## 3 Dynamic Analysis and Design for General VSM with Energy Recovery Control
### Frequency, Power and Energy Dynamics  
As energy recovery control is a slower process, it is first ignored first. The power-frequency dynamics due to the power disturbance \(\Delta P_{L}\) can be estimated by Equations (1)-(4), where \(H\) and \(D\) denote the inertia and damping, \(k_{p}\) and \(k_{i}\) are the governor gain and integral gain in frequency control, the subscripts \"SG\" and \"VSM\" represent synchronous generator and virtual synchronous machine, \(\Delta P_{\text{SG}}\), \(\Delta P_{\text{HD}}\), \(\Delta P_{\text{Gov}}\) represent the power of SG, virtual inertia and damping, and virtual governor, respectively.  
\[\Delta f(s)=G_{f}(s)\Delta P_{L}=-\frac{(T_{\text{VSM}}{}^{8}+1)(T_{\text{SG}} s+1)s}{\left\{\begin{array}{l}((H_{\text{VSM}}+H_{\text{BG}})s+D_{\text{VSM}}+D_{ \text{SG}})(T_{\text{VSM}}s+1)(T_{\text{SG}}s+1)s\\ +k_{p,\text{VSM}}{}^{8}(T_{\text{SG}}s+1)+(k_{p,\text{SG}}s+k_{i,\text{SG}})(T_ {\text{VSM}}s+1)\end{array}\right\}\Delta P_{L} \tag{1}\]  
\[\Delta P_{\text{SG}}(s)=-\bigg{(}\frac{k_{p,\text{SG}}+k_{i,\text{SG}}/s}{T_{ \text{SG}}s+1}+(H_{\text{SG}}s+D_{\text{SG}})\bigg{)}\Delta f(s) \tag{2}\]  
\[\Delta P_{\text{HD}}(s)=-(H_{\text{VSM}}s+D_{\text{VSM}})\Delta f(s) \tag{3}\]  
\[\Delta P_{\text{Gov}}(s)=-\frac{k_{p,\text{VSM}}}{T_{\text{VSM}}s}\Delta f(s) \tag{4}\]  
As observed, the power dynamics of virtual inertia and damping control behave like a high-pass filter (HPF) process, while the power dynamics due to virtual governor control exhibits a low-pass filter (LPF) behavior. The VSM control with adjustable inertia, damping and governor control may outperform conventional synchronous generator with fixed inertia and little damping. Though larger inertia value is preferred to improve frequency performance, too large inertia will result in oscillation and slow frequency regulation [34]. Moreover, the inertia value is limited by the energy capacity of storage [1].  
The steady-state energy variation of ESS in virtual inertia and damping (\(\Delta E_{\text{HD}}\)), and governor control (\(\Delta E_{\text{Gov}}\)), is estimated by Equation (5), based on final value theorem. If both control loops are implemented on one ESS, the total steady-state energy variation of VSM is given by Equation (6). The energy variation can be used as a criterion to size ESS in VSM application. Compared with the research work [21], the integral frequency control gain of synchronous generator (\(k_{i,\text{SG}}\)), is found related with the energy variation of VSM.  
\[\Delta E_{\text{HD}}=\underset{s\to 0}{\lim}\Delta E_{\text{HD}}(s)= \left.\Delta P_{\text{HD}}(s)\right|_{s\to 0}=\frac{D_{\text{VSM}}}{k_{i,\text{SG}}} \Delta P_{L} \tag{5}\] \[\Delta E_{\text{Gov}}=\underset{s\to 0}{\lim}\Delta E_{\text{Gov}}(s)= \left.\Delta P_{\text{Gov}}(s)\right|_{s\to 0}=\frac{k_{p,\text{ VSM}}}{k_{i,\text{SG}}}\Delta P_{L}\]  
\[\Delta E_{\text{VSM}}=\frac{D_{\text{VSM}}+k_{p,\text{VSM}}}{k_{i,\text{SG}}} \Delta P_{L} \tag{6}\]  
Figure 6: Example power network studied.  
The above analysis indicates that both virtual damping and governor will cause energy dissipation, for which the ESS must be sized properly. The energy recovery control, however, can reduce the size of storage needed.