Solving the Multi-Objective Optimal Power Flow Problem Using the Multi-Objective Firefly Algorithm with a Constraints-Prior Pareto-Domination Approach
Abstract
:1. Introduction
2. Mathematical Modeling of the MOOPF Problem
2.1. Objective Functions
2.2. Equality Constraints
2.3. Inequality Constraints
- Limits of Pg
- Limits of Vg
- Limits of T
- Limits of Qc
- Limits of Pgref
- Limits of Vl
- Limits of Qg
- Limits of S
2.4. Constraints Processing
2.4.1. Penalty Function Approach
2.4.2. Proposed Constraints-Prior Pareto-Domination Approach
3. MOFA Algorithm
3.1. Basic Firefly Algorithm
3.2. MOFA-PFA Algorithm
3.3. MOFA-CPA Algorithm
3.4. Stopping Condition
4. Simulation Results and Analysis
4.1. Algorithm Parameters
4.2. PF Computation of IEEE30
4.3. Analysis of the BCSs for Cases 1–5
4.4. PF Computation of IEEE57
4.5. Analysis of the Constraints Processing
4.6. Performance Evaluation
5. Conclusions
Supplementary Materials
Author Contributions
Funding
Conflicts of Interest
Appendix A
References
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1 population initialization; |
2 while t ≤ maximum iteration Tmax |
3 read the algorithm parameters; |
4 set penalty factors KV, KQ, KS, KL; |
5 calculate the value of the objective functions as Section 2.1 shown; |
6 calculate the penalty according to the Equation (20), and added to the objective functions; |
7 for i = 1 to NF do |
8 for j = 1 to NF do |
9 if cjold Pareto-dominates ciold according to Equation (22) |
10 generate new solution cinew as Equation (24) shown; |
11 if cinew Pareto-dominates ciold |
12 select cinew as the new solution in the FP; |
13 end |
14 else |
15 select ciold as the new solution in the FP; |
16 end |
17 end |
18 end |
19 end |
20 update and output the Pareto optimal solutions to next iterations; |
21 sort and find the current best approximation to the Pareto front; |
22 t++; |
23 end |
1 population initialization; |
2 while t ≤ maximum iteration Tmax |
3 read the algorithm parameters; |
4 calculate the value of the objective functions as Section 2.1 shown; |
5 for i = 1 to NF do |
6 for j = 1 to NF do |
7 if cjold constraints-prior Pareto-dominates ciold using proposed CPA in Section 2.4.2; |
8 generate new solution cinew as Equation (24) shown; |
9 if cinew constraints-prior Pareto-dominates ciold |
10 select cinew as the new solution in the FP; |
11 end |
12 else |
13 select ciold as the new solution in the FP; |
14 end |
15 end |
16 end |
17 end |
18 update and output the Pareto optimal solutions to next iterations; |
19 sort and find the current best approximation to the Pareto front; |
20 t++; |
21 end |
Test Cases | Objective Combinations | Test System |
---|---|---|
Case 1 | min f1 = fcost&f2 = fPloss | IEEE 30 |
Case 2 | min f1 = fcost_vp&f2 = fPloss | |
