Noninferior Solution Grey Wolf Optimizer with an Independent Local Search Mechanism for Solving Economic Load Dispatch Problems
Abstract
:1. Introduction
- The first three local optimal solutions of the current iteration are used to replace the alpha, beta, and delta of the standard GWO algorithm for searching the population.
- A local independent search mechanism for noninferior solutions is introduced in the standard GWO algorithm to avoid local optimization and to find more promising solutions.
- The NGWO algorithm is proposed based on 1 and 2 and is applied to solve complex ELD problems.
2. The Proposed NGWO Algorithm
- The basic GWO algorithm: The standard GWO algorithm is chosen as a comparison algorithm to compare the performance in solving different ELD cases with the other three improved versions.
- The compared GWOI algorithm: The standard GWO algorithm is improved by changing the strategy of searching the population, but without considering the case of a noninferior solution.
- The compared GWOII algorithm: The standard GWO algorithm is improved by only introducing the local independent search mechanism for the noninferior solution.
- The proposed NGWO algorithm: The standard GWO algorithm integrated with both the GWOI and GWOII methods.
2.1. The Basic GWO Algorithm
2.2. The Compared GWOI Algorithm
2.3. The Compared GWOII Algorithm
2.4. The Proposed NGWO Algorithm
Algorithm 1. Pseudo code of the NGWO algorithm |
Begin Initialize the prey wolf population Xi (i = 1, 2, …, n) and set the maximum number of iterations T. Initialize a, A, and C Calculate fitness function value of each search agent f(Xi) Xα = the global best search agent; = the local best search agent Xβ = the global second search agent; = the local second search agent Xδ = the global third search agent; = the local third search agent while t < T do for each search agent Update the position of the current search agent by Equation (8) end for Update a, A, and C Calculate the fitness function value of all search agents for each search agent if Equation (9) is ture Update the position of the current search agent by Equation (11) end if end for Update Xα, Xβ, Xδ, , and t = t + 1 end while return Xα and f(Xα) |
3. Economic Load Dispatch Formulations
3.1. Objective Function
3.2. Constraints and Variables
3.2.1. Power Balance Constraints and Variables
3.2.2. Generating Capacity Limits and Variables
3.2.3. Ramp Rate Limits and Variables
3.2.4. Prohibited Operating Zones Constraints and Variables
4. Implementation of NGWO Method in Solving the ELD Problem
4.1. Constraints Handling in ELD Problems with NGWO Approache
4.1.1. ELD Problem without the Valve-Point Loading Effects
4.1.2. ELD Problem with Considering the Valve-Point Loading Effects
4.2. Implementation Steps of NGWO to ELD Problem
Algorithm2. Pseudo code of NGWO algorithm employed to solve the ELD problem |
Begin Initialize the prey wolf population Xi (I = 1, 2, …, n) and set the maximum number of iterations T. Input the relevant constraint parameters of the generator unit. If the ELD problem has no valve-point loading effects, then map the initialized grey wolf individuals to the feasible domain according to Equation (22) to obtain P(Xi). Otherwise, map the initialized grey wolf individuals to the feasible domain according to Equation (26) to obtain P(Xi), and then calculate the transmission loss Ploss by using Equation (17). Initialize a, A, and C Calculate fitness function value of each search agent F(P(Xi)) according to Equation (27) for considering the valve-point loading effects. Otherwise, calculate F(P(Xi)) by using Equation (23). Xα = the global best search agent; = the local best search agent Xβ = the global second search agent; = the local second search agent Xδ = the global third search agent; = the local third search agent while t < T do for each search agent Update the position of the current search agent by Equation (8) end for Update a, A, and C Calculate the fitness function value for each search agents according to Equation (27) for considering the valve-point loading effects. Otherwise calculate the fitness function value of all search agents by utilizing Equation (23). for each search agent if Equation (9) is satisfied Update the position of the current search agent by Equation (11) end if end for Update Xα, Xβ, Xδ, , and t = t + 1 end while return P(Xα) and F(P(Xα)) |
5. Numerical Simulation Results and Analysis
- Case I.
- A 3-generator system for load demand of 850 MW, and valve-point loading effects are considered.
- Case II.
- A 13-generator system for a load demand of 2520 MW, and valve-point loading effects are considered.
- Case III.
- A 40-generator system for a load demand of 10500 MW, and valve-point loading effects are considered.
