## 2 Methodology
### Fault Detection and Isolation (FDI)
#### 2.3.2 Statistical Process Control (SPC) Approach  
Statistical process control (SPC) is a quality-control method used to monitor and control processes to ensure they operate consistently within specified limits. It involves the \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**MCD Test No.** & \begin{tabular}{c} **Degradation Level** \\ **(Component 1)** \\ **(in Percentage of** \\ **DPV Opening)** \\ \end{tabular} & \begin{tabular}{c} **Degradation Level** \\ **(Component 2)** \\ **(in Percentage of** \\ **DPV Opening)** \\ \end{tabular} & \begin{tabular}{c} **Operational Speed** \\ **Range (rpm)** \\ \end{tabular} & **Fault Combination** \\ \hline T13 (1) & Pump (medium severity) & Constant degradation of & & \\  & at 45\% & nozzle (30–100\% severity) & 700 & FC2 and FC4 \\ T14 (1) & Pump (medium severity) & Constant degradation of & & \\  & at 45\% & nozzle (30–100\% severity) & 950 & FC2 and FC4 \\ T15 (1) & Filter (high severity) at 68\% & Healthy condition & & \\  & & (0\% severity) & & \\ T16 (1) & Pump (medium severity) & Nozzle (medium severity) & 700 to 950 & FC2 and FC4 \\ T17 (1) & \begin{tabular}{c} Constant degradation of \\ pump (0–100\% severity) \\ \end{tabular} & \begin{tabular}{c} Constant degradation of \\ pipe (0–100\% severity) \\ \end{tabular} & 800 & FC2 and FC5 \\ T18 (1) & \begin{tabular}{c} Intermittent faults for the \\ pump between a 45\% and \\ 60\% level of severity \\ \end{tabular} & \begin{tabular}{c} Healthy condition \\ (0\% severity) \\ \end{tabular} & 850 & FC2 only \\ T19 (1) & \
\begin{tabular}{c} Constant degradation of \\ pump (0–100\% severity) \\ \end{tabular} &
\begin{tabular}{c} Constant degradation of \\ valve (30–100\% severity) \\ Nozzle (high severity) \\ at 55\% \\ \end{tabular} & 850 & FC2 and FC3 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Test degradation scenarios (the fault in component 1 is initiated before that of component 2). Figure 4: Deviation of process parameters from healthy condition for T13. Figure 5: Deviation of process parameters from healthy condition for T17. use of statistical techniques to analyse process data, identify variations, and take corrective actions when necessary. Threshold testing is a critical aspect of statistical process control (SPC) used to determine whether a process is operating within acceptable limits. In SPC, control charts are employed to monitor process data over time, and threshold testing involves setting limits or thresholds on these charts to distinguish between common cause variation and special cause variation [25]. In this report, Equation (7) was used to calculate the upper and lower bounds for pressure and flow measurements at each pump speed for each faulty condition scenario, where \(\mu\) and \(\sigma\) are the mean and standard deviation of the sensor measurements, respectively. Equation (8) was also used to develop the SPC table at each pump speed for each faulty condition scenario, which was then used to set the thresholds for each component failure. \(T_{1}\) and \(T_{2}\) represent the lower and upper bounds, respectively, of the process dynamic faulty condition scenarios FC1, FC2, FC3, FC4, and FC5, and the logic condition returned 1 when true and 0 when false using Equation (9). The logic condition of each faulty condition at time \(t_{1}\) returned a response of \"true\" when \(T_{1}\leq x_{1}\leq T_{2}\), where \(x_{1}\) is the process parameter at \(t_{1}\). An SPC surface was then developed for every faulty condition scenario at pump speeds of between 700 and 950 rpm at intervals of 50 rpm and stored in an SPC model repository for FDI predictions. For instance, the SPC model repository contained an SPC surface such as the one shown in Figure 6 for sensor measurements at P3 for the healthy condition scenario of the system. It was then used to define the distribution of the data at that sensor measurement for a particular component condition. \[Upper_{bound}(UB)=\mu+3\sigma \tag{7}\] \[Lower_{bound}(LB)=\mu-3\sigma\]  
Equation (7): lower and upper bounds.