## 2 Methodology
### Solid Oxide Fuel Cell (SOFC) Model  
A polarisation curve model has been also used to model the operation of a monocell pure hydrogen solid oxide fuel cell (SOFC) with MATLAB/SIMULINK. The model used has been presented in a recent paper written by [PERSON] et al. [13]. This model takes into consideration the physical parameters of the cell such as diffusion coefficient and electrode thicknesses. It includes the Nerst potential, ohmic, activation, and mass transport losses, which are calculated as follows:  
\[\mathrm{E_{nerst}}=(1.253-0.000245*\mathrm{T})+\frac{\mathrm{RT}}{2\mathrm{F }}\log\left(\frac{\mathrm{P_{H2}^{\prime}*P_{O2}^{\prime 0.5}}}{\mathrm{P_{water}^{ \prime}}}\right) \tag{14}\]  
\[\mathrm{V_{act}}=\frac{\mathrm{RT}}{2\mathrm{F}}\mathrm{arcsinh}\left(\frac{ \mathrm{i}}{2\mathrm{i_{0,anode}}}\right)+\frac{\mathrm{RT}}{2\mathrm{F}} \mathrm{arcsinh}\left(\frac{\mathrm{i}}{2\mathrm{i_{0,cathode}}}\right) \tag{15}\]  
where i_(0,cathode) and i_(0,cathode) are the limiting current densities at the cathode and the anode, respectively. The mass transport losses are given by:  
\[\mathrm{V_{conc}}=-\frac{\mathrm{RT}}{2\mathrm{F}}\ln\left(\frac{\mathrm{P_{ H2}^{\prime}}}{\mathrm{P_{water}^{\prime}}}*\frac{\mathrm{P_{W,in}}}{\mathrm{P_{H,in}}} \right)-\frac{\mathrm{RT}}{4\mathrm{F}}\ln\left(\frac{\mathrm{P_{O2}^{\prime}} }{\mathrm{P_{O,in}}}\right) \tag{16}\]  
where \(\mathrm{P_{H,in}}\), \(\mathrm{P_{W,in}}\) and \(\mathrm{P_{O,in}}\) are the stack inlet pressures of hydrogen, water, and oxygen, respectively. \(\mathrm{P_{H2}^{\prime}}\), \(\mathrm{P_{water}^{\prime}}\) and \(\mathrm{P_{O2}^{\prime}}\) are the hydrogen, water, and oxygen pressures on the electrode's surface. They are given by the following equations:  
\[\mathrm{P_{O2}^{\prime}}=\mathrm{P_{O,in}}-\left(\frac{\mathrm{i *RT}}{\mathrm{P_{A,in}*4F}}\right)\left[\left(\  
still the electric efficiency of the stack, whereas the efficiency \(\mathrm{R_{eff,GT}}\) was the thermodynamic efficiency of the system {burner + gas turbine + heat exchanger}, which was calculated as follows:  
\[\mathrm{R_{eff,\ GT}}=\frac{\mathrm{P_{generator}+P_{heat}}}{\mathrm{Dm_{A,in}*C_{ p}*(T_{burner,out}-T_{burner,in})}} \tag{20}\]  
where \(\mathrm{C_{p}}\) is the inlet burner air's calorific capacity. \(\mathrm{T_{burner,out}}\) and \(\mathrm{T_{burner,in}}\) are the outlet and inlet burner temperatures, respectively, \(\mathrm{P_{generator}}\) is the electric power produced by the motor generator, and \(\mathrm{P_{heat}}\) is the power transferred to the stack inlet air through the heat exchanger. Many papers have referred to \"system efficiency\". It must be noticed that \(\mathrm{R_{eff,GT}}\) is a thermodynamic efficiency, whereas \(\mathrm{R_{eff}}\) is not. Nevertheless, it is possible to use a formulation as the ratio of the useful power and the theoretical maximum power. As a result, system efficiency can be formulated as follows:  
\[\mathrm{R_{eff,system}}=\frac{\mathrm{P+P_{generator}+P_{heat}}}{(1.481* \mathrm{N*I})+\left[\mathrm{Dm_{A,in}*C_{p}*(T_{burner,out}-T_{burner,in})} \right]} \tag{21}\]  
As seen in Figure 5, the GT-SUITE model of the hybrid system {SOFC + gas turbine} was similar to the PEMFC stack model. It must be noticed that this model included three proportional-integral-derivative (PID) controllers. The first one controlled the output power by taking into consideration the output power of the SOFC stack and the recovered power through the turbine. The second PID controlled the injection of the fuel into the burner. It adjusted the hydrogen mass flow rate to reach the targeted temperature. In an actual SOFC system, the hydrogen is not injected; rather, it is collected at the stack's output [14]. Nevertheless, this model assumed a total consumption of the hydrogen at inlet. So, in order to consider the global consumption of hydrogen, an injector equipped with a controller was used. A classical burner output temperature for an SOFC stack using such a system is 1200 \({}^{\circ}\)C [14].