Table 7 Eigenvalue analysis of four loading conditions
| Â | Loading condition 1 | Loading condition 2 | Loading condition 3 | Loading condition 4 | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
NO-PSS | 0.91 ± 8.55i | − 0.10 | 8.60 | 1.84 ± 9.59i | − 0.188 | 9.77 | 1.75 ± 8.90i | − 0.19 | 9.07 | 2.88 ± 12.8i | − 0.220 | 13.1 |
|  | − 26.1 ± 13.3 | 0.89 | 29.3 | − 27.1 ± 15.1i | 0.87 | 31.0 | 22.0 | 1.00 | 22.0 | − 28.1 ± 24.3i | 0.757 | 37.2 |
MPSS | − 0.51 | 1.00 | 0.51 | − 0.515 | 1.00 | 0.51 | − 0.51 | 1.00 | 0.51 | − 0.59 | 1.00 | 0.52 |
|  | − 1.34 ± 6.39i | 0.20 | 6.53 | − 1.24 ± 6.27i | 0.19 | 6.39 | − 1.03 ± 6.73i | 0.15 | 6.81 | − 1.71 ± 6.07i | 0.27 | 6.30 |
|  | − 15.6 | 1.00 | 15.6 | − 16.9 | 1.00 | 16.9 | − 17.8 | 1.00 | 1.78 | − 8.74 | 1.00 | 8.74 |
|  | − 44.3 | 1.00 | 44.3 | − 56.2 ± 15.6i | 0.96 | 58.4 | − 49.0 | 1.00 | 49.1 | − 29.8 | 1.00 | 29.8 |
|  | − 76.6 | 1.00 | 76.6 |  |  |  | − 66.6 | 1.00 | 66.6 | − 116 | 1.00 | 116 |
PSO-MPSS | − 0.5 | 1.00 | 0.52 | − 0.51 | 1.00 | 0.51 | − 0.51 | 1.00 | 0.51 | − 0.55 | 1.00 | 0.55 |
|  | − 2.03 ± 6.44i | 0.30 | 6.75 | − 1.58 ± 6.55i | 0.23 | 6.73 | − 1.63 ± 6.79, | 0.23 | 6.99 | − 3.66 ± 5.71i | 0.54 | 6.78 |
|  | − 17.7 | 1.00 | 17.7 | − 19.5 | 1.00 | 19.5 | − 20.1 | 1.00 | 20.1 | − 8.11 | 1.00 | 8.11 |
|  | − 45.3 ± 4.50i | 0.99 | 45.5 | − 45.5 ± 12.5i | 0.96 | 47.2 | − 42.3 ± 7.18i | 0.98 | 42.9 | − 28.2 | 1.00 | 28.2 |
PSO-PID-MPSS | − 1.70e−15 | 1.00 | 1.7e−15 | − 2.1e−16 | 1.002 | 1e−16 | − 1.00e−15 | 1.00 | 1.0e−15 | − 2.1e−15 | 1.00 | 2.1e−15 |
|  | − 0.26 | 1.00 | 0.26 | -0.37 | 1.00 | 0.37 | − 0.39 | 1.00 | 0.39 | − 0.02 | 1.00 | 0.02 |
|  | − 6.16 ± 10.1i | 0.52 | 11.8 | − 3.53 ± 7.42i | 0.43 | 8.22 | − 3.86 ± 8.10i | 0.43 | 8.97 | − 3.73 ± 10.0i | 0.34 | 10.7 |
|  | − 13.6 | 1.00 | 13.6 | − 21.3 | 1.00 | 21.3 | − 19.1 | 1.00 | 19.1 | − 11.6 | 1.00 | 11.6 |
|  | − 109 | 1.00 | 109 | − 106 | 1.00 | 106 | − 107 | 1.00 | 107 | − 158 | 1.00 | 158 |
DE-MPSS | 0.51 | 1.00 | 0.51 | 48.1 | 1.00 | 48.1 | − 0.44 | 1.00 | 0.44 | − 3.32 ± 5.42i | 0.52 | 6.35 |
|  | − 1.89 ± 6.46 | 0.28 | 6.73 | − 1.70 ± 6.061i | 0.26 | 7.24 | − 1.60 ± 6.84i | 0.22 | 7.03 | − 8.84 | 1.00 | 8.84 |
|  | − 17.6 | 1.00 | 17.6 | − 20.2 | 1.00 | 20.2 | − 19.9 | 1.00 | 19.9 | − 27.7 | 1.00 | 27.7 |
|  | 43.9 | 1.00 | 43.9 | − 40.5 | 1.00 | 40.5 | − 38.7 | 1.00 | 38.7 | − 67.8 | 1.00 | 67.8 |
|  | − 61.9 | 1.00 | 61.9 | − 41.4 | 1.00 | 41.4 | − 46.5 | 46.5 | 1.00 |  |  |  |
Proposed WOA-MPSS | − 0.10 | 1.00 | 0.10 | − 0.10 | 1.00 | 0.10 | − 0.10 | 1.00 | 0.10 | − 0.10 | 1.00 | 0.10 |
|  | − 2.06 ± 1.57i | 0.79 | 2.59 | − 1.85 ± 1.41i | 0.80 | 2.33 | − 1.98 ± 2.26i | 0.70 | 3.00 | − 1.91 ± 1.36 | 0.81 | 2.34 |
|  | − 7.25 | 1.00 | 7.25 | − 14.0 ± 7.85i | 0.90 | 0.16 | − 8.26 | 1.00 | 8.26 | − 14.6 ± 1.36i | 0.98 | 15.00 |
|  | − 35.10 ± 10.00i | 0.96 | 36.50 | − 117.00 | 1.00 | 117 | − 27.10 | 1.00 | 27.1 | − 128.00 | 1.00 | 128.0 |