## 3 Model Development
### Dynamic Mechanistic Model
#### 3.2.1 Stack Voltage Model  
The stack voltage can be analytically expressed by Nernst voltage and three voltage losses: activation, ohmic, and concentration [33; 34; 35]. Since boundary conditions limit some mathematical formulas, empirical formulas are used instead. The operating voltage of a single fuel cell is expressed as  
\[v_{fc}=E-v_{act}-v_{ohm}-v_{conc} \tag{1}\]  
where \(v_{fc}\) is the output voltage of a single cell, the total voltage of the stack is calculated by multiplying the cell number by the output voltages of single cells, \(E\) is the Nernst voltage, \(v_{act}\) is the activation polarization voltage, \(v_{ohm}\) is the ohmic loss, and \(v_{conc}\) is the concentration polarization loss.  
Based on the heating value at standard conditions, the Nernst voltage is expressed as [33]  
\[\begin{array}{l}E=1.229-0.85\times 10^{-3}(T_{st}-298.15)\\ +4.3085\times 10^{-5}T_{st}[\ln(p_{H_{2},in})+\frac{1}{2}\ln(p_{\mathrm{O_{2}},in })]\end{array} \tag{2}\]  
In this paper, the activation polarization voltage uses an empirical equation, expressed as [20]  
\[v_{act}=v_{o}+v_{a}(1-e^{-c_{1}I}) \tag{3}\]  
\[c1=10 \tag{4}\]  
Figure 3: Structural diagrams of the dynamic mechanistic model.  
The empirical voltage parameters \(\
u_{a}\) and \(\
u_{o}\) are expressed using empirical fitting equations as follows:  
\[\begin{array}{l}\mathcal{V}_{0}=0.279-8.5\times 10^{-4}(T_{st}-298.15)+4.308 \times 10^{-5}T_{st}[\ln(\frac{P_{0,in}-P_{out}}{1.01325})\\ \qquad+\frac{1}{2}\ln(\frac{0.1173(p_{out}-p_{out})}{10.1325}))]\\ \\ \mathcal{V}_{a}=(-1.618\times 10^{-5}T_{st}+1.618\times 10^{-2}){(\frac{P_{0,in}}{ 0.1173}+p_{sat})}^{2}+(1.8\times 10^{-4}T_{st}\\ \qquad-0.166)(\frac{P_{0,in}}{0.1173}+p_{sat})+(-5.8\times 10^{-4}T_{st}+0.5736) \\ p_{sat}=(2.6032\times 10^{-6}T_{st}{}^{5}+3.1485\times 10^{-4}T_{st}{}^{4}+2.336 5\times 10^{-2}T_{st}{}^{3}\\ \qquad+1.55\times T_{st}{}^{2}+42.8T_{st}+615.62)/1000\end{array} \tag{5}\]  
where \(p_{sat}\) is the saturated vapor pressure.  
The ohmic loss is the voltage loss due to the membrane impedance and is calculated from [PERSON]'s law:  
\[\
u_{ohm}=I\cdot R_{ohm} \tag{8}\]  
Membrane impedance is expressed as a function of membrane conductivity \(\sigma_{m}\) and membrane thickness [36]:  
\[R_{ohm}=\frac{t_{m}}{\sigma_{m}} \tag{9}\]  
The membrane conductivity \(\sigma_{m}\) is correlated with water content \(\lambda_{m}\) and stack temperature \(T_{fc}\), varying between 0 and 14 according to relative humidity [37], expressed as  
\[\sigma_{m}=b_{1}\exp(b_{2}(\frac{1}{303}-\frac{1}{T_{st}})) \tag{10}\]  
\[b_{1}=(0.005139\lambda_{m}-0.00326) \tag{11}\]  
\[b_{2}=350 \tag{12}\]  
The concentration voltage loss is caused by a drop in the reactant gas concentration, responsible for the rapid drop in the output voltage at high current density. The concentration voltage loss \(\
u_{conc}\) is expressed as  
\[\
u_{conc}=I\cdot(c_{2}\cdot\frac{I}{I_{\max}})^{c_{3}} \tag{13}\]  
\[I_{\max}=2.2 \tag{14}\]  
\[c_{2}=\begin{cases}(7.16\times 10^{-4}T_{st}-0.622)(\frac{P_{0,in}}{0.1173}+p_{ sat})&(\frac{P_{0,in}}{0.1173}+p_{sat})<2\\ \qquad+(-1.45\times 10^{-3}T_{st}+1.68)&(\frac{P_{0,in}}{0.1173}+p_{sat})<2\\ (8.66\times 10^{-5}T_{st}-0.068)(\frac{P_{0,in}}{0.1173}+p_{sat})&(\frac{P_{0,in }}{0.1173}+p_{sat})>2\\ \qquad+(-1.6\times 10^{-4}T_{st}+0.54)&(\frac{P_{0,in}}{0.1173}+p_{sat})>2\end{cases} \tag{15}\]  
\[c_{3}=2 \tag{16}\]  
where \(I_{\max}\) [A/cm\({}^{2}\)] is the current density threshold that causes the output voltage to drop rapidly with the increase in current density. Its value is set using experiment results. \(c_{2}\) is set using parametric nonlinear fitting, and \(c_{3}\) is an empirical parameter [19].