Improved Krill Herd Algorithm with Novel Constraint Handling Method for Solving Optimal Power Flow Problems
Abstract
:1. Introduction
- An improved krill herd algorithm is proposed, namely IKHA, which introduces the onlooker search mechanism to reduce the probability of falling into local optimum. Moreover, the parameter values of the proposed algorithm including inertia weight ω and step-length scale factor Ct are varied according to the iteration of evolutionary process.
- A novel constraint handling method, which contains two parts of control variable constraint and state variable constraint, is proposed to guide the individual to the feasible space and ensure the optimal solutions satisfy the security constraints, especially in larger systems.
- The OPF problem is successfully implemented on standard IEEE 30 bus, IEEE 57 bus and IEEE 118 bus systems to solve 10 different cases by using the proposed method.
2. OPF Problem Formulation
2.1. Control Variable
2.2. State Variable
2.3. Equality Constraint
2.4. Inequality Constraint
- Generator constraints:
- Transformer constraints:
- Shunt reactive compensator constraints:
- Security constraints:
3. Improved Krill Herd Algorithm (IKHA)
3.1. Brief on Krill Herd Algorithm (KHA)
- Movement induced by other krill individuals
- Foraging motion
- Random diffusion
3.1.1. Movement Induced by Other Krill Individuals
3.1.2. Foraging Motion
3.1.3. Random Diffusion
3.1.4. Position Update
3.1.5. Genetic Operators
3.1.6. The Process of Krill Herd Algorithm
- Step 1:
- Initialization: The algorithm parameters, the power system data, limits of variables.
- Step 2:
- The generation of initial population: The individual is randomly generated in the search space of the optimization problem. Random values are assigned to each D-dimensional individual according to:
- Step 3:
- Fitness evaluation: Evaluate each krill individual according to its position and memorise the global best solution.
- Step 4:
- Motion calculation:
- Movement induced by other krill individuals
- Foraging motion
- Random diffusion
- Step 5:
- Perform the genetic operators including crossover and mutation.
- Step 6:
- Update the population and repeat the Step 3.
- Step 7:
- Stop and display the final solution if the stop criteria is reached, else go back to Step 4.
3.2. Onlooker Search Mechanism
3.3. Parameter Improvement
3.4. Implementation of IKHA Algorithm
4. Novel Constraint Handling Method
4.1. State Variable Constraint
4.2. Control Variable Constraint
5. Application and Simulation Results
5.1. IEEE 30 Bus System
