## 2 Case Study and System Model
### Linearized Model of the Components
Each ORC group may operate at nominal load, with constant electrical efficiency, if the hot water at 90 \({}^{\circ}\)C is not completely consumed by the other users, otherwise, the ORC groups remain in stand-by. The electrical efficiency is set consistently with the technical data supplied by CLIMEON group [55]:  
* _eta_orc_ = 7% in the summer cruise, considering an average cooling temperature of 30 \({}^{\circ}\)C,
* _eta_orc_ = 10% in the winter cruise, considering an average cooling temperature of 15 \({}^{\circ}\)C.  
The variables describing the operation of the Stirling and ORC groups in the linear model are the following, where \(g\), \(i\) and \(j\) are generic day, hour and Stirling/ORC, respectively:  
* _Pel_str_ (\(g\),\(i\),\(j\)), electrical power produced by each Stirling [kW];
* _y_str_ (\(g\),\(i\),\(j\)), ON/OFF condition of each Stirling;
* _X_str_ (\(j\)), existence of each Stirling;
* _Pel_orc_ (\(g\),\(i\),\(j\)), electrical power produced by each ORC [kW];
* _y_orc_ (\(g\),\(i\),\(j\)), ON/OFF condition of each ORC;
* _X_orc_ (\(j\)), existence of each ORC. A group of three constraints similar to Equations (1)-(3), allows to determine the heat recovered from the ICE gases (_Hexg_) and destined for the production of steam or for feeding the Stirling, while a single linear relationship is sufficient to express the heat recovered (_HTh_), by the HTW vector, vs the load of each ICE, given its intrinsically close to a linear trend (Figures 6 and 7):  
Figure 7:0  
Figure 7:1  
Figure 7:2  
Figure 7:3  
As described above, the Stirling and ORC groups are considered to operate at a constant power value, close to the nominal one, therefore, the optimizer is called to define the optimal number of modules that will come into operation at any hour, thus operating with a \"step\" modality. The thermal input of the Stirling modules is considered directly derived by the exhaust gases line, without passing through the production of steam or hot water. For this reason, the energy exploitation by the Stirling groups is considered to result in a lower reduction in the availability of steam, compared to the exploitation by the users who actually use the steam. For this reason, the consumption of the Stirling groups (_Hexg_str_) appears multiplied by a coefficient equal to 0.9 in the subsequent thermal balance of the steam vector (Equation (39)). The thermal input of the ORC modules (_HTh_orc_) is considered instead to directly derive by the HTW line, at about 90 \({}^{\circ}\)C. The electric power produced by each Stirling group (_Pel_str_) and by each ORC group (_Pel_orc_) in each hour of the cruise, is then obtained from the following relations:  
Figure 7:4  
Figure 7:5  
Figure 7:6  
Figure 7:7  
In the following the numerical values of the efficiencies and of the COP regarded as constant, are reported for all the remaining components considered in the super-structure:  
* SB. Steam generator. It produces the integration heat when the heat recovery from the ICE is not enough. In the linearized model, a constant efficiency of approx. 70% is considered, accordingly to the average value indicated by the ship manufacturer [53]. * SS. Steam storage. Its possible presence and dimensions will be optimized according to the thermal requirements and the overall constraints of the system. Keeping in mind that all the storage will be characterized by a short operating cycle (few hours) the losses due to dispersion have been neglected, as an acceptable approximation. * HTW. Boilers, divided according to 3 different types of input power considered: liquid fuel (B), on board steam (S) and solar thermal panels (ST). For the latter, it will be necessary to carefully check the definition of a penalty function relating to the dimensions associated with a system of this type.