Table 14 Compromise solution values obtained by each algorithm. Significant values are in bold.

Control variables

 \(P_{{G_{2} }} \;({\text{MW}})\)

20

80

55.0827

67.7103

57.3602

21.8571

22.5218

47.7580

 \(P_{{G_{5} }} \;({\text{MW}})\)

0

75

51.5612

71.7705

61.5017

38.9172

66.2482

46.7818

 \(P_{{G_{8} }} \;({\text{MW}})\)

0

40

22.5658

28.8133

25.0488

22.7267

21.2179

28.1285

 \(P_{{G_{11} }} \;({\text{MW}})\)

0

60

60

38.0716

38.3362

42.6629

41.4882

32.9828

 \(P_{{G_{13} }} \;({\text{MW}})\)

0

50

44.8031

9.2449

32.6630

34.2024

42.5803

31.0533

 \(V_{1} \;({\text{P.U.}})\)

0.95

1.10

1.0593

1.0591

1.0400

1.0797

1.0456

1.0547

 \(V_{2} \;({\text{P.U.}})\)

0.95

1.10

1.0398

1.0595

1.0353

0.9658

1.0187

1.0456

 \(V_{5} \;({\text{P.U.}})\)

0.95

1.10

1.0557

1.0560

1.0794

1.0998

1.0979

1.0926

 \(V_{8} \;({\text{P.U.}})\)

0.95

1.10

1.0999

1.0704

1.0356

0.9907

1.0303

1.0162

 \(V_{11} \;({\text{P.U.}})\)

0.95

1.10

1.0891

1.0813

1.0923

1.0999

1.0780

1.0786

 \(V_{13} \;({\text{P.U.}})\)

0.95

1.10

1.0684

1.0217

1.0705

1.0707

1.0959

1.0727

State variables

 \(P_{{G_{1} }} \;({\text{MW}})\)

52.3824

71.2731

71.9532

129.0667

93.0083

101.2521

 \(Q_{{G_{1} }} \;({\text{MVAr}})\)

7.1748

− 16.8915

− 13.1505

50.4159

6.7013

− 7.6273

 \(Q_{{G_{2} }} \;({\text{MVAr}})\)

− 20

18.8164

− 5.2117

− 20

− 20

17.2120

 \(Q_{{G_{5} }} \;({\text{MVAr}})\)

35

35

34.2552

− 5.6715

33.0471

13.7113

 \(Q_{{G_{8} }} \;({\text{MVAr}})\)

40

40

40

40

40

40

 \(Q_{{G_{11} }} \;({\text{MVAr}})\)

27.7525

25.3068

30

30

27.4323

24.9460

 \(Q_{{G_{13} }} \;({\text{MVAr}})\)

23.0593

7.9208

25

25

25

25

Objectives

 F1: Generation Cost [$/h)

852.2493

903.5947

841.0084

790.0268

828.3344

823.4770

 F2: Emission [ton/h]

0.0486

0.0726

0.0715

1.1849

0.1598

0.2340

 F3: Real power loss [MW]

3.0052

3.4297

3.4722

6.0329

3.6648

4.5565

 F4: Voltage deviation [p.u]

0.5193

0.4388

0.4111

0.4158

0.4310

0.4147