## 2 Fuel Cell
#### Fuel Cell Losses  
Due to the losses that exist within the fuel cells, achieving the ideal cell potential is not possible, and the amount of voltage generated is always less than the theoretical value, which can be defined as voltage drop as shown in Equation (8):  
\[V=E-E_{losses} \tag{8}\]  
Typically, the losses occurring within a fuel cell can be divided into two main categories: ohmic polarization loss and activation loss. The main cause of the activation losses is the slow response of chemical reactions at the surface of electrodes. This group of fuel cell losses accounts for the most contribution to the total losses, which is why it is well-known as the most important factor in the voltage drop (\(V_{act}\)). A portion of the potential is consumed for emitting electrons to electrodes; therefore, it will reduce the total actual potential [21]. This means that the electrons are continuously moving which causes a current called the Exchange Current (\(i_{0}\)). The direct relationship between the exchange current and the activation voltage drop can be obtained by the electrochemical kinetics equation, and its simplified formula is shown in Equation (9) [22]. \[i_{fc}=i_{0}e^{\frac{z\pi Fy_{act}}{RT}} \tag{9}\]  
By taking the logarithm of both sides of the above equation, the activation voltage loss for the fuel cell can be determined as follows [23]:  
\[V_{act}=-\mathrm{A}\ln i_{fc} \tag{10}\]  
\[A=\frac{RT}{z}\alpha F \tag{11}\]  
\[V_{act}=\frac{RT}{z\alpha F}\ln\left(\frac{i_{fc}}{i_{0}}\right)=A\ln\left( \frac{i_{fc}}{i_{0}}\right) \tag{12}\]  
where \(i_{fc}\) is the fuel cell current and \(A\) is the slope of the Tafel curve. It should be noted that hydrogen oxidation at the anode is much faster than oxygen reduction at the cathode, and the cathode is the major factor in limiting the dynamic behavior of the activation voltage drop [24]. When the load suddenly changes, the amplitude of the current is directly proportional to the changes. These current variations instantaneously lead to altering the ohmic losses, but the cell voltage needs time to sense the changes and reaches its steady state. This problem can be solved by considering a delay based on the cell response time (\(T_{d}\)) in the activation losses equation [25]. Therefore, the modified equation of the activation voltage drop can be found as follows:  
\[V_{act}=\frac{RT}{z\alpha F}\ln(\frac{i_{fc}}{i_{0}})e^{-sT_{d}}=\frac{RT}{z \alpha F}\ln(\frac{i_{fc}}{i_{0}})\frac{1}{sT_{d}+1} \tag{13}\]  
In addition to the activation losses, another important loss is the ohmic or resistive loss, which is related to the transportation of protons and electrons within the cell. This loss is due to the resistance of the electrons in the anode and cathode and the resistance to the membrane. The total ohmic loss can be described by multiplying the current by an internal resistance (\(R_{ohm}\)) as follows:  
\[V_{\text{ohm}}=V_{\text{ohm.a}}+V_{\text{ohm.membrane}}+V_{\text{ohm.c}}=I_{fc }\times R_{ohm} \tag{14}\]  
The exchange current \(i_{0}\) based on the hydrogen and oxygen concentration or partial pressures is derived as follows [26]:  
\[i_{0}=\frac{zFk(P_{H_{2}}+P_{O_{2}})}{Rh}e^{\frac{-\alpha C}{RT}} \tag{15}\]  
where \(k\) is the [PERSON]'s constant and \(h\) is the [PERSON]'s constant. When the current does not flow, the voltage will only depend on the inlet pressures because the utilization of hydrogen and oxygen is zero. For the currents greater than zero, the hydrogen and oxygen utilization will also affect the Nernst voltage. Thus, the open-circuit voltage would be as follows [19]:  
\[E_{\infty}= \begin{cases}K_{i}E_{n}&i_{fc}\leq 0\\ K_{c}E_{n}&i_{fc}>0\end{cases}  \tag{16}\]  
where \(K_{i}\) and \(K_{c}\) are the ratios of open circuit voltage to the Nernst voltage is zero and nominal hydrogen and oxygen utilization, respectively. According to the mentioned equations, a detailed model of the PEM fuel cell can be extracted as shown in Figure 1. Additionally, the complete model is simulated in the MATLAB Simulink environment.