Modified Quasi-Opposition-Based Grey Wolf Optimization for Mathematical and Electrical Benchmark Problems
Abstract
:1. Introduction
2. Problem Formulation
2.1. Economic Dispatch Problem
- (i)
- Power Balance
- (ii)
- Generator Capacity Limit
- (iii)
- Ramp RateLimit (RRL)
- (iv)
- ProhibitedOperating Zones (POZs)
2.2. The Microgrid Scheduling Problem
3. Optimization
3.1. Grey Wolf Optimizer
- I.
- Entrapment of Prey
- II.
- Hunting of Prey
- III.
- Attacking the Prey
- Search agent position vectors are initialized randomly within the lower and upper limits.
- The fitness value of each agent is evaluated based on three categories of wolves (alpha, beta, and delta) among the population. They adjust their position to catch the prey using , and , as per (22).
- Search agents update their position by (23).
- The steps of fitness calculation and update mechanism are repeated to reach the specified termination criteria.
3.2. Intelligent Grey Wolf Optimizer
3.3. Modified Quasi-Opposition-Based Grey Wolf Optimization (mQOGWO)
- (i)
- Quasi-Opposition-Based Learning
- (ii)
- Non-Linear Decreasing Function
4. Simulation Results
4.1. Mathematical Benchmark Functions
4.2. Economic Dispatch Problems
- Test System I: A 15 Unit System
- Test System II: A 40 Unit System
- Test System III: A 140 Unit System (Korean Power System)
- Test System IV: Simulation Results of the Dynamic ED Problem of a Microgrid
4.3. A Comparative Study
- (i)
- Standard deviation
- (ii)
- Wilcoxon’s p-value
- (iii)
- Data dispersion and skewness in the box plot
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A
30 | [−100,100] | 0 | |||
30 | [−10,10] | 0 | |||
30 | [−100,100] | 0 | |||
30 | [−100,100] | 0 | |||
30 | [−30,30] | 0 | |||
30 | [−100,100] | 0 | |||
30 | [−1.28,1.28] | 0 | |||
30 | [−500,500] | −12,569.5 | |||
30 | [−5.12,5.12] | 0 | |||
30 | [−32,32] | 0 | |||
30 | [−600,600] | 0 | |||
30 | [−50,50] | 0 | |||
were | 30 | [−50,50] | 0 |
Name | Function | dim | Range | ||
---|---|---|---|---|---|
Fixed Dimension Benchmark | |||||
Shekel’s Foxholes | where | 2 | [−65.536, 65,536] | 1 | |
Kowalik | 4 | [−5,5] | 0.003075 | ||
Six Hump Camel Back | 2 | [−5,5] | −1.0316285 | ||
2 | [0,10] | 0.398 | |||
2 | [−2,2] | 3 | |||
4 | [0,1] | −3.86 | |||
6 | [0,1] | −3.32 | |||
4 | [0,10] | −10 | |||
4 | [0,10] | −10 | |||
] | 4 | [0,10] | −10 |
Units(MW) | mQOGWO | IGWO | GWO |
---|---|---|---|
P1 | 455.000 | 454.994 | 455.000 |
P2 | 380.000 | 380.003 | 380.000 |
P3 | 130.000 | 130.000 | 130.000 |
P4 | 130.000 | 130.000 | 130.000 |
P5 | 170.000 | 169.942 | 170.000 |
P6 | 460.000 | 459.999 | 460.000 |
P7 | 430.000 | 429.978 | 430.000 |
P8 | 69.476 | 87.951 | 116.554 |
P9 | 60.108 | 41.841 | 45.883 |
P10 | 160.000 | 160.000 | 127.412 |
P11 | 80.000 | 80.000 | 80.000 |
P12 | 80.000 | 80.000 | 80.000 |
P13 | 25.000 | 25.000 | 25.000 |
P14 | 15.000 | 15.001 | 15.000 |
P15 | 15.000 | 15.004 | 15.000 |
Ploss (MW) | 29.585 | 29.717 | 29.849 |