5.1.1. Case 1: Minimization of Quadratic Fuel Cost Function
5.1.2. Case 2: Minimization of Voltage Magnitude Deviation
5.1.3. Case 3: Minimization of Fuel Cost Emission
5.1.4. Case 4: Minimization of Transmission Real Power Losses
5.1.5. Case 5: Minimization of Quadratic Cost and Voltage Magnitude Deviation
5.1.6. Case 6: Minimization of Quadratic Cost and Transmission Real Power Losses
5.2. IEEE 57 Bus System
5.2.1. Case 7: Minimization of Quadratic Fuel Cost Function
5.2.2. Case 8: Minimization of Voltage Magnitude Deviation
5.2.3. Case 9: Minimization of Quadratic Cost and Voltage Magnitude Deviation
5.3. IEEE 118 Bus System
Case 10: Minimization of Quadratic Fuel Cost Function
6. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
References
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Test System | Name | Objective Function | Constraints |
---|---|---|---|
IEEE 30 | Case 1 | Quadratic fuel cost function | Equality/non-equality |
Case 1-a | Fuel cost function with multiple fuel sources | Equality/non-equality | |
Case 1-b | Fuel cost function with valve point effect | Equality/non-equality | |
Case 2 | Voltage magnitude deviation | Equality/non-equality | |
Case 3 | Fuel cost emission | Equality/non-equality | |
Case 4 | Transmission real power losses | Equality/non-equality | |
Case 5 | Quadratic cost considering voltage deviation | Equality/non-equality | |
Case 6 | Quadratic cost considering power losses | Equality/non-equality | |
IEEE 57 | Case 7 | Quadratic fuel cost function | Equality/non-equality |
Case 8 | Voltage magnitude deviation | Equality/non-equality | |
Case 9 | Quadratic cost considering voltage deviation | Equality/non-equality | |
IEEE 118 | Case 10 | Quadratic fuel cost function | Equality/non-equality |
Algorithm | NP | gmax | Nmax | Vf | Dmax | Ct |
---|---|---|---|---|---|---|
IKHA | 30 | 500 | 0.01 | 0.02 | 0.005 | 0.4/0.7 |
KHA | 30 | 500 | 0.01 | 0.02 | 0.005 | 0.4 |
Control Variables | Case 1 | Case 1-a | Case 1-b | Case 2 | Case 3 | Case 4 | Case 5 | Case 6 |
---|---|---|---|---|---|---|---|---|
P1 (MW) | 177.0460 | 139.9931 | 199.2307 | 53.7862 | 64.0580 | 51.4880 | 176.4745 | 102.5066 |
P2 (MW) | 48.7423 | 54.9987 | 51.9824 | 79.8421 | 67.5612 | 79.9973 | 48.8341 | 55.6717 |
P5 (MW) | 21.3782 | 24.1051 | 15.0001 | 49.8070 | 50.0000 | 50.0000 | 21.6357 | 38.0835 |
P8 (MW) | 21.3084 | 34.9883 | 10.0001 | 34.7705 | 35.0000 | 35.0000 | 22.0723 | 34.9998 |
P11 (MW) | 11.9203 | 18.4108 | 10.0002 | 30.0000 | 30.0000 | 29.9998 | 12.2127 | 29.9980 |
P13 (MW) | 12.0020 | 17.6480 | 12.0002 | 39.0733 | 40.0000 | 40.0000 | 12.0005 | 26.6695 |