- Case IV.
- A 6-generator system with a quadratic cost function, POZs and transmission loss, and a load demand of 1263 MW.
- Case V.
- A 15-generator system with a quadratic cost function, POZs and transmission loss, and a load demand of 2630 MW.
5.1. Case I: 3-Generator System
5.2. Case II: 13-Generator System
5.3. Case III: 40-Generator System
5.4. Case IV: 6-Generator System
5.5. Case V: 15-Generator System
6. Conclusions and Future Work
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Generator | Pimin (MW) | Pimax (MW) | ai | bi | ci | ei | fi |
---|---|---|---|---|---|---|---|
1 | 100 | 600 | 0.001562 | 7.92 | 561 | 300 | 0.0315 |
2 | 50 | 200 | 0.004820 | 7.97 | 78 | 150 | 0.063 |
3 | 100 | 400 | 0.001940 | 7.85 | 310 | 200 | 0.042 |
Unit | GA [46] | PSO [46] | MP-CJAYA [37] | CJAYA [37] | EP [47] | GWO | GWOI | GWOII | NGWO |
---|---|---|---|---|---|---|---|---|---|
1 | 398.700 | 300.268 | 350.2464 | 350.0254 | 300.264 | 299.838 | 300.618 | 300.248 | 300.562 |
2 | 399.600 | 400.000 | 400.000 | 400.000 | 400.000 | 399.600 | 399.600 | 399.600 | 399.600 |
3 | 50.100 | 149.732 | 99.7576 | 99.9511 | 149.736 | 150.57 | 149.803 | 150.153 | 149.843 |
Ptotal(MW) | 848.400 | 850.000 | 850.004 | 849.977 | 850.000 | 850.011 | 850.021 | 850.010 | 850.005 |
Fmean ($/h) | 8234.72 | 8234.09 | 8232.06 | 8289.41 | 8234.16 | 8305.91 | 8284.772 | 8240.313 | 8233.567 |
Fbest ($/h) | 8222.07 | 8234.07 | 8223.29 | 8226.18 | 8234.07 | 8223.61 | 8223.367 | 8223.197 | 8223.104 |
Generator | Pimin (MW) | Pimax (MW) | ai | bi | ci | ei | fi |
---|---|---|---|---|---|---|---|
1 | 00 | 680 | 0.00028 | 8.10 | 550 | 300 | 0.035 |
2 | 00 | 360 | 0.00056 | 8.10 | 309 | 200 | 0.042 |
3 | 00 | 360 | 0.00056 | 8.10 | 307 | 200 | 0.042 |
4 | 60 | 180 | 0.00324 | 7.74 | 240 | 150 | 0.063 |
5 | 60 | 180 | 0.00324 | 7.74 | 240 | 150 | 0.063 |
6 | 60 | 180 | 0.00324 | 7.74 | 240 | 150 | 0.063 |
7 | 60 | 180 | 0.00324 | 7.74 | 240 | 150 | 0.063 |
8 | 60 | 180 | 0.00324 | 7.74 | 240 | 150 | 0.063 |
9 | 60 | 180 | 0.00324 | 7.74 | 240 | 150 | 0.063 |
10 | 40 | 120 | 0.00284 | 8.6 | 126 | 100 | 0.084 |
11 | 40 | 120 | 0.00284 | 8.6 | 126 | 100 | 0.084 |
12 | 55 | 120 | 0.00284 | 8.6 | 126 | 100 | 0.084 |
13 | 55 | 120 | 0.00284 | 8.6 | 126 | 100 | 0.084 |
Unit | GA [46] | CPSO [48] | CJAYA [37] | JAYA [37] | SA [46] | GWO | GWOI | GWOII | NGWO |
---|---|---|---|---|---|---|---|---|---|
1 | 627.05 | 628.32 | 628.3185 | 628.3185 | 668.40 | 647.3842 | 645.5569 | 630.9811 | 630.9951 |
2 | 359.40 | 299.83 | 299.1992 | 299.2009 | 359.78 | 306.3995 | 306.9539 | 300.8038 | 297.9355 |
