## 3 Methodology
### Economic Performance Indicators  
The following performance indicators were evaluated for the three cases: investment costs, operating costs, payback time and profit after a lifetime of 20 years. Since the reference system for the proposed system is simply the proposed HP without the RPW-HEX, solely the additional costs to install the RPW-HEX to the HP were considered for the investment costs. They consist of the costs for the RPW-HEX including the insulation, the additional three-way valve and the piping needed for the integration in the HP cycle. All three costs can be separated in fixed and variable costs:  
\[C_{\rm invest}=C_{\rm invest,fix}+\tilde{c}_{\rm invest,var}\times Q_{\rm RPW,max}, \tag{1}\]where \(Q_{\text{RPW,max}}\) is the maximum storage capacity of the RPW-HEX. The fixed and variable costs were estimated after manufacturing a first version of the RPW-HEX for [25]. The following costs and properties were taken into account for a series production: 5.00 EUR/kg for PCM, 4.55 EUR/kg for aluminum, 6.00 EUR/m\({}^{2}\) for mineral wool (10 cm thickness), a latent phase change energy of the PCM of 250 kJ/kg, a specific heat capacity of 2.00 kJkg\({}^{-1}\)K\({}^{-1}\) for the PCM and 0.900 kJkg\({}^{-1}\)K\({}^{-1}\) for aluminum, a PCM/aluminum ratio of 1:2.44 and a temperature difference of 10 K between the charging and discharging of the RPW-HEX during operation. The so-calculated fixed and variable costs for (1) are summarized in Table 2. Please note that the costs of the insulation do not correlate linearly with the storage capacity as the surface does not linear dependent on the volume. Nevertheless, for the sake of simplicity and the comparatively small share of the insulation material to the total costs, a linear dependency was also used for the costs of the insulation material.  
As the maintenance costs of the RPW-HEX are included in the service costs incurred anyway, solely the differences in electric energy consumption were the basis for calculating the operating costs. The payback time \(t_{\text{payback}}\) was calculated as the time when the net present value becomes equal to the investment costs:  
\[C_{\text{interest}}\ =\sum\
olimits_{tp=1}^{N(t_{\text{payback}})}\frac{\left(W_{el }^{\text{ref}}-W_{d}^{\text{RPW}}\right)\times\tilde{c}_{el}(tp)}{(1+i)^{tp}}, \tag{2}\]  
where the electric energy price/kWh\({}_{\text{el}}\)\(\tilde{c}_{\text{el}}\) was assumed to be 0.22 EUR/kWh at the time of the investment (average energy price per kWh for households in the Euro area in 2018) with an increase of 0.006 EUR/year [26]. The discount rate \(i\) per time period \(tp\) was correlated with a fixed rate of 2 %/year and \(N\) is the number of periods at the payback time. Please note that waste disposal costs and profits from recycling were not considered. Using a standard Newton-solver, the payback time \(t_{\text{payback}}\) was calculated numerically from (2).  
The annual cost savings are calculated from the difference between the annual electric energy consumption with and without the RPW-HEX (\(W_{\text{el}}^{\text{ref}}\), \(W_{\text{el}}^{\text{RPW}}\)) and the electric energy price/kWh\({}_{\text{el}}\) (\(\tilde{c}_{\text{el}}\)) in the year of operation \(t\):  
\[C_{\text{annual savings}}(t)=\left(W_{\text{el}}^{\text{ref}}-W_{\text{el}}^{ \text{RPW}}\right)\times\tilde{c}_{\text{el}}(t). \tag{3}\]  
The economic profit \(C_{\text{profit}}\) after \(t_{\text{lifetime}}\) years follows from:  
\[C_{profit}\ =\left(ceil\big{(}t_{\text{payback}}\big{)}-t_{\text{payback}} \right)\times C_{\text{annual savings}}\big{(}floor\big{(}t_{\text{payback}} \big{)}\big{)}+\sum\
olimits_{t=ceil(t_{\text{payback}})}^{t_{\text{lifetime}}} C_{\text{annual savings}}(t). \tag{4}\]  
Recent investigations with a PCM suitable for the proposed concept (RT64 HC) did not show relevant degradation up to 8000 cycles [27]. This would imply an operating time of the proposed system of about 20 years. Therefore, no degradation effects of the PCM were considered in the calculations.