## 2 Materials and Methods
### Description of the Methodology Path  
The goal of this work is the yearly analysis of a solar-assisted heat pump heating system. A building in Athens, Greece (37\({}^{\circ}\)59\({}^{\prime}\) N, 23\({}^{\circ}\)43\({}^{\prime}\) E), with a 225 m\({}^{2}\) floor area is examined and its heating loads were calculated with a model in the TRNSYS tool [22]. A high-temperature heat pump was examined thermodynamically with a created model in Engineering Equation Solver [23]. Data from the Bitzer manufacturer were used to determine the isentropic efficiency of the compressor for the different operating scenarios. It is remarkable that the isentropic efficiency was found to range from 50% to 67%. A comparison of the heat pump COP with the Bitzer data proved a mean deviation of 0.8% which indicates the high accuracy of the developed mode. More details regarding the validity of the present model for the heat pump can be found in Ref. [33]. The next step was the COP approximation with a polynomial in order to develop an accurate formula for further use. Section 3.1 includes the developed formula and its evaluation procedure.  
The final step was the development of a dynamic code in a homemade program for simulating the dynamic behavior of the total system. The weather data from the TRNSYS were used, and the heating demand of the simulation was inserted in the code. An assumption of the model is based on the satisfaction of the building loads during the heating period (October to April according to the simulation analysis). The dynamic model was based on the solution of the differential equation of the storage tank's energy balance. The heat pump performance was modeled by the use of the approximation formula, as it was previously described. The goal is to connect the high-temperature heat pump with radiators and thus the condenser temperature was selected at 70 \({}^{\circ}\)C. This temperature level makes the operation of radiators possible, with a slightly lower temperature level compared to other systems. However, the selection of the condenser temperature at 70 \({}^{\circ}\)C and not at a greater temperature makes the efficient heat pump operation possible. It is also useful to add that there is an auxiliary electrical thermal resistance inside the storage tank (5 kW capacity) which make it possible to keep the mean tank temperature over 12 \({}^{\circ}\)C and also helps the heat pump to operate with a greater evaporator's temperature and consequently with higher COP. The activation of the auxiliary resistance is seldom, and it takes place during the coldest days of the year with restricted solar potential.  
The dynamic part of this study is presented in detail for the representative case with A\({}_{\text{col}}\) = 30 m\({}^{2}\) and V = 3 m\({}^{3}\). Moreover, the yearly evaluation indexes of the system are given for different combinations of solar field areas and tank volumes. More specifically, the collecting area is varied from 10 to 50 m\({}^{2}\), while the tank volume is from 1 to 5 m\({}^{3}\). The system that leads to the lowest LCC was found to be the most appropriate one. Moreover, an extra multi-objective optimization procedure was applied by using two goals: (i) the minimization of the LCC and the maximization of the SCOP. The global optimum choice was found after using the criterion of the minimum dimensionless geometrical distance between the ideal point and the Pareto Front points. More specifically, the goal of the multi-objective optimization procedure was to minimize the following expression:  
\[\mathrm{F}=\sqrt{\left(\frac{\mathrm{SCOP_{max}-SCOP}}{\mathrm{SCOP_{max}-SCOP_{ min}}}\right)^{2}+\left(\frac{\mathrm{LCC-LCC_{min}}}{\mathrm{LCC_{max}-LCC_{ min}}}\right)^{2}} \tag{24}\]  
where the (min) and the (max) subscripts concern the minimum and the maximum obtained values, respectively, among the different studied scenarios. Every case is a different combination of collecting area and storage tank volume. Practically, these two parameters are the optimization parameters of the present analysis.