6.4 Simulation Results
6.4.2 Test Case B: IEEE 39-Bus 10-Generator System
Figure 6.5: Variation of fitness value against iteration for Case A.2
obtained by the both GWO and IGWO algorithms for the production cost minimiza- tion objective function are presented in Table 6.3. This table also includes the results obtained by EPNN [121], ABC [124], CABC [124], WOA [125] and CWOA [125] al- gorithms.
The production cost is calculated by using optimal parameters using Equation 6.4.
By inspecting the calculated production costs obtained by different algorithms it is observed that the production cost 576.9377 $/hwhich is obtained by using proposed IGWO parameters is the minimum. The comparative convergence analysis, obtained by both GWO and the proposed IGWO, is shown in Figure 6.5. This figure presents that IGWO based objective function value for this case converges smoothly and reaches the near global optimal value. The relative rotor angle trajectories are also shown in Figures 6.6 and 6.7 by using GWO and IGWO respectively. As seen from these figures all the generators are stable and the rotor angles of all the generators do not cross the value δmax. Moreover, the statistical comparison of best, worst and mean fuel cost values as obtained using different algorithms are listed in Table 6.4.
Figure 6.6: Relative rotor angles obtained by the GWO for Case A.2
Figure 6.7: Relative rotor angles obtained by the IGWO for Case A.2
Table 6.4: Comparative results of IEEE 30-Bus system for Case A.2
Cost ($/h) EP [121]
EPNN [121]
ABC [124]
CABC [124]
WOA [125]
CWOA
[125] GWO IGWO Minimum 585.15 585.12 577.71 577.47 577.3819 577.257 577.20 576.93
Maximum 586.86 586.73 583.26 580.74 - - 583.48 579.87
Average 585.83 585.84 580.21 579.10 - - 578.79 577.69
Table 6.5: Cost-coefficient data of 10-Generator 39-Bus System
Generator Number
Bus Number
For Cases B.1 & B.2 as per Ref. [221, 287] For Case B.3 as per Ref. [250]
αi βi γi αi βi γi
1 30 0.0193 6.9 0 0.01 0.3 0.2
2 31 0.0111 3.7 0 0.01 0.3 0.2
3 32 0.0104 2.8 0 0.01 0.3 0.2
4 33 0.0088 4.7 0 0.01 0.3 0.2
5 34 0.0128 2.8 0 0.01 0.3 0.2
6 35 0.0094 3.7 0 0.01 0.3 0.2
7 36 0.0099 4.8 0 0.01 0.3 0.2
8 37 0.0113 3.6 0 0.01 0.3 0.2
9 38 0.0071 3.7 0 0.01 0.3 0.2
10 39 0.0065 3.9 0 0.01 0.3 0.2
system are given in Appendix A. The system data such as bus data, line data, and initial values of control variables are taken from [286]. The fuel cost coefficients data and the rating of generators are the same as in [221, 275, 287]. The upper and lower limits of all the bus voltages magnitudes are available in [275]. The total load demand for the operating condition considered for this test system arePLoad = 6098 MW and QLoad = 1409 MVAr. Three widely used case studies, including the base load condition, are used to exhibit the comparison of proposed approach with the contemporary approaches.
• Case B.1: A 3-phase to ground fault is considered near bus 29 and in-between lines 28-29 and is cleared after 0.1s [218].
• Case B.2: A 3-phase to ground fault is considered near bus 17 and in-between lines 17-18 and is cleared after 0.2s [124, 221].
• Case B.3: A 3-phase to ground fault is considered near bus 17 and in-between lines 17-18 and is cleared after 0.2s [124].
The fuel cost coefficient data and the rating of the generators are considered as given in [221, 287] for case studies B.1 and B.2, whereas for case B.3, these values are taken from [250] as shown in Table 6.5. For these cases, population size and maximum number of iterations for optimization are considered as 50 and 100 respectively. Furthermore, the results of the GWO and IGWO algorithms are obtained after carrying out 30 independent run for different cases.
