## 2 Case Study and System Model
### Energy Balances and Objective Functions  
On the basis of the documentation made available by the manufacturer [53], the electrical and thermal demand curves have been identified for the two type of cruises (summer and winter).  
The MILP model calculates, for each hour of the cruise, the balances of the nodes where the energy vectors produced by the system have to satisfy the user demands (Equations (38)-(41)): the electric demand (RICEL), the steam demand (RICTER_St), the hot water demand (RICTER_HTW) and the chilled water demand (RICCOOL).  
\[\sum_{j=1}^{N_{ICE}}pel(g,i,j)+\sum_{j=1}^{N_{Sat}}pel\_str(g,i,j)+\sum_{j=1}^{ N_{Sat}}pel\_orc(g,i,j)+Ppv(g,i)-pel\_com\ (g,i)\geq RICEL\ (g,i) \tag{38}\]  
\[\sum_{j=1}^{N_{ICE}}\textit{Hexg}(g,i,j)+ Int\_St(g,i)+ Output\_St\ (g,i)-Input\_St\ (g,i)-SttoHTW\ (g,i)- \tag{39}\] \[0.9\times\sum_{j=1}^{N_{Sup}}\textit{Hexg\_str}(g,i)\geq RICTER \_St\ (g,i)\]\[\sum_{j=1}^{N_{ICE}}HTh(g,i,j)+Int\_HTW\left(g,i\right)+SttoHTW\left(g,i\right)+ Output\_HTW\left(g,i\right)- \tag{40}\] \[Input\_HTW\left(g,i\right)-HTh\_abs\left(g,i\right)-\sum_{j=1}^{N_{ NC}}HTh\_or\left(g,i,j\right)+Pst\left(g,i\right)\geq RICTER\_HTW\left(g,i\right)\] \[Cool\_abs\left(g,i\right)+Cool\_com\left(g,i\right)+Output\_CW \left(g,i\right)-Input\_CW\left(g,i\right)\geq RICCOOL\left(g,i\right) \tag{41}\]  
Stirling engines are powered directly by the ICE flue gases, but are considered within the steam carrier balance, by introducing an appropriate reduction coefficient of consumption (0.90), which takes into account the direct transfer of thermal energy, without requiring the intervention of recovery boilers (EGRS).  
As previously mentioned, the thermal demands classified as low temperature (40-45 \({}^{\circ}\)C) are considered to be certainly satisfied by the low temperature hot water produced by the ICE, without the need for integrations, so that the H\({}_{2}\)O LT balance is not considered in the optimization procedure.  
In this paper, the MILP problem is solved by means of a multi-objective optimization, considering to minimize fuel consumption during the cruise and, at the same time, the size of the additional energy recovery system. This second objective is quantified by the PENALTY function which assigns 1 pt of penalty equivalent to approximately 10 m\({}^{3}\) of additional volume occupied.  
For thermal storage, absorption chillers, photovoltaic panels and solar collectors, penalties proportional to the optimized size are introduced.  
A general objective function (\(OB\)/\(Func\)) is also defined for the MILP problem, consisting of the weighted sum of the fuel consumption during the cruise, and of the PENALTY function.  
\[Fuel\_TOT=\sum_{g=1,\ i=1}\sum_{j=1}^{N_{ICE}}Fuel(g,i,j)+Fuel\_BSt(g,i)+Fuel\_BHt (g,i) \tag{42}\]  
\[PENALTY=\sum_{j=1}^{N_{NI}}PEN\_or(j)\times X\_or(j)+\sum_{j=1}^{N_{HP}}PEN\_ str(j)\times X\_str(j)+Apv\times PENpv+ \tag{43}\]  
\[Ast\times PENst+Cool\_abs\_size\times PENabs+Lev\_St\_size\times PEN\_St+Lev\_HTW\_size \times PEN\_HTW\] \[OB \tag{44}\]  
The \(PEN\)_conv_ conversion factor has the purpose of making the two terms of the objective function homogeneous [kWh] and to allow changing the relative importance of one objective with respect to the other.  
Note that, in the perspective of multi-objective optimization, a specific value of the \(PEN\)_conv_ conversion factor simply identifies a point on the Pareto front, so that the possibility offered by the evolutionary approach with the software modeFRONTIER(r) to obtain the complete front, regardless of the value of \(PEN\)_conv_, makes a careful definition of this coefficient absolutely non-critical.