Case 3 | min f1 = fcost_vp&f2 = femission | |
Case 4 | min f1 = fcost&f2 = fPloss&f3 = femission | |
Case 5 | min f1 = fcost_vp&f2 = fPloss&f3 = femission | |
Case 6 | min f1 = fcost&f2 = fPloss&f3 = femission | IEEE 57 |
Parameters | Case 1 | Case 2 | Case 3 | Case 4 | Case 5 | Case 6 |
---|---|---|---|---|---|---|
FP Size | 100 | 100 | 100 | 100 | 100 | 100 |
Tmax | 300 | 300 | 300 | 500 | 500 | 500 |
γ | 1 | 1 | 1 | 1 | 1 | 1 |
α | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 |
β0 | 1 | 1 | 1 | 1 | 1 | 1 |
Trials | 30 | 30 | 30 | 30 | 30 | 30 |
Variables | Case 1 | Case 2 | Case 3 | Case 4 | Case 5 | |||||
---|---|---|---|---|---|---|---|---|---|---|
Alg1 1 | Alg2 2 | Alg1 | Alg2 | Alg1 | Alg2 | Alg1 | Alg2 | Alg1 | Alg2 | |
Pg2 (MW) | 52.171 | 54.788 | 53.613 | 49.098 | 64.174 | 57.294 | 57.647 | 57.890 | 61.750 | 59.497 |
Pg5 (MW) | 32.314 | 34.118 | 30.305 | 29.139 | 23.342 | 24.599 | 39.791 | 36.290 | 30.736 | 31.945 |
Pg8 (MW) | 35.000 | 34.515 | 34.428 | 35.000 | 34.687 | 33.050 | 35.000 | 35.000 | 35.000 | 34.633 |
Pg11 (MW) | 28.381 | 30.000 | 22.187 | 23.853 | 17.439 | 22.333 | 28.504 | 29.271 | 29.457 | 30.000 |
Pg13 (MW) | 22.252 | 25.079 | 13.928 | 17.249 | 15.483 | 18.837 | 35.413 | 40.000 | 27.389 | 27.781 |
Vg1 (p.u.) | 1.100 | 1.1000 | 1.1000 | 1.1000 | 1.1000 | 1.0402 | 1.1000 | 1.0985 | 1.0965 | 1.0920 |
Vg2 (p.u.) | 1.089 | 1.0936 | 1.0902 | 1.0923 | 1.0933 | 1.0262 | 1.0887 | 1.0869 | 1.0904 | 1.0848 |
Vg5 (p.u.) | 1.063 | 1.0769 | 1.0646 | 1.0631 | 1.0804 | 1.0053 | 1.0690 | 1.0625 | 1.0687 | 1.0642 |
Vg8 (p.u.) | 1.074 | 1.0822 | 1.0804 | 1.0811 | 1.0688 | 1.0084 | 1.0777 | 1.0767 | 1.0743 | 1.0744 |
Vg11 (p.u.) | 1.100 | 1.1000 | 1.1000 | 1.0714 | 1.0713 | 1.0663 | 1.0951 | 1.0857 | 1.0892 | 1.0901 |
Vg13 (p.u.) | 1.100 | 1.0894 | 1.1000 | 1.0422 | 1.1000 | 1.0668 | 1.0941 | 1.0386 | 1.0517 | 1.0413 |
T11 (p.u.) | 0.9832 | 1.0160 | 1.0210 | 1.0760 | 0.9290 | 0.9780 | 1.0140 | 1.0860 | 1.0760 | 1.0730 |
T12 (p.u.) | 0.9319 | 0.9650 | 0.9270 | 0.9850 | 1.0390 | 0.9920 | 0.9240 | 0.9930 | 0.9400 | 1.0030 |
T15 (p.u.) | 1.0068 | 0.9960 | 0.9870 | 1.0440 | 1.0170 | 1.0560 | 0.9930 | 1.0520 | 0.9650 | 1.0400 |
T36 (p.u.) | 0.9696 | 0.9710 | 0.9650 | 1.0110 | 1.0240 | 0.9690 | 0.9640 | 1.0770 | 0.9890 | 1.0200 |
Qc10 (p.u.) | 0.0258 | 0.0350 | 0.0230 | 0.0060 | 0.0150 | 0.0240 | 0.0350 | 0.0140 | 0.0460 | 0.0160 |
Qc12 (p.u.) | 0.0389 | 0.0000 | 0.0250 | 0.0140 | 0.0130 | 0.0240 | 0.0350 | 0.0220 | 0.0020 | 0.0110 |
Qc15 (p.u.) | 0.0301 | 0.0280 | 0.0320 | 0.0170 | 0.0210 | 0.0250 | 0.0320 | 0.0080 | 0.0470 | 0.0000 |
Qc17(p.u.) | 0.0032 | 0.0410 | 0.0110 | 0.0310 | 0.0110 | 0.0080 | 0.0120 | 0.0250 | 0.0410 | 0.0210 |
Qc20 (p.u.) | 0.0407 | 0.0220 | 0.0350 | 0.0400 | 0.0310 | 0.0360 | 0.0420 | 0.0390 | 0.0340 | 0.0040 |
