# Modelica-Based Energy Management of PEMFC Hybrid Power System of Vehicle
## 2 Mathematic Model of Hybrid Power System
### PEMFC Model
#### 2.1.1 Stack Voltage  
For PEMFC, the cell voltage is produced by the electrochemical reaction of hydrogen and oxygen [24]. The actual output circuit voltage can be stated as Equation (1):  
\[V_{oc}=V_{Nernst}-V_{act}-V_{ohmic}-V_{conc} \tag{1}\]  
where \(V_{Nernst}\) indicates the Nernst open circuit voltage, \(V_{act}\) is the activation polarization voltage, \(V_{ohmic}\) is the ohmic polarization voltage, and \(V_{conc}\) is the concentration polarization voltage. In the model, \(V_{Nernst}\) is the theoretical reversible voltage at a certain temperature effect and pressure of hydrogen and oxygen [25]. Equation (2) gives the numerical expression while the liquid water is produced by the electrochemical reaction. \[E_{Nernst}=1.229-8.5\times 10^{-5}(T_{FC}-T_{0})+4.3085\times 10^{-5}T_{FC} \big{[}\ln\big{(}P_{H_{2},an}\big{)}+0.5\ln\big{(}P_{O_{2},ca}\big{)}\big{]} \tag{2}\]  
where the temperature of fuel cell \(T_{FC}\) is provided by the temperature calculation model described in Section 2.1.2, \(T_{0}\) denotes the reference temperature (\(T_{0}\) = 298.15 K), and \(P_{H_{2},an}\) and \(P_{O_{2},ca}\) are the partial pressure of anode and cathode reactants, respectively. The activation polarization voltage mainly originates from the energy required to transfer electrons during the electrochemical reaction, as well as the energy involved  
Figure 2: Structure diagram of the PEMFC. in breaking or forming chemical bonds at the anode and cathode sides [26]. Therefore, activation losses exist on both the cathode and anode sides. Generally, the chemical reaction of oxygen is slower compared to the reduction reaction of hydrogen. Equation (3) is a theoretical expression according to the Butler-Volmer equation. \[V_{act}=\xi_{1}+(\xi_{2}+\xi_{3}\ln(CO_{2})+\xi_{4}\ln(I_{FC}))T_{FC} \tag{3}\]  
where \(\xi_{1}\), \(\xi_{2}\), \(\xi_{3}\), and \(\xi_{4}\) are the empirical coefficient, \(CO_{2}\) is the oxygen concentration of cathode catalyst interface, and \(I_{FC}\) is the output current of the external circuit. The ohmic polarization voltage losses consist of the electrode resistance and the internal resistance of the PEM [27]. As the equivalent membrane impedance of the PEM is the main contributor to the ohmic polarization voltage, it can be described using Equation (4):  
\[V_{olmic}=I_{FC}\times R_{ohmic}=I_{FC}\times(R_{mem}+R_{electrode})=I_{FC}\times R_{mem} \tag{4}\]  
where \(R_{ohmic}\), \(R_{mem}\), and \(R_{electrode}\) denote the total resistance, PEM resistance, and the electrode resistance, respectively. Theoretically, the PEM resistance strongly depends on the moisture of PEM, current density, and working temperature of PEMFC. \[R_{mem}=\frac{l_{mem}}{\sigma_{mem}} \tag{5}\]  
where \(l_{mem}\) is the thickness of the membrane, and \(\sigma_{mem}\) is the conductivity of PEM as the following formula:  
\[\sigma_{mem}=\frac{181.6\times\left[1+0.03\frac{I_{FC}}{S_{FC}}+0.062{(\frac{ T_{FC}}{303})}^{2}{\left(\frac{I_{FC}}{S_{FC}}\right)}^{2.5}\right]}{\left[ \lambda_{mem}-0.634-\frac{3I_{FC}}{S_{FC}}\right]\exp{\left[4.18{(\frac{T_{FC }-303}{I_{FC}})}\right]}} \tag{6}\]  
where \(S_{FC}\) is the effective area of PEM, and \(\lambda_{mem}\) is the membrane water content indicating the following equation:  
\[\lambda_{mem}=\left\{ \begin{array}{l}0.0043+17.81a_{w}-39.85a_{w}^{2}+36a_{w }^{3},0\leq a_{w}\leq 1\\ 14+1.4(a_{w}-1),a_{w}>1\end{array} \right. \tag{7}\]  
where \(a_{w}\) is the water activity. The PEM must maintain a certain level of hydration to ensure efficient proton transport through the membrane. Therefore, its conductivity is related to the water content. When the water content in the membrane is at an optimal level, it not only maximizes conductivity but also effectively reduces the internal resistance of PEMFC. The concentration polarization voltage arises from the potential difference caused by the reduction in reactant concentration as they are consumed [28]. This loss is the primary reason for the rapid voltage drop of the fuel cell stack at high current densities.