## 2 Case Study and System Model
### Linearized Model of the Components

Table 1: Performance curve data of a vacuum tube collector Sunwood SW15/20 [56]. The last two equations allow us to identify the minimum size of the two types of chillers that are capable of producing the required cooling powers at any time of the day, according to the optimal management. The size of the absorption chillers allows their penalty function to be identified and used during the optimization of the objective function. The integrative thermal powers produced by the steam and the water boilers (_Int_St (g,i)_, _Int_HTW (g,i)_) and the related fuel consumption (_Fuel_BSt (g,i)_ and _Fuel_BHt (g,i)_), are expressed by the following equations:  
\[Fuel\_BSt\ (g,i)=Int\_St\ (g,i)/eta\_St \tag{23}\]  
\[Fuel\_BHt\ (g,i)=Int\_HTW\ (g,i)/eta\_HTW \tag{24}\]  
\[Int\_St\ (g,i)\leq Int\_St\_max \tag{25}\]  
\[Int\_HTW\ (g,i)\leq Int\_HTW\_max \tag{26}\]  
The variables describing the state of two thermal storages (steam and water HTW) are the thermal input and output powers, the quantity of energy stored and the binary variables that indicate the incoming or out coming flow condition from each storage:  
* _Lev_St (g,i)_, energy stored in the steam storage [kWh];
* _Input_St (g,i)_, input power in the steam storage [kW];
* _Output_St (g,i)_, output power in the steam storage [kW];
* _Lev_HTW (g,i)_, energy stored in HTW storage [kWh];
* _Input_HTW (g,i)_, power input into the HTW storage [kW];
* _Output_HTW (g,i)_, output power in the HTW storage [kW];
* _y_Input_St (g,i)_, ON/OFF of the flow entering the steam storage;
* _y_Output_St (g,i)_, ON/OFF of the output stream in the steam storage;
* _y_Input_HTW (g,i)_, ON/OFF of the flow entering the HTW storage;
* _y_Output_HTW (g,i)_, ON/OFF of the output stream in the HTW storage. The storage energy balance in the generic operating hour calculates the level of energy stored as the level of the previous hour, increased by the net input during the current hour. It is necessary to ensure the continuity of the levels of energy stored during the passage from one cruise day to the next. In correspondence with the first hour of each day, it is therefore, necessary to consider the conditions of the last hour of the previous day. To do this, the following IF cycle has been introduced:  
\[\text{if ($g=1$ and $I=1$) then}\]  
\[\textit{Lev\_St (1,1)=Lev\_St (7,24)-Output\_St (1,1)+Input\_St (1,1)} \tag{27}\]  
\[\text{elif ($g>1$ and $i=1$) then}\]  
\[\textit{Lev\_St (g,i)=Lev\_St (g-1,24)-Output\_St (g,1)+Input\_St (g,1)} \tag{28}\]  
else  
\[\textit{Lev\_St (g,i)=Lev\_St (g,i-1)-Output\_St(g,i)+Input\_St (g,i)} \tag{29}\]  
\[Fuel\_BSt\ (g,i)=Int\_St\ (g,i)/eta\_St \tag{23}\]0  
The description of the steam storage potentially present inside the MILP model has to be completed by a group of constraints that allow one phase only (charge or discharge) to be active in each hour, with a power limited by a maximum value, previously defined. It is, therefore, necessary to introduce binary variables that indicate whether, in each hour, a storage charging or discharging phase is taking place; these variables allow a constraint to be introduced, which excludes at least one of the two phases, preventing the solver from potentially evaluating infinite pairs of input and output values to achieve a certain variation of the stored energy. The optimal size of the storage will result as the upper limit of the energy levels accumulated during the cruise, while the choice of not using it will be highlighted by a null value of this upper limit. \[Fuel\_BSt\ (g,i)=Int\_St\ (g,i)/eta\_St \tag{23}\]1  
\[Fuel\_BSt\ (g,i)=Int\_St\ (g,i)/eta\_St \tag{23}\]2  
\[Fuel\_BSt\ (g,i)=Int\_St\ (g,i)/eta\_St \tag{23}\]3  
\[Fuel\_BSt\ (g,i)=Int\_St\ (g,i)/eta\_St \tag{23}\]4  
The energy levels of the thermal storage at the beginning and at the end of the cruise are the same and they are chosen by the optimization algorithm. A completely similar group of relations describes the accumulation of HTW hot water.