Hybrid Imperialist Competitive and Grey Wolf Algorithm to Solve Multiobjective Optimal Power Flow with Wind and Solar Units
Abstract
:1. Introduction
2. OPF Problem Formulation
2.1. Objective Functions
2.1.1. Wind Cost Function
- and symbolizes speed and rated speed of WE generators,
- and symbolizes cut-in and cut-out speed of WE generators,
- , symbolizes shape and scale parameters of the Weibull distribution.
2.1.2. PV Cost Function
- Solar cells or PV cells are hypersensitive to the amount of solar radiation. The PDF of solar radiation can be modeled by a beta distribution [12]:
- The relation between power output of PV and output power of solar cell generator which is related to the solar radiation can be calculated as follows:
2.1.3. Basic Fuel Cost Function
2.1.4. Piecewise Quadratic Fuel Cost Function
2.1.5. Piecewise Quadratic Fuel Cost with Valve Point Loading
2.1.6. Emission Cost Function
2.1.7. Power Loss Cost Function
2.1.8. Fuel Cost and Active Power Loss Cost Function
2.1.9. Fuel Cost and Voltage Deviation
2.1.10. Fuel Cost and Voltage Stability Enhancement
2.1.11. Fuel Cost and Voltage Stability Enhancement during Contingency Condition
2.1.12. Fuel Cost, Emission, Voltage Deviation, and Active Power Loss
2.2. Constraints
3. New Hybrid Optimization Algorithm
3.1. Imperialist Competitive Algorithm (ICA)
3.1.1. Creation of Initial Empires
3.1.2. Assimilation
3.1.3. Revolution
3.1.4. Exchanging Positions of a Colony and the Imperialist
3.1.5. Union of Empires
3.1.6. Total Empire Power
3.1.7. Imperialistic Competition
3.2. Grey Wolf Optimizer (GWO)
3.2.1. Social Hierarchy
3.2.2. Encircling Prey
3.2.3. Hunting
3.2.4. Attacking Prey (Exploitation)
3.2.5. Search for Prey (Exploration)
3.3. Hybrid IC-GWA Optimization Approach
- The power system data is specified. The HIC-GWA parameters are determined.
- Initialize the countries randomly, calculate their costs, and use assimilation.
- Revolution.
- Exchange positions between imperialist and colony if it has a lower cost.
- Unite similar empires.
- Calculate the total cost of all empires.
- Imperialist competition.
- Discard powerless empires.
- Use solution obtained by ICA as initial condition for GWA.
- The lower solution between ICA and GWA is saved as best solution.
- Go to step (ii) if the stop condition is not satisfied, otherwise, finish simulation.
4. Simulation Results
4.1. IEEE 30 Bus Test System
4.1.1. Simple Objective OPF
4.1.2. Multiobjective OPF
4.2. The IEEE 118 Bus Power System
4.3. HIC-GWA Robustness Analysis
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
Nomenclature
direct cost of WE and PV($/h) | |
underestimating penalty cost of ith WE and PV ($/h) | |
