# Adaptive Feedback Control for Four-Phase Interleaved Boost Converter Used with PEM Fuel Cell +
## 2 Four-Phase IBC and PEM Fuel Cell Modeling
### PEM Fuel Cell Modeling  
Historically, the principle of the fuel cell was discovered in 1839 by the British physicist [PERSON] [24]. The PEM fuel cell is widely used in industrial applications, particularly in vehicles. In fact, the PEM fuel cell is an electrochemical device that converts hydrogen energy into electrical energy. Figure 1 shows an equivalent electrical model of a single cell, where \(C_{cell}\) represents the equivalent capacitor, \(E\) indicates the reversible potential, and \(R_{act}\), \(R_{conc}\), and \(R_{ohmic}\) correspond to the equivalent resistances of activation, concentration, and ohmic effects, respectively. The reactions that occur at the anode and cathode of the PEM fuel cell, respectively, are the oxidation of \(H_{2}\) and the reduction of \(O_{2}\), which can be expressed as follows [24]:  
\[Anode:2H_{2}\Rightarrow 4H^{+}+4e^{-} \tag{1}\]  
\[Cathode:O_{2}+4H^{+}+4e^{-}\Rightarrow 2H_{2}O \tag{2}\]  
The overall reaction at the heart of a fuel cell is given by  
\[O_{2}+2H_{2}\Rightarrow 2H_{2}O+electric+heat \tag{3}\]  
Achieving an accurate PEM fuel cell model necessitates incorporating multidisciplinary knowledge encompassing thermodynamics, electrochemistry, and fluid mechanics. The analysis of energy conversion demands a mathematical representation of the PEMFC's output voltage. By considering all voltage drops related to losses within the PEMFC model, the operational voltage of an individual cell within a PEM stack can be expressed as follows [24,25]:  
\[v_{coll}=E-v_{act}-v_{ohm}-v_{conc} \tag{4}\]  
Figure 1: The equivalent electrical circuit of one single cell. where \(E\) is the Nerst voltage, and \(v_{act}\), \(v_{ohm}\), and \(v_{conc}\), correspond to the activation loss, ohmic loss, and concentration loss, respectively. The open-circuit voltage \(E\) of the fuel cell can be found in [26]:  
\[E=1.229-8,5.10^{-4}(T_{cf}-298.15)+4,3085.10^{-5}T_{cf}ln(P_{H_{2}}P_{O_{2}}^{ \frac{1}{2}}) \tag{5}\]  
where \(T_{cf}\) is the fuel cell temperature in \((Kelvin)\), and \(P_{O_{2}}\) and \(P_{H_{2}}\) are the partial pressures of oxygen and hydrogen, respectively. The activation overvoltage can be approximated by the Tafel equation:  
\[v_{act}=v_{0}+v_{a}(1-e^{-c_{i}}) \tag{6}\]  
with \(v_{0}\), \(v_{a}\), and \(c\) being parameters that depend on the partial pressure of oxygen and the temperature of the system; their values can be determined from a nonlinear formula based on experimental data. \(i\) is the fuel cell current. For the ohmic voltage, which represents the voltage drop due to the internal resistance of the fuel cell, it includes both the voltage drop across the ionic resistance in the electrolyte and the electronic resistance in the cell components. The ohmic voltage can be approximated by the following equation:  
\[v_{ohm}=R_{ohm}.i \tag{7}\]  
where \(R_{ohm}\) is the ohmic resistance, representing the sum of the resistance of the polymer membrane to the transfer of protons and the resistance of the electrodes and collector plates to the transfer of electrons. For more information, refer to [26]. The nominal values of the system are provided in Table 1. The concentration overvoltage arises from increased losses at high current densities, such as a significant reduction in reactant concentration due to both high consumption of reactants and pressure drop at high flow rates. Some models, like [27], omit this term, likely because operating the fuel cell stack in regions where concentration overvoltage is high leads to lower efficiency. However, if the stack is intended to operate at high current densities, this term becomes essential and must be accounted for. An equation to estimate concentration losses is provided in [27]:  
\[v_{conc}=i(c_{1}\frac{i}{I_{max}})^{c2} \tag{8}\]  
with \(c_{1}\) and \(c_{2}\) being constants that depend on the system temperature and the partial pressure of the reactants, which can be determined empirically. \(I_{max}\) represents the maximum current of the fuel cell.