4.2 Generating control data for FDI schemes
4.2.2 Distance parameter method
interval, of which the first 20 graphs are still under NOC.
2. Compare an operational attributed graph of a specific system condition (GOP) with each one of the 21-reference attributed graphs in the database by using the HEOM function as described in Figure 4.4. This will produce 21 cost matrices. These cost matrices are then recorded in one row of an array of cost matrices where each row represents a specific sample number, and each column represents a reference condition. An example of this can be seen in Figure 4.5.
3. Calculate all the distance parameters of the 21 cost matrices and insert these 21 param- eters into a row of the distance array displayed in Figure 4.6. Each row in the distance array represents the sample number, and each column represents one of the 21 refer- ence conditions. It should be noted that the array of cost matrices and distance array displayed in Figure 4.6 represent a single operational condition. This means that these arrays will have to be generated for each of the 21 conditions.
4. To isolate a fault, determine the column index of the minimum distance parameter of the specific row (sample number) in the distance array. This method isolates the faulty operational condition (anyone of the last 481 operational attributed graphs in the set of 501 operational attributed graphs) to the reference condition with the same column index as the minimum distance parameter in a specific row (sample number).
5. A fault is detected whenever this method does not isolate an operational fault condition to the column representing the reference NOC. Thus, a fault is detected when the column index of the minimum distance parameter is not 1 (See Figure 4.6).
6. Repeat steps 2 - 5 for each of the 501 sampled operational attributed graphs produced for each of the 21 system conditions.
As an example of how faults are detected and isolated, consider Figure 4.7 and Figure 4.8 below. Figure 4.7 contains the plotted results of the distance parameters that are produced by comparing all 501 of the sampled operational attributed graphs of Fault 2 (GF T2) with all 21 of the reference attributed graphs. It is clear from Figure 4.7 that the distance parameters pro- duced by comparing the operational attributed graphs of Fault 2 with the reference attributed graph of Fault 2 are, for the most part, smaller than any of the other distance parameters.
The method, therefore, can detect and isolate Fault 2 for most of the 501 samples.
This is, however, not the case when the results plotted in Figure 4.8 are analysed. Most of the time, the distance parameters produced by comparing the operational attributed graphs of Fault 8 with the reference attributed graph of Fault 8 are not smaller than any of the other distance parameters. This FDI method, therefore, cannot uniquely isolate Fault 8. However, since the distance parameters produced by comparing the operational attributed graphs of Fault 8 with the reference normal attributed graph are, for the most part, not smaller than
the other distance parameters, the method can detect the presence of a fault in the system.
Figure 4.6: Illustration of how the distance array is produced from the array of cost matrices.
Table 4.3 contains the detection and isolation rates obtained from implementing the distance parameter FDI method. The top row contains the fault detection rates, while the rest contain the fault isolation rates. Each isolation rate entry represents the percentage of times the reference condition represented by that entry’s row index was isolated to the operational condition represented by that entry’s column index. The main column represents the diagonal entries of the isolation rates, representing the percentage of times an operational condition was correctly isolated to its reference condition.
Figure 4.7: Plot of distance parameters produced by compar- ing the operational attributed graph of Fault 2 with all the reference graphs.
Figure 4.8: Plot of distance parameters produced by compar- ing the operational attributed graph of Fault 8 with all the reference graphs.
Table 4.3: Detection and isolation rates of the distance parameter FDI method.
Main F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14 F15 F16 F17 F18 F19 F20
DR (%) 98 99 86 95 91 100 92 89 86 87 87 90 92 84 84 84 86 96 84 85
IR (%) F1 75 75 0 2 0 0 0 3 8 1 2 0 1 0 1 1 1 1 1 0 1
IR (%) F2 85 0 85 1 0 0 0 1 0 1 0 0 0 0 1 3 2 1 0 1 1
IR (%) F3 15 0 0 15 0 0 0 3 0 11 1 1 0 0 4 12 7 7 2 10 10
IR (%) F4 33 0 0 4 33 0 0 2 0 3 1 11 0 0 9 4 3 19 0 2 4
IR (%) F5 21 0 0 8 0 21 0 2 0 9 1 1 0 1 5 11 8 6 2 8 8
IR (%) F6 59 0 0 0 0 5 59 7 9 0 0 0 11 3 0 0 0 0 6 0 0
IR (%) F7 8 0 0 9 0 1 0 8 1 10 4 1 0 5 10 10 9 5 2 7 9
IR (%) F8 1 0 0 11 0 0 0 5 1 10 5 1 1 2 5 9 9 7 2 10 11
IR (%) F9 10 0 0 14 0 0 0 4 1 10 2 2 0 0 7 11 7 5 1 9 12
IR (%) F10 2 0 0 10 0 0 0 6 0 11 2 2 1 1 7 12 8 3 4 11 9
IR (%) F11 5 0 0 8 0 0 0 3 0 6 1 5 0 0 13 10 6 18 1 6 8
IR (%) F12 8 0 0 9 0 0 0 7 0 10 2 3 8 2 8 10 7 5 2 10 6
IR (%) F13 5 0 0 6 0 0 0 8 0 12 2 1 1 5 8 9 12 5 2 8 10
IR (%) F14 7 0 0 11 0 0 0 4 0 10 1 1 0 0 7 14 7 4 1 10 11
IR (%) F15 14 0 0 9 0 0 0 6 1 14 2 1 0 1 3 14 8 4 3 11 8
IR (%) F16 10 0 0 8 0 0 0 4 0 12 2 2 0 1 5 11 10 5 4 9 10
IR (%) F17 13 0 0 9 0 0 0 4 0 10 1 4 1 0 10 11 6 13 1 6 10
IR (%) F18 8 0 0 4 0 12 0 12 8 5 1 2 10 15 4 4 5 5 8 1 2
IR (%) F19 9 0 0 12 0 0 0 5 1 10 1 2 0 1 7 13 7 4 2 9 11
IR (%) F20 10 0 0 9 0 0 0 6 1 12 2 1 0 1 4 14 8 3 3 12 10
51
4.2.2.1 Detection
The confusion matrix of the distance parameter FDI method can be seen in Table 4.4. All 501 sample attributed graphs of all the 21 system conditions (a total of 10521 samples) were assessed with the FDI method to produce this confusion matrix. The overall detection accu- racy of the method is expressed as the percentage of times that the operational condition (the actual system condition) was a fault condition and the method successfully detected that a fault was present in the system. This is expressed as the true positive (TP) value divided by the sum of the true positive (TP) and false-negative (FN) values d+bd
. From the matrix, this method achieved an overall detection rate accuracy (true positive rate) of 89.81 % and a false negative rate of 10.19 %. This method is, therefore, quite proficient in detecting faults from the TEP data.
Table 4.4: Confusion matrix of distance parameter FDI method.
CONFUSION MATRIX DETECTION RATES
True condition
Fault-free Fault Rate %
a TN b FN R FN 10.19
Fault-free
119 980 R TP 89.81
c FP d TP Accuracy 89.91
Detected condition
Fault
782 8640
4.2.2.2 Isolation
To determine the overall isolation rate of this FDI method, the diagonal entries of the isolation rate rows of Table 4.3 are considered. Since each of these diagonal entries represents the rate at which a specific operational condition was correctly isolated to the condition’s corresponding reference condition, the overall isolation rate of the method can be determined by calculating the average value of the diagonal entries. The overall isolation rate for the distance parameter FDI method applied to the TEP data is calculated as 19.90 %.