## 1 Introduction
### Overview of the General Model  
The upper layer of the model of the system under study is shown in Figure 4. It represents a small isolated power system, comprehending three kind of generating plants: a reheated steam turbine, wind turbines and photovoltaic units. It has to be specified that the total generated power (that in equilibrium condition equals the load power), as well as the power of the three generating blocks, vary with the magnitude of the participation factors. In fact, in this kind of study, the importance is related to the relative weight of the three generating blocks rather than to the absolute magnitude of the system, since it is based on per unit variations in a reference common basis. This concept is particularly relevant for simulations at different participation factors, as the ones presented in Section 4.  
Before going deeper into the characteristics of the different subsystems present in the model, it can be useful to understand the main connections between them, to facilitate their study. Each of the three generation blocks produces a signal of variation of power in per  
Figure 4: General layout of the model.  
unit with respect to the steady state value (\(dP_{i}\)), which is the power corresponding to the initial condition:  
\[P_{i,0};\quad\Sigma_{i}\left(P_{i,0}\right)=P_{load,0} \tag{4}\]  
Indeed, all the simulations are initialized to have a \(dP_{i}\) [\(pu\)] = \(dP_{load}\) [\(pu\)] = 0, corresponding to a frequency value equal to the rated one, which is 50 Hz (i.e., \(\Delta f=0\)).  
The three signals of variation of power of the generating blocks are initially in per unit with respect to the generator nominal power, so they require to be weighted to obtain the equivalent variation on a common basis (the sum of the nominal powers of the three generators blocks): each of them passes through a gain block in which it is multiplied by its participation factor, as in Equation (5). The \(dP_{load}\), on the contrary, is already defined in per unit common basis.  
\[dP_{i[pu\_common\_basis]}=p_{i}\ dP_{i[pu]} \tag{5}\]  
For simplicity, from now on, \"common basis\" is omitted in the equations. Then, the four \(dP\) signals are summed up to obtain the instantaneous power unbalance of the entire system:  
\[dP_{tot[pu]}=\Sigma_{i}\left(dP_{i[pu]}\right)-dP_{load[pu]} \tag{6}\]  
A non-negative power unbalance leads, through a transfer function, to a frequency deviation from the nominal value. The transfer function describes the frequency response behavior of the system. The meaning is captured by two parameters: \(H_{eq}\) is the equivalent inertia constant of the synchronous generator present in the system, and \(D\) is the damping constant of the load, which represents the sensitivity of the load power to the frequency variations. The function output is the signal representing the frequency variation in per unit with respect to the nominal value, which is also one of the metrics to evaluate the system behavior in the analysis presented in Section 4. The frequency variation signal is also the main input to the controllers of the three generators. Indeed, the synchronous one is traditionally able to respond to a change in system frequency by adapting its power output, but in this study also photovoltaics and wind turbines are, thanks to the presence of the power reserve control block (PRC) and the inertial control block, respectively.  
After this brief introduction, a more detailed analysis of each block is required, to understand the modeling of the generators as well as the functioning of the above-mentioned controllers.