Qc21 (p.u.) | 0.0319 | 0.0170 | 0.0050 | 0.0100 | 0.0270 | 0.0460 | 0.0080 | 0.0270 | 0.0290 | 0.0310 |
Qc23 (p.u.) | 0.0225 | 0.0300 | 0.0330 | 0.0450 | 0.0400 | 0.0360 | 0.0250 | 0.0100 | 0.0060 | 0.0130 |
Qc24 (p.u.) | 0.0396 | 0.0260 | 0.0480 | 0.0250 | 0.0490 | 0.0170 | 0.0460 | 0.0170 | 0.0240 | 0.0360 |
Qc29 (p.u.) | 0.0243 | 0.0230 | 0.0140 | 0.0090 | 0.0390 | 0.0290 | 0.0230 | 0.0500 | 0.0220 | 0.0420 |
Fuel cost | 833.94 | 845.01 | 858.50 | 860.37 | 852.02 | 859.47 | 878.13 | 879.91 | 916.59 | 918.57 |
Power loss | 5.0075 | 4.6727 | 5.9031 | 5.9547 | - | - | 3.9232 | 4.2179 | 4.7780 | 4.7949 |
Emission | - | - | - | - | 0.2788 | 0.2733 | 0.2161 | 0.2165 | 0.2322 | 0.2323 |
Algorithms | Population Size | Maximum Iteration | Trials | Approach | Fuel Cost | Power Loss |
---|---|---|---|---|---|---|
MOFA-CPA | 100 | 300 | 30 | CPA | 833.94 | 5.0075 |
MOFA-PFA | 100 | 300 | 30 | PFA | 845.01 | 4.6727 |
MOEA/D [13] | 100 | 500 | 31 | PFA | 835.36 | 4.9099 |
NSGA-II [13] | 100 | 500 | 31 | PFA | 833.57 | 5.199 |
MOACSA [30] | 50 | 500 | - | PFA | 837.7994 | 4.9342 |
MODE [30] | 100 | 300 | - | PFA | 828.59 | 5.69 |
MOHS [31] | 20 | 1000 | - | - | 832.6709 | 5.3143 |
NSGA-II [31] | 20 | 1000 | - | - | 837.416 | 5.2397 |
MOABC/D [32] | 100 | 1000 | 20 | PFA | 827.636 | 5.2451 |
MOTLA/D [32] | 100 | 1000 | 20 | PFA | 826.446 | 5.3074 |
NS_CPSO [33] | 100 | 1000 | 50 | PFA | 837.8715 | 4.8173 |
SWTC_NSPSO [33] | 100 | 1000 | 50 | PFA | 841.1731 | 4.6846 |
MO-DEA [34] | 50 | 300 | 30 | PFA | 820.8802 | 5.5949 |
ICA [35] | 120 | 500 | 50 | PFA | 848.0544 | 4.5603 |
Algorithms | Population Size | Maximum Iteration | Trials | Approach | Fuel Cost | Power Loss | Emission |
---|---|---|---|---|---|---|---|
MOFA-CPA | 100 | 300 | 30 | CPA | 878.13 | 3.9232 | 0.2171 |
MOFA-PFA | 100 | 300 | 30 | PFA | 879.91 | 4.2179 | 0.2165 |
MOEA/D [13] | 100 | 500 | 31 | PFA | 902.54 | 3.4594 | 0.2107 |
MOPSO [13] | 100 | 500 | 31 | PFA | 897.48 | 3.9557 | 0.2144 |
WA [36] | 80 | 300 | - | - | 897.2797 | 4.6211 | 0.2175 |
Variables | Alg1 1 | Alg2 2 | Variables | Alg1 | Alg2 | variables | Alg1 | Alg2 |
---|---|---|---|---|---|---|---|---|
Pg2 (MW) | 72.0680 | 90.9355 | Vg9 (p.u.) | 1.0696 | 1.0237 | T58 (p.u.) | 0.9820 | 0.9550 |
Pg3 (MW) | 78.0025 | 87.0173 | Vg12 (p.u.) | 1.0592 | 1.0131 | T59 (p.u.) | 1.0010 | 0.9410 |
Pg6 (MW) | 100.0000 | 100.0000 | T19 (p.u.) | 1.0310 | 0.9150 | T65 (p.u.) | 0.9930 | 0.9550 |
Pg8 (MW) | 328.7255 | 334.1000 | T20 (p.u.) | 1.0210 | 0.9380 | T66 (p.u.) | 1.0020 | 0.9040 |
Pg9 (MW) | 100.0000 | 100.0000 | T31 (p.u.) | 1.0780 | 1.0310 | T71 (p.u.) | 1.0000 | 0.9380 |
Pg12 (MW) | 395.0391 | 406.3353 | T35 (p.u.) | 0.9670 | 0.9260 | T73 (p.u.) | 1.0410 | 0.9380 |
