilities and frequency response performance on the APL controller.  
This article primarily focuses on improving the dynamics and the transient stability of the system under active power disturbances and grid frequency fluctuations. However, this proposed approach has not been experimentally validated in a real microgrid system to obtain more reliable data. Therefore, future work will involve conducting experiments on an actual microgrid testbed and performing tests under VSG parallel operations and off-grid operations.  
Conceptualization, Y.X. and Y.C.; methodology, Y.X.; validation, Y.X., Y.C. and Y.W.; formal analysis, Y.X. and Y.W.; investigation, Y.X. and Y.C.; resources, R.C. and K.L.; writing--original draft preparation, Y.C.; writing--review and editing, Y.X., J.S. and Y.Y.; supervision, J.S. and Y.Y.; project administration, Y.X., R.C. and K.L. All authors have read and agreed to the published version of the manuscript.  
This research was funded by Hydrogen Energy and Multi-energy Complementary Microgrid Engineering Technology Research Center of Sichuan Province (2024 DWNV006); Sichuan Science and Technology Program (2022 SZYZF01); and Innovation Fund of Postgraduate, Sichuan University of Science and Engineering (Grant No. Y2023292).  
Contained within the text and cited references.  
This work was supported by the School of Automation and Information Engineering and Sichuan University of Science & Engineering.  
[PERSON] has received research grants from Zonergy Co., Ltd. The authors declare no conflicts of interest.