3 | 358.95 | 299.17 | 299.1993 | 306.9105 | 358.20 | 309.6117 | 306.5356 | 302.7475 | 299.9253 |
4 | 158.93 | 159.70 | 159.7330 | 159.7339 | 104.28 | 175.1400 | 169.6878 | 160.1702 | 157.9267 |
5 | 159.73 | 159.64 | 159.7331 | 159.7337 | 60.36 | 66.8791 | 168.4922 | 161.0252 | 159.6433 |
6 | 159.68 | 159.67 | 159.7331 | 159.7338 | 110.64 | 162.7466 | 174.9721 | 160.9845 | 159.2335 |
7 | 159.53 | 159.64 | 159.7330 | 109.8673 | 162.12 | 174.3111 | 167.1394 | 159.1231 | 159.7630 |
8 | 158.89 | 159.65 | 159.7330 | 159.7342 | 163.03 | 61.2250 | 116.8800 | 110.4278 | 159.6615 |
9 | 110.15 | 159.78 | 159.7331 | 159.7340 | 161.52 | 175.1400 | 116.8800 | 159.7720 | 159.4265 |
10 | 77.27 | 112.46 | 110.0403 | 114.8012 | 117.09 | 116.7600 | 116.8800 | 116.8577 | 76.8790 |
11 | 75.00 | 74.00 | 114.7994 | 114.8001 | 75.00 | 116.7600 | 109.9096 | 77.0418 | 79.5038 |
12 | 60.00 | 56.50 | 55.0000 | 92.4018 | 60.00 | 99.9167 | 59.0347 | 91.4990 | 86.8040 |
13 | 55.41 | 91.64 | 55.0000 | 55.0027 | 119.58 | 108.5598 | 66.5129 | 88.6915 | 94.1941 |
Ptotal (MW) | 2520 | 2520 | 2519.96 | 2519.97 | 2520 | 2520.83 | 2521.74 | 2520.13 | 2520.21 |
Fmean ($/h) | -- | -- | 24,385.7604 | 24,476.2547 | -- | 24,442.08 | 24,471.53 | 24,395.58 | 24,366.12 |
Fbest ($/h) | 24,418.99 | 24,211.56 | 24,178.8040 | 24,220.7529 | 24,970.91 | 24,231.18 | 24,244.69 | 24,198.47 | 24,185.45 |
Generator | Pimin (MW) | Pimax (MW) | ai | bi | ci | ei | fi |
---|---|---|---|---|---|---|---|
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 | 309.54 | 100 | 0.084 |
4 | 80 | 190 | 0.00942 | 8.18 | 369.03 | 150 | 0.063 |
5 | 46 | 97 | 0.01140 | 5.35 | 148.89 | 120 | 0.077 |
6 | 68 | 140 | 0.01142 | 8.05 | 222.33 | 100 | 0.084 |
7 | 110 | 300 | 0.00357 | 8.03 | 287.71 | 200 | 0.042 |
8 | 135 | 300 | 0.00492 | 6.99 | 391.98 | 200 | 0.042 |
9 | 135 | 300 | 0.00573 | 6.60 | 455.76 | 200 | 0.042 |
10 | 130 | 300 | 0.00606 | 12.9 | 722.82 | 200 | 0.042 |
11 | 94 | 375 | 0.00515 | 12.9 | 635.20 | 200 | 0.042 |
12 | 94 | 375 | 0.00569 | 12.8 | 654.69 | 200 | 0.042 |
13 | 125 | 500 | 0.00421 | 12.5 | 913.40 | 300 | 0.035 |
14 | 125 | 500 | 0.00752 | 8.84 | 1760.4 | 300 | 0.035 |
15 | 125 | 500 | 0.00708 | 9.15 | 1728.3 | 300 | 0.035 |
16 | 125 | 500 | 0.00708 | 9.15 | 1728.3 | 300 | 0.035 |
17 | 220 | 500 | 0.00313 | 7.97 | 647.85 | 300 | 0.035 |
18 | 220 | 500 | 0.00313 | 7.95 | 649.69 | 300 | 0.035 |
19 | 242 | 550 | 0.00313 | 7.97 | 647.83 | 300 | 0.035 |
20 | 242 | 550 | 0.00313 | 7.97 | 647.81 | 300 | 0.035 |
21 | 254 | 550 | 0.00298 | 6.63 | 785.96 | 300 | 0.035 |
22 | 254 | 550 | 0.00298 | 6.63 | 785.96 | 300 | 0.035 |
