Research on Economic Load Dispatch Problem of Microgrid Based on an Improved Pelican Optimization Algorithm
Abstract
:1. Introduction
2. Materials and Methods
2.1. Relationship Work
2.1.1. Electric Power Economic Load Dispatch (ELD)
2.1.2. Pelican Optimization Algorithm
- Population initialization
- Exploration phase
- Development phase
2.2. Improved Pelican Optimization Algorithm
2.2.1. Fusion of Improved Crisscross Optimization Algorithm for Local Search
2.2.2. Improved Global Search
2.2.3. Dimensional Variation Strategy
Algorithm 1. Mutates Dimensionally |
1: Generate d random numbers of T-distribution with 25 degrees of freedom parameter. 2: for i = 1: d 4: boundary condition procedure 5: if fnew < fbest 8: end if 9: end for 10: Return the best fitness value and the best individual |
2.2.4. IPOA Implementation Process
2.3. IPOA Algorithm Performance Test and Analysis
2.3.1. Experimental Environment and Test Function
2.3.2. Comparisons with POA, PSO, SSA, and WOA
2.4. IPOA Solves the Problem of Economic Dispatch
Algorithm 2. IPOA for ELD |
1: Input: Population size, Dimension, variable bounds Maximum failure count 2: Initialization: Initialize population X and Calculate fitness value using Equation (5) 3: for i = 1: Max_iterations 4: for j = 1: N 5: Randomly select an individual 6: if fit(p) < fit(j) 7: Update positions by Equation (7) 8: else 9: Update positions by Equation (13) 10: end if 11: Update positions by Equation (14) 12: Use algorithm1 update the global optimum solution 13: Handling boundary conditions 14: Calculating individual fitness values using Equation (5) 15: Update the global optimum solution 16: end for 17: end for 18: Calculate fuel cost using Equation (1) 19: Output: Optimal cost, Unit’s output |
3. Experimental Results and Discussion
3.1. 10 Units
3.2. 40 Units
3.3. 80 Units
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Functions | Best Value | Types | |
---|---|---|---|
Shifted and Rotated Bent Cigar | 100 | Unimodal | |
Shifted and Rotated Rastrigin’s | 400 | Simple Multimodal | |
Shifted and Rotate Lunacek Bi_Rastrigin | 600 | ||
Shifted and Rotated Schwefel’s | 900 | ||
Hybrid Function 2 (N = 3) | 1100 | Hybrid | |
Hybrid Function 6 (N = 5) | 1600 | ||
Composition Function 1 (N = 3) | 2000 | Composition | |
Composition Function 7 (N = 6) | 2600 |
Function | Index | Algorithm | ||||
---|---|---|---|---|---|---|
IPOA | POA | PSO | WOA | SSA | ||
Best | 100.7108 | 6097 | 57,929,542 | 592,934.6706 | 133.5179 | |
Worst | 1226.4633 | 1,944,227,684 | 2,850,258,819 | 74,471,522.3021 | 12,381.7875 | |
Mean | 4768.0518 | 219,781,765 | 792,680,137 | 9,361,246.7600 | 4438.0798 | |