Case B.1: A 3-phase to ground fault is considered near bus 29 and in- between lines 28-29 and is cleared after 0.1s
The study is shown with base case loading and 3-φ fault near bus-29 and in-between the line 28-29. Fault clearing time (FCT) is taken as 0.1s. The optimal parameters obtained by the both GWO and IGWO algorithms for the production cost mini- mization objective function are presented in Table 6.6. This table also includes the results obtained by Dynamic Simulation Algorithm (DSA) [288], generator Classical Model(CM) [289], generator Detailed Model (DM) [289], algorithms.
The production cost is calculated by using optimal parameters using Equation 6.4.
By inspecting the calculated production costs obtained by different algorithms it is observed that the production cost 60906.32 $/hwhich is obtained by using proposed
Table 6.6: Best control variables and production cost for IEEE 39-Bus system (Case B.1)
Parameter DSA [288] CM [289] DM [289] GWO IGWO
PG30 (MW) 247.83 248.73 249.45 226.27 235.47
PG31 (MW) 577.23 577.84 578.36 558.11 549.93
PG32 (MW) 653.41 654.47 654.35 613.28 627.82
PG33 (MW) 643.28 645.00 641.76 629.68 619.26
PG34 (MW) 517.78 518.82 517.41 506.42 499.73
PG35 (MW) 662.46 664.32 660.73 630.07 635.26
PG36 (MW) 569.59 571.37 568.18 544.63 547.51
PG37 (MW) 543.88 547.81 547.69 518.58 525.40
PG38 (MW) 774.54 752.02 754.61 798.71 785.37
PG39 (MW) 1000.35 995.60 1003.18 1100.00 1100.00
V30(p.u.) 0.98 1.01 1.01 1.014 1.014
V31(p.u.) 1.07 1.08 1.08 1.04 1.05
V32(p.u.) 1.00 1.02 1.02 1.04 1.05
V33(p.u.) 1.01 1.01 1.01 1.05 1.05
V34(p.u.) 1.01 1.02 1.02 1.04 1.05
V35(p.u.) 1.06 1.06 1.06 1.05 1.05
V36(p.u.) 1.08 1.09 1.09 1.04 1.05
V37(p.u.) 1.01 1.04 1.04 1.04 1.03
V38(p.u.) 1.05 1.03 1.03 1.05 1.05
V39(p.u.) 1.01 1.05 1.05 1.04 1.02
Minimum Cost($/h) 61799.68 61600.76 61597.76 60912.81 60906.32
Maximum Cost($/h) - - - 60984.22 61243.33
Average Cost($/h) - - - 66941.25 61064.78
Figure 6.8: Variation of fitness value against iteration for Case B.1
Figure 6.9: Relative rotor angles obtained by IGWO for Case B.1
IGWO parameters is the minimum. The comparative convergence analysis, obtained by both GWO and the proposed IGWO, is shown in Figure 6.8. This figure presents that IGWO based objective function value for this case converges smoothly and reaches the near global optimal value. The relative rotor angle trajectories are also shown in Figure 6.9 by using IGWO. As seen from this figure all the generators are stable and the rotor angles of all the generators do not cross the value δmax.
Case B.2: A 3-phase to ground fault is considered near bus 17 and in- between lines 17-18 and is cleared after 0.2s
The study is shown with base case loading and 3-φ fault near bus-17 and in-between the line 17-18. Fault clearing time (FCT) is taken as 0.2s. The optimal parameters obtained by the both GWO and IGWO algorithms for the production cost mini- mization objective function are presented in Table 6.7. This table also includes the results obtained by TS [221], ABC [124], CABC [124], WOA [290] and CWOA [290]
algorithms. The comparative convergence analysis, obtained by both GWO and the proposed IGWO, is shown in Figure 6.10. This figure presents that IGWO based objective function value for this case converges smoothly and reaches the near global optimal value.
