# Virtual Synchronous Machine Control of RES Plants in Isolated Power Systems
## 2 Structure of the System and Proposed Converter Topology  
The power system of a small island not supplied by the main transmission grid can be represented, for the aim of this study, as shown in Figure 1. The island's generation system is characterized by a central diesel power plant, represented in Figure 1 as an equivalent SG identified as a \"GRID\" and connected in series to the grid impedance denoted as \"Grid-Z\". All VSCs are represented by an equivalent VSC with an internal VSM-structure with VCC and an RL filter. The details of the proposed topology for the VSC are described in this section, and Table 2 reports a description of all of the symbols used in the following equations. Figure 1: Equivalent model of the considered case study. Based on the swing equation, the VSC model shows a power balance synchronization mechanism [23]:  
\[2T_{\text{vsm}}\frac{d\dot{\omega}_{\text{vsm}}}{dt}=P_{ref}-P_{\text{vsm}}-K_{d, \text{vsm}}\dot{\omega}_{\text{vsm}} \tag{1}\]  
\[\frac{d\theta_{\text{vsm}}}{dt}=\omega_{o}(\dot{\omega}_{\text{vsm}}-1) \tag{2}\]  
\[P_{\text{vsm}}=\frac{V_{\text{vsc}}V_{g}}{X_{f}}sin(\theta_{\text{vsm}}) \tag{3}\]  
Based on the aforementioned equations, a small-signal analysis of the system yields the following expression. \[2T_{\text{vsm}}\cdot\frac{d\Delta\dot{\omega}_{\text{vsm}}}{dt}=\Delta P_{ref} -\frac{\text{d}P_{\text{vsm}}}{\text{d}\theta_{\text{vsm}}}\cdot\Delta\theta_{ \text{vsm}}-K_{d,\text{vsm}}\cdot\Delta\dot{\omega}_{\text{vsm}} \tag{4}\]  
\[\frac{d\Delta\theta_{\text{vsm}}}{dt}=\omega_{o}\cdot\Delta\dot{\omega}_{ \text{vsm}} \tag{5}\]  
\[\frac{dP_{\text{vsm}}}{dt\theta_{\text{vsm}}}=\frac{V_{\text{vsc}}\cdot V_{ \text{PCC}}}{X_{f}}\cdot_{\text{cos}}\theta_{\text{vsm}0}=K_{p,\text{vsm}} \tag{6}\]  
Studying the controller response based on Equations (1) and (6) provides a relationship between the variation in output power \(\Delta P_{\text{vsm}}\) and the variation in reference power \(\Delta P_{ref}\). Hence, the linear model of the system can be expressed as follows:  
\[\frac{d}{dt} \begin{bmatrix}\Delta\dot{\omega}_{\text{vsm}}\\ \Delta\theta_{\text{vsm}}\end{bmatrix} = \begin{bmatrix}-K_{d,\text{vsm}}/2T_{ \text{vsm}}&-K_{p,\text{vsm}}/2T_{\text{vsm}}\\ \omega_{o}&0\end{bmatrix}\begin{bmatrix}\Delta\dot{\omega}_{\text{vsm}}\\ \Delta\theta_{\text{vsm}}\end{bmatrix} + \begin{bmatrix}1/2T_{\text{vsm}}\\ 0\end{bmatrix} \Delta P_{ref} \tag{7}\]  
Considering the Laplace Transform for Equations (3)-(7), the following equation is obtained:  
\[\frac{\Delta P_{\text{vsm}}}{\Delta P_{ref}}=\frac{\omega_{o}K_{d,\text{vsm}} }{s^{2}+\frac{\omega_{o}K_{d,\text{vsm}}}{2T_{\text{vsm}}}s+\frac{\omega_{o}K_ {p,\text{vsm}}}{2T_{\text{vsm}}}}=\frac{\omega_{n}^{2}}{(s^{2}+2\zeta\omega_{ n}s+\omega_{n}^{2})} \tag{8}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Nomenclature** & **Description** \\ \hline \(T_{\text{vsm}}\) & The virtual inertia of the VSC [s] \\ \hline \(\omega_{o}\) & The rated angular frequency of the system [rad/s] \\ \hline \(\dot{\omega}_{\text{vsm}}\) & The angular frequency at the VSC [p.u.] \\ \hline \(P_{ref}\) & The reference active power of the VSC [p.u.] \\ \hline \(P_{\text{vsm}}\) & The actual active power output of the VSC [p.u.] \\ \hline \(K_{d,\text{vsm}}\) & The virtual damping coefficient of the VSC \\ \hline \(\theta_{\text{vsm}}\) & The angle position of the converter voltage concerning a reference frame synchronized with the grid and rotating at a constant frequency in steady-state [rad] \\ \hline \(\delta_{o}\) & The angle between the grid voltage \(V_{g}\) at the PCC and the output voltage of the VCS \(V_{\text{vsc}}\) [rad] \\ \hline \(X_{f}\) & The VSC filter reactance [p.u.] \\ \hline \(\theta_{\text{vsm}0}\) & The steady-state virtual angle [rad] \\ \hline \(K_{p,\text{vsm}}\) & The synchronizing torque coefficient \\ \hline \(\xi\) & The damping ratio \\ \hline \(\omega_{n}\) & The natural frequency of the system [p.u.] \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Nomenclatures employed in Equations (1)–(3).