23 | 254 | 550 | 0.00284 | 6.66 | 794.53 | 300 | 0.035 |
24 | 254 | 550 | 0.00284 | 6.66 | 794.53 | 300 | 0.035 |
25 | 254 | 550 | 0.00277 | 7.10 | 801.32 | 300 | 0.035 |
26 | 254 | 550 | 0.00277 | 7.10 | 801.32 | 300 | 0.035 |
27 | 10 | 150 | 0.52124 | 3.33 | 1055.1 | 120 | 0.077 |
28 | 10 | 150 | 0.52124 | 3.33 | 1055.1 | 120 | 0.077 |
29 | 10 | 150 | 0.52124 | 3.33 | 1055.1 | 120 | 0.077 |
30 | 47 | 97 | 0.01140 | 5.35 | 148.89 | 120 | 0.077 |
31 | 60 | 190 | 0.00160 | 6.43 | 222.92 | 150 | 0.063 |
32 | 60 | 190 | 0.00160 | 6.43 | 222.92 | 150 | 0.063 |
33 | 60 | 190 | 0.00160 | 6.43 | 222.92 | 150 | 0.063 |
34 | 90 | 200 | 0.0001 | 8.95 | 107.87 | 200 | 0.042 |
35 | 90 | 200 | 0.0001 | 8.62 | 116.58 | 200 | 0.042 |
36 | 90 | 200 | 0.0001 | 8.62 | 116.58 | 200 | 0.042 |
37 | 25 | 110 | 0.0161 | 5.88 | 307.45 | 80 | 0.098 |
38 | 25 | 110 | 0.0161 | 5.88 | 307.45 | 80 | 0.098 |
39 | 25 | 110 | 0.0161 | 5.88 | 307.45 | 80 | 0.098 |
40 | 242 | 550 | 0.00313 | 7.97 | 647.83 | 300 | 0.035 |
Unit | NPSO [49] | PSO-LRS [37] | MPSO [50] | CJAYA [37] | IGA [50] | GWO | GWOI | GWOII | NGWO |
---|---|---|---|---|---|---|---|---|---|
1 | 113.9891 | 111.9858 | 114.000 | 114.0000 | 110.97 | 109.0947 | 109.7268 | 107.6544 | 111.3177 |
2 | 113.6334 | 110.5273 | 114.000 | 111.6651 | 110.88 | 112.0471 | 111.7342 | 109.2161 | 112.7551 |
3 | 97.5500 | 98.5560 | 120.000 | 119.9876 | 98.17 | 115.4584 | 119.2197 | 94.7874 | 118.6377 |
4 | 180.0059 | 182.9622 | 182.222 | 188.2606 | 178.85 | 179.8333 | 181.6041 | 182.3441 | 183.3649 |
5 | 97.0000 | 87.7254 | 97.000 | 96.9763 | 87.78 | 46.1649 | 89.8836 | 86.9731 | 91.8097 |
6 | 140.0000 | 139.9933 | 140.000 | 139.9488 | 140.00 | 83.1571 | 125.1816 | 109.1907 | 104.3697 |
7 | 300.0000 | 259.6628 | 300.000 | 264.0949 | 260.37 | 261.6345 | 265.0775 | 259.4910 | 297.6533 |
8 | 300.0000 | 297.7912 | 299.021 | 299.9814 | 286.83 | 292.4025 | 290.2216 | 284.1803 | 289.4349 |
9 | 284.5797 | 284.8459 | 300.000 | 284.9042 | 285.14 | 284.7149 | 285.2586 | 285.1526 | 298.4044 |
10 | 130.0517 | 130.0000 | 130.000 | 130.0908 | 204.86 | 132.9049 | 134.9231 | 129.3500 | 129.3500 |
11 | 243.7131 | 94.6741 | 94.000 | 94.0011 | 165.98 | 101.6726 | 167.9983 | 317.4787 | 241.9702 |
12 | 169.0104 | 94.3734 | 94.000 | 94.0000 | 167.75 | 319.8174 | 183.6314 | 157.3563 | 166.9113 |
13 | 125.0000 | 214.7369 | 125.000 | 125.1028 | 214.31 | 215.0746 | 219.5396 | 300.6095 | 214.8490 |
14 | 393.9662 | 394.1370 | 304.485 | 394.2529 | 305.65 | 394.9259 | 394.9259 | 305.0848 | 215.6690 |
15 | 304.7586 | 483.1816 | 394.607 | 484.1262 | 393.66 | 398.1829 | 212.7154 | 395.3099 | 305.6922 |