Table 6.7: Best control variables and production cost for IEEE 39-Bus system (Case B.2)
Parameter TS [221]
ABC [124]
CABC [124]
WOA [290]
CWOA
[290] GWO IGWO PG30 (MW) 243.61 313.03 300.00 329.60 306.89 226.24 242.98 PG31 (MW) 568.34 616.15 570.30 527.49 520.56 551.78 562.68 PG32 (MW) 643.81 610.23 653.26 601.54 616.80 630.07 641.80 PG33 (MW) 644.57 617.57 600.00 647.77 600.76 612.02 629.26 PG34 (MW) 243.58 455.23 443.87 400.44 395.67 499.80 507.35 PG35 (MW) 658.27 602.95 645.14 605.55 645.78 641.10 655.09 PG36 (MW) 565.44 507.08 503.38 513.34 510.12 533.96 562.49 PG37 (MW) 538.17 600.00 600.00 568.86 598.46 534.27 533.69 PG38 (MW) 533.19 702.48 719.36 778.89 801.56 797.71 807.38 PG39 (MW) 1200.00 1110.00 1100.00 1168.69 1149.68 1098.84 983.09
V30(p.u.) - 1.00 1.01 1.01 1.02 1.04 1.00
V31(p.u.) - 1.01 1.00 1.00 1.02 1.05 1.05
V32(p.u.) - 1.02 1.01 1.02 1.03 1.03 1.05
V33(p.u.) - 1.03 1.03 0.99 1.01 1.04 1.05
V34(p.u.) - 1.01 1.02 0.97 1.04 1.03 1.05
V35(p.u.) - 1.03 1.03 1.02 1.00 1.05 1.05
V36(p.u.) - 1.03 1.03 1.00 1.00 1.02 1.05
V37(p.u.) - 1.02 1.02 1.01 0.98 1.04 1.03
V38(p.u.) - 1.03 1.04 1.02 1.01 1.05 1.05
V39(p.u.) - 0.95 0.96 0.99 1.00 1.00 1.01
Cost($/h) 62261.28 61485.48 61369.19 61126.23 61106.27 60917.17 60783.22
Figure 6.10: Variation of fitness value against iteration for Case B.2
Figure 6.11: Relative rotor angles obtained by IGWO for Case B.2
Table 6.8: Comparative Results of IEEE 39-bus Test System for Case B.2
Cost ($/h) TS [221]
ABC [124]
CABC [124]
WOA [290]
CWOA
[290] GWO IGWO
Minimum 62261.28 61485.48 61369.19 61126.23 61106.27 60917.17 60783.22 Maximum - 61703.42 61602.53 61198.23 61176.67 61134.71 61091.46 Average - 61594.45 61485.86 61132.55 61126.67 61094.63 60984.44
The production cost is calculated by using optimal parameters using Equation 6.4.
By inspecting the calculated production costs obtained by different algorithms it is observed that the production cost 60783.22 $/hwhich is obtained by using proposed IGWO parameters is the minimum. The relative rotor angle trajectories are also shown in Figure 6.11 by using IGWO. As seen from this figure all the generators are stable and the rotor angles of all the generators do not cross the value δmax. Moreover, the statistical comparison of best, worst and mean fuel cost values as obtained using different algorithms are listed in Table 6.8.
Case B.3: A 3-phase to ground fault is considered near bus 17 and in- between lines 17-18 and is cleared after 0.2s
The study is shown with base case loading and 3-φ fault near bus-17 and in-between the line 17-18. Fault clearing time (FCT) is taken as 0.2s. The optimal parameters obtained by the both GWO and IGWO algorithms for the production cost minimiza- tion objective function are presented in Table 6.9. This table also includes the results obtained by ABC [124], CABC [124], WOA [290] and CWOA [290] algorithms.
The production cost is calculated by using optimal parameters using Equation 6.4.
