# Model-Based Fault Analysis and Diagnosis of PEM Fuel Cell Control System
## 2 PEMFC System Model and Verification
### PEMFC Model  
#### 2.1.1 Air Supply System  
Sufficient reactants must be provided for the fuel cell stack to produce the required amount of electric energy. Therefore, air from the atmosphere is compressed using an air compressor so that the air flows into the fuel cell stack for power generation. The air flow \begin{table}
\begin{tabular}{c c c} \hline \hline
**Subsystem** & **Type** & **Component** \\ \hline \multirow{4}{*}{Air supply} & Sensor & Air flow sensor \\  & & Air pressure sensor \\  & Actuator & Air compressor \\  & & Air shut-off valve \\ \hline \multirow{4}{*}{Fuel supply} & Sensor & Fuel flow sensor \\  & & Fuel pressure sensor \\  & Actuator & Fuel flow valve \\  & & Ejector \\ \hline Fuel cell Stack & Sensor & Voltage sensor \\ \hline \multirow{4}{*}{Sensor} & Sensor & Coolant temp. sensor \\  & Actuator & Coolant pump \\  & & Coolant flow valve \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of sensors and actuators in subsystems. rate is determined by the motor speed and the pressure difference between the atmosphere and supply manifold. Though the relationship can often be determined from a 3-D look-up table, faults cannot be mathematically analyzed with the look-up table. Therefore, in this paper, a mathematical model of the air compressor is used for the model-based fault analysis. Table 2 shows the coefficient values of air compressor equations. The equations are obtained from a surface-fitted function with [PERSON] and [PERSON]'s method from [PERSON]'s model [17]. \[M=\frac{U_{c}}{\sqrt{\gamma R_{a}T_{cp,IN}}} \tag{1}\]  
\[U_{c}=\frac{d_{c}}{2\sqrt{\theta}}\omega_{cp} \tag{2}\]  
\[\phi_{\max}=a_{4}M^{4}+a_{3}M^{3}+a_{2}M^{2}+a_{1}M+a_{0} \tag{3}\]  
\[\beta=b_{2}M^{2}+b_{1}M+b_{0} \tag{4}\]  
\[\psi_{\max} =c_{5}M^{5}+c_{4}M^{4} \tag{5}\] \[+c_{3}M^{3}+c_{2}M^{2}+c_{1}M+c_{0}\]  
\[\psi=C_{p}T_{cp,IN}\frac{\left[\left(\frac{p_{cp,IN}}{P_{cp,IN}}\right)^{ \frac{\gamma-1}{\gamma}}-1\right]}{\left(\frac{U_{c}L_{c}^{2}}{2}\right)} \tag{6}\]  
\[\phi=\phi_{max}\left[1-exp\left(\beta\left(\frac{\psi}{\phi_{max}}-1\right) \right)\right] \tag{7}\]  
\[W_{cp}=\frac{\delta}{\sqrt{\theta}}\frac{\pi\Phi\rho_{a}d_{c}{}^{2}U_{c}}{4} \tag{8}\]  
The air mass flow rate through a compressor \(W_{cp}\) is derived from the air compressor motor speed \(\omega_{cp}\), temperature of the air in the supply manifold \(T_{cp,IN}\), and pressure of the air in the supply manifold \(p_{cp,OUT}\), but the mass flow rate is also regulated by the air shut-off valve. \(M\) is the inlet Mach number and \(\phi_{\max}\), \(\beta\), and \(\psi_{\max}\) are polynomial functions of the \(M\). \(\psi\) is the dimensionless head parameter and \(\phi\) is the normalized compressor flow rate. \(U_{c}\) is compressor blade tip speed, \(\gamma\) is the ratio of the specific heats of the gas at constant pressure, \(R_{a}\) is the air gas constant, \(d_{c}\) is the compressor diameter, \(\theta\) is corrected temperature, \(\delta\) is corrected pressure, and \(\rho_{a}\) is air density. The air shut-off valve is normally open, and a fault can occur. A fault is specified as in the proportional electric valve described by [PERSON] [18]. The amount of air that passes through the valve is \begin{table}
\begin{tabular}{c c} \hline \hline
**Coefficient** & **Value** \\ \hline \(a_{4}\) & \(-3.69906^{-5}\) \\ \(a_{3}\) & \(2.70399^{-4}\) \\ \(a_{2}\) & \(-5.36235^{-4}\) \\ \(a_{1}\) & \(-4.63685^{-5}\) \\ \(a_{0}\) & \(2.21195^{-3}\) \\ \(b_{2}\) & \(1.76567\) \\ \(b_{1}\) & \(-1.34837\) \\ \(b_{0}\) & \(2.44419\) \\ \(c_{5}\) & \(-9.78755^{-3}\) \\ \(c_{4}\) & \(0.10581\) \\ \(c_{3}\) & \(-0.42937\) \\ \(c_{2}\) & \(0.80121\) \\ \(c_{1}\) & \(-0.68344\) \\ \(c_{0}\) & \(0.43331\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Coefficient values of air compressor equations. defined as the air flow rate ratio \(\phi_{air,flow}\) and is controlled by its input \(u_{value,air}\). The fault variable \(f_{value,air}\) describes the fault in the valve, and \(d_{1}\) is the characteristic coefficient of the valve.