16 | 304.5120 | 304.5381 | 305.323 | 304.5950 | 394.60 | 304.1546 | 484.5572 | 203.9544 | 394.6479 |
17 | 489.6024 | 489.2139 | 490.272 | 490.8265 | 489.22 | 490.0842 | 494.3478 | 489.6721 | 494.7618 |
18 | 489.6087 | 489.6154 | 500.000 | 489.3438 | 489.25 | 493.2515 | 491.2367 | 492.3490 | 493.1559 |
19 | 511.7903 | 511.1782 | 511.404 | 511.3775 | 511.23 | 511.4229 | 514.3755 | 514.3882 | 512.7416 |
20 | 511.2624 | 511.7336 | 512.174 | 512.1395 | 510.69 | 511.9422 | 514.3755 | 511.7323 | 520.8929 |
21 | 523.3274 | 523.4072 | 550.000 | 523.6621 | 524.74 | 532.3762 | 522.6016 | 532.2046 | 526.1137 |
22 | 523.2196 | 523.4599 | 523.655 | 523.3534 | 525.52 | 532.2484 | 523.6988 | 527.3193 | 532.1443 |
23 | 523.4707 | 523.4756 | 534.661 | 524.9677 | 522.98 | 530.7732 | 523.6988 | 527.3193 | 536.8421 |
24 | 523.0661 | 523.7032 | 550.000 | 524.2850 | 522.22 | 526.1112 | 536.1385 | 539.9336 | 524.4669 |
25 | 523.3978 | 523.7854 | 525.057 | 522.9279 | 523.26 | 524.4545 | 523.5451 | 526.6306 | 525.2461 |
26 | 523.2897 | 523.2757 | 549.155 | 523.2298 | 523.32 | 523.4934 | 524.0780 | 524.8658 | 529.3289 |
27 | 10.0208 | 10.0000 | 10.000 | 10.0000 | 10 | 11.5028 | 14.8568 | 9.9500 | 9.9500 |
28 | 10.0927 | 10.6251 | 10.000 | 10.0047 | 10 | 9.9541 | 21.0962 | 9.9500 | 9.9500 |
29 | 10.0621 | 10.0727 | 10.000 | 10.0000 | 10 | 10.3272 | 13.1286 | 9.9500 | 9.9500 |
30 | 88.9456 | 51.3321 | 97.000 | 97.0000 | 88.86 | 91.6019 | 88.5089 | 90.3385 | 88.4106 |
31 | 189.9951 | 189.8048 | 190.000 | 190.0000 | 162.30 | 188.8475 | 188.0180 | 159.6875 | 188.9088 |
32 | 190.0000 | 189.7386 | 190.000 | 189.9503 | 177.94 | 165.2531 | 166.2968 | 188.9923 | 188.8126 |
33 | 190.0000 | 189.9122 | 190.000 | 190.0000 | 160.18 | 188.9197 | 182.0808 | 173.1974 | 186.9624 |
34 | 165.9825 | 199.3258 | 200.000 | 169.8860 | 166.54 | 189.2968 | 164.9636 | 189.6808 | 195.0897 |
35 | 172.4153 | 199.3065 | 200.000 | 199.8549 | 164.80 | 180.4605 | 172.6948 | 192.1671 | 171.5047 |
36 | 191.2978 | 192.8977 | 200.000 | 199.9896 | 170.68 | 184.2693 | 191.0765 | 157.5027 | 176.1085 |
37 | 109.9893 | 110.0000 | 110.000 | 109.9712 | 108.17 | 89.6748 | 108.8942 | 104.4095 | 89.5297 |
38 | 109.9521 | 109.8628 | 110.000 | 109.9977 | 100.68 | 90.1485 | 100.8804 | 86.74132 | 89.3589 |
39 | 109.8733 | 92.8751 | 110.000 | 109.9871 | 109.34 | 57.0464 | 27.8744 | 100.2970 | 109.3222 |
40 | 511.5671 | 511.6883 | 512.964 | 511.2250 | 511.28 | 514.3622 | 511.7717 | 512.43873 | 512.5412 |
Ptotal (MW) | 10,499.9989 | 10,499.9452 | 10,500 | 10,499.97 | 10,500 | 10,499.97 | 10,499.96 | 10,499.97 | 10,499.93 |
