# Virtual Synchronous Machine Control of RES Plants in Isolated Power Systems
## 2 Structure of the System and Proposed Converter Topology
Based on the aforementioned transfer function, the active power controller poles can be determined as follows:  
\[PL_{s1-2}=-\zeta\cdot\omega_{n}\pm\omega_{n}\cdot\sqrt{\zeta^{2}-1} \tag{9}\]  
The VSM control guarantees the VSC has behavior similar to SG, but does not guarantee the protection of the converter from overcurrent in the presence of a short-circuit in the grid. A solution to this problem is to limit the converter current using the VCC. The VCC is one of the most common methods used for VSCs because of its simple implementation and robust performance in ideal-grid conditions. Figure 2 represents the block diagram of VSM with VCC with the reference current (RC). Seven blocks are visible:  
* A measurement block at the PCC, taking the measurements of \(P_{g}\), \(I_{g}\), and \(V_{g}\), which are the input of the RC block and the abc/dq block;
* The abc/dq and dq/abc blocks, using \(\theta_{\texttt{VSC}}\) for the transformations of voltages and currents;
* A damping controller denominated as DC used to improve the response in the active power of the VSC;
* A VCC used to protect the converter from overcurrent;
* An RC used to obtain the converter input of the current control and coordinate of the abc/dq and dq/abc transformations; \(P_{ref}\) and \(T_{\texttt{VSC}}\) are the reference variables, while \(L_{f}\) and \(\omega_{0}\) are the complementary variables. VCC is defined by Equation (10). \[V_{\texttt{vsc}}^{dq*}=V_{g}^{dq}+CC\cdot I_{f}^{dq}+\text{PI}\cdot\left(I_{f} ^{dq*}-I_{f}^{dq}\right) \tag{10}\]  
In this paper, the input reference current \(I_{f}^{dq*}\) appearing in Equation (10) is calculated using two different methods:  
1. Option I (RC-1): \[I_{f}^{dq*}=\left[\frac{V_{g}^{dq*}-V_{g}^{dq}+(PI-CC)\cdot I_{f}^{dq}}{PI} \right]CL\] (11)  
Figure 2: Block diagram of the VSM model and the main control loops. In particular, the dashed line is the daily power profile generated by the VSC, which represents the energy mix composed of wind and photovoltaic plants. According to the power profiles in Figure 3, the inertia \(T_{S}\) of the isolated power system in each instant is evaluated as follows [26]:  
\[T_{S}=\frac{\sum_{1}^{Ng}T_{i}\cdot A_{n,i}}{P_{load}} \tag{13}\]  
where \(T_{i}\) is the inertia and \(A_{n,i}\) is the rated apparent power of the \(i\)-th synchronous generator connected to the grid, and \(P_{load}\) is the total active power requested by the load at each instant. The worst case, defined as the working condition presenting the least inertia, is defined in Table 4 for both summer (scenario A) and winter (scenario B). In Table 4, NSPL refers to the non-synchronous penetration level. VSPL can be defined as the rate between the total power generated by RESs and the total power generated at a given hour [27]. The load at the PCC is modelled as a fixed impedance, where the real power \(P_{load}\) and the reactive power \(Q_{load}\) are evaluated as follows:  
\[P_{load}=P_{load-0}\cdot\left(\frac{V_{base}}{V_{n}}\right)^{NP}\cdot\left(1+ K_{PF}\cdot\Delta F\right) \tag{14}\]  
\[Q_{load}=Q_{load-0}\cdot\left(\frac{V_{base}}{V_{n}}\right)^{NQ}\cdot\left(1+ K_{QF}\cdot\Delta F\right) \tag{15}\]  
where \(\Delta\) is the frequency variation; \(V_{n}\) is the rated voltage, \(P_{load-0}\) and \(Q_{load-0}\) are the active and reactive power of the load at the rated voltage and frequency, respectively; and \(NP\), \(K_{PF}\), \(NQ\), and \(K_{QF}\) are the indexes expressing the dependence of active and reactive power from the actual values of the voltage and frequency [20]. The grid has been simulated in the case of a phase-to-phase fault. The flowchart in Figure 4 represents the methodology of the proposed control strategy. The proposed methodology has the ability to deal with fault conditions and to check the VSM behavior in normal and fault operations.