## 3 MFCS Model
### PEM Fuel Cell Stack  
PEM fuel cell models of 20 kW, 70 kW, and 120 kW are obtained by fitting based on data from a 70 kW fuel cell system. The effective area of a single cell is 280 cm\({}^{2}\), and the number of cells is 130, 440, and 750, respectively. The PEM fuel cell outputs electrical energy, which can be simplified to an equivalent circuit model consisting of multiple resistors and capacitors connected in series or in parallel. The output voltage of a single cell can be expressed as  
\[V_{\text{cell}}=E_{\text{cell}}-V_{\text{act}}-V_{\text{ohm}}-V_{\text{conc}}\] (2) s.t. \[\left\{\begin{array}{l}E_{\text{cell}}=1.229-8.5\times 10^{-4}(T_{ \text{st}}-298)+\\ 4.308\times 10^{-5}T_{\text{st}}(\ln\frac{Pt_{\text{th}}}{1.013}+\frac{1}{2}\ln \frac{P_{\text{0}_{\text{th}}}}{1.013})\\ V_{\text{act}}=V_{0}+V_{\text{a}}(1-e^{-\zeta_{\text{i}}t})\\ V_{\text{ohm}}=IR_{\text{ohm}}\\ V_{\text{conc}}=m\exp C_{2}i\end{array}\right.,\]  
where \(V_{\text{cell}}\) denotes the single-cell output voltage; \(E_{\text{cell}}\) denotes the single-cell reversible potential, which is a temperature-dependent function; \(T_{\text{st}}\) denotes the PEM fuel cell temperature; \(p_{\text{H}_{2}}\) and \(p_{\text{O}_{2}}\) denote the pressure of hydrogen on the anode channel and oxygen on the cathode channel, respectively; \(V_{\text{act}}\) denotes the activation voltage drop; \(V_{0}\) and \(V_{\text{a}}\) are related to the fuel cell stack temperature and the cathode oxygen pressure, respectively; \(i\) denotes the current density; \(V_{\text{ohm}}\) denotes the ohmic voltage drop; \(R_{\text{ohm}}\) denotes the resistance of the PEM fuel cell; \(V_{\text{conc}}\) denotes the concentration voltage drop; \(m\) denotes an empirical parameter that is negatively correlated with stack temperature; and \(C_{1}\) and \(C_{2}\) are data-based fitting constants.  
There is transmembrane transport of liquid and nitrogen within the PEM fuel cell. Transmembrane transport of liquid is classified as proton electrical drag, concentration diffusion, and pressure diffusion. Transmembrane transport of nitrogen is caused by concentration diffusion.  
The mass flow rate of liquid water transmembrane transport in a single PEM fuel cell can be expressed as  
\[\begin{array}{l}\dot{m}_{\text{liquid}}=(N_{\text{liq,diff}}-N_{\text{liq, drag}})\times M_{\text{liq}}\times A_{\text{mem}}\\ \text{s.t}\Bigg{\{}\begin{array}{l}N_{\text{liq,diff}}=D_{\text{liq}}\frac{c _{\text{co,ca}}-c_{\text{co,an}}}{l_{\text{mem}}}\\ N_{\text{liq,drag}}=n_{\text{d}}\frac{i}{F}\end{array}\end{array}, \tag{3}\]  
where \(N_{\text{liq,diff}}\) denotes the liquid molar flow rate of the concentration diffusion; \(D_{\text{liq}}\) denotes the liquid diffusion coefficient; \(c_{\text{v,ca}}\) and \(c_{\text{v,an}}\) denote the liquid water concentration on the PEM surface of the cathode and anode, respectively; \(l_{\text{mem}}\) denotes the PEM thickness; \(N_{\text{liq,drag}}\) denotes the liquid molar flow rate of the proton electrical drag; \(n_{\text{d}}\) denotes the proton electric drag coefficient; \(F\) denotes a Faraday constant; \(M_{\text{liq}}\) denotes the molar mass of H\({}_{2}\)O; and \(A_{\text{mem}}\) denotes the effective PEM area. The liquid pressure diffusion is of a very small order of magnitude, and this phenomenon is ignored when calculating transmembrane transport of liquid.  
According to [PERSON]'s law, the mass flow rate of nitrogen that is transported across the membrane into the anode flow channel can be expressed as  
\[\dot{m}_{\text{N}_{2},\text{mem}}=M_{\text{N}_{2}}k_{\text{N}_{2}}\times \frac{\Delta p_{\text{N}_{2}}\times A_{\text{mem}}\times l_{\text{mem}}}{RT_{ \text{st}}}, \tag{4}\]  
where \(M_{\text{N}_{2}}\) denotes the nitrogen molar mass, \(k_{\text{N}_{2}}\) denotes the nitrogen diffusion coefficient, \(\Delta p_{\text{N}_{2}}\) denotes the nitrogen pressure difference between the cathode and anode, and \(R\) denotes the gas constant.