By inspecting the calculated production costs obtained by different algorithms it is observed that the production cost 35256.50 $/hwhich is obtained by using proposed
Figure 6.12: Variation of fitness value against iteration for Case B.3
IGWO parameters is the minimum. The comparative convergence analysis, obtained by both GWO and the proposed IGWO, is shown in Figure 6.12. This figure presents that IGWO based objective function value for this case converges smoothly and reaches the near global optimal value. The relative rotor angle trajectories are also shown in Figure 6.13 by using IGWO. As seen from this figure all the generators
Table 6.9: Best control variables and production cost for IEEE 39-Bus system (Case B.3)
Parameter ABC [124]
CABC [124]
WOA [290]
CWOA
[290] GWO IGWO
PG30 (MW) 336.96 350.00 393.60 364.56 569.02 574.84 PG31 (MW) 562.59 564.26 525.56 529.89 586.12 574.30 PG32 (MW) 549.17 577.14 560.19 587.78 582.10 576.21 PG33 (MW) 627.34 600.00 612.34 609.98 569.95 557.32 PG34 (MW) 489.38 491.62 445.56 446.62 508.00 508.00 PG35 (MW) 535.59 556.22 525.55 555.55 588.68 566.71 PG36 (MW) 577.91 564.63 567.61 543.78 573.67 554.28 PG37 (MW) 580.68 568.49 610.45 598.23 562.35 545.17 PG38 (MW) 759.98 763.34 745.68 788.67 596.36 717.60 PG39 (MW) 1119.70 1100.00 1149.70 1109.26 989.56 951.37
V30(p.u.) 1.03 1.00 1.03 1.03 0.95 0.95
V31(p.u.) 1.05 0.98 1.02 1.01 0.99 1.04
V32(p.u.) 0.99 1.01 1.01 1.02 0.98 1.00
V33(p.u.) 1.05 1.06 1.00 1.01 0.97 1.01
V34(p.u.) 1.02 1.06 1.00 1.00 1.01 1.00
V35(p.u.) 1.05 1.06 1.01 0.96 1.02 0.99
V36(p.u.) 1.03 1.06 0.99 0.99 0.98 0.98
V37(p.u.) 1.01 1.01 0.99 1.00 0.95 0.96
V38(p.u.) 0.95 1.06 1.05 1.01 1.05 1.05
V39(p.u.) 1.03 1.00 1.00 0.97 0.95 0.96
Cost($/h) 36058.69 35869.23 35930.45 35857.71 35680.63 35256.50
Table 6.10: Comparative results of IEEE 39-Bus 10-Generator system for Case B.3
Cost ($/h) ABC [124]
CABC [124]
WOA [290]
CWOA
[290] GWO IGWO
Minimum 36058.69 35869.23 35930.45 35857.71 35680.63 35256.50 Maximum 36678.11 36258.53 36882.78 36987.09 35874.44 35604.01 Average 36368.4 36063.88 36828.45 36619.34 35617.61 35403.46
Figure 6.13: Relative rotor angles obtained by IGWO for Case B.3
are stable and the rotor angles of all the generators do not cross the value δmax. Moreover, the statistical comparison of best, worst and mean fuel cost values as obtained using different algorithms are listed in Table 6.10.
The following interpretations can be drawn from the results shown in this section:
• The optimal setting of control variables obtained from the both GWO and proposed IGWO for fuel cost minimization.
• The results obtained from proposed IGWO-TSSCOPF are compared with the other obtained results offered by different published techniques for both IEEE 30-bus 6-generator system and IEEE 39-bus 10-generator system.
• It is observed from the comparison that the total cost from IGWO is less than that obtained by all other published algorithms for both IEEE 30-bus 6-generator system and IEEE 39-bus 10-generator system..
• Moreover, the statical comparison of best, worst and mean fuel cost values of different algorithms are listed in results. From the statical results, it is clear that the difference among the best, worst and mean objective cost values, as obtained by IGWO, are very much insignificant.
• It is observed from the results the worst value of IGWO is even much batter than the best values obtained by the other indicated methods.
• Results clearly suggest that the proposed IGWO-TSSCOPF method produces similar results in most of the trials and robustness of the proposed method is thus proved.