Table 12 Compromise solution values obtained by each algorithm. Significant values are in bold.

Control variables

 \(P_{{G_{2} }} \;({\text{MW}})\)

20

80

57.4243

51.9143

80

21.5395

61.4589

58.7567

 \(P_{{G_{5} }} \;({\text{MW}})\)

0

75

74.3827

61.7280

63.3067

52.1651

53.7995

62.3518

 \(P_{{G_{8} }} \;({\text{MW}})\)

0

40

34.1595

17.0056

35

20.8785

30.8963

23.5868

 \(P_{{G_{11} }} \;({\text{MW}})\)

0

60

39.6052

49.5539

26.0829

33.9943

49.5359

46.6194

 \(P_{{G_{13} }} \;({\text{MW}})\)

0

50

28.8977

49.8478

0

34.5303

29.6939

18.5482

 \(V_{1} \;({\text{P.U.}})\)

0.95

1.10

1.0471

1.0557

1.0614

1.0258

0.9938

1.0119

 \(V_{2} \;({\text{P.U.}})\)

0.95

1.10

1.0819

0.9676

1.0931

1.0323

1.0497

1.0421

 \(V_{5} \;({\text{P.U.}})\)

0.95

1.10

1.0807

1.0481

1.0326

1.0771

1.0890

1.0405

 \(V_{8} \;({\text{P.U.}})\)

0.95

1.10

0.9500

1.0873

0.9500

1.0998

0.9916

1.0081

 \(V_{11} \;({\text{P.U.}})\)

0.95

1.10

1.0998

1.0840

1.0961

1.0999

1.0790

1.0771

 \(V_{13} \;({\text{P.U.}})\)

0.95

1.10

1.0797

1.0787

1.1000

1.1000

1.0734

1.0859

State variables

 \(P_{{G_{1} }} \;({\text{MW}})\)

52.3740

56.3713

84.1607

125.9656

65.9542

78.3861

 \(Q_{{G_{1} }} \;({\text{MVAr}})\)

− 13.5846

5.2632

− 6.9106

− 20

− 20

− 20

 \(Q_{{G_{2} }} \;({\text{MVAr}})\)

60

− 20

60

6.4275

54.3186

35.1202

 \(Q_{{G_{5} }} \;({\text{MVAr}})\)

− 30

35

− 30

35

− 13.4506

5.4266

 \(Q_{{G_{8} }} \;({\text{MVAr}})\)

40

40

40

40

40

40

 \(Q_{{G_{11} }} \;({\text{MVAr}})\)

30

25.7018

30

30

27.5689

27.7867

 \(Q_{{G_{13} }} \;({\text{MVAr}})\)

25

25

25

25

25

25

Objectives

 F1: Generation cost [$/h]

884.6613

854.9569

794.2250

793.6699

876.0810

860.6207

 F2: Emission [ton/h]

0.0489

0.0509

1.9315

0.9777

0.0622

0.0874

 F4: Voltage deviation [p.u]

0.4037

0.4836

0.4094

0.4159

0.4075

0.4058