Fcost (USD/h) | 32,692.230 | 32,693.190 | 32,702.120 |
Units(MW) | QOGWO | IGWO | GWO | Units | QOGWO | IGWO | GWO |
---|---|---|---|---|---|---|---|
P1–P3 | 114.00 | 114.00 | 114.00 | P21 | 523.28 | 523.32 | 523.63 |
P4 | 179.73 | 181.37 | 180.12 | P22 | 523.30 | 550.00 | 550.00 |
P5 | 87.80 | 88.18 | 88.81 | P23–P24 | 523.28 | 523.28 | 523.28 |
P6 | 140.00 | 140.00 | 140.00 | P25 | 523.28 | 523.30 | 524.44 |
P7–P8 | 300.00 | 300.00 | 300.00 | P26 | 523.28 | 523.30 | 523.37 |
P9 | 289.89 | 300.00 | 300.00 | P27–P29 | 10.00 | 10.00 | 10.00 |
P10 | 279.60 | 279.60 | 279.60 | P30 | 87.80 | 87.80 | 87.80 |
P11 | 243.60 | 243.60 | 243.42 | P31 | 190.00 | 190.00 | 190.00 |
P12 | 94.00 | 94.00 | 94.00 | P34–P35 | 200.00 | 200.00 | 200.00 |
P13–P16 | 484.04 | 484.04 | 484.04 | P36 | 164.80 | 164.80 | 164.80 |
P17–P18 | 489.28 | 489.28 | 489.28 | P37–P39 | 110.00 | 110.00 | 110.00 |
P19–P20 | 511.28 | 511.28 | 511.30 | P40 | 550.00 | 511.90 | 511.34 |
Ploss (MW) | 972.20 | 973.00 | 973.22 | ||||
Fcost (USD/h) | 136,437.81 | 136,444.30 | 136,447.39 |
Units | QOGWO | IGWO | GWO | Units | QOGWO | IGWO | GWO |
---|---|---|---|---|---|---|---|
P1 | 119.00 | 119.00 | 119.00 | P82 | 56.00 | 56.00 | 56.00 |
P2 | 164.00 | 164.00 | 164.00 | P83–P85 | 115.00 | 115.00 | 115.00 |
P3–P6 | 190.00 | 190.00 | 190.00 | P86–P87 | 207.00 | 207.00 | 207.00 |
P7–P8 | 490.00 | 490.00 | 490.00 | P88–P89 | 175.00 | 175.00 | 175.00 |
P9–P12 | 496.00 | 496.00 | 496.00 | P90 | 180.43 | 180.49 | 180.49 |
P13 | 506.00 | 506.00 | 506.00 | P91 | 175.00 | 175.00 | 175.00 |
P14 | 509.00 | 509.00 | 509.00 | P92 | 575.40 | 575.40 | 575.40 |
P15 | 506.00 | 506.00 | 506.00 | P93 | 547.50 | 547.50 | 547.50 |
P16 | 505.00 | 505.00 | 505.00 | P94 | 836.80 | 836.80 | 836.80 |
P17–P18 | 506.00 | 506.00 | 506.00 | P95 | 837.50 | 837.50 | 837.50 |
P19–P24 | 505.00 | 505.00 | 505.00 | P96 | 682.00 | 682.00 | 682.00 |
P25–26 | 537.00 | 537.00 | 537.00 | P97 | 720.00 | 720.00 | 720.00 |
P27–28 | 549.00 | 549.00 | 549.00 | P98 | 718.00 | 718.00 | 718.00 |
P29 | 501.00 | 501.00 | 501.00 | P99 | 720.00 | 720.00 | 720.00 |
P30 | 499.00 | 499.00 | 499.00 | P100 | 964.00 | 964.00 | 964.00 |
P31–P34 | 506.00 | 506.00 | 506.00 | P101 | 958.00 | 958.00 | 958.00 |
P35–P36 | 500.00 | 500.00 | 500.00 | P102 | 947.90 | 947.90 | 947.90 |
P37–P38 | 241.00 | 241.00 | 241.00 | P103 | 934.00 | 934.00 | 934.00 |
P39 | 774.00 | 774.00 | 774.00 | P104 | 935.00 | 935.00 | 935.00 |
P40 | 769.00 | 769.00 | 769.00 | P105 | 876.50 | 876.50 | 876.50 |
P41–P42 | 3.00 | 3.00 | 3.00 | P106 | 880.90 | 880.90 | 880.90 |
P43–P50 | 250.00 | 250.00 | 250.00 | P107 | 873.70 | 873.70 | 873.70 |
P51–54 | 165.00 | 165.00 | 165.00 | P108 | 877.40 | 877.40 | 877.40 |
P55–P56 | 180.00 | 180.00 | 180.00 | P109 | 871.70 | 871.70 | 871.70 |
P57 | 103.00 | 103.00 | 103.00 | P110 | 864.80 | 864.80 | 864.80 |
P58 | 198.00 | 198.00 | 198.00 | P111 | 882.00 | 882.00 | 882.00 |
P59 | 312.00 | 312.00 | 312.00 | P112–P114 | 94.00 | 94.00 | 94.00 |
