# Model-Based Fault Analysis and Diagnosis of PEM Fuel Cell Control System
## 3 Fault Analysis  
Fault analysis can be conducted with a mathematical model of PEM fuel cells. When a fault occurs, it propagates through variables that are related by mathematical equations. On the basis of this principle, the direction of the fault must be clarified by examining the relationship among equations. When the values of the known variables (e.g., control input, sensor measurement) are determined, the values of unknown variables can also be obtained with equations. Therefore, an incidence matrix, which is a logical matrix used to present the relationship between variables, is used. There are known variables and unknown variables, which are the variables that cannot be measured, and faults. After clarifying the relationship between the variables of the model, the Dulmage-Mendelsohn decomposition is applied to the incidence matrix to determine the equation set with redundancy. With the Dulmage-Mendelsohn decomposition, equations are classified into three parts: the overdetermined part, the determined part, and the underdetermined part [22]. In the overdetermined case, the number of equations is greater than the number of unknown variables by more than one. To specify the unknown variables, the same number of unknown variables and unique equations are required. When there are more equations than unknown variables, the redundant equations can be used to determine the fault. The value of the fault variable will be zero in the case where there is no fault. The number of redundant equations is called the degree of redundancy. As the degree of redundancy increases, there are more methods to obtain the values of fault variables. In the determined case, the number of equations is the same as the number of unknown variables. Therefore, unknown variables can be specified, but there are no redundant equations and there is no possibility of determining the value of the fault variable. In the underdetermined case, the number of equations is less than the number of unknown variables that are related. There is no possibility of unknown variables nor of fault variables being determined. As the number of sensors in the system increases, the degree of redundancy tends to get higher since only a single fault is considered. When multi-fault is considered, the degree of redundancy will remain the same, even though additional sensors are applied in the system because of the possibility of the fault of added sensors. Hardware redundancy and the structural conversion of the system for increasing the number of relations of variables are the solutions to increase the degree of redundancy in multi-fault cases. Prior to the fault analysis, components that are vulnerable to faults are chosen: faults in sensors, faults in actuators, and faults chosen from the study of the fault tree analysis (FTA) on PEM fuel cells [23; 24]. Faults in sensors and actuators from Table 1 are both considered. From FTA, hydrogen leakage of PEMCs and membrane degradation shows the highest mean failure rate among others. The failure rate was calculated from a Weibull distribution, which is used for lifespan data analysis. Furthermore, the stack voltage drop arises from PEMFC degradation. Fault equations are added into the system equation set, and fault analysis is conducted. The incidence matrix shows the relationship between variables. With the incidence matrix, the overdetermined part of the result of the Dulmage-Mendelsohn decomposition is shown in Figure 4. Analytical redundancy exists in the overdetermined part, and single faults in this area are detectable. Faults not included in the overdetermined part were undetectable due to a lack of redundancy, and appropriate sensor placement is required to diagnose those faults. In the overdetermined case, residuals are obtained from a bi-partite graph. Bi-partite graphs link variables and show how they are affected by other related variables. The residual is calculated from the difference between measured outputs, between estimated values of a variable, or between the measured output and estimated values.