## 1 Introduction
### Photovoltaic Generators Model  
One of the three generating blocks represents the photovoltaic modules present in the system. The modeling of this block is quite challenging, requiring a lot of different literature sources as well as customized solutions to improve its functioning. Figure 5 gives an overview of the model. The model is divided into two subsystems: the upper one contains the electrical model of a photovoltaic string, while the lower block represents the control system of the PV string modified to implement the power reserve control. Observing the inputs, some observations can be performed:  
* Irradiance and cell temperature are the two ambient conditions that mainly influence the performance of the photovoltaic modules. They are assumed to be measured in real time: this is a strong assumption, which implies higher costs with respect to other solutions such as non-linear least squares fitting methods. This assumption comes from the fact that the aim of this work is centered on the control logic, not on the determination of the irradiance and cell temperature. * The reserve signal is produced by another subsystem that is presented in the following. It is sensible to frequency variation: if the frequency is at its nominal value, the reserve level is fixed at a certain value imposed by the steady state conditions. When the frequency oscillates, the reserve level is adapted automatically to face this variation. Of course, this signal is required by the control system to calculate the operating PV string voltage corresponding to a curtailed operation. * The last external input, indicated with _Side_, represents the desired operating side of the power-voltage curve for implementing the reserve control. In this work, the innovative approach proposed in [17] has been adopted, which allows, depending on the requirements, to operate on both sides of the power curve, by simply switching this input value. The control system receives these four inputs, as well as the measurement of the actual voltage applied to the PV string, and calculates the maximum power point, the corresponding voltage, the reference voltage to operate in the desired curtailed condition and, above all, a duty cycle signal for the boost converter necessary to impose that desired operation to the PV string. The general diagram of the voltage control subsystem is shown in Figure 6.  
This subsystem is mainly composed of a function and a PID controller. The function is the most important element of the PV system, since it contains almost all the features that characterize the PV model proposed in this work. For this reason, it is appropriate to explain how the function works to obtain \(v_{\textit{ref}}\), i.e., the control variable of the PRC. Figure 5: Photovoltaic system model: (**a**) General flowchart showing the electrical model (left block) and the voltage control (right block); (**b**) Simulink implementation, including the electrical model (upper subsystem) and the voltage control (bottom subsystem). The first aim of the function is the determination in real-time of the power-voltage characteristic of the module. As already observed in Figure 1, there is a strong dependence of the shape of the power-voltage curve on the environmental conditions. This means that the model has to properly account for this fact. Two of the characteristic parameters present in the datasheet of all the photovoltaic modules, the open-circuit voltage (\(V_{\alpha c}\)) and the short-circuit current (\(I_{sc}\)), participate in solving the equations and are impacted by irradiance and temperature. This means that, to be precise, they have to be adapted to the actual environmental conditions. In the literature, different approaches are proposed to do so, which are reviewed in [34]. In this work, the following correlations have been used:  
\[I_{sc}=I_{sc,pf}\ G/G_{nf}\ [1+\alpha_{sc}\ (T_{cell}-T_{cell,nf})] \tag{7}\]  
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