P60 | 308.60 | 308.59 | 308.59 | P115–P117 | 244.00 | 244.00 | 244.00 |
P61 | 163.00 | 163.00 | 163.00 | P118–P119 | 95.00 | 95.00 | 95.00 |
P62 | 95.00 | 95.00 | 95.00 | P120 | 116.00 | 116.00 | 116.00 |
P63 | 511.00 | 503.05 | 503.05 | P121 | 175.00 | 175.00 | 175.00 |
P64 | 511.00 | 511.00 | 511.00 | P122 | 2.00 | 2.00 | 2.00 |
P65 | 490.00 | 490.00 | 490.00 | P123 | 4.00 | 4.00 | 4.00 |
P66 | 256.84 | 256.80 | 256.80 | P124 | 15.00 | 15.00 | 15.00 |
P67–P68 | 490.00 | 490.00 | 490.00 | P125 | 9.00 | 9.00 | 9.00 |
P69 | 130.00 | 130.00 | 130.00 | P126 | 12.00 | 12.00 | 12.00 |
P70 | 294.56 | 294.58 | 294.58 | P127 | 10.00 | 10.00 | 10.00 |
P71 | 141.59 | 141.67 | 141.67 | P128 | 112.00 | 112.00 | 112.00 |
P72 | 365.92 | 365.95 | 365.95 | P129 | 4.00 | 4.00 | 4.00 |
P73 | 195.00 | 195.00 | 195.00 | P130–P131 | 5.00 | 5.00 | 5.00 |
P74 | 217.10 | 204.67 | 204.67 | P132 | 50.00 | 50.00 | 50.00 |
P75 | 217.89 | 241.27 | 241.27 | P133 | 5.00 | 5.00 | 5.00 |
P76 | 258.68 | 257.86 | 257.86 | P134–P135 | 42.00 | 42.00 | 42.00 |
P77 | 403.29 | 400.96 | 400.96 | P136 | 41.00 | 41.00 | 41.00 |
P78 | 330.00 | 330.00 | 330.00 | P137 | 17.00 | 17.00 | 17.00 |
P79–80 | 531.00 | 531.00 | 531.00 | P138–P139 | 7.00 | 7.00 | 7.00 |
P81 | 542.00 | 542.00 | 542.00 | P140 | 26.00 | 26.00 | 26.00 |
Fcost (USD/h) | 1,655,679.43 | 1,655,679.57 | 1,655,685.80 |
Units | QOGWO | IGWO | GWO | Units | QOGWO | IGWO | GWO |
---|---|---|---|---|---|---|---|
P1 | 119.00 | 119.00 | 119.00 | P82 | 56.00 | 56.00 | 56.00 |
P2 | 164.00 | 164.00 | 164.00 | P83–P84 | 115.00 | 115.00 | 115.00 |
P3–P4 | 190.00 | 190.00 | 190.00 | P86–P87 | 207.00 | 207.00 | 207.00 |
P5 | 168.54 | 168.54 | 168.54 | P88–P89 | 175.00 | 175.00 | 175.00 |
P6 | 190.00 | 190.00 | 190.00 | P90 | 180.41 | 180.42 | 180.62 |
P7–P8 | 490.00 | 490.00 | 490.00 | P91 | 175.00 | 175.00 | 175.00 |
P9–P12 | 496.00 | 496.00 | 496.00 | P92 | 575.40 | 575.40 | 575.40 |
P13 | 506.00 | 506.00 | 506.00 | P93 | 547.50 | 547.50 | 547.50 |
P14 | 509.00 | 509.00 | 509.00 | P94 | 836.80 | 836.80 | 836.80 |
P15 | 506.00 | 506.00 | 506.00 | P95 | 837.50 | 837.50 | 837.50 |
P16 | 505.00 | 505.00 | 505.00 | P96 | 682.00 | 682.00 | 682.00 |
P17–P18 | 506.00 | 506.00 | 506.00 | P97 | 720.00 | 720.00 | 720.00 |
P19–P24 | 505.00 | 505.00 | 505.00 | P98 | 718.00 | 718.00 | 718.00 |
P25–P26 | 537.00 | 537.00 | 537.00 | P99 | 720.00 | 720.00 | 720.00 |
P27–P28 | 549.00 | 549.00 | 549.00 | P100 | 964.00 | 964.00 | 964.00 |
P29 | 501.00 | 501.00 | 501.00 | P101 | 958.00 | 958.00 | 958.00 |
P30 | 499.00 | 499.00 | 499.00 | P102 | 947.90 | 947.90 | 947.90 |
P31–P34 | 506.00 | 506.00 | 506.00 | P103 | 934.00 | 934.00 | 934.00 |
P35–P36 | 500.00 | 500.00 | 500.00 | P104 | 935.00 | 935.00 | 935.00 |
P37–P38 | 241.00 | 241.00 | 241.00 | P105 | 876.50 | 876.50 | 876.50 |
P39 | 774.00 | 774.00 | 774.00 | P106 | 880.90 | 880.90 | 880.90 |
P40 | 769.00 | 769.00 | 769.00 | P107 | 873.70 | 873.70 | 873.70 |
