## 4 Discussion
### Numeric Optimization  
These two Equations (5) and (6) are in accordance with commons sense expectative. In this way, the equation of cost shows a base value of 254,641 euros which is more reduced when the variables ACH, UF, UC experiment their highest values related with the actual conditions and low-cost materials. It is interesting to highlight that it happens when the boiler is not changed (0) due to, in this condition, the photovoltaic electrical production will exert an inverse effect increasing the cost at the time that the PV production increases. When a change of boiler is done, this previous effect of the PV production is modified by new terms of this Equation (1) that now are not neglected. If we now analyze the second equation, Equation (2), the same conclusions cannot be obtained so easily about the effect of each variable in the energy consumption. With the mathematical models defined by Equations (5) and (6), it can be now obtained the energy consumption and economic investment of each possible retrofits process (columns 4-8 of Tables 6 and 7). If all these possibilities are ordered in accordance with their economic investment (row number), it is obtained Table 6. In consequence, Table 6 starts in row 1 showing the more expensive actions in the building and its related energy consumption. In consequence, on one hand, it was observed that the investment with the highest economical cost does not provide the lowest energy consumption. In this sense, a selection of carpentry category 1 (the best), transmittance of facade 0.59 (the best), transmittance of cover 0.45 (the best), natural gas and nonincorporation of photovoltaic electrical production corresponds to an investment of 215,345 \(\upmu\) and reduces consumption to 134.800 kW/m\({}^{2}\) as we can see in Table 6 in its row 1. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**Toutdoor** & **Jan** & **Feb** & **Mar** & **Apr** & **May** & **Jun** & **Jul** & **Ago** & **Sep** & **Oct** & **Nov** & **Dec** \\ (**°C**)/month** & & & & & & & & & & & & & \\ \hline standard & 8.19 & 9.31 & 10.82 & 12.33 & 15.60 & 16.09 & 19.29 & 19.28 & 17:15 & 16.08 & 11.31 & 8.87 \\
2017 & 7.33 & 9.65 & 10.59 & 13.53 & 15.94 & 18.18 & 18.41 & 18.68 & 15.71 & 15.92 & 10.03 & 8.36 \\ Deviation (\%) & 10.44 & 3:67 & \(-\)2.15 & 9.72 & 2.20 & 12.97 & \(-\)4.56 & \(-\)3.15 & \(-\)8.39 & \(-\)1.00 & 11.30 & \(-\)5.76 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Monthly average outdoor temperature (from standards and sampled values). In this Table 6, it can be observed that the effect to change the boiler from the diesel boiler (0) to a natural gas boiler (1) will exert a higher effect over the final cost with independence of the level of photovoltaic energy generated. This maximum cost is obtained for changing the boiler and carpentry and transmittances in its better values, as we can see in rows 1 to 5. At the same time, the PV production exerts in these initial combinations a slight decrement of the cost at the time PV value increases. From this same table, it can be observed that, only when the maximum available photovoltaic energy production is combined with a carpentry level 2, without changing the diesel boiler, the cost will be as higher than the previous combinations after changing to a natural gas boiler, see row 13. Despite this similar economical cost of retrofit combinations, it can be observed that the change to PV production instead a boiler implies a so high-energy consumption of 201.000 kWh/m\({}^{2}\) in this row 13 which is nearly the double of the energy consumption of the previous combination showed in the row 12 for nearly the same economical investment. Another conclusion that can be observed form this Table 6 is that the maximum investments are always obtained remaining the transmittance of the facade and the cover in its minimum value of 0.59 (W/m\({}^{2}\cdot\)K) and 0.45 (W/m\({}^{2}\cdot\)K), respectively.