## 2 Materials and Methods
### D Compact Model  
Considering the LHTSs dynamics, the thermal power released or absorbed by LHTSs varies over time, as the thermal conductivity of the liquid and solid material is different. Indeed, as the solidification or liquefaction of the PCM proceeds, the power exchanged decreases, since the melting front is further from the fins. For this reason, it was assumed that the most relevant parameter influencing the thermal power input/output of an LHTS is represented by its state of charge (SOC). The SOC can be defined as the ratio between the actual energy stored inside the LHTS (\(E_{stored}\)) and its reference value when fully charged (7).  
\[SOC=\ \frac{E_{stored}(t)-\ E_{min}}{E_{max}-E_{min}} \tag{7}\]  
where \(E_{max}\) is the energy contained in the PCM when the LHTS is fully charged. Similarly, \(E_{min}\) is the energy contained in the PCM when fully discharged.  
A mathematical correlation between the current state of charge of the LHTS (\(SOC\)) and its thermal power input/output was identified using the results of the 2D model as shown by Equations (8)-(10). These were obtained observing the shape of the 2D model results and searching for their best fit through the sum of exponential functions. Each expression refers to a different operational phase (i.e., charge, discharge, or idle phase).  
\[\dot{Q_{dis}}=\ Ae^{B*SOC_{n}}+Ce^{D*SOC_{n}}+K(1-SOC_{0})e^{-\left(\frac{ SOC_{n}-E}{F}\right)^{2}} \tag{8}\]  
\[\dot{Q_{clrg}}=\ Ae^{B*SOC_{n}}+Ce^{D*SOC_{n}}+K*SOC_{0}e^{-\left(\frac{ SOC_{n}-E}{F}\right)^{2}} \tag{9}\]  
\[\dot{Q_{idle}}=0 \tag{10}\]  
\(SOC_{n}\) is the normalized state of charge, and it is equal to \(\left(\frac{SOC}{SOC_{0}}\right)\) during discharge and \(\left(\frac{SOC-SOC_{0}}{1-SOC_{0}}\right)\) during charge, \(SOC_{0}\) is the initial state of charge, \(\dot{Q_{dis}}\) is the thermal power released by each tube during discharge, \(\dot{Q_{clrg}}\) is the thermal power requested by each tube during the charging phase, and \(\dot{Q_{idle}}\) is the thermal power when the LHTS is not operated. Therefore, depending on the operational phase of the LHTS, the value of the thermal power released or requested by each LHTS tube can be expressed as follows (11):  
\[\dot{Q_{LHTS}}=x_{1}\dot{Q_{dis}}-\ x_{2}\dot{Q_{dis}}+\ x_{3}\dot{Q_{idle}} \tag{11}\]  
where \(x_{1},x_{2},x_{3}\in[0;1]\) and \(x_{1}+x_{2}+x_{3}=1\).  
Equations (8) and (9) were obtained fitting with exponential and Gaussian functions the LHTS thermal power and states of charge resulting from the simulations of the 2D model. The initial conditions of these simulations were, respectively, a fully charged LHTS and a fully discharged LHTS. Moreover, assuming that the thermal losses from the PCM to the HTF are negligible, the thermal power is always zero during the idle phase. Although these expressions were retrieved considering a specific LHTS design with fixed operating parameters (such as water mass flow rate and temperatures), the proposed methodology can be easily tailored to different LHTS concepts and operating conditions. Indeed, the evolution of the LHTS heat rate highlighted by several studies [12; 15; 22] is comparable with the one expressed by Equations (8) and (9).