P41–P42 | 3.00 | 3.00 | 3.00 | P108 | 877.40 | 877.40 | 877.40 |
P43–P50 | 250.00 | 250.00 | 250.00 | P109 | 871.70 | 871.70 | 871.70 |
P51–P54 | 165.00 | 165.00 | 165.00 | P110 | 864.80 | 864.80 | 864.80 |
P55–P56 | 180.00 | 180.00 | 180.00 | P111 | 882.00 | 882.00 | 882.00 |
P57 | 103.00 | 103.00 | 103.00 | P112–P114 | 94.00 | 94.00 | 94.00 |
P58 | 198.00 | 198.00 | 198.00 | P115–P117 | 244.00 | 244.00 | 244.00 |
P59 | 312.00 | 312.00 | 312.00 | P118–P119 | 95.00 | 95.00 | 95.00 |
P60 | 308.59 | 308.59 | 308.73 | P120 | 116.00 | 116.00 | 116.00 |
P61 | 163.00 | 163.00 | 163.00 | P121 | 175.00 | 175.00 | 175.00 |
P62 | 95.00 | 95.00 | 95.00 | P122 | 2.00 | 2.00 | 2.00 |
P63–P64 | 511.00 | 511.00 | 511.00 | P123 | 4.00 | 4.00 | 4.00 |
P65 | 490.00 | 490.00 | 490.00 | P124 | 15.00 | 15.00 | 15.00 |
P66 | 256.75 | 256.81 | 257.47 | P125 | 9.00 | 9.00 | 9.00 |
P67–P68 | 490.00 | 490.00 | 490.00 | P126 | 12.00 | 12.00 | 12.00 |
P69 | 130.00 | 130.00 | 130.00 | P127 | 10.00 | 10.00 | 10.00 |
P70 | 339.44 | 339.44 | 339.44 | P128 | 112.00 | 112.00 | 112.00 |
P71 | 141.59 | 141.59 | 141.82 | P129 | 4.00 | 4.00 | 4.00 |
P72 | 388.33 | 388.33 | 388.33 | P130–P131 | 5.00 | 5.00 | 5.00 |
P73 | 195.00 | 195.00 | 195.00 | P132 | 50.00 | 50.00 | 50.00 |
P74 | 196.23 | 214.74 | 195.10 | P133 | 5.00 | 5.00 | 5.00 |
P75 | 196.10 | 175.00 | 175.00 | P134–P135 | 42.00 | 42.00 | 42.00 |
P76 | 257.97 | 258.57 | 262.69 | P136 | 41.00 | 41.00 | 41.00 |
P77 | 400.95 | 402.89 | 417.18 | P137 | 17.00 | 17.00 | 17.00 |
P78 | 330.00 | 330.00 | 330.00 | P138–P139 | 7.00 | 7.00 | 7.00 |
P79–P80 | 531.00 | 531.00 | 531.00 | P140 | 26.00 | 26.00 | 26.00 |
P81 | 542.00 | 542.00 | 542.00 | Fcost (USD/h) | 1,657,962.73 | 1,657,962.76 | 1,657,962.89 |
D1 (kW) | D2 (kW) | FC1 (kW) | FC2 (kW) | FC3 (kW) | W1 (kW) | W2 (kW) |
---|---|---|---|---|---|---|
134.76 | 83.08 | 72.96 | 55.29 | 40.87 | 133.52 | 133.52 |
81.21 | 105.20 | 54.34 | 47.58 | 57.03 | 102.32 | 102.32 |
101.16 | 153.55 | 82.55 | 45.61 | 46.40 | 107.87 | 107.87 |
119.89 | 152.48 | 72.88 | 35.49 | 49.74 | 128.76 | 128.76 |
157.25 | 248.01 | 57.25 | 46.59 | 49.20 | 142.35 | 142.35 |
183.20 | 408.02 | 88.68 | 47.62 | 64.13 | 163.17 | 163.17 |
285.25 | 405.09 | 89.80 | 52.50 | 67.15 | 212.61 | 212.61 |
241.02 | 451.45 | 89.76 | 68.34 | 45.75 | 248.84 | 248.84 |
248.55 | 519.03 | 80.72 | 55.62 | 42.84 | 240.62 | 240.62 |
244.58 | 452.48 | 109.14 | 66.71 | 57.25 | 231.42 | 231.42 |
200.82 | 473.70 | 74.38 | 56.16 | 44.68 | 194.13 | 194.13 |
216.78 | 315.30 | 80.32 | 56.78 | 48.31 | 183.26 | 183.26 |
229.70 | 312.38 | 62.48 | 42.34 | 50.21 | 167.45 | 167.45 |
169.26 | 363.63 | 68.49 | 38.10 | 57.06 | 150.73 | 150.73 |
249.75 | 393.50 | 77.99 | 42.64 | 49.32 | 135.41 | 135.41 |
210.96 | 328.56 | 68.45 | 56.19 | 53.62 | 157.11 | 157.11 |
234.01 | 404.81 | 80.35 | 54.71 | 58.82 | 142.66 | 142.66 |
251.32 | 545.20 | 90.02 | 55.13 | 59.57 | 187.38 | 187.38 |
