## 4 Cost Functions
### Latent Heat Thermal Energy Storage (LHTS) and Concrete Storage  
Both the LHTS system and the concrete storage considered in this work consist of a tube bundle surrounded with thermal storage material, as shown in Figure 3. For both charging and discharging, the heat transfer fluid flows through the same tubes. It is assumed that the heat transfer fluid is liquid water or steam, respectively. When the thermal storage is charged, steam flows through the pipes and condenses, whereas in the case of discharging, liquid water evaporates within the tubes. It is assumed that the mass flow of the heat transfer fluid is controlled to ensure full evaporation or condensation within the storage tubes. Figure 4 (left) shows the flow-chart for the calculation of the different storage configurations for LHTS and concrete storages. The tube diameter \(d_{tube}\) and the heat storage material layer \(s_{mat}\) are varied within user-defined ranges. For each combination of tube diameter and storage material layer a charging cycle is simulated. Since the dynamic behavior of the concrete storage and even more so of the LHTS is highly complex and a rigorous transient simulation model would result in excessively long computation time, a simple quasi-stationary node model illustrated in Figure 5 using the so-called enthalpy approach is used for simulation. \begin{table}
\begin{tabular}{c c c} \hline
**Type** & **Quantity Per Storage (pcs.)** & **Total Costs (€)** \\ \hline \begin{tabular}{c} boundon pressure gauge incl. ring type syphon tube, liquid damping \\ bimetallic temperature gauge incl. thermo wells \\ \end{tabular} & 3 & 1260.- \\ \begin{tabular}{c} Drain valve DN50 PN40 \\ Vacuum breaker DN15 PN40 \\ \end{tabular} & 1 & 830.- \\ \
\begin{tabular}{c} Relief valve \\ Pressure reducing valve \\ \end{tabular} & 1 & * \\
\begin{tabular}{c} Safety valve \\ Float ball valve \\ \end{tabular} & 1 & * \\ \hline \end{tabular}
* calculated for each storage configuration, depends on storage requirements.  
\end{table} 
Table 2: Valves and instrumentation considered for Ruths steam storage. Prices are according to [18; 23; 24]. Figure 4: Flow-charts for the calculation of storage parameters for LHTS and concrete storages (**left**) and for the calculation of average heat loads (**right**). Figure 3: Schematic drawings of the tube surrounded by heat storage material for both LHTS and concrete storage (**top**), the LHTS system (**left**) and the concrete storage system (**right**) considered in this work. Both TES systems are represented without thermal insulation material. In this model, the storage material layer is divided to discrete volumes with index \(i\). These volumes are defined by:  
\[v_{i}=\left(\left(\frac{d_{i}}{2}\right)^{2}-\left(\frac{d_{i-1}}{2}\right)^{2} \right)\pi\,l,\quad d_{i-1=0}=d_{tube}. \tag{28}\]  
To account for the fact that a sufficient temperature difference between storage material and HTF is necessary to obtain sufficient heat loads, an effective temperature range is specified that depicts the useful temperature range for storage of sensible heat. For LHTS, the total storage capacity \(cap_{total}\) considering the effective temperature range \(\Delta T^{eff}\) is calculated:  
\[cap_{total}=v_{mat}\,\left(h_{lat}+c_{p}\,\,\Delta T^{eff}\right). \tag{29}\]  
whereas for concrete, the storage capacity calculation simplifies to:  
\[cap_{total}=v_{mat}\,\,c_{p}\,\,\Delta T^{eff} \tag{30}\]  
with  
\[\Delta T^{eff}=(T_{max}-T_{min})\eta_{T} \tag{31}\]  
where \(\eta_{T}\) is the temperature efficiency factor, which was set to 0.8 in this work. This factor reduces the theoretically available temperature range to a more realistic range where reasonable driving temperature differences are ensured.