## 4 Results and Discussion
### Influence of Thermal Storage Size  
The thermal energy contained in the tank is subjected to convective losses, which are modeled in Section 3.3. Such losses depend, among other factors, on the exchange surface between the environment and the tank (the latter being already minimized by imposing \(H=D\)). Hence, the bigger the tank, the larger the losses, all the rest being equal. However, a larger water tank allows more thermal heat storage, allowing the system, theoretically, to operate less often. The effect of these two counteracting phenomena is investigated in this section.  
\begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Parameter** & **Sodium–Sulfur** & **Li-Ion** & **Lead Acid** & **Sodium Metal** \\ \hline Capital Cost-Energy Capacity (USD/kWh) & 661 & 271 & 260 & 700 \\ Power Conversion System (PCS) (USD/kW) & 350 & 288 & 350 & 350 \\ Balance of Plant (BOP) (USD/kW) & 100 & 100 & 100 & 100 \\ Construction and Commissioning (USD/kWh) & 133 & 101 & 176 & 115 \\ \hline Total Project Cost (USD/kWh) & 907 & 469 & 549 & 928 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Summary of battery types and costs for CHP applications in 2018 according to [24]. Data from [24]. Copyright 2019, U.S. Department of Energy (DOE) Pacific Northwest National Lab. (PNNL-28866).  
Figure 9: Battery charge/discharge status at every time interval of the operation period. Climate Zone 13, typical day WSB, battery 2 kWh, thermal storage 1200 L, operation partial-load.  
Figure 10 shows the influence of thermal storage size on operative cost for volumes up to 2000 L. V0 is the relative amount of water at the beginning of the day, and simulations were conducted for V0 = full, half, and empty tank. This parameter adds a further variable to the analysis of [PERSON] [17], who only considered the case with a full initial tank. The results show a clear decreasing trend between operative cost and thermal storage size. This fact suggests that heat losses have a marginal role when compared to the gain of a larger tank. The importance of having hot water ready for use is further underlined by the three different configurations: when the day starts with a tank completely empty, more energy will be spent in the first hours to fill the tank and fulfill the demand at the same time. Finally, no valid operative solutions could be found for tanks smaller than 400 L with this setup.  
It must be observed that, over long periods of time, the water tank will most probably start every day with different values of V0; moreover, in real applications, maximum and minimum thresholds both of water content and temperature are generally set. Nevertheless, this analysis shows that a minimum of 400 L is highly recommended and that larger water tanks are desirable in order to be progressively more thermally self-sufficient, depending on building size. Moreover, the German Federal Office of Economics and Export Control grants financing for those who desire installing CHP systems up to 20 kWel [26]. The MTT Enertwin, with its 3.2 kWel, fulfills the requirements and can benefit from a financing of up to EUR 3200, provided a thermal storage of minimum 936 L is installed.