V1 (p.u.) | 1.0827 | 1.0734 | 0.9784 | 1.0054 | 1.0636 | 1.0609 | 1.0409 | 1.0675 |
V2 (p.u.) | 1.0635 | 1.0593 | 0.9607 | 1.0070 | 1.0575 | 1.0569 | 1.0244 | 1.0571 |
V5 (p.u.) | 1.0325 | 1.0341 | 0.9799 | 1.0189 | 1.0381 | 1.0377 | 1.0127 | 1.0338 |
V8 (p.u.) | 1.0374 | 1.0401 | 0.9631 | 1.0114 | 1.0444 | 1.0441 | 1.0010 | 1.0421 |
V11 (p.u.) | 1.0897 | 1.0972 | 1.0992 | 0.9880 | 1.0877 | 1.0805 | 1.0428 | 1.0773 |
V13 (p.u.) | 1.0470 | 1.0411 | 0.9831 | 1.0049 | 1.0487 | 1.0564 | 0.9920 | 1.0511 |
T11 (p.u.) | 1.0400 | 1.0300 | 0.9000 | 1.0000 | 1.0800 | 1.0800 | 1.0600 | 1.0500 |
T12 (p.u.) | 0.9300 | 0.9800 | 0.9000 | 0.9000 | 0.9000 | 0.9000 | 0.9000 | 0.9100 |
T15 (p.u.) | 0.9700 | 0.9700 | 1.1000 | 0.9700 | 0.9800 | 0.9900 | 0.9500 | 0.9900 |
T36 (p.u.) | 0.9700 | 0.9700 | 0.9200 | 0.9500 | 0.9700 | 0.9800 | 0.9600 | 0.9800 |
QC10 (p.u.) | 0.0020 | 0.0440 | 0.0480 | 0.0500 | 0.0000 | 0.0060 | 0.0500 | 0.0010 |
QC12 (p.u.) | 0.0130 | 0.0440 | 0.0430 | 0.0000 | 0.0070 | 0.0000 | 0.0130 | 0.0430 |
QC15 (p.u.) | 0.0410 | 0.0460 | 0.0050 | 0.0500 | 0.0440 | 0.0430 | 0.0500 | 0.0460 |
QC17 (p.u.) | 0.0500 | 0.0350 | 0.0450 | 0.0000 | 0.0500 | 0.0500 | 0.0000 | 0.0500 |
QC20 (p.u.) | 0.0390 | 0.0400 | 0.0120 | 0.0500 | 0.0390 | 0.0400 | 0.0500 | 0.0390 |
QC21 (p.u.) | 0.0500 | 0.0460 | 0.0000 | 0.0500 | 0.0500 | 0.0500 | 0.0500 | 0.0500 |
QC23 (p.u.) | 0.0280 | 0.0410 | 0.0470 | 0.0500 | 0.0280 | 0.0290 | 0.0500 | 0.0280 |
QC24 (p.u.) | 0.0500 | 0.0480 | 0.0100 | 0.0500 | 0.0500 | 0.0500 | 0.0500 | 0.0500 |
QC29 (p.u.) | 0.0220 | 0.0190 | 0.0040 | 0.0070 | 0.0190 | 0.0240 | 0.0170 | 0.0260 |
Fuel cost | 800.4143 | 646.5126 | 929.901 | 965.5317 | 944.3314 | 967.6201 | 803.5879 | 859.0579 |
Voltage deviations | 0.9215 | 0.9256 | 0.6826 | 0.0892 | 0.9226 | 0.8814 | 0.0984 | 0.9083 |
Emission | 0.3660 | 0.2835 | 0.4410 | 0.2077 | 0.204818 | 0.2073 | 0.3642 | 0.2287 |
Power loss | 8.9972 | 6.7439 | 14.8136 | 3.8792 | 3.2192 | 3.0850 | 9.8297 | 4.5291 |
Algorithms | Min ($/h) | Simulation Time (s)/gmax |
---|---|---|
IKHA | 800.4143 | 75.11/500 |
KHA | 801.4675 | 74.06/500 |
MSLFA [30] | 802.2870 | NA/100 |
ABC [31] | 800.6600 | 130.16/200 |
MSA [22] | 800.5099 | NA/100 |
MGBICA [32] | 801.1409 | NA/NA |
ARCBBO [34] | 800.5159 | NA/200 |
Jaya [33] | 800.479 | 72.4/100 |
GSA [35] | 798.6751 a | 10.7582/200 |
BBO [36] | 799.1116 a | NA/200 |
Algorithms | Case 1-a Min ($/h) | Case 1-b Min ($/h) | Average Time (s)/gmax |
---|---|---|---|
IKHA | 646.5126 | 929.9010 | 80.75/500 |
