# Simulation of Self-Consumption Photovoltaic Installations: Profitability Thresholds
## 3 Results
\begin{table}
\begin{tabular}{|r|r r r r r r r r r r r r|} \multicolumn{1}{c}{Inclination(\({}^{\circ}\))} & & & & & & & & & & & & \\ \hline
90 & 14.7 & 13.9 & 12.5 & 11.3 & 10.4 & 9.9 & 9.9 & 10.0 & 10.5 & 11.3 & 12.5 & 13.9 & 14.7 \\
75 & 12.7 & 11.8 & 10.7 & 9.8 & 9.1 & 8.6 & 8.5 & 8.7 & 9.2 & 9.9 & 10.8 & 11.8 & 12.7 \\
60 & 10.5 & 10.1 & 9.5 & 8.8 & 8.3 & 7.9 & 7.7 & 7.9 & 8.3 & 8.9 & 9.6 & 10.2 & 10.5 \\
45 & 9.1 & 9.0 & 8.6 & 8.1 & 7.7 & 7.4 & 7.3 & 7.5 & 7.8 & 8.2 & 8.7 & 9.0 & 9.1 \\
30 & 8.2 & 8.1 & 7.9 & 7.7 & 7.4 & 7.2 & 7.1 & 7.2 & 7.4 & 7.7 & 8.0 & 8.1 & 8.2 \\
15 & 7.6 & 7.6 & 7.5 & 7.3 & 7.2 & 7.1 & 7.1 & 7.1 & 7.2 & 7.4 & 7.5 & 7.6 & 7.6 \\
0 & 7.2 & 7.2 & 7.2 & 7.2 & 7.2 & 7.2 & 7.2 & 7.2 & 7.2 & 7.2 & 7.2 & 7.2 & 7.2 \\ \hline  & \(-180\) & \(-150\) & \(-120\) & \(-90\) & \(-60\) & \(-30\) & 0 & 30 & 60 & 90 & 120 & 150 & 180 \\ \end{tabular}
\end{table} 
Table 8: \(Pay-back\) of a 1000 Wp PV installation and an obstruction angle of 30\({}^{\circ}\) as a function of the azimuth and inclination of collectors. \begin{table}
\begin{tabular}{|r|r r r r r r r r r r r r|} \multicolumn{1}{c}{Inclination(\({}^{\circ}\))} & & & & & & & & & & & \\ \hline
90 & 14.7 & 13.9 & 12.5 & 11.3 & 10.4 & 9.9 & 9.9 & 10.0 & 10.5 & 11.3 & 12.5 & 13.9 & 14.7 \\
75 & 12.7 & 11.8 & 10.7 & 9.8 & 9.1 & 8.6 & 8.5  
The optimal achievable \(NPV\) values for each power and degree of obstruction are represented in Figure 7 (circles) as well as the minimum \(NPV\) values in the most unfavourable orientation (squares). In this graph it is observed that the maximum value of \(NPV\) (EUR 1626.0), is achieved for the value of \(PP=1500\) Wp in non-obstruction circumstances. Thus, this figure shows the variation interval of \(NPV\) for each power and degree of obstruction. In it, it is observed that, for the most unfavourable orientation (\(\gamma=180^{\circ}\) and \(\beta=90^{\circ}\)), the increase in installed power implies a worsening of the \(NPV\), regardless of the angle of obstruction. This graph also shows the inconvenience of increasing the peak power of an installation with a low \(NPV\) value. However, installations with lower powers may be of great interest in cases where investment capacity is limited and, therefore, the \(NPV/\textit{investment}\) indicator is considered. Figure 8 shows this indicator as a function of peak power and degree of obstruction. It shows the generalised reduction in unit profitability as installed power grows. Along the same lines, Figure 9 shows the greater ease of recovering the investment in low-power and well-oriented installations, given that the cost is proportional to the installed power and, in addition, they consume all their production. Figures 7-9 show, as in Figure 6, the  
Finally, to simplify and generalize the set of results obtained, a way to characterise them mathematically has been sought. In this sense, it should be remembered that the model described and used (for each of the 2548 cases) can be considered as a mathematical function to obtain the study variables from the initial data. However, the complexity of the model is not known to be much smaller than the number of variables, and the complexity of the model is not known to be much smaller than the number of variables. Figure 8: \(NPV\) /_investment_ depending on peak power and degree of obstruction. Figure 9: Achievable \(Pay-back\) depending on peak power and degree of obstruction. model and the need to add results on the different representative days makes it difficult, in principle, to know the weight or influence of each variable on the final results. Thus, to overcome this difficulty, in the present work the proposal is to replace this complex function by a mathematical function with a simple expression that reproduces the result of the complex model with the least possible error.