329.19 | 672.45 | 103.10 | 56.86 | 49.57 | 228.42 | 228.42 |
296.41 | 640.66 | 95.03 | 55.11 | 51.63 | 256.58 | 256.58 |
264.04 | 645.94 | 91.95 | 74.59 | 65.01 | 246.24 | 246.24 |
355.50 | 517.98 | 89.65 | 53.49 | 54.89 | 195.25 | 195.25 |
268.85 | 461.29 | 141.94 | 65.76 | 53.27 | 175.44 | 175.44 |
222.58 | 370.71 | 90.00 | 56.34 | 55.16 | 135.61 | 135.61 |
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Method | Modification | Domain of Application |
---|---|---|
GWO | N.A. | Mathematical benchmark, welded beam design, pressure vessel design, optical buffer design [13], economic dispatch (ED) [23], short-term hydro-thermal scheduling (STHTS) [24], combined heat and power (CHP) with ED [25], microgrids [26], distribution generator (DG) placement [27], controller design [28,29,30], wireless networks [31,32], image processing [33,34], regular design [35], and parameter estimation [36]. |
Complex-Valued Encoding Grey Wolf Optimizer (CGWO) | DE/best/2 mutation strategy is embedded with GWO. | Mathematical benchmark and infinite impulse response (IIR) model identification [37]. |
Powell Local Optimization-Based Grey Wolf Optimizer (PGWO) | Powell method is embedded with GWO. | Mathematical benchmark and Data Clustering [38]. |
Hybrid Grey Wolf Optimizer (HGWO) | DE/best/1 and dynamic crossover rate are implanted in GWO. | ED [39]. |
Modified Grey Wolf Optimizer (mGWO) | Exponential decay of ‘a’ is utilized here in place of linear decay. | Mathematical benchmark and cluster head selection problem in Wireless Sensor Networks (WSNs) [40]. |
Chaos-Based Grey Wolf Optimizer (Ch-GWO) | Tent and Singer map is used to enhance global search capability. | Position control of a robotic manipulator [41]. |
Mean Grey Wolf Optimizer (MGWO) | Encircling the pray phase of GWO is carried out by considering the mean distance of grey wolves from the prey. | Mathematical benchmark and real-life dataset problems [42]. |
Ameliorated Grey Wolf Optimizer (Am-GWO) | Exploratory search mechanism to ensure the right direction of each wolf; opposite-based learning (OBL) maintains a good and diverse population; local search mechanism for fine-tuning is unlisted. | ED [43]. |
Opposition-Based Grey Wolf Optimizer (OGWO) | OBL is incorporated to find a better candidate solution. | Mathematical benchmark and ED [44]. |
Inspired Grey Wolf Optimizer (In-GWO) | Logarithmic decay characteristics of parameter are introduced and the position updating mechanism is carried out based on and of PSO. | Mathematical benchmark, pressure vessel design, welded beam design, spring design, and load forecasting [45] |
Binary Hybrid GWO and PSO (BGWOPSO) | A binary version of hybrid GWO and PSO is utilized here. | 18 standard University of California Irvine (UCI) benchmark datasets [46]. |
Β–GWO | GWO is hybridized with β–hill climbing. | ED [47]. |
Orthogonal Grey Wolf Optimizer (Or-GWO) | Orthogonal Array Design (OAD) is incorporated for updating the position of leader wolves. | Mathematical benchmark and clustering datasets [48]. |