KHA | 647.0264 | 932.1784 | 78.22/500 |
MSA [22] | 646.8364 | 930.7441 | NA/100 |
Algorithms | Min | Simulation Time (s)/gmax |
---|---|---|
IKHA | 0.0892 | 70.40/500 |
KHA | 0.1029 | 68.02/500 |
MGBICA [32] | 0.1239 | NA/NA |
LTLBO [23] | 0.0974 | 20.17/100 |
BBO [36] | 0.0951 | NA/200 |
Algorithms | Min (ton/h) | Simulation Time (s)/gmax |
---|---|---|
IKHA | 0.204818 | 76.68/500 |
KHA | 0.205082 | 74.89/500 |
DSA [9] | 0.2058255 | NA/500 |
ABC [31] | 0.204826 | NA/200 |
MSA [22] | 0.20482 | NA/100 |
MGBICA [32] | 0.2048 | NA/NA |
MSLFA [30] | 0.2056 | NA/100 |
Algorithms | Min (MW) | Simulation Time (s)/gmax |
---|---|---|
IKHA | 3.0805 | 72.32/500 |
KHA | 3.1409 | 70.64/500 |
ABC [31] | 3.1078 | NA/200 |
Combined approach [28] | 3.2601 | 3.3109/NA |
DSA [9] | 3.0945 | NA/500 |
MSA [22] | 3.1005 | NA/100 |
MGBICA [32] | 4.937 | NA/NA |
Proposed EEA [38] | 3.2823 | 5.7167/94 |
ALC-PSO [13] | 3.1700 | 10.2345/500 |
Jaya [33] | 3.1035 | NA/100 |
Algorithms | Fuel Cost ($/h) | Voltage Deviations | Total | Time (s)/gmax |
---|---|---|---|---|
IKHA | 803.5879 | 0.0984 | 813.4279 | 78.36/500 |
KHA | 803.8889 | 0.0987 | 813.7589 | 75.68/500 |
PSOGSA [29] | 804.43123 | 0.09638 | 814.06923 | NA/200 |
The proposed KHA [39] | 804.6337 | 0.0996 | 814.5937 | NA/100 |
ABPPO [40] | 804.7339 | 0.09232 | 813.9659 | NA/300 |
MSA [22] | 803.3125 | 0.10842 | 814.1545 | NA/100 |
LTLBO [23] | 803.7431 | 0.0974 | 813.4831 | 20.17/100 |
GABC [24] | 803.5785 | 0.1007 | 813.6485 | 2.98/100 |
ICBO [12] | 803.3978 | 0.1014 | 813.5378 | NA/500 |
Algorithms | Fuel Cost ($/h) | Power Loss (MW) | Total | Simulation Time (s)/gmax |
---|---|---|---|---|
IKHA | 859.0579 | 4.5291 | 1040.2219 | 77.29/500 |
KHA | 859.4961 | 4.5246 | 1040.4801 | 75.04/500 |
MSA [22] | 859.1915 | 4.5404 | 1040.8075 | NA/100 |
MDE [22] | 868.7138 | 4.3891 | 1044.2778 | NA/100 |
PSOGSA [29] | 822.40631 | 5.46816 | 1041.13271 | NA/200 |
ABPPO [40] | 822.7693 | 5.452 | 1040.8493 | NA/300 |
Control Variables | Case7 | Case8 | Case9 | ||
---|---|---|---|---|---|
IKHA | IKHA | IKHA | DSA [9] | MSA [22] | |
P1 (MW) | 143.0334 | 355.4995 | 142.8777 | 142.6780 | 143.8661 |
P2 (MW) | 85.3299 | 2.3831 | 88.5835 | 89.6450 | 85.34818 |
P3 (MW) | 44.8387 | 124.4492 | 45.1741 | 45.6795 | 45.85493 |
P6 (MW) | 75.1387 | 86.9520 | 70.3558 | 73.1394 | 71.30797 |
P8 (MW) | 461.8476 | 212.7213 | 460.6468 | 461.7316 | 462.4092 |
P9 (MW) | 95.0960 | 99.1914 | 96.7565 | 92.1106 | 94.08068 |
P12 (MW) | 360.3747 | 387.2737 | 361.9443 | 361.4796 | 363.8543 |
V1 (p.u.) | 1.0528 | 1.0082 | 1.0269 | 1.0212 | 1.022121 |