Std | 1560.1471 | 495,103,154 | 831,763,818 | 15,616,489.4653 | 3567.2322 | |
Best | 400.0002 | 400.7755 | 411.1083 | 400.6303 | 400.1664 | |
Worst | 473.2955 | 496.9909 | 825.9439 | 563.8685 | 472.0999 | |
Mean | 404.1049 | 418.4436 | 505.0500 | 440.8641 | 404.6719 | |
Std | 13.1237 | 21.6410 | 102.3337 | 47.2803 | 12.7660 | |
Best | 600.0000 | 607.4886 | 608.6856 | 610.2782 | 600.0000 | |
Worst | 601.4796 | 638.7122 | 635.2207 | 657.3534 | 613.0539 | |
Mean | 600.1154 | 621.6511 | 618.6479 | 634.4335 | 602.8292 | |
Std | 0.33723 | 9.2482 | 6.2520 | 11.8559 | 3.8600 | |
Best | 900.0000 | 906.0017 | 931.4729 | 921.8618 | 900.0000 | |
Worst | 929.6692 | 1387.7745 | 1293.9060 | 3566.0554 | 1829.3862 | |
Mean | 903.8361 | 1092.0306 | 1008.2076 | 1612.5818 | 1116.2890 | |
Std | 7.1136 | 141.2868 | 65.9039 | 643.7356 | 308.2101 | |
Best | 1100.0366 | 1109.0200 | 1171.9598 | 1123.9527 | 1103.0719 | |
Worst | 1137.6526 | 1315.8967 | 1903.2485 | 1568.3389 | 1258.4352 | |
Mean | 1116.7835 | 1171.1777 | 1345.1482 | 1208.2003 | 1145.7665 | |
Std | 10.4314 | 49.0942 | 165.8870 | 90.6109 | 41.5469 | |
Best | 1600.7438 | 1607.6178 | 1636.0000 | 1622.9391 | 1601.4464 | |
Worst | 1960.8268 | 1938.6274 | 2246.0736 | 2304.1432 | 2139.5167 | |
Mean | 1689.9921 | 1763.1663 | 1804.9491 | 1913.4186 | 1832.8280 | |
Std | 115.7566 | 106.2803 | 157.1715 | 187.8456 | 138.5551 | |
Best | 2000.0000 | 2024.1393 | 2057.8897 | 2043.5482 | 2005.5991 | |
Worst | 2140.3403 | 2162.7398 | 2244.1384 | 2450.8169 | 2278.7248 | |
Mean | 2037.2731 | 2083.7765 | 2127.0903 | 2191.1716 | 2088.4404 | |
Std | 26.1220 | 40.1675 | 58.2918 | 88.8603 | 67.2939 | |
Best | 2600.0043 | 2608.7520 | 2967.3881 | 2628.1123 | 2800.0000 | |
Worst | 3165.7513 | 3904.5444 | 4257.0824 | 4786.9540 | 3395.5483 | |
Mean | 2966.1859 | 3023.9175 | 3329.9615 | 3585.4080 | 4483.6695 | |
Std | 134.8942 | 241.6267 | 450.6247 | 574.6367 | 532.1813 |
Units | Pmin | Pmax | a | b | c | e | f |
---|---|---|---|---|---|---|---|
1 | 100 | 250 | 0.002176 | −0.3975 | 26.97 | 0.02697 | −3.975 |
2 | 50 | 230 | 0.004194 | −1.269 | 118.4 | 0.1184 | −12.69 |
3 | 200 | 500 | 0.00001176 | 0.4864 | −95.14 | −0.05914 | 4.864 |
4 | 99 | 265 | 0.005935 | −2.338 | 266.8 | 0.2668 | −23.38 |
5 | 190 | 490 | 0.0001498 | 0.4462 | −53.99 | −0.05399 | 4.462 |
6 | 85 | 265 | 0.005935 | −2.338 | 266.8 | 0.2668 | −23.38 |
7 | 200 | 500 | 0.0002454 | 0.3559 | −43.35 | −0.04335 | 3.559 |
8 | 99 | 265 | 0.005935 | −2.338 | 266.8 | 0.2668 | −23.38 |
9 | 130 | 440 | 0.0006121 | −0.0182 | 14.23 | 0.01423 | −0.1817 |
10 | 200 | 490 | 0.0000416 | 0.5084 | −61.13 | −0.06113 | 5.084 |
Units | Algorithms | ||||
---|---|---|---|---|---|
IPOA | POA | HHO | SSA | WOA | |