Accelerated Grey Wolf Optimizer (A-GWO) | An acceleration factor and uniform distribution are used to boost exploration and exploitation. | Mathematical benchmark, gear design, frequency modulated, beam design, and cost minimization of a life support system [49]. |
Grey Wolf Optimizer Based on Weighted Distance (GWO-WD) | The weighted distance concept is used to modify the position-updating mechanism; elimination and repositioning strategy is employed to reposition the worst search agents. | Mathematical benchmark, pressure vessel design, welded beam design, and gear design [50]. |
mQOGWO | IGWO [52] | GWO [13] | PSO [13] | GSA [13] | DE [13] | |||||||
Ave | SD | Ave | SD | Ave | SD | Ave | SD | Ave | SD | Ave | SD | |
0 | 0 | 5.55 × 10−26 | 1.00 × 10−25 | 6.59 × 10−28 | 6.34 × 10−5 | 0.000136 | 0.000202 | 5.30 × 10−17 | 9.67 × 10−17 | 8.20 × 10−14 | 5.90 × 10−14 | |
1.39 × 10−200 | 0 | 7.75 × 10−16 | 7.82 × 10−16 | 7.18 × 10−17 | 0.029014 | 0.042144 | 0.045421 | 0.055655 | 0.194074 | 1.50 × 10−9 | 9.90 × 10−10 | |
0 | 0 | 9.93 × 10−5 | 0.000763 | 3.29 × 10−6 | 79.14958 | 70.12562 | 22.11924 | 896.5347 | 318.9559 | 6.80 × 10−11 | 7.40 × 10−11 | |
3.08 × 10−176 | 0 | 1.08 × 10−6 | 1.05 × 10−6 | 5.61 × 10−7 | 1.315088 | 1.086481 | 0.317039 | 7.35487 | 1.741452 | 0 | 0 | |
25.7884 | 0.1090 | 27.0042 | 0.642515 | 26.81258 | 69.90499 | 96.71832 | 60.11559 | 67.54309 | 62.2253 | 0 | 0 | |
0.7541 | 0.0818 | 0.6677 | 0.31327 | 0.816579 | 0.000126 | 0.000102 | 8.28 × 10−5 | 2.50 × 10−16 | 1.74 × 10−16 | 0 | 0 | |
7.983 × 10−5 | 1.27 × 10−5 | 0.00182 | 0.001074 | 0.002213 | 0.100286 | 0.122854 | 0.044957 | 0.089441 | 0.04339 | 0.00463 | 0.0012 | |
−4614.8 | 261.8496 | −5991.23 | 950.3294 | −6123.1 | −4087.44 | −4841.29 | 1152.814 | −2821.07 | 493.0375 | −11,080.1 | 574.7 | |
0 | 0 | 1.270284 | 2 2.7344 | 0.310521 | 47.35612 | 46.70423 | 11.62938 | 25.96841 | 7.47006 | 69.2 | 38.8 | |
2.664 × 10−15 | 3.243 × 10−16 | 1.64 × 10−13 | 44.31 × 10−14 | 1.06 × 10−13 | 0.020734 | 0.276015 | 0.50901 | 0.062087 | 0.23628 | 9.70 × 10−8 | 4.20 × 10−8 | |
0 | 0 | 0.001994 | 0.005099 | 0.004485 | 0.006659 | 0.009215 | 0.007724 | 27.70154 | 5.040343 | 0 | 0 | |
0.0481 | 0.0036 | 0.042402 | 0.052673 | 0.053438 | 0.020734 | 0.006917 | 0.026301 | 1.799617 | 0.95114 | 7.90 × 10−15 | 8.00 × 10−15 | |
0.7591 | 0.0618 | 0.551296 | 0.21782 | 0.654464 | 0.004474 | 0.006675 | 0.008907 | 8.899084 | 7.126241 | 5.10 × 10−14 | 4.80 × 10−14 |
mQGWO | IGWO [52] | GWO [13] | PSO [13] | GSA [13] | DE [13] | |||||||
Ave | SD | Ave | SD | Ave | SD | Ave | SD | Ave | SD | Ave | SD | |
4.6806 | 0.8197 | 4.038 | 3.7415 | 4.0424 | 4.2527 | 3.6271 | 2.5608 | 5.8598 | 3.8312 | 0.99 | 3.3 × 10−16 | |
4.054 × 10−4 | 1.760 × 10−5 | 4.158 × 10−4 | 1.813 × 10−5 | 0.000337 | 0.000625 | 0.000577 | 0.000222 | 0.003673 | 0.001647 | 4.50 × 10−14 | 0.00033 | |
