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Table 3 The eigenvalues, damping ratios and associated states of the IEEE 14-bus system after optimization of damping ratios

From: An enhanced particle swarm optimization algorithm for improving the renewable energy penetration and small signal stability in power system

Eigennumbers Eigenvalues Damping ratio Most associated states and % participation
1 − 1000 1 vm_Exc_1 (100)
2 − 1000 1 vm_Exc_2(100)
3 − 92.859 1 iqr_Dfig_1 (99.34)
4 − 68.3848 1 iq_Spv_1 (91.43)
5 − 56.1733 1 id_Spv_1 (87.85)
6 and 7 − 36.8065 ± 0.48647i 1 e2d_Syn_1 (44.55), e2q_Syn_1(33.61), btx1_Spv_1 (11.96)
8 − 31.6088 1 e2q_Syn_2 (69.99), e2q_Syn_1 (25.907)
9 − 25.5361 1 e2q_Syn_3 (91.47)
10 − 22.2968 1 e2q_Syn_4 (93.12)
11 − 17.7616 1 e2d_Syn_2(62.395), e1d_Syn_2(23.52)
12 and 13 − 2.9492 ± 8.75306i 0.32 omega_Syn_2 (32.30), delta_Syn_2 (31.98), e2d_Syn_2 (8.667)
14 and 15 − 3.1789 ± 9.9343i 0.305 omega_Syn_3 (31.437), delta_Syn_3 (31.26), e2d_Syn_3 (18.58)
16 and 17 − 1.5058 ± 7.6511i 0.193 omega_Syn_4 (29.485), delta_Syn_4(29.35), e2d_Syn_4 (12.10), omega_Syn_1(10.872), delta_Syn_1 (6.35)
18 − 11.8927 1 e2d_Syn_3 (51.98), omega_Syn_3 (10.573), delta_Syn_3 (10.397), e2d_Syn_4 (8.778), btx1_Spv_1(7.17)
19 − 11.0105 1 btx1_Spv_1(52.368), e2d_Syn_4(18.063), id_Spv_1 (7.79), e2d_Syn_3(5.746), omega_Syn_4(5.193), delta_Syn_4 (5.1)
20 − 9.9229 1 tg_Tg_2(96.44)
21 − 8.6679 1 e2d_Syn_4 (51.276), btx1_Spv_1(5.006), omega_Syn_4 (8.692), delta_Syn_4 (8.494), e2d_Syn_2 (6.209)
22 − 8.4751 1 e1d_Syn_2 (52.62), e2d_Syn_2 (23.58), e2d_Syn_4 (14.13)
23 and 24 − 1.8633 ± 3.0534i 0.521 omega_Syn_1(35.656), tg_Tg_1 (25.193), delta_Syn_1(16.667)
25 and 26 − 3.3114 ± 0.03738i 1 tg_Tg_1 (30.535), omega_Syn_1(25.6), delta_Syn_1 (16.95)
27 − 0.18505 1 e1q_Syn_1 (96.726)
28 − 0.68776 1 e1q_Syn_4 (82.208), e1d_Syn_4 (12.12)
29 − 0.79842 1 e1q_Syn_2 (79.609), e1q_Syn_3 (5.727), e1d_Syn_4(7.635)
30 − 0.83005 1 e1d_Syn_4 (73.221), e1q_Syn_4 (11.83)
31 − 0.85987 1 omega_m_Dfig_1(67.974), delta_Syn_1(15.07)
32 − 0.90239 1 e1q_Syn_3(86.856)
33 − 0.89951 1 e1d_Syn_3 (94.42)
34 − 30.303 1 e1d_Syn_1(100)
35 − 1 1 vr2_Exc_1(100)
36 − 5.0106 1 vf_Exc_1(100)
37 − 50 1 vr1_Exc_1(100)
38 − 50 1 vr1_Exc_2(100)
39 − 1 1 vr2_Exc_2 (100)
40 − 0.78592 1 vf_Exc_2 (100)
41 − 0.25 1 vw_Wind_1 (100)
42 − 0.33333 1 theta_p_Dfig_1(100)