Abstract: To better interface renewable energy resources（RERs）with the distribution network,the virtual synchronous generators（VSGs）are highlighted.An insightful analysis on the completely mathematical model and control scheme for the VSG is proposed in different virtual inertia and damping conditions.Additionally,according to the mathematical model,the optimal design of energy storage（ES）device of the VSG is achieved.It is found that,to emulate the inertia and damping features of the synchronous generator,an extra energy storage cell should be embedded in it.Besides,the optimal capacity of the ES is the power fluctuation of RERs VSG should suppress,and the desired energy of the ES is analytically decided by the virtual inertia and damping parameters of the VSG.Furthermore,the desired dynamic response time of ES is also obtained for VSG application.Finally,the experimental results obtained on a 50 kVA VSG prototype have verified the proposed model.