## 3 Results
### Parametric Analysis and Optimization
It is remarkable to state that the optimum collecting area of 35 m\({}^{2}\) has a significant difference from that of the value of 50 m\({}^{2}\). This fact indicates that the selection of the optimization criteria plays an important role in the final selection of the installed collecting area. In the case where the economic parameter is the most valuable index, a lower collecting area of 35 m\({}^{2}\) has to be selected, while in the case where energy efficiency is important, a higher collecting area of 50 m\({}^{2}\) has to be chosen. Practically, a higher collecting area makes the efficiency be enhanced and this fact indicates that in the cases that the efficiency plays a critical role in the decision-making process, then higher collecting areas have to be selected. Table 1 includes the results for the two optimum scenarios. It is interesting to state that the optimum design according to the multi-objective optimization procedure leads to a significantly lower electricity demand for both the compressor and auxiliary electrical resistance. Moreover, in terms of the LCC, the difference is not high, and it is about 1390 EUR. In other words, the optimum solution from the multi-objective optimization procedure leads to a 6.1% cost increase, while there is a significant reduction in the electricity demand of 27.5%. Thus, it can be said that the sacrifice of 6.1% in cost leads to energy savings of 27.5% which is a satisfactory percentage. To conclude, the addition of a greater solar field seems to be a reasonable choice because the increase in the LCC is not so high compared to the gain in the system's energy performance. However, land restrictions and other issues have to be considered in the final selection of the installed solar field in every case. Thus, the final selection of the optimum design has to be based on the multi-objective optimization procedure and by considering any specific constraints of the examined case study. For the present system, as an empirical rule, it could be said that the final heat pump COP could be estimated by using equation (25) for T\({}_{\text{cond}}\) = 70 \({}^{\circ}\)C and T\({}_{\text{evap}}\) = 31 \({}^{\circ}\)C in order to reach the value of SCOP = 4.651 approximately. This is an empirical rule which can estimate in a quick way the total SCOP of the system and it can be used for performing quick calculations for the design of future systems. Practically, in future solar-assisted heat pump heating systems, the final optimum SCOP could be calculated by using the inputs (T\({}_{\text{evap}}\) = 31 \({}^{\circ}\)C and T\({}_{\text{cond}}\) = 70 \({}^{\circ}\)C) for having a preliminary reasonable estimation. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & \
\begin{tabular}{c} **Economic Optimization** \\ **(LCC Minimization)** \\ \end{tabular} &
\begin{tabular}{c} **Multi-Objective** \\ **Optimization** \\ \end{tabular} \\ \hline Optimum collecting area & 35 m\({}^{2}\) & 50 m\({}^{2}\) \\ \hline Optimum storage tank volume & 2 m\({}^{3}\) & 3 m\({}^{3}\) \\ \hline Yearly solar thermal efficiency & 31.60\% & 25.33\% \\ \hline System SCOP & 2.947 & 4.066 \\ \hline Heat pump SCOP & 4.125 & 4.651 \\ \hline LCC & 22,694 EUR & 24,084 EUR \\ \hline Heating demand & 6785 kWh & 6785 kWh \\ \hline Total electricity demand & 2302 kWh & 1669 kWh \\ \hline Compressor electricity demand & 1645 kWh & 1459 kWh \\ \hline Auxiliary electricity demand & 657 kWh & 210 kWh \\ \hline Percentage of auxiliary demand & 28.55\% & 12.59\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of the results for the optimum designs.