Figure 1:
Figure 1: Scatter plot of main types of observations.
Figure 2:
Figure 2: Residual pattern of the regression observations.
Figure 3:
Figure 3: The topological diagram of the IEEE 30-bus system.
Table 1.
Numerical characteristic of the test systems.
| Buses of | ||||
|---|---|---|---|---|
| Test systems | High SE errors | Outlier measurements | GLCR | BLCR |
| IEEE 30-bus | 26, 29, 30 | 2, 4, 9, 10, 11, 12, 18, 19, 21, 22, 23, 24, 25, 26, 29 | 4, 9, 11, 18, 19 | 21, 22, 23, 24, 25, 26 |
| UK 18-bus | 13, 14 | 1, 5, 7, 9, 12, 14, 16, 17 | 1, 7 | 14, 16, 17 |
| IEEE 118-bus | 10, 35, 36, 60, 61, 64 | 1, 3, 5, 9, 12, 17, 31, 32, 34, 52, 54, 59, 61, 63, 64, 66, 78, 79, 86, 89, 99, 110, 113, 114, 116, 117 | 3, 12, 32, 54, 76, 96, 110, 114 | 9, 25, 34, 35, 52, 59, 64, 66, 68, 86, 113 (11 meters) |
Table 2.
Comparison of the numerical performance of the SE of the test systems.
| Total SE errors | Condition numbers | No. of outliers | ||||
|---|---|---|---|---|---|---|
| Test systems | Before | After | Before | After | Before | After |
| IEEE 30-bus | 8.14 × 10−4 | 5.06 × 10−4 | 1.84 × 106 | 3.87 × 104 | 15 | 2 |
| UK 18-bus | 9.73 × 10−4 | 6.26 × 10−4 | 2.86 × 107 | 2.02 × 104 | 8 | 2 |
| IEEE 118-bus | 5.93 × 10−6 | 9.48 × 10−4 | 1.27 × 1017 | 1.27 × 109 | 26 | 5 |
Figure 4:
Figure 4: Scatter pattern of DRGP vs LTSR for 30-bus system.
Table 3.
Comparison of PMU placement methods.
| Systems | IEEE 30-Bus | IEEE 118-Bus |
|---|---|---|
| GA (Xu and Abur, 2004) | 7 | 29 |
| IP (Liu et al., 2014) | 7 | 29 |
| BPSO (Ahmadi et al., 2011) | 7 | 29 |
| PSO (Saleh et al., 2017) | 7 | 28 |
| Proposed CSO-based | (3 PMU + 3 CM) = 6 | (8 PMU + 11 CM) = 19 |
Figure 5:
Figure 5: The UK 18-bus test system.
Figure 6:
Figure 6: Convergence of the cuckoo search of the 118-bus SE.