V2 (p.u.) | 1.0494 | 1.0001 | 1.0246 | 1.0740 | 1.019646 |
V3 (p.u.) | 1.0455 | 1.0125 | 1.0185 | 1.0646 | 1.013444 |
V6 (p.u.) | 1.0581 | 1.0032 | 1.0313 | 0.9913 | 1.025691 |
V8 (p.u.) | 1.0748 | 1.0086 | 1.0505 | 1.0519 | 1.044968 |
V9 (p.u.) | 1.0587 | 1.0192 | 1.0312 | 1.0808 | 1.014033 |
V12 (p.u.) | 1.0397 | 1.0232 | 1.0115 | 1.0103 | 1.010858 |
T4-18 (p.u.) | 1.0449 | 0.9935 | 0.9690 | 0.9688 | 0.9101725 |
T4-18 (p.u.) | 0.9495 | 0.9472 | 0.9901 | 0.9952 | 1.075124 |
T21-20 (p.u.) | 1.0223 | 0.9765 | 0.9933 | 1.0248 | 0.9854176 |
T24-25 (p.u.) | 0.9577 | 1.0504 | 0.9640 | 1.0010 | 0.9872317 |
T24-25 (p.u.) | 1.0798 | 1.0465 | 1.0665 | 1.0025 | 1.053424 |
T24-26 (p.u.) | 1.0190 | 0.9985 | 1.0179 | 0.9452 | 1.016568 |
T7-29 (p.u.) | 0.9970 | 0.9985 | 1.0059 | 0.9000 | 1.00709 |
T34-32 (p.u.) | 0.9672 | 0.9237 | 0.9405 | 0.9443 | 0.9348021 |
T11-41 (p.u.) | 0.9028 | 0.9000 | 0.9005 | 0.9542 | 0.900021 |
T15-45 (p.u.) | 0.9712 | 0.9317 | 0.9624 | 0.9772 | 0.9479471 |
T14-46 (p.u.) | 0.9616 | 0.9740 | 0.9611 | 0.9252 | 0.9608689 |
T10-51 (p.u.) | 0.9795 | 1.0097 | 0.9799 | 0.9665 | 0.9781408 |
T13-49 (p.u.) | 0.9355 | 0.9013 | 0.9343 | 1.0116 | 0.9182851 |
T11-43 (p.u.) | 0.9833 | 0.9648 | 0.9610 | 0.9343 | 0.9509346 |
T40-56 (p.u.) | 0.9945 | 1.0188 | 1.0210 | 1.0130 | 0.9941227 |
T39-57 (p.u.) | 0.9605 | 0.9010 | 0.9274 | 0.9861 | 0.9361633 |
T9-55 (p.u.) | 1.0093 | 1.0186 | 1.0098 | 1.0214 | 0.998129 |
QC18 (p.u.) | 0.1037 | 0.0025 | 0.0667 | 0.1095 | 0.1188253 |
QC25 (p.u.) | 0.1437 | 0.1890 | 0.1648 | 0.1370 | 0.1678665 |
QC53 (p.u.) | 0.1243 | 0.2889 | 0.1482 | 0.1326 | 0.1828455 |
Fuel cost ($/h) | 41,663.3910 | 48,834.0293 | 41,697.5456 | 41,699.4 | 41,714.9851 |
V-deviatios | 1.5494 | 0.5520 | 0.7233 | 0.7620 | 0.67818 |
Total | 41,818.3310 | 48,889.22930 | 41,769.8815 | 41,775.6 | 41,782.8031 |
Algorithms | Min ($/h) | Simulation Time (s)/gmax |
---|---|---|
IKHA | 41,663.3910 | 136.34/500 |
KHA | 41,687.8183 | 130.85/500 |
MSA [22] | 41,673.7231 | NA/NA |
LTLBO [23] | 41,679.5451 | NA/150 |
ICBO [12] | 41,697.3324 | NA/1500 |
DSA [9] | 41,686.82 | NA/500 |
ARCBBO [34] | 41,686 | NA/500 |
GABC [42] | 41,684.2011 | NA/100 |
Variables | Value | Var. | Value | Var. | Value | Var. | Value | Var. | Value |
---|---|---|---|---|---|---|---|---|---|
PG4 | 75.5892 | PG65 | 1.1061 | PG116 | 2.6329 | V61 | 1.0136 | V112 | 1.0220 |
PG6 | 2.3944 | PG66 | 346.5375 | V1 | 0.9873 | V62 | 1.0103 | V113 | 0.9817 |
PG8 | 13.2117 | PG69 | 311.5309 | V4 | 1.0092 | V65 | 1.0497 | V116 | 1.0243 |
PG10 | 1.8752 | PG70 | 2.5131 | V6 | 0.9970 | V66 | 1.0292 | T8-5 | 0.0960 |
