## 1 Introduction
### Photovoltaic Generators Model
\begin{array}{c}V_{\alpha c}=V_{\alpha c,rf}\ [1+\beta_{\alpha c}\ (T_{cell}-T_{cell,nf})]+54.68511 \cdot 10^{-3}\ \log(G/G_{nf})+\\ 5.973869\cdot 10^{-3}\ \log(G/G_{nf})^{2}+761.6178\cdot 10^{-6}\ \log(G/G_{nf})^{3} \end{array}  \tag{8}\]  
The _Single-Diode Model_ (SDM) of the photovoltaic cell [35] adopted in this work is characterized by five parameters: \(I_{ph,cell}\) (also called \(I_{L}\)) is the photocurrent, \(I_{s,cell}\) is the saturation current of the diode (also called \(I_{0}\)), \(n\) is the diode ideality factor, and \(R_{s,cell}\) and \(R_{sh,cell}\) are the series and shunt resistances, respectively. The electrical variables of interest are the PV cell terminal voltage (\(V_{p,cell}\)) and current (\(I_{p,cell}\)) [35]. The parameters solve the implicit equation describing the current-voltage characteristic of the cell (which is extendable to the module):  
\[I=I_{L}-I_{0}\bigg{\{}\exp\bigg{(}\frac{V+R_{s}\ I}{n\ V_{th}}\bigg{)}-1\bigg{\}} -\frac{V+R_{s}\ I}{R_{slt}}, \tag{9}\]  
where \(V_{th}\) is the thermal voltage. In principle, all the parameters necessary to solve the equation can be calculated numerically as in [35]. On the other hand, to estimate \(I_{L}\) and \(I_{0}\), Equations (10) and (11) can be used, respectively [35]:  
Figure 6: Voltage control subsystem: (**a**) General flowchart; (**b**) Simulink implementation. \[I_{L}\approx\frac{R_{s}+R_{sh}}{R_{sh}}I_{sc}, \tag{10}\]  
\[I_{0}\approx\frac{(R_{s}+R_{sh})I_{sc}-V_{oc}}{R_{sh}}\exp\left(\frac{-V_{oc}}{n \ V_{th}}\right). \tag{11}\]  
Once the current is known for each voltage between 0 and the actual \(V_{oc}\), the power curve is easily obtained as the product between it and the corresponding voltage. Once the power-voltage characteristic is known, the maximum power point can be easily extracted, as well as the corresponding voltage level. Indeed, the MPP is found as the maximum of the power-voltage curve. In the last part of the function, the reference voltage for the power reserve control is found. In particular, knowing the desired reserve amount, it is possible to calculate the corresponding power. Once this is known, an operational side of the power curve is imposed through an external signal, determining the reference voltage (\(v_{ref}\)) thanks to a simple minimization problem and finding the voltage vector element on the desired side, which corresponds to the calculated curtailed power. The reference voltage signal is then compared with the actual voltage applied to the module. The associated PI controller acts on the duty cycle signal to minimize the difference. Indeed, the output signal of the voltage control subsystem is the duty cycle, which is imposed to the DC-DC boost converter in order to set the module (the series) voltage to the desired value. The second subsystem forming the photovoltaic system block is the one containing its electrical model, which is in practice the object of the control subsystem's operation. It is basically composed of a PV array in parallel with a capacitor and a boost converter. The DC-DC converter connects these elements to a DC-link, whose voltage for the aims of this work is assumed to be fixed to 230 V, as common practice in the literature when studying the power reserve control of a PV system [17]. In practice, the single-stage PV model already mentioned (as shown in Figure 2) has been used. The corresponding Simulink implementation is the subsystem diagram presented in Figure 7.  
To fully describe the photovoltaic system, it is necessary to specify another aspect: how the adaptation of the active power output happens in response to a frequency change. This is implemented partially in the voltage control block, but to close the loop, it is important to notice how the _reserve_ input signal is generated. The reserve adaptation block (PRC in Figure 4) receives the per unit frequency deviation as an input. The reserve is adapted through a droop characteristic so that the output variable is changed proportionally to the input one.