## 2 Methodology  
### Proton Exchange Membrane Fuel Cell (PEMFC) Model  
A polarisation curve model has been used to model the operation of a monocell pure hydrogen PEMFC with MATLAB/SIMULINK. The model, which was proposed by [PERSON] [5], and was used in the MATLAB/SIMULINK environment, is described by the following equations:  
\[\mathrm{V_{cell}=E_{nerst}-V_{act}-V_{conc}-V_{ohm}} \tag{1}\]  
\[\mathrm{E_{nerst}=1.229-0.85*10^{-3}(T-298.15)+\frac{RT}{2F}\log\left(P_{H2}*P_ {O2}^{0.5}\right)} \tag{2}\]  
\[\mathrm{V_{act}=v_{0}+v_{a}[1-exp(-c_{1}*i)]} \tag{3}\]  
\[\mathrm{V_{0}=0.279-0.85*10^{-3}(T-298.15)+\frac{RT}{2F}\log\left[\ \left(P_{H,in}-P_{sat}\right)*0.1173*\left(P_{A,in}-P_{sat}\right)^{0.5}\right]} \tag{4}\]  
Figure 1: Influence of the operating pressure on the fuel cell output power. \[\mathrm{V_{a}}=\left[-1.618*10^{-5}*\mathrm{T}+1.168*10^{-2}\right]\left[\frac{ \mathrm{P_{O2}}}{0.1173}+\mathrm{P_{sat}}\right]^{2}+\left[1.8*10^{-4}*\mathrm{ T}-0.166\right] \tag{5}\] \[\left[\frac{\mathrm{P_{O2}}}{0.1173}+\mathrm{P_{sat}}\right]+ \left[-5.8*10^{-4}*\mathrm{T}+0.5736\right]\] \[\mathrm{V_{conc}}=\mathrm{i*}\left(\mathrm{c_{2}*\frac{i}{i_{L}}} \right)^{\mathrm{c_{3}}}\] (6) \[\mathrm{c_{2}}=\left[8.66*10^{-5}*\mathrm{T}-0.068\right)\left[ \frac{\mathrm{P_{O2}}}{0.1173}+\mathrm{P_{sat}}\right]-\left(1.6*10^{-4}* \mathrm{T}+0.54\right)\] (7) \[\mathrm{V_{ohm}}=\mathrm{R_{i}*i} \tag{8}\]  
where \(\mathrm{V_{cell}}\) is the output tension of the monocell. \(\mathrm{E_{nerst}}\) is the Nerst potential. \(\mathrm{V_{act}}\), \(\mathrm{V_{conc}}\) and \(\mathrm{V_{ohm}}\) are respectively the activation, mass transfer, and ohmic losses. C1 and C3 are given by a recent paper concerning air supply system control [6] as C1 = 10 and C3 = 2. The total output power of the N-cells' stack is calculated as:  
\[\mathrm{E_{nerst}=1.229-0.85*10^{-3}(T-298.15)+\frac{RT}{2F}\log\left(P_{H2}*P_ {O2}^{0.5}\right)} \tag{2}\]0  
The electrochemical reaction is considered as stoichiometric. The system is supposed to run with an excess of air. The current is calculated from the hydrogen mass flow rate, and the excess of air is included in the calculation of oxygen partial pressure. As a result, the PO2 and PH2 from previous equations are calculated by taking the average between the inlet and outlet stack pressure as follows:  
\[\mathrm{E_{nerst}=1.229-0.85*10^{-3}(T-298.15)+\frac{RT}{2F}\log\left(P_{H2}*P_ {O2}^{0.5}\right)} \tag{2}\]1  
\[\mathrm{E_{nerst}=1.229-0.85*10^{-3}(T-298.15)+\frac{RT}{2F}\log\left(P_{H2}*P_ {O2}^{0.5}\right)} \tag{2}\]2  
where \(\mathrm{Dm_{O,in}}\) and \(\mathrm{Dm_{H,in}}\) are the inlet mass flow rates, and \(\mathrm{Dm_{O,out}}\) and \(\mathrm{Dm_{H,out}}\) are the outlet mass flow rates of oxygen and hydrogen, respectively. \(\mathrm{P_{A,in}}\) and \(\mathrm{P_{H,in}}\) are the inlet pressures of air and hydrogen. As seen in Figure 2, the MATLAB/SIMULINK model runs as a black box in the GT-SUITE environment. The \"PEMFCs_model\" refers to the MATLAB function using the model described. The inputs are the inlet mass flow rates (\(\mathrm{Dm_{A,in}}\), \(\mathrm{Dm_{H,in}}\)), the inlet pressures (\(\mathrm{P_{A,in}}\), \(\mathrm{P_{H,in}}\)), and the required power by the air supply system (\(\mathrm{P_{comp}}\)). The outputs include the outlet mass flow rates (\((\mathrm{Dm_{A,out}}\), \(\mathrm{Dm_{H,out}}\)), the outlet air pressure (\(\mathrm{P_{A,out}}\)), the output produced power, the current, and the electric efficiencies (\(\mathrm{P}\), \(\mathrm{I}\), \(\mathrm{R_{eff}}\), \(\mathrm{R_{eff,system}}\)). There are different ways of calculating the electric efficiency.