−1.0316 | 1.4024 × 10−5 | −1.0316 | 7.765 × 10−12 | −1.03163 | −1.03163 | −1.03163 | 6.25 × 10−16 | −1.03163 | 4.88 × 10−16 | −1.03163 | 3.1 × 10−13 | |
0.3982 | 1.7518 × 10−4 | 0.3993 | 5.156 × 10−6 | 0.397889 | 0.397887 | 0.397887 | 0 | 0.397887 | 0 | 0.397887 | 9.9 × 10−9 | |
3.0000 | 3.9096 × 10−6 | 3.0000 | 7.693 × 10−6 | 3.000028 | 3 | 3 | 1.33 × 10−15 | 3 | 4.17 × 10−15 | 3 | 2 × 10−15 | |
−3.8616 | 3.6947 × 10−4 | −3.8614 | 4.054 × 10−4 | −3.86263 | −3.86278 | −3.86278 | 2.58 × 10−15 | −3.86278 | 2.29 × 10−15 | N/A | N/A | |
−3.2735 | 0.0119 | −3.2453 | 0.0139 | −3.28654 | −3.25056 | −3.26634 | 0.060516 | −3.31778 | 0.023081 | N/A | N/A | |
−10.1532 | 3.6926 × 10−11 | −10.1532 | 2.583 × 10−8 | −10.1514 | −9.14015 | −6.8651 | 3.019644 | −5.95512 | 3.737079 | −10.1532 | 0 | |
−10.4029 | 3.2452 × 10−11 | −10.4029 | 4.353 × 10−8 | −10.4015 | −8.58441 | −8.45653 | 3.087094 | −9.68447 | 2.014088 | −10.4029 | 3.9 × 10−7 | |
−10.5364 | 2.0493 × 10−11 | −10.3561 | 0.1772 | −10.5343 | −8.55899 | −9.95291 | 1.782786 | −10.5364 | 2.60 × 10−15 | −10.5364 | 1.9 × 10−7 |
Functions | Unimodal | F1 | F2 | F3 | F4 | F5 | F6 | F7 | ||
IGWO | 7.06 × 10−18 | 5.01 × 10−11 | 9.06 × 10−8 | 1.12 × 10−10 | 3.59 × 10−5 | 3.50 × 10−9 | 0.0156 | |||
GWO | 2.56 × 10−34 | 3.02 × 10−11 | 3.82 × 10−9 | 7.44 × 10−9 | 4.64 × 10−5 | 1.43 × 10−8 | 6.07 × 10−11 | |||
Functions | Multimodal | F8 | F9 | F10 | F11 | F12 | F13 | |||
IGWO | 0.0079 | 0.011 | 7.93 × 10−13 | 0.0214 | 1.29 × 10−9 | 4.08 × 10−11 | ||||
GWO | 3.52 × 10−7 | 1.19 × 10−12 | 1.15 × 10−12 | 0.0215 | 5.00 × 10−9 | 3.16 × 10−10 | ||||
Functions | Fixed Dimension | F14 | F15 | F16 | F17 | F18 | F19 | F20 | F21 | F22 |
IGWO | 0.0029 | 0.0212 | 2.62 × 10−5 | 0.0173 | 0.0292 | 0.03478 | 0.0121 | 0.0344 | 0.0044 | |
GWO | 0.0059 | 0.0042 | 5.35 × 10−6 | 0.0221 | 0.0109 | 0.02789 | 0.0288 | 1.24 × 10−7 | 6.47 × 10−8 |
Methods | Min Cost (USD/h) | Ave Cost (USD/h) | Max Cost (USD/h) | SD | Ave CPU Time (s) |
---|---|---|---|---|---|
AIS [63] | 32,854.00 | 32,873.25 | 32,892.00 | 10.81 | NA |
SA [64] | 32,786.40 | 32,869.51 | 33,038.95 | 112.32 | 71.25 |
GA [64] | 32,779.81 | 32,841.21 | 33,041.64 | 81.22 | 48.17 |
TSA [64] | 32,762.12 | 32,822.84 | 33,041.64 | 60.59 | 26.41 |
PSO [64] | 32,724.17 | 32,807.45 | 32,841.38 | 21.24 | 13.25 |
MTS [64] | 32,716.87 | 32,767.21 | 32,796.15 | 17.51 | 3.65 |
DSPSO-TSA [65] | 32,715.06 | 32,724.63 | 32,730.39 | 8.40 | 2.30 |
Jaya [66] | 32,712.65 | 32,743.46 | 32,822.99 | 47.03 | 3.80 |
Jaya-SML [66] | 32,706.36 | 32,706.68 | 32,707.29 | 2.32 | 5.14 |
CJaYa [67] | 32,710.08 | 32,740.07 | 32,828.66 | NA | NA |
MP-Cjaya [67] | 32,706.52 | 32,706.72 | 32,708.87 | NA | NA |
GWO | 32,702.12 | 32,703.31 | 32,704.58 | 1.48 | 8.23 |
IGWO | 32,693.19 | 32,694.74 | 32,695.61 | 1.38 | 7.16 |
mQOGWO | 32,692.23 | 32,692.40 | 32,692.60 | 1.09 | 3.28 |
Methods | Min Cost (USD/h) | Ave Cost (USD/h) | Max Cost (USD/h) | SD | Ave CPU Time (s) |
---|---|---|---|---|---|
