## 2 Methods and Models
### The Models of the Main Subsystems
\begin{table}
\begin{tabular}{c c c} \hline
**OM** & **Subsystems** & **Operating Condition** \\ \hline \(-1\) & LFR (defocused) + ORC & Oil flow rate = 0.22 kg/s \\  & LFR recirculation & If \(\mathrm{P_{LFR,out}}<15\) kWh and \(\mathrm{T_{LFR,out}}<\mathrm{T_{TES,AV}}\), \\  & & oil flow rate = 1 kg/s \\  & & If 15 kWh \(<\mathrm{P_{LFR,out}}<26\) kWh, \\  & & 0.11 kg/s \(<\mathrm{oil}\) flow rate \(<\) 0.22 kg/s \\  & & If \(\mathrm{P_{LFR,out}}=0\) kWh \\  & & \\
3 & LFR + LHTES & If \(\mathrm{P_{LFR,out}}<15\) kWh and \(\mathrm{T_{LFR,out}}=\mathrm{T_{TES,AV}}+10\)\({}^{\circ}\)C \\  & & If \(\mathrm{P_{LFR,out}}>26\) kWh and \(\mathrm{T_{TES,AV}}<280\)\({}^{\circ}\)C, \\  & & 0.22 kg/s \(<\) oil flow rate \(<3\) kg/s, \\  & & otherwise OM\(-1\) \\
5 & TES + ORC & If \(\mathrm{P_{LFR,out}}=0\) kWh and \(\mathrm{T_{TES,AV}}>\mathrm{T_{ORC,off}}\), \\  & & oil flow rate = 3 kg/s \\
6 & LFR + LHTES + ORC & If \(\mathrm{P_{LFR,out}}<15\) kWh and \(\mathrm{T_{TES,AV}}>\mathrm{T_{ORC,off}}\), \\  & & oil flow rate = 3 kg/s \\ \hline \end{tabular}
\end{table} 
Table 1: Operation modes of the Innova Microsolar plant. the accuracy of the model, an internal time step lower than that of the overall simulation is considered. As regard the thermal losses of the TES unit, they were equal to 0.4 W/m\({}^{2}\cdot\)K.  
The pipelines connecting all the different subsystems were modeled by means of a one-dimensional longitudinal model, as discussed in detail in [26]. In summary, the simplified advection Equation (4) was solved for each tube:  
\[\partial\left(\rho c_{\mathrm{p}}\mathrm{\,TA}\right)/\partial\mathrm{t}+ \partial\left(\rhouc_{\mathrm{p}}\mathrm{\,TA}\right)/\partial\mathrm{x}=-P_{ \mathrm{exch}} \tag{4}\]  
where \(\rho\), \(c_{\mathrm{p}}\), T, and \(\mathrm{u}\) are, respectively, the density, specific heat, temperature, and axial velocity of the oil; A is the internal cross-sectional area of the tube; and \(P_{\mathrm{exch}}\) is the exchanged power between the fluid and the environment (positive for losses, negative for gain). The equation was solved using the finite difference method with the explicit first-order upwind scheme. With reference to the ORC unit, its electric and thermal power outputs were calculated as in Equations (5) and (6):  
\[P_{\mathrm{el}}=\dot{m}_{\mathrm{f}}\cdot\left[\eta_{\mathrm{m}}\cdot\eta_{ \mathrm{el}}\cdot\Delta h_{\mathrm{e}}-\Delta h_{\mathrm{p}}/\left(\eta_{ \mathrm{m}}\cdot\eta_{\mathrm{el}}\right)\right] \tag{5}\]  
\[P_{\mathrm{th}}=\dot{m}_{\mathrm{c}}\cdot c_{\mathrm{p},\mathrm{c}^{\ast}}(T_ {\mathrm{out}}-T_{\mathrm{in}}) \tag{6}\]  
where \(\dot{m}_{\mathrm{f}}\) is the mass flow rate of the working fluid; \(\Delta h_{\mathrm{e}}\) and \(\Delta h_{\mathrm{p}}\) are the expander and pump enthalpy difference, respectively; and \(\eta_{\mathrm{m}}\) and \(\eta_{\mathrm{el}}\) the mechanical and electrical efficiencies assumed constant and equal to 0.95 and 0.9, respectively, for both the expander and the pump. In addition, \(m_{\mathrm{c}}\) is the mass flow rate of the cooling water, \(c_{\mathrm{p},\mathrm{c}}\) its specific heat, and \(T_{\mathrm{in}}\) and \(T_{\mathrm{out}}\) the inlet and outlet temperatures, respectively, of the cooling water at the condenser. The turbine isentropic efficiency varies with the operating conditions, and its values are based on the experimental data provided by the manufacturer ENOGIA [27]. In particular, under nominal operating conditions, the maximum isentropic efficiency is about 0.53, and it decreases with a pressure ratio, as better detailed in [28]. For the sake of the present analysis, a gear pump was considered to ensure the required refrigerant mass flow rate. Despite its related small size, the isentropic efficiency of this kind of pump can be assumed constant over all the considered operating range and equal to 0.7 according to [29]. The heat exchangers were modeled by means of the simplified \(\varepsilon\)-NTU method, according to which the overall heat transfer coefficient is considered constant.