Vg1 (p.u.) | 1.0784 | 1.0464 | T36 (p.u.) | 0.9810 | 0.9800 | T76 (p.u.) | 1.0090 | 1.0130 |
Vg2 (p.u.) | 1.0705 | 1.0384 | T37 (p.u.) | 0.9980 | 1.0550 | T80 (p.u.) | 0.9820 | 1.0240 |
Vg3 (p.u.) | 1.0686 | 1.0252 | T41 (p.u.) | 0.9810 | 0.9860 | Qc18 (p.u.) | 0.1820 | 0.0690 |
Vg6 (p.u.) | 1.0717 | 1.0331 | T46 (p.u.) | 1.0010 | 0.9000 | Qc25 (p.u.) | 0.1330 | 0.0560 |
Vg8 (p.u.) | 1.0750 | 1.0279 | T54 (p.u.) | 0.9020 | 0.9720 | Qc53 (p.u.) | 0.1710 | 0.0640 |
Fuel cost | 42639.96 | 42665.51 | ||||||
Emission | 11.2686 | 11.7785 | ||||||
Power loss | 1.4928 | 1.5234 |
Algorithms | Success Rate | |||||
---|---|---|---|---|---|---|
Case 1 | Case 2 | Case 3 | Case 4 | Case 5 | Case 6 | |
MOFA-CPA | 30/30 | 30/30 | 30/30 | 30/30 | 30/30 | 30/30 |
MOFA-PFA | 13/30 | 15/30 | 20/30 | 16/30 | 9/30 | 0/30 |
Algorithm | Quality Indictor | Case 1 | Case 2 | Case 3 | Case 4 | Case 5 | Case 6 | |
---|---|---|---|---|---|---|---|---|
MOFA-CPA | HV | Mean | 759.14 | 1060.53 | 35.32 | 110.96 | 205.93 | 602,013.21 |
Std | 6.93 | 8.22 | 0.43 | 0.54 | 3.16 | 53,158.19 | ||
SPREAD | Mean | 0.74 | 0.73 | 0.88 | 0.76 | 0.77 | 0.73 | |
Std | 0.02 | 0.02 | 0.02 | 0.03 | 0.04 | 0.03 | ||
MOFA-PFA | HV | Mean | 739.70 | 1050.39 | 35.09 | 88.33 | 204.07 | 598,973.46 |
Std | 6.93 | 7.62 | 0.39 | 3.69 | 3.66 | 54,066.57 | ||
p | 1.73 × 10−6 | 5.75 × 10−6 | 0.0495 | 1.73 × 10−6 | 0.0571 | 0.6733 | ||
h | 1 | 1 | 1 | 1 | 0 | 0 | ||
SPREAD | Mean | 0.85 | 0.86 | 0.89 | 0.94 | 0.79 | 0.83 | |
Std | 0.01 | 0.01 | 0.02 | 0.02 | 0.04 | 0.02 | ||
p | 1.73 × 10−6 | 1.73 × 10−6 | 0.2369 | 1.73 × 10−6 | 0.0687 | 1.73 × 10−6 | ||
h | 1 | 1 | 0 | 1 | 0 | 1 |
Algorithms | Mean CPU Time (sec)/Tmax | |||||
---|---|---|---|---|---|---|
Case 1 | Case 2 | Case 3 | Case 4 | Case 5 | Case 6 | |
MOFA-CPA | 145.93/300 | 166.53/300 | 168.71/300 | 315.21/500 | 332.13/500 | 519.25/500 |
MOFA-PFA | 153.74/300 | 169.68/300 | 170.38/300 | 319.54/500 | 336.14/500 | 525.12/500 |
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Chen, G.; Yi, X.; Zhang, Z.; Lei, H. Solving the Multi-Objective Optimal Power Flow Problem Using the Multi-Objective Firefly Algorithm with a Constraints-Prior Pareto-Domination Approach. Energies 2018, 11, 3438. https://doi.org/10.3390/en11123438
Chen G, Yi X, Zhang Z, Lei H. Solving the Multi-Objective Optimal Power Flow Problem Using the Multi-Objective Firefly Algorithm with a Constraints-Prior Pareto-Domination Approach. Energies. 2018; 11(12):3438. https://doi.org/10.3390/en11123438
Chicago/Turabian StyleChen, Gonggui, Xingting Yi, Zhizhong Zhang, and Hangtian Lei. 2018. "Solving the Multi-Objective Optimal Power Flow Problem Using the Multi-Objective Firefly Algorithm with a Constraints-Prior Pareto-Domination Approach" Energies 11, no. 12: 3438. https://doi.org/10.3390/en11123438