overestimating penalty cost of ith WE and PV ($/h) | |
total cost of ith WE and PV ($/h) | |
direct cost coefficient of WE and PV ($/MW) | |
underestimating coefficient cost of ith WE and PV ($/MW) | |
overestimating coefficients cost of ithWE and PV ($/MW) | |
power of the ith WE (MW) | |
power of the ith PV (MW) | |
rated power of the ith WE (MW) | |
rated power of the ith PV(MW) | |
number of WEs | |
number of PVs |
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Characteristics | IEEE 30 | IEEE 118 | ||
---|---|---|---|---|
Buses | 30 | [47] | 118 | [48] |
Branches | 41 | 186 | ||
Load voltage | 24 | [0.95, 1.05] | [0.94, 1.06] | |
Control variables | 24 | 130 |
ICA Parameters | GWA Parameters | ||||||
---|---|---|---|---|---|---|---|
30 bus | 118 bus | 30 bus | 118 bus | 30 bus | 118 bus | ||
1.02 | 15 | 100 | 0.90 | 5 | 20 | 5 | 10 |
Acronym | Reference | Simple Objective | Multiobjective | Fuel Cost | Emission | VD | L Index | |
---|---|---|---|---|---|---|---|---|
MSA | [19] | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |
MPSO | [20] | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |
MDE | [21] | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |
MFO | [22] | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |
FPA | [23] | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |
ARCBBO | [24] | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | |
RCBBO | [24] | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | |
GWO | [25] | ✓ | ✓ | ✓ | ✓ | |||
DE | [25] | ✓ | ✓ | ✓ | ✓ | |||
MGBICA | [26] | ✓ | ✓ | ✓ | ✓ | |||
ABC | [27] | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |
HSFLA-SA | [28] | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | |
LTLBO | [29] | ✓ | ✓ | ✓ | ✓ | ✓ | ||
TLBO | [30] | ✓ | ✓ | ✓ | ✓ | ✓ | ||
HMPSO-SFLA | [31] | ✓ | ✓ | ✓ | ✓ | |||
PSO | [31] | ✓ | ✓ | ✓ | ✓ | ✓ | ||
GABC | [32] | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | |
DSA | [33] | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |
EEA | [34] | ✓ | ✓ | ✓ | ✓ | ✓ | ||
EGA | [34] | ✓ | ✓ | ✓ | ✓ | ✓ | ||
ALC-PSO | [35] | ✓ | ✓ | ✓ | ✓ |
Solutions | Case 1 | Case 2 | Case 3 | Case4 | Case 5 |
---|---|---|---|---|---|
Fuel cost ($/h) | 798.20 | 645.85 | 902.25 | 959.54 | 1000.30 |
Emission (t/h) | 0.37 | 0.28 | 0.45 | 0.20 | 0.21 |
(MW) | 8.86 | 6.59 | 11.18 | 2.67 | 2.61 |
VD (p.u.) | 1.15 | 1.25 | 0.96 | 1.68 | 1.41 |
L index | 0.13 | 0.13 | 0.17 | 0.13 | 0.12 |
Algorithm | Fuel Cost ($/h) | Emission (t/h) | VD (p.u.) | L Index | |
---|---|---|---|---|---|
MSA | 800.51 | 0.37 | 9.03 | 0.90 | 0.14 |
MPSO | 800.52 | 0.37 | 9.04 | 0.90 | 0.14 |
MDE | 800.84 | 0.36 | 808365.00 | 0.78 | 0.14 |
MFO | 800.69 | 0.37 | 9.15 | 0.76 | 0.14 |