HGWO [39] | 136,681.00 | 136,684 | NA | NA | NA |
OGWO [44] | 136,440.62 | 136,442.26 | 136,445.98 | 0.1003 | NA |
BBO [68] | 137,026.82 | 137,116.58 | 137,587.82 | NA | 40.00 |
DE/BBO [68] | 136,950.77 | 136,966.77 | 137,150.77 | NA | 32.00 |
ORCCRO [69] | 136,855.19 | 136,855.19 | 136,855.19 | NA | 14.00 |
SCA [69] | 136,653.02 | 136,653.02 | 136,653.10 | NA | 28.00 |
OIWO [70] | 136,452.68 | 136,452.68 | 136,452.68 | NA | 10.70 |
GWO | 136,447.39 | 136,541.34 | 136,588.22 | 25.48 | 11.34 |
IGWO | 136,444.30 | 136,462.93 | 136,510.15 | 17.12 | 10.78 |
QOGWO | 136,437.81 | 136,440.70 | 13,663.40 | 5.46 | 9.86 |
Methods | Min Cost (USD/h) | Ave Cost (USD/h) | Max Cost (USD/h) | SD | Ave CPU Time (s) |
---|---|---|---|---|---|
Test System III-A (Convex Characteristics) | |||||
FPA [73] | 1,655,685.80 | 1,655,709.06 | 1,655,732.32 | 24.86 | 10.24 |
MFPA [73] | 1,655,679.39 | 1,655,679.42 | 1,655,679.43 | 0.02 | 5.57 |
CCPSO [62] | 1,655,685.00 | 1,655,685.00 | 1,655,685.00 | NA | 42.90 |
CQGSO [72] | 1,655,679.43 | 1,655,679.43 | 1,655,679.43 | NA | 18.61 |
HHE [71] | 1,655,679.41 | NA | NA | NA | 8.23 |
GWO | 1,655,685.80 | 1,656,187.78 | 1,656,575.94 | 20.02 | 5.45 |
IGWO | 1,655,679.57 | 1,655,965.60 | 1,656,498.17 | 24.9 | 5.54 |
QOGWO | 1,655,679.43 | 1,655,869.64 | 1,656,018.68 | 2.96 | 6.76 |
Test System III-B (Non-Convex Characteristics) | |||||
GSO [72] | 1,728,151.17 | 1,745,515.00 | 1,753,229.56 | NA | NA |
BBO [68] | 1,665,478.25 | 1,667,548.32 | 1,669,536.35 | NA | NA |
DE/BBO [68] | 1,660,215.65 | 1,661,257.35 | 1,662,349.58 | NA | NA |
ORCCRO [68] | 1,659,654.83 | 1,659,725.96 | 1,659,823.97 | 0.16 | NA |
SCA [69] | 1,658,384.88 | 1,658,384.25 | 1,658,386.57 | 0.1 | NA |
CQGSO [72] | 1,657,962.73 | 1,657,962.74 | 1,657,776.00 | NA | 31.67 |
CCPSO [64] | 1,657,962.73 | 1,657,962.73 | 1,657,962.73 | 0.00 | 150.00 |
HHE [71] | 1,657,962.71 | NA | NA | NA | 8.80 |
FPA [73] | 1,657,962.77 | 1,658,051.90 | 1,658,570.77 | 228.84 | 12.67 |
MFPA [73] | 1,657,962.69 | 1,657,962.75 | 1,657,962.82 | 0.06 | 5.71 |
GWO | 1,657,962.89 | 1,658,612.89 | 1,659,262.89 | 40.11 | 5.70 |
IGWO | 1,657,962.76 | 1,658,027.76 | 1,658,092.76 | 25.31 | 5.75 |
QOGWO | 1,657,962.73 | 1,657,969.23 | 1,657,975.73 | 4.03 | 6.89 |
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Dubey, S.M.; Dubey, H.M.; Salkuti, S.R. Modified Quasi-Opposition-Based Grey Wolf Optimization for Mathematical and Electrical Benchmark Problems. Energies 2022, 15, 5704. https://doi.org/10.3390/en15155704
Dubey SM, Dubey HM, Salkuti SR. Modified Quasi-Opposition-Based Grey Wolf Optimization for Mathematical and Electrical Benchmark Problems. Energies. 2022; 15(15):5704. https://doi.org/10.3390/en15155704
Chicago/Turabian StyleDubey, Salil Madhav, Hari Mohan Dubey, and Surender Reddy Salkuti. 2022. "Modified Quasi-Opposition-Based Grey Wolf Optimization for Mathematical and Electrical Benchmark Problems" Energies 15, no. 15: 5704. https://doi.org/10.3390/en15155704