Fmean ($/h) | 122,221.3697 | 122,558.4565 | -- | 122,581.85 | 122,811.41 | 124,796.61 | 125,155.07 | 123,314.39 | 122,787.77 |
Fbest ($/h) | 121,704.7391 | 122,035.7946 | 122,252.265 | 121,799.88 | 121,915.93 | 122,602.37 | 122,678.91 | 122,430.74 | 121,881.81 |
Generator | Pimin (MW) | Pimax (MW) | ai | bi | ci | URi | DRi | Prohibited Zones | |
---|---|---|---|---|---|---|---|---|---|
1 | 100 | 500 | 0.0070 | 7.0 | 240 | 440 | 80 | 120 | [210, 240], [350, 380] |
2 | 50 | 200 | 0.0095 | 10.0 | 200 | 170 | 50 | 90 | [90, 110], [140, 160] |
3 | 80 | 300 | 0.0090 | 8.5 | 220 | 200 | 65 | 100 | [150, 170], [210, 240] |
4 | 50 | 150 | 0.0090 | 11.0 | 200 | 150 | 50 | 90 | [80, 90], [110, 120] |
5 | 50 | 200 | 0.0080 | 10.5 | 220 | 190 | 50 | 90 | [90, 110], [140, 150] |
6 | 50 | 120 | 0.0075 | 12.0 | 190 | 110 | 50 | 90 | [75, 85], [100, 105] |
B = 1 × 10−2 | 0.0017 | 0.0012 | 0.0007 | −0.0001 | −0.0005 | −0.0002 |
---|---|---|---|---|---|---|
0.0012 | 0.0014 | 0.0009 | 0.0001 | −0.0006 | −0.0001 | |
0.0007 | 0.0009 | 0.0031 | 0 | −0.001 | −0.0006 | |
−0.0001 | 0.0001 | 0 | 0.0024 | −0.0006 | −0.0008 | |
−0.0005 | −0.0006 | −0.001 | −0.0006 | 0.0129 | −0.0002 | |
−0.0002 | −0.0001 | −0.0006 | −0.0008 | −0.0002 | 0.015 | |
B0 = 1×10−3 | −0.3908 | −0.1297 | 0.7047 | 0.0591 | 0.2161 | −0.6635 |
B00 = 10× | 0.0056 |
Generator | SA [46] | GA [51] | MTS [52] | NPSO [49] | PSO [46] | JAYA [37] | GWO | GWOI | GWOII | NGWO |
---|---|---|---|---|---|---|---|---|---|---|
1 | 478.1258 | 462.0444 | 448.1277 | 447.4734 | 447.5823 | 457.9858 | 446.6281 | 447.2399 | 446.9060 | 448.7973 |
2 | 163.0249 | 189.4456 | 172.8082 | 173.1012 | 172.8387 | 176.8785 | 171.7686 | 175.0336 | 172.1000 | 174.4309 |
3 | 261.7146 | 254.8535 | 262.5932 | 262.6804 | 261.3300 | 250.0717 | 264.6710 | 262.6065 | 263.8918 | 262.9964 |
4 | 125.7665 | 127.4296 | 136.9605 | 139.4156 | 138.6812 | 129.3748 | 141.3356 | 138.8324 | 139.8172 | 138.2484 |
5 | 153.7056 | 151.5388 | 168.2031 | 165.3002 | 169.6781 | 172.8886 | 166.5389 | 167.2797 | 164.4018 | 164.9710 |
6 | 93.7965 | 90.7150 | 87.3304 | 87.9761 | 85.8963 | 88.4618 | 85.0000 | 85.0000 | 88.8170 | 86.46551 |
Ptotal (MW) | 1276.1339 | 1276.0270 | 1276.0232 | 1275.96 | 1276.0066 | 1275.6611 | 1276.3156 | 1276.0229 | 1276.0155 | 1275.4658 |
Ploss (MW) | 13.1317 | 13.0268 | 13.0205 | 12.9470 | 13.0066 | 12.6665 | 13.3099 | 13.0222 | 13.0066 | 12.8486 |
Fbest ($/h) | 15,461.10 | 15,457.96 | 15,450.06 | 15,450.00 | 15,450.14 | 15,447.09 | 15,450.07 | 15,443.25 | 15,449.96 | 15,449.17 |
Algorithm | Fbest ($/h) | Fworst ($/h) | Fmean ($/h) |
---|---|---|---|
SA [46] | 15,461.10 | 15,545.50 | 15,488.98 |
