## 5 Experimental Results and Discussion
### Real-World Application: Circulating Fluidized Bed Boiler

Table 2: _Cont._Figure 11 compares the bandwidth parameter selection results for AAKR, AASKR-NP, and AASKR. AAKR and AASKR-NP were set to 0.37 as the proper bandwidth for fault detection in CFBC. AASKR-NP is the same as AAKR due to the absence of distance readjustment based on shared neighbors. In contrast, AASKR set the bandwidth parameter (\(h\)) to 1 as distance values were readjusted. This shows that variations in distance values can influence the selection of the bandwidth parameter. Figure 12 illustrates the fault detection times of conventional and proposed methods for an unplanned boiler shutdown. The shutdown occurred on September 9 at 14:25, corresponding to the 1094 th sample in the dataset. As shown in Figure 12a, PCA detected the fault at 14:53:25, 18 min and 20 s after the shutdown. This indicates that PCA failed to detect the fault in a timely manner, as the detection occurred post-shutdown. As demonstrated in previous studies, PCA performance tends to degrade in complex processes. In Figure 12b,c, the distance-based method kNN detected the fault 59 min prior to the boiler shutdown. In contrast, LOF detected the fault at 14:36, 28 s after the shutdown, showing a delay comparable to that of PCA. LOF's reliance on neighbor density necessitates a method for identifying reliable neighbors. Consequently, LOF is highly sensitive to outliers and noise, not only in TEPs but also in real-world processes, highlighting the need for structural improvements. Although kNN detects faults later than the proposed method, it demonstrates the capability to identify anomalies before boiler shutdown, suggesting its potential applicability in real-world processes. In Figure 12d, AAKR detected the anomaly \begin{table}
\begin{tabular}{c c c} \hline
**Variable** & **Description** & **Unit** \\ \hline \(x_{30}\) & Outlet output of feedwater pipe 2 & \% \\ \(x_{44}\) & Inlet temperature inlet of lower place furnace (sensor C) & \({}^{\circ}\)C \\ \(x_{35}\) & Inlet temperature of lower place furnace (sensor C) & \({}^{\circ}\)C \\ \(x_{36}\) & Outlet temperature of 1 st superheater & \({}^{\circ}\)C \\ \(x_{37}\) & Outlet temperature of 1 st superheater & \({}^{\circ}\)C \\ \(x_{38}\) & Inlet temperature of 2 nd superheater (sensor A) & \({}^{\circ}\)C \\ \(x_{39}\) & Inlet temperature of 2 nd superheater (sensor B) & \({}^{\circ}\)C \\ \(x_{100}\) & Temperature of steam supplied of boiler silencer & \({}^{\circ}\)C \\ \(x_{101}\) & Inlet temperature of 1 st superheater (sensor A) & \({}^{\circ}\)C \\ \(x_{102}\) & Inlet temperature of 1 st superheater (sensor B) & \({}^{\circ}\)C \\ \(x_{103}\) & Steam drum level of feedwater tank & mm \\ \(x_{104}\) & Outlet pressure 2 nd superheater & MPa \\ \(x_{105}\) & Outlet pressure steam supplied of 2 nd superheater & MPa \\ \(x_{106}\) & Inlet pressure 2 nd superheater & MPa \\ \(x_{107}\) & Amount of outlet steam flow 2 nd superheater & t/h \\ \(x_{108}\) & Amount of inlet steam flow 2 nd superheater & t/h \\ \(x_{109}\) & Steam output of steam drum & t/h \\ \hline \end{tabular}
\end{table} 
Table 2: \(Cont.\)  
Figure 11: Comparison of the bandwidth parameters AAKR, AASKR-NP, and AASKR: (a) AAKR/AASKR-NP; (b) AASKR. earlier than PCA, kNN, and LOF. The detection time of AAKR was 12:32 (356 th data sample), identifying a problem with the CFBC boiler 2 h and 3 min before shutdown. In Figure 12e, AASKR-NP also detected the fault at 12:32 (356 th data sample), matching AAKR's detection time and confirming the anomaly 2 h and 3 min prior to shutdown. The similarity in detection times is due to the identical structure of AASKR-NP and AAKR. In Figure 12f, the proposed method showed improved performance in early fault detection compared to conventional methods. The anomaly was detected at the 247 th data sample, 2 h and 21 min prior to the boiler shutdown, which was 28 min earlier than both AAKR and AASKR-NP. Additionally, when compared to kNN, a distance-based method, the proposed approach detected the fault 1 h and 23 min earlier.