203.5350 | 202.8740 | 211.5970 | 202.7439 | 220.4145 | |
210.4219 | 210.4247 | 215.8357 | 210.9169 | 207.4651 | |
200.6466 | 200.0152 | 206.2645 | 200.0000 | 224.0729 | |
238.8801 | 237.3994 | 238.8798 | 239.5520 | 242.8912 | |
185.0712 | 194.4705 | 215.2235 | 190.0000 | 200.0707 | |
236.0326 | 238.9872 | 238.5807 | 238.3172 | 235.6278 | |
273.2652 | 269.0280 | 267.1062 | 282.0928 | 226.3169 | |
238.3423 | 238.6122 | 245.7352 | 237.8052 | 239.6864 | |
423.9302 | 418.0496 | 405.2114 | 408.7595 | 413.4965 | |
489.8126 | 489.9749 | 454.6139 | 489.8125 | 489.9581 |
Algorithms | Statistics | |||
---|---|---|---|---|
Min Cost | Max Cost | Mean Cost | SD | |
IPOA | 651.8784 | 655.5161 | 652.6444 | 1.0014 |
POA | 652.0687 | 654.4392 | 659.458 | 1.7685 |
HHO | 653.4787 | 662.7219 | 679.2167 | 6.3263 |
SSA | 651.9516 | 653.2228 | 656.5612 | 1.614 |
WOA | 653.7402 | 672.8395 | 699.5087 | 12.5738 |
Units | Pmin | Pmax | a | b | c | e | f |
---|---|---|---|---|---|---|---|
1 | 36 | 114 | 0.00690 | 6.73 | 94.705 | 100 | 0.084 |
2 | 36 | 114 | 0.00690 | 6.73 | 94.705 | 100 | 0.084 |
3 | 60 | 120 | 0.02028 | 7.07 | 309.540 | 100 | 0.084 |
4 | 80 | 190 | 0.00942 | 8.18 | 369.030 | 150 | 0.063 |
5 | 47 | 97 | 0.01140 | 5.35 | 148.890 | 120 | 0.077 |
6 | 68 | 140 | 0.01142 | 8.05 | 222.330 | 100 | 0.084 |
7 | 110 | 300 | 0.00357 | 8.03 | 287.710 | 200 | 0.042 |
8 | 135 | 300 | 0.00492 | 6.99 | 391.980 | 200 | 0.042 |
9 | 135 | 300 | 0.00573 | 6.60 | 455.760 | 200 | 0.042 |
10 | 130 | 300 | 0.00605 | 12.90 | 722.820 | 200 | 0.042 |
11 | 94 | 375 | 0.00515 | 12.90 | 635.200 | 200 | 0.042 |
12 | 94 | 375 | 0.00569 | 12.80 | 654.690 | 200 | 0.042 |
13 | 125 | 500 | 0.00421 | 12.50 | 913.400 | 300 | 0.035 |
14 | 125 | 500 | 0.00752 | 8.84 | 1760.400 | 300 | 0.035 |
15 | 125 | 500 | 0.00708 | 9.15 | 1728.300 | 300 | 0.035 |
16 | 125 | 500 | 0.00708 | 9.15 | 1728.300 | 300 | 0.035 |
17 | 220 | 500 | 0.00313 | 7.97 | 647.850 | 300 | 0.035 |
18 | 220 | 500 | 0.00313 | 7.95 | 649.690 | 300 | 0.035 |
19 | 242 | 550 | 0.00313 | 7.97 | 647.830 | 300 | 0.035 |
20 | 242 | 550 | 0.00313 | 7.97 | 647.810 | 300 | 0.035 |
21 | 254 | 550 | 0.00298 | 6.63 | 785.960 | 300 | 0.035 |
22 | 254 | 550 | 0.00298 | 6.63 | 785.960 | 300 | 0.035 |
23 | 254 | 550 | 0.00284 | 6.66 | 794.530 | 300 | 0.035 |
24 | 254 | 550 | 0.00284 | 6.66 | 794.530 | 300 | 0.035 |
25 | 254 | 550 | 0.00277 | 7.10 | 801.320 | 300 | 0.035 |
26 | 254 | 550 | 0.00277 | 7.10 | 801.320 | 300 | 0.035 |
27 | 10 | 150 | 0.52124 | 3.33 | 1055.100 | 120 | 0.077 |
28 | 10 | 150 | 0.52124 | 3.33 | 1055.100 | 120 | 0.077 |
29 | 10 | 150 | 0.52124 | 3.33 | 1055.100 | 120 | 0.077 |
