Table 10 Compromise solution values obtained by each algorithm. Significant values are in bold.

Control variables

 \(P_{{G_{2} }} \;({\text{MW}})\)

20

80

28.6566

54.4651

44.2135

61.6351

53.2922

55.3038

 \(P_{{G_{5} }} \;({\text{MW}})\)

0

75

74.5949

60.5986

63.4783

60.3973

59.9942

59.9810

 \(P_{{G_{8} }} \;({\text{MW}})\)

0

40

24.3144

22.9219

23.6391

21.9237

24.4547

21.6535

 \(P_{{G_{11} }} \;({\text{MW}})\)

0

60

57.8024

41.9632

48.1911

48.0717

57.8974

34.9888

 \(P_{{G_{13} }} \;({\text{MW}})\)

0

50

47.2424

33.9319

40.8802

30.2714

38.6188

32.3837

 \(V_{1} \;({\text{P.U.}})\)

0.95

1.10

1.0428

1.0653

1.0073

1.0551

1.0624

1.0464

 \(V_{2} \;({\text{P.U.}})\)

0.95

1.10

1.0514

1.0575

1.0296

0.9500

1.0297

1.0368

 \(V_{5} \;({\text{P.U.}})\)

0.95

1.10

1.0779

1.0676

1.0897

1.0336

1.0309

1.0572

 \(V_{8} \;({\text{P.U.}})\)

0.95

1.10

1.0478

1.0399

1.0488

1.1000

1.0453

1.0390

 \(V_{11} \;({\text{P.U.}})\)

0.95

1.10

1.0828

1.0162

1.0898

1.1000

1.0527

1.0718

 \(V_{13} \;({\text{P.U.}})\)

0.95

1.10

1.0997

1.0651

1.0779

1.1000

1.0223

1.0655

State variables

 \(P_{{G_{1} }} \;({\text{MW}})\)

53.3042

72.9842

67.0141

64.3562

52.0941

82.9726

 \(Q_{{G_{1} }} \;({\text{MVAr}})\)

− 20

0.2124

− 20

9.8130

28.9735

− 3.6311

 \(Q_{{G_{2} }} \;({\text{MVAr}})\)

13.5855

15.9054

2.0945

− 20

− 20

− 8.8179

 \(Q_{{G_{5} }} \;({\text{MVAr}})\)

26.6269

24.3422

35

35

35

35

 \(Q_{{G_{8} }} \;({\text{MVAr}})\)

40

40

40

31.4350

34.9244

40

 \(Q_{{G_{11} }} \;({\text{MVAr}})\)

24.8168

4.2196

30

30

21.5755

23.7276

 \(Q_{{G_{13} }} \;({\text{MVAr}})\)

25

25

25

25

12.1666

25

Objectives

 F1: Generation cost [$/h]

863.2328

838.6775

845.0190

847.8680

857.7920

834.2038

 F2: Emission [ton/h]

0.0508

0.0732

0.0622

0.0601

0.0482

0.1024

 F3: Real power loss [MW]

2.3446

3.4651

3.1340

3.2564

2.9515

3.8834