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Sustainable Energy Research

Table 2 The eigenvalues, damping ratios and associated states of the IEEE 14 bus system before and after maximization with solar and wind power

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

Before maximization with solar and wind power

After maximization with wind and solar power

Eigen numbers

Eigenvalues

Damping ratio

Most associated states

Eigen numbers

Eigen values

Damping ratio

Associated states

1

− 1000

1

vm_Exc_2

1

− 1000

1

vm_Exc_2

2

− 1000

1

vm_Exc_1

2

− 1000

1

vm_Exc_1

3

− 99.2985

1

iqr_Dfig_1

3

− 92.859

1

iqr_Dfig_1

4

− 73.0801

1

iq_Spv_1

4

− 68.3848

1

iq_Spv_1

5

− 53.4892

1

id_Spv_1

5

− 56.1733

1

id_Spv_1

6

− 49.9798

1

vr1_Exc_2

6 and 7

− 44.4131 ± 7.7136i

0.985

vr1_Exc_1, e2q_Syn_1, vf_Exc_1

7 and 8

− 44.0346 ± 8.2427i

0.983

vr1_Exc_1, e2q_Syn_2

8

− 36.7227

1

e2d_Syn_1, e2d_Syn_2

9

− 37.0999

1

e2d_Syn_1

9

− 33.2848

1

e2q_Syn_2

10

− 33.1002

1

e2q_Syn_2

10

− 25.6455

1

e2q_Syn_3

11

− 25.6419

1

e2q_Syn_3

11

− 22.3182

1

e2q_Syn_4

12

− 21.3858

1

e2q_Syn_4

12

− 17.855

1

e2d_Syn_2, e1d_Syn_2

13

− 16.7754

1

e2d_Syn_2

13 and 14

− 2.2759 ± 11.6701i

0.191

omega_Syn_2, delta_Syn_2

14 and 15

− 3.3659 ± 9.8278i

0.324

omega_Syn_3, delta_Syn_3

15 and 16

− 3.1892 ± 9.8849i

0.307

omega_Syn_3, delta_Syn_3

16 and 17

− 1.1061 ± 9.3197i

0.118

e1q_Syn_1, vf_Exc_1

17 and 18

− 0.74009 ± 9.4357i

0.078

e1q_Syn_1, vf_Exc_1

18 and 19

− 2.9434 ± 8.7538i

0.319

omega_Syn_2, delta_Syn_2

19 and 20

− 1.3519 ± 7.3681i

0.18

omega_Syn_4, delta_Syn_4, e2d_Syn_4

20 and 21

− 1.0872 ± 7.5138i

0.143

omega_Syn_4, delta_Syn_4

21

− 11.8806

1

e2d_Syn_3, delta_Syn_3,, omega_Syn_3

22

− 12.0538

1

e2d_Syn_3

22

− 11.0187

1

btx1_Spv_1, e2d_Syn_4,

23 and 24

− 10.459 ± 0.03002i

1

btx1_Spv_1, e2d_Syn_4

23

− 9.6637

1

tg_Tg_2,e1d_Syn_2, e2d_Syn_2

25

− 9.5669

1

tg_Tg_2

24

− 9.1454

1

tg_Tg_1, e2d_Syn_4, omega_Syn_

26

− 8.9995

1

tg_Tg_1

25

− 8.5301

1

e2d_Syn_4, e1d_Syn_2

27

− 7.4985

1

e1d_Syn_2

26

− 8.3408

1

e1d_Syn_2, e2d_Syn_2,

28 and 29

− 0.50581 ± 1.9977i

0.245

omega_Syn_1, delta_Syn_1

27 and 28

− 0.34253 ± 2.2326i

0.0757

omega_Syn_1, delta_Syn_1

30

− 2.5454

1

idr_Dfig_1

29

− 2.8708

1

idr_Dfig_1

31

− 0.24538

1

e1q_Syn_4

30

− 0.68888

1

e1q_Syn_4, e1d_Syn_4

32 and 33

− 0.56184 ± 0.38565

0.824

e1q_Syn_2, vf_Exc_2

31

− 0.99054

1

omega_m_Dfig_1

34

− 0.79035

1

omega_m_Dfig_1

32

− 1.0116

1

vr2_Exc_1

35

− 0.89037

1

e1d_Syn_3

33

− 0.82144

1

e1q_Syn_2, e1d_Syn_4, e1q_Syn_3

36

− 0.89365

1

e1q_Syn_3

34

− 0.83365

1

e1q_Syn_2, e1d_Syn_4

37

− 1.0228

1

vr2_Exc_1

35

− 0.89925

1

e1d_Syn_3

38

− 1.0181

1

e1d_Syn_4

36

− 0.90227

1

e1q_Syn_3

39

− 1.0092

1

vr2_Exc_2

37

− 30.303

1

e1d_Syn_1

40

− 1.25

1

e1d_Syn_1

38

− 50

1

vr1_Exc_2

41

− 0.25

1

vw_Wind_1

39

− 1

1

vr2_Exc_2

42

− 0.33333

1

theta_p_Dfig_1

40

− 0.78592

1

vf_Exc_2

    

41

− 0.25

1

vw_Wind_1

    

42

− 0.33333

1

theta_p_Dfig_1

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