GA [51] | 15,457.96 | 15,524.69 | 15,477.71 |
MTS [52] | 15,450.06 | 15,453.64 | 15,451.17 |
NPSO [49] | 15,450.00 | 15,454.00 | 15,452.00 |
PSO [46] | 15,450.14 | 15,491.71 | 15,465.83 |
JAYA [37] | 15,477.09 | 15,622.16 | 15,500.11 |
GWO | 15,450.07 | 15,487.14 | 15,453.41 |
GWOI | 15,450.15 | 15,455.17 | 15,451.13 |
GWOII | 15,449.96 | 15,452.41 | 15,450.48 |
NGWO | 15,449.17 | 15,460.10 | 15,449.86 |
Generator | Pimin (MW) | Pimax (MW) | ai | bi | ci | URi | DRi | Prohibited Zones | |
---|---|---|---|---|---|---|---|---|---|
1 | 150 | 455 | 0.000299 | 10.1 | 671 | 400 | 80 | 120 | |
2 | 150 | 455 | 0.000183 | 10.2 | 574 | 300 | 80 | 120 | [185, 225], [305, 335], [420, 450] |
3 | 20 | 130 | 0.001126 | 8.8 | 374 | 105 | 130 | 130 | |
4 | 20 | 130 | 0.001126 | 8.8 | 374 | 100 | 130 | 130 | |
5 | 150 | 470 | 0.000205 | 10.4 | 461 | 90 | 80 | 120 | [180, 200], [305, 335], [390, 420] |
6 | 135 | 460 | 0.000301 | 10.1 | 630 | 400 | 80 | 120 | [230, 255], [365, 395], [430, 455] |
7 | 135 | 465 | 0.000364 | 9.8 | 548 | 350 | 80 | 120 | |
8 | 60 | 300 | 0.000338 | 11.2 | 227 | 95 | 65 | 100 | |
9 | 25 | 162 | 0.000807 | 11.2 | 173 | 105 | 60 | 100 | |
10 | 25 | 160 | 0.001203 | 10.7 | 175 | 110 | 60 | 100 | |
11 | 20 | 80 | 0.003586 | 10.2 | 186 | 60 | 80 | 80 | |
12 | 20 | 80 | 0.005513 | 9.9 | 230 | 40 | 80 | 80 | [30, 40], [55, 65] |
13 | 25 | 85 | 0.000371 | 13.1 | 225 | 30 | 80 | 80 | |
14 | 15 | 55 | 0.001929 | 12.1 | 309 | 20 | 55 | 55 | |
15 | 15 | 55 | 0.004447 | 12.4 | 323 | 20 | 55 | 55 |
Generator | SA [46] | GA [47] | MTS [53] | TSA [54] | PSO [46] | AIS [55] | SPSO [48] | GWO | GWOI | GWOII | NGWO |
---|---|---|---|---|---|---|---|---|---|---|---|
1 | 453.6646 | 445.5619 | 453.9922 | 440.500 | 454.7167 | 441.159 | 439.12 | 455.0000 | 455.0000 | 455.0000 | 455.0000 |
2 | 377.6091 | 380.0000 | 379.7434 | 346.800 | 376.2002 | 409.587 | 407.97 | 380.0000 | 380.0000 | 380.0000 | 380.0000 |
3 | 120.3744 | 129.0605 | 130.0000 | 110.880 | 129.5547 | 117.298 | 119.63 | 130.0000 | 130.0000 | 130.0000 | 130.0000 |
4 | 126.2668 | 129.5250 | 129.9232 | 122.460 | 129.7083 | 131.258 | 129.99 | 130.0000 | 130.0000 | 130.0000 | 130.0000 |
5 | 165.3048 | 169.9659 | 168.0877 | 177.740 | 169.4407 | 151.011 | 151.07 | 170.0000 | 165.3122 | 167.8379 | 160.5430 |
6 | 459.2455 | 458.7544 | 460.0000 | 459.110 | 458.8153 | 466.258 | 460.00 | 159.0815 | 460.0000 | 460.0000 | 460.0000 |
7 | 422.8619 | 417.9041 | 429.2253 | 406.410 | 427.5733 | 423.368 | 425.56 | 430.0000 | 430.0000 | 430.0000 | 430.0000 |
8 | 126.4025 | 97.8230 | 104.3097 | 107.550 | 67.2834 | 99.948 | 98.57 | 102.8806 | 60.8698 | 78.4634 | 84.1915 |