PG12 | 350.2058 | PG72 | 0.3332 | V8 | 0.9740 | V69 | 0.9943 | T26-25 | 0.1070 |
PG1 | 82.1866 | PG73 | 2.9797 | V10 | 0.9918 | V70 | 1.0007 | T30-17 | 0.1000 |
PG18 | 5.6590 | PG74 | 79.1184 | V12 | 0.9949 | V72 | 1.0113 | T38-37 | 0.1010 |
PG19 | 55.9071 | PG76 | 26.5457 | V15 | 0.9646 | V73 | 1.0440 | T63-59 | 0.1010 |
PG24 | 6.0264 | PG77 | 47.3801 | V18 | 0.9688 | V74 | 0.9738 | T64-61 | 0.1020 |
PG25 | 2.9631 | PG80 | 410.2660 | V19 | 0.9628 | V76 | 0.9415 | T65-66 | 0.1020 |
PG26 | 182.0172 | PG85 | 53.3588 | V24 | 1.0099 | V77 | 0.9617 | T68-69 | 0.1080 |
PG27 | 253.7070 | PG87 | 7.1808 | V25 | 1.0084 | V80 | 0.9765 | T81-80 | 0.1040 |
PG31 | 24.5511 | PG89 | 440.0013 | V26 | 1.0130 | V85 | 0.9786 | QC5 | 0.4000 |
PG32 | 8.1844 | PG90 | 0.5161 | V27 | 1.0219 | V87 | 0.9654 | QC34 | 0.0000 |
PG34 | 4.8480 | PG91 | 0.1538 | V31 | 0.9965 | V89 | 1.0127 | QC37 | 0.0500 |
PG36 | 12.1483 | PG92 | 35.7542 | V32 | 1.0063 | V90 | 0.9842 | QC44 | 0.0000 |
PG40 | 0.3641 | PG99 | 0.6210 | V34 | 0.9659 | V91 | 0.9894 | QC45 | 0.1000 |
PG42 | 22.5623 | PG100 | 182.5943 | V36 | 0.9593 | V92 | 0.9893 | QC46 | 0.0500 |
PG46 | 45.1902 | PG103 | 34.1980 | V40 | 0.9612 | V99 | 0.9465 | QC48 | 0.1500 |
PG49 | 18.9517 | PG104 | 24.3619 | V42 | 0.9970 | V100 | 0.9761 | QC74 | 0.1200 |
PG54 | 187.8779 | PG105 | 74.1968 | V46 | 1.0045 | V103 | 0.9798 | QC79 | 0.1000 |
PG55 | 49.2065 | PG107 | 0.8261 | V49 | 1.0119 | V104 | 0.9770 | QC82 | 0.1500 |
PG56 | 48.1089 | PG110 | 55.5449 | V54 | 1.0201 | V105 | 0.9818 | QC83 | 0.1000 |
PG59 | 26.1531 | PG111 | 32.5718 | V55 | 1.0154 | V107 | 1.0051 | QC105 | 0.0000 |
PG61 | 94.0442 | PG112 | 0.4718 | V56 | 1.0152 | V110 | 1.0039 | QC107 | 0.0000 |
PG62 | 135.9548 | PG113 | 0.2558 | V59 | 1.0106 | V111 | 1.0078 | QC110 | 0.0600 |
Fuel cost ($/h) | 131,427.2636 | ||||||||
PG1(MW) | 442.1525 |
© 2018 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
Chen, G.; Lu, Z.; Zhang, Z. Improved Krill Herd Algorithm with Novel Constraint Handling Method for Solving Optimal Power Flow Problems. Energies 2018, 11, 76. https://doi.org/10.3390/en11010076
Chen G, Lu Z, Zhang Z. Improved Krill Herd Algorithm with Novel Constraint Handling Method for Solving Optimal Power Flow Problems. Energies. 2018; 11(1):76. https://doi.org/10.3390/en11010076
Chicago/Turabian StyleChen, Gonggui, Zhengmei Lu, and Zhizhong Zhang. 2018. "Improved Krill Herd Algorithm with Novel Constraint Handling Method for Solving Optimal Power Flow Problems" Energies 11, no. 1: 76. https://doi.org/10.3390/en11010076