## 2 Case Study and System Model
### Linearized Model of the Components  
As mentioned before, to reduce the computational effort in optimizing the on-board energy system during the whole cruise, linearized models have been used for all the components potentially present in the super-structure. In particular, the linearized models of the 12 V and 8 L Diesel engines are obtained directly from the operating curves provided by ship manufacturer [53]. To evaluate the enthalpy content, which can be extracted from the gases, a minimum temperature of 170 \({}^{\circ}\)C has been considered. The characteristic curves of the ICE are shown in Figures 6 and 7, together with the linearization equations. They show the fuel consumption, the thermal power recovered by the exhaust gases (\(Q_{evh,gas}\)) and the thermal power recovered by the H\({}_{2}\)O HT line (\(Q_{HT\text{-}TOTrecov}\)), as a function of the engine load (\(MCR\%\)). Figure 6: Operating curves of the Internal Combustion Engine (ICE) 12V, with the linearized equations. For each ICE, the variables describing the load of in the linearized models are shown below. They include binary variables that describe the ON/OFF status of each group:  
* _MCR1(g,i,j)_, % of the load in the low-load part of the operating curve (Figures 6 and 7);
* _MCR2(g,i,j)_, % of the load in the high-load part of the operating curve (Figures 6 and 7);
* _y1(g,i,j)_, ON/OFF in the low-load part of the operating curve (Figures 6 and 7);
* _y2(g,i,j)_, ON/OFF in the high-load part of the operating curve (Figures 6 and 7);  
Figure 7: Operating curves of the ICE 8L, with the linearized equations. where \(g\), \(i\) and \(j\) are generic day, hour and ICE, respectively. The constraints affecting ICE operation are:  
\[y1\ (g\),\(i\),\(j)\times MCRinf\leq MCR1\ (g\),\(i\),\(j)\leq y1\ (g\),\(i\),\(j)\times MCRm \tag{1}\]  
\[y2\ (g\),\(i\),\(j)\times MCRm\leq MCR2\ (g\),\(i\),\(j)\leq y2\ (g\),\(i\),\(j)\times MCRsup \tag{2}\]  
\[y1\ (g\),\(i\),\(j)+y2\ (g\),\(i\),\(j)\leq 1 \tag{3}\]  
The constraints expressed by Equations (1) and (2), allow the choice variables \(MCR1\) and \(MCR2\) to take positive values only if the \(j\)-th motor is ON in the \(i\)-th hour of the \(g\)-th day and the engine load is within the permitted operating ranges. The last constraint (Equation (3)) states that only one of the two choice variables \(MCR1\) and \(MCR2\) can actually be non-zero. The actual load--\(MCR\) (\(g\), \(i\), \(j\))--and the electrical power produced by each motor--_Pel (\(g\), \(i\), \(j\))--_can therefore be calculated as:  
\[MCR\ (g\),\(i\),\(j)=MCR1\ (g\),\(i\),\(j)+MCR2\ (g\),\(i\),\(j) \tag{4}\]  
\[Pel\ (g\),\(i\),\(j)=MCR\ (g\),\(i\),\(j\))/100 \(\times Pmax\ (j)\ \times eta\_el \tag{5}\]  
Engine fuel consumption (fuel) is obtained from the linearized characteristic curves in a similar way to Equations (1)-(5) and are calculated using the following Equations (6)-(8), where it is necessary to specify a pair of coefficients (\(A\), \(B\)) for each linearization part:  
\[Fuel1\ (g\),\(i\),\(j)=A1\ (j)\times(MCR1\ (g\),\(i\),\(j)\ -\ MCRinf\times y1\ (g\),\(i\), \(j))+B1\ (i)\times y1\ (g\),\(i\),\(j) \tag{6}\]  
\[Fuel2\ (g\),\(i\),\(j)=A2\ (j)\times(MCR2\ (g\),\(i\),\(j)\ -\ MCRinf\times y2\ (g\),\(i\),\(j))+B2\ (i)\times y2\ (g\),\(i\),\(j) \tag{7}\]  
\[Fuel\ (g\),\(i\),\(j)=Fuel1\ (g\),\(i\),\(j)+Fuel2\ (g\),\(i\),\(j) \tag{8}\]  
The power produced and the thermal flows consumed at the different loads by the ORC groups have been modeled on the basis of the data available for the Cimeon group [55]. They show an efficiency dependent on the temperatures of the supply thermal vector and of the cooling flow, therefore the efficiency can be regarded as constant if the group is fed by the hot water line at 85-90\({}^{\circ}\)C (H\({}_{2}\)O HT), and the cooling flow is sea water, which temperature depends only by the winter/summer conditions. The model considers one or more ORC groups, all of the same nominal size (150 kW), powered by the recovery circuit on high temperature with a constant flow rate of 40 L/s.