## 3 Results
### Parametric Analysis and Optimization
The range of the total electrical demand is calculated to be from 1503 kWh (for A\({}_{\text{col}}\) = 50 m\({}^{2}\) and V = 5 m\({}^{3}\)) to 6112 kWh (for A\({}_{\text{col}}\) = 10 m\({}^{2}\) and V = 1 m\({}^{3}\)). The next step is the presentation of the auxiliary electricity percentage to the total electricity demand for the different cases, as Figure 13 depicts. It is obvious that an increase in the collecting area leads to a dramatic decrease in the auxiliary electricity percentage. Moreover, an increase in the storage tank volume leads to a significant decrease in the auxiliary electricity percentage. The range of the auxiliary electricity percentage is found to be from 4.92% (for A\({}_{\text{col}}\) = 50 m\({}^{2}\) and V = 5 m\({}^{3}\)) to 62.26% (for A\({}_{\text{col}}\) = 10 m\({}^{2}\) and V = 1 m\({}^{3}\)). The system seasonal coefficient of performance (SCOP) is illustrated in Figure 14. It is clear that higher values of the collecting area and the tank's volume lead the SCOP to receive higher values. It can be said that the design with a tank capacity of 1 m\({}^{3}\) leads to a very small efficiency, while the use of collecting areas up to 20-25 m\({}^{2}\) lead also to low performance. The range of the SCOP is found to be from 1.11 (for A\({}_{\text{col}}\) = 10 m\({}^{2}\) and V = 1 m\({}^{3}\)) to 4.513 (for A\({}_{\text{col}}\) = 50 m\({}^{2}\) and V = 5 m\({}^{3}\)). The financial evaluation of the unit is based on the calculation of the life cycle cost index, as it is given in Figure 15. The aim is to determine the design which minimizes the LCC in order to have an economically feasible design. For every storage tank capacity, there is a specific collecting area that minimizes the LCC, and it is 35 m\({}^{2}\) for all the tank volumes. The global minimum LCC is found at 22,694 EUR (for A\({}_{\text{col}}\) = 35 m\({}^{2}\) and V = 2 m\({}^{3}\)). So, this design is the optimal configuration according to the financial criterion. Figure 12: Total electricity demand (compressor and auxiliary electrical resistance) for different combinations of (A\({}_{\text{col}}\)) and (V). Figure 14: System seasonal COP for different combinations of (A\({}_{\rm col}\)) and (V). Figure 13: Percentage of the auxiliary electricity compared to the total electricity demand for different combinations of (A\({}_{\rm col}\)) and (V). The last step is the multi-objective optimization of the unit by using both energy and economic criteria. More specifically, the energy criterion is the maximization of the SCOP, and the economic criterion is the minimization of the LCC. The ideal point is the one that presents the maximum SCOP and the minimum LCC [SCOP = 4.513, LCC = 22,694 EUR] and the objective is to determine the design that has the lowest dimensionless distance from this point. Figure 16 depicts the examined designs, and it can be said that there are many designs close to the ideal one. The most appropriate design, after conducting the evaluation procedure, was found to be the one with (A\({}_{\rm col}\) = 50 m\({}^{2}\) and V = 3 m\({}^{3}\)) which leads to [SCOP = 4.066, LCC = 24,084 EUR]. Figure 8:0 Illustration of the multi-objective optimization process of the system aiming for the maximization of the system SCOP and the minimization of the LCC. Figure 8:1 Life cycle cost of the heating system for different combinations of (A\({}_{\rm col}\)) and (V). The previous analysis indicates that different criteria lead to different optimum solutions. The economic optimization indicates that the best design is the one with (A\({}_{\text{col}}\) = 35  m\({}^{2}\) and V = 2 m\({}^{3}\)), while the multi-objective optimization (energy and economic) indicates that the optimum scenario is (A\({}_{\text{col}}\) = 50 m\({}^{2}\) and V = 3 m\({}^{3}\)). The ratio of the collectors' area to the tank's volume was found at 17.5 m\({}^{2}\)/m\({}^{3}\) and 16.7 m\({}^{2}\)/m\({}^{3}\) which are similar values, and this fact indicates that the storage tank volume \"follows\" the selection of the collecting area value.