## 3 Results
### Analysis of Stand-Alone Photovoltaic Systems  
According to the present work purposes, four stand-alone configurations (HPWH (2.1), (2.2), (2.3), and (2.4)) were considered to outline electrical storage behavior. Preliminary simulations were useful for determining PV plant features to be tested. Looking at the results of simulations, January and December appear as the most critical months with average daily electrical consumption equal to 1.4 kWh per day (Figure 10). Figure 10: Daily energy consumption for the system. Figure 9: Comparison between electrical storage water heater (ESWH) and HPWH energy consumption. Moreover, the HPWH was switched on mainly in conjunction with tapping events in the early morning and in the evening. As mentioned above, the median value of irradiation was calculated. Based on these data, a day for each month characterized by irradiation figures closest to the median was selected. For these 12 days (one per month), the daily frequency of on and off operation of the heat pump in the selected time range was evaluated. Figure 11 shows the percentage of time that the HPWH was \"on\" in an hour during the year with respect to the adopted tapping profile. To check the thermal storage behavior in operating conditions, further simulations were run. In this case, the energy produced by the PV plant was exclusively considered to drive the HP, without any electrical storage support. Whenever the temperature tank dropped to the setpoint value and, simultaneously, whenever the power from the PV string was suitable to drive the cycle without relating to the electrical grid or battery, the model allowed the switching on of the HPWH. As described above, four different PV plant sizes were considered: 720 Wp, 1200 Wp, 1440 Wp, and 1680 Wp. They correspond to three, five, six, and seven PV panels, respectively. Table 4 shows the plant configuration with respect to the following features: number of days of autonomy (N\({}_{\text{aut}}\)) for stand-alone (SA) plants, inverter (P\({}_{\text{Inv}}\)) and plant power (PPV), battery capacity (C\({}_{\text{St}}\)), and number of PV panels, for which characteristics could be inferred from Tables 2 and 3. It can be noted that systems 2.3 and 2.4 were definitively oversized in order to use them as comparison terms with respect to system 2.1 and system 2.2 considering that no battery storage is available. In Table 4, the characteristics of the four selected PV plants (systems 2.1, 2.2, 2.3. and 2.4) are listed. The last two plants were definitely oversized in order to use them as comparison terms. It is possible to see in Figure 12 that all plant configurations could match the summer DHW demand (from May until October) without discomfort events (monthly DI and PDI equal to 0%); moreover, as seen by looking at the irradiance value (Figure 5 in Section 2) and the adopted tapping profile (Figures 7 and 9 in Section 2), winter months appeared the most critical. The next step was to calculate the discomfort indices DI and PDI (Figure 12). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**PV-plant** & **Type; Naut** & **PInv (W)** & **PPV (Wp)** & **CSt (Wh)** & **No. of PV panels** & **Ppanel (Wp)** \\ \hline System 2.1 & SA; 2-4 & 1500 & 720 & 2400(a)/4800(b) & 3 & 240 \\ System 2.2 & SA; 2-4 & 750 & 1200 & 2400(a)/4800(b) & 5 & 240 \\ System 2.3 & SA; 0 & 1500 & 1440 & 0 & 6 & 240 \\ System 2.4 & SA; 0 & 2000 & 1680 & 0 & 7 & 240 \\ System 2.5 & GC & 750 & 300 & 0 & 1 & 300 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: PV plant features. Figure 11: Percentage of time that the HPWH was “on” in an hour during the year with respect to the adopted tapping profile. As in this case, the most critical results were recorded in December for the 720-Wp plant (system 2.1) because of the small size of PV generator compared to the HP nominal power and to the low values of irradiation. A higher PV plant power obviously leads to better results (Figure 13).