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