30 | 47 | 97 | 0.01140 | 5.35 | 148.890 | 120 | 0.077 |
31 | 60 | 190 | 0.00160 | 6.43 | 222.920 | 150 | 0.063 |
32 | 60 | 190 | 0.00160 | 6.43 | 222.920 | 150 | 0.063 |
33 | 60 | 190 | 0.00160 | 6.43 | 222.920 | 150 | 0.063 |
34 | 90 | 200 | 0.00010 | 8.95 | 107.870 | 200 | 0.042 |
35 | 90 | 200 | 0.00010 | 8.62 | 116.580 | 200 | 0.042 |
36 | 90 | 200 | 0.00010 | 8.62 | 116.580 | 200 | 0.042 |
37 | 25 | 110 | 0.01610 | 5.88 | 307.450 | 80 | 0.098 |
38 | 25 | 110 | 0.01610 | 5.88 | 307.450 | 80 | 0.098 |
39 | 25 | 110 | 0.01610 | 5.88 | 307.450 | 80 | 0.098 |
40 | 242 | 550 | 0.00313 | 7.97 | 647.830 | 300 | 0.035 |
Units | Outputs | Unit | Outputs | Unit | Outputs | Unit | Outputs |
---|---|---|---|---|---|---|---|
113.1496 | 243.6059 | 523.2740 | 190.0000 | ||||
114.0000 | 94.00949 | 523.2890 | 190.0000 | ||||
97.40526 | 304.5174 | 523.2808 | 190.0000 | ||||
179.7357 | 304.5203 | 523.2905 | 200.0000 | ||||
94.50869 | 304.5219 | 523.2831 | 167.4762 | ||||
140.0000 | 304.5212 | 523.2792 | 200.0000 | ||||
259.6008 | 489.2985 | 10.00649 | 110.0000 | ||||
284.6023 | 489.2820 | 10.00295 | 110.0000 | ||||
284.6312 | 511.2877 | 10.00000 | 110.0000 | ||||
130.0066 | 511.2906 | 97.00000 | 511.2834 |
Algorithms | Statistics | |||
---|---|---|---|---|
Min Cost | Max Cost | Mean Cost | SD | |
IPOA | 121,591.3068 | 123,933.37 | 122,659.9709 | 654.9886 |
POA | 124,907.4276 | 129,260.1887 | 126,473.6095 | 937.3753 |
HHO | 123,387.6705 | 128,381.2468 | 125,532.5618 | 1075.9279 |
SSA | 122,693.0062 | 127,500.1279 | 124,321.0393 | 1124.6677 |
WOA | 125,880.0464 | 134,779.6761 | 129,308.2354 | 1817.6021 |
Algorithms | Statistics | |||
---|---|---|---|---|
Min Cost | Max Cost | Mean Cost | SD | |
IPOA | 244,105.2816 | 249,955.5348 | 247,043.7003 | 1493.4631 |
POA | 253,073.4467 | 258,114.9399 | 255,577.6569 | 1279.7300 |
HHO | 249,554.8627 | 257,240.0592 | 252,846.8087 | 1948.0503 |
SSA | 246,532.3112 | 251,589.9268 | 248,650.6662 | 1167.5877 |
WOA | 258,734.1637 | 271,925.7694 | 263,327.7722 | 3230.8254 |
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Zhang, Y.; Li, H. Research on Economic Load Dispatch Problem of Microgrid Based on an Improved Pelican Optimization Algorithm. Biomimetics 2024, 9, 277. https://doi.org/10.3390/biomimetics9050277
Zhang Y, Li H. Research on Economic Load Dispatch Problem of Microgrid Based on an Improved Pelican Optimization Algorithm. Biomimetics. 2024; 9(5):277. https://doi.org/10.3390/biomimetics9050277
Chicago/Turabian StyleZhang, Yi, and Haoxue Li. 2024. "Research on Economic Load Dispatch Problem of Microgrid Based on an Improved Pelican Optimization Algorithm" Biomimetics 9, no. 5: 277. https://doi.org/10.3390/biomimetics9050277