9 | 54.4742 | 54.2933 | 35.0358 | 107.270 | 75.2673 | 110.684 | 113.49 | 43.5154 | 69.1738 | 48.3551 | 57.7845 |
10 | 149.0879 | 144.2214 | 155.8829 | 140.560 | 155.5899 | 100.229 | 101.11 | 125.8636 | 158.4501 | 148.3092 | 146.7789 |
11 | 77.9594 | 77.3302 | 79.8994 | 78.470 | 79.9522 | 32.057 | 33.91 | 80.0000 | 80.0000 | 80.0000 | 80.0000 |
12 | 93.9489 | 77.0371 | 79.9037 | 74.170 | 79.8947 | 78.815 | 79.96 | 80.0000 | 80.0000 | 80.0000 | 80.0000 |
13 | 25.0022 | 31.1537 | 25.0220 | 31.950 | 25.2744 | 23.568 | 25.00 | 33.2702 | 30.1938 | 30.5576 | 32.7497 |
14 | 16.0636 | 15.0233 | 15.2586 | 37.380 | 16.7318 | 40.258 | 41.41 | 26.4876 | 17.6755 | 18.5833 | 17.2977 |
15 | 15.0196 | 33.6125 | 15.0796 | 22.470 | 15.1967 | 36.906 | 35.61 | 15.0116 | 15.1082 | 23.69718 | 15.4832 |
Ptotal (MW) | 2663.29 | 2661.23 | 2661.36 | 2663.70 | 2661.19 | 2662.04 | 2662.4 | 2662.2318 | 2662.1458 | 2660.96 | 2660.54 |
Ploss (MW) | 33.2737 | 31.2363 | 31.3523 | 33.8110 | 31.1697 | 32.4075 | 32.431 | 32.2317 | 31.0156 | 30.8550 | 30.0148 |
Fbest ($/h) | 32,786.40 | 32,779.81 | 32,716.87 | 32,918.00 | 32,724.17 | 32,854.00 | 32,858 | 32,743.2959 | 32,733.8961 | 32,734.6249 | 32,712.6131 |
Algorithm | Fbest ($/h) | Fworst ($/h) | Fmean ($/h) |
---|---|---|---|
SA [46] | 32,786.40 | 33,028.95 | 32,869.51 |
GA [47] | 32,779.81 | 33,041.64 | 32,841.21 |
MTS [53] | 32,716.87 | 32,796.15 | 32,767.21 |
TSA [54] | 32,917.87 | 33,245.54 | 33,066.76 |
PSO [46] | 32,724.17 | 32,841.38 | 32,807.45 |
SPSO [48] | 32,858.00 | 33,331.00 | 33,039.00 |
AIS [55] | 32,854.00 | 32,892.00 | 32,873.25 |
GWO | 32,743.30 | 32,857.96 | 32,784.96 |
GWOI | 32,733.90 | 32,889.63 | 32,783.22 |
GWOII | 32,734.62 | 32,817.91 | 32,774.82 |
NGWO | 32,712.61 | 32,830.61 | 32,752.78 |
© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Xu, J.; Yan, F.; Yun, K.; Su, L.; Li, F.; Guan, J. Noninferior Solution Grey Wolf Optimizer with an Independent Local Search Mechanism for Solving Economic Load Dispatch Problems. Energies 2019, 12, 2274. https://doi.org/10.3390/en12122274
Xu J, Yan F, Yun K, Su L, Li F, Guan J. Noninferior Solution Grey Wolf Optimizer with an Independent Local Search Mechanism for Solving Economic Load Dispatch Problems. Energies. 2019; 12(12):2274. https://doi.org/10.3390/en12122274
Chicago/Turabian StyleXu, Jianzhong, Fu Yan, Kumchol Yun, Lifei Su, Fengshu Li, and Jun Guan. 2019. "Noninferior Solution Grey Wolf Optimizer with an Independent Local Search Mechanism for Solving Economic Load Dispatch Problems" Energies 12, no. 12: 2274. https://doi.org/10.3390/en12122274