FPA | 802.80 | 0.36 | 9.54 | 0.37 | 0.15 |
ARCBO | 800.52 | 0.37 | 9.03 | 0.89 | 0.14 |
HSFLA-SA | 801.79 | ||||
HIC-GWA | 798.20 | 0.37 | 8.86 | 1.15 | 0.13 |
Algorithm | Fuel Cost ($/h) | Emission (t/h) | VD (p.u.) | L Index | |
---|---|---|---|---|---|
MSA | 646.84 | 0.28 | 6.80 | 0.84 | 0.14 |
MPSO | 646.73 | 0.28 | 6.80 | 0.77 | 0.14 |
MDE | 650.28 | 0.28 | 6.98 | 0.58 | 0.14 |
MFO | 649.27 | 0.28 | 7.29 | 0.47 | 0.14 |
FPA | 651.38 | 0.28 | 7.24 | 0.31 | 0.15 |
LTLBO | 647.43 | 0.28 | 6.93 | 0.89 | |
TLBO | 647.92 | 7.11 | 1.42 | 0.12 | |
HIC-GWA | 645.85 | 0.28 | 6.59 | 1.25 | 0.13 |
Algorithm | Fuel Cost ($/h) | Emission (t/h) | VD (p.u.) | L Index | |
---|---|---|---|---|---|
MSA | 930.74 | 0.43 | 13.14 | 0.45 | 0.16 |
MPSO | 952.30 | 0.30 | 7.30 | 0.72 | 0.14 |
MDE | 930.94 | 0.43 | 12.73 | 0.45 | 0.16 |
MFO | 930.72 | 0.44 | 13.18 | 0.47 | 0.16 |
FPA | 931.75 | 0.43 | 12.11 | 0.47 | 0.15 |
HIC-GWA | 902.25 | 0.45 | 11.18 | 0.96 | 0.17 |
Algorithm | Fuel Cost ($/h) | Emission (t/h) | VD (p.u.) | L Index | |
---|---|---|---|---|---|
MSA | 944.50 | 0.2048 | 3.24 | 0.87 | 0.14 |
MPSO | 879.95 | 0.2325 | 7.05 | 0.57 | 0.14 |
MDE | 927.81 | 0.2093 | 4.85 | 0.40 | 0.15 |
MFO | 945.46 | 0.2049 | 3.43 | 0.71 | 0.14 |
FPA | 948.95 | 0.2052 | 4.49 | 0.43 | 0.14 |
ARCBO | 945.16 | 0.2048 | 3.26 | 0.86 | 0.14 |
MGBICA | 942.84 | 0.2048 | |||
GBICA | 944.65 | 0.2049 | |||
ABC | 944.44 | 0.2048 | 3.25 | 0.85 | 0.14 |
DSA | 944.41 | 0.2583 | 3.24 | 0.13 | |
HMPSO-SFLA | 0.2052 | ||||
HIC-GWA | 959.54 | 0.2009 | 2.67 | 1.68 | 0.13 |
Algorithm | Fuel Cost ($/h) | Emission (t/h) | VD (p.u.) | L Index | |
---|---|---|---|---|---|
MSA | 967.66 | 0.2073 | 3.10 | 0.89 | 0.14 |
MPSO | 967.65 | 0.2073 | 3.10 | 0.96 | 0.14 |
MDE | 967.65 | 0.2073 | 3.16 | 0.77 | 0.14 |
MFO | 967.68 | 0.2073 | 3.11 | 0.92 | 0.14 |
FPA | 967.11 | 0.2076 | 6.57 | 0.39 | 0.14 |
ARCBO | 967.66 | 0.2073 | 3.10 | 0.89 | 0.14 |
GWO | 968.38 | 3.41 | |||
DE | 968.23 | 3.38 | |||
ABC | 967.68 | 0.2073 | 3.11 | 0.90 | 0.14 |
DSA | 967.65 | 0.2083 | 3.09 | 0.13 | |
EEA | 952.38 | 3.28 | |||
EGA | 967.93 | 3.24 | |||
ALC-PSO | 967.77 | 3.17 | |||
HIC-GWA | 1000.30 | 0.2080 | 2.61 | 1.41 | 0.12 |
Solutions | Case 6 | Case 7 | Case 8 | Case 9 | Case 10 |
---|---|---|---|---|---|
Fuel cost ($/h) | 856.99 | 802.45 | 797.80 | 802.00 | 817.59 |
Emission (t/h) | 0.23 | 0.36 | 0.37 | 0.36 | 0.27 |
(MW) | 4.04 | 9.95 | 8.75 | 9.67 | 5.29 |
VD (p.u.) | 1.78 | 0.10 | 1.98 | 1.97 | 0.23 |
L index | 0.12 | 0.13 | 0.11 | 0.11 | 0.15 |
Solutions | MSA | MDE | MPSO | FPA | MFO | HIC-GWA |
---|---|---|---|---|---|---|
Fuel cost ($/h) | 859.19 | 868.71 | 859.58 | 855.27 | 858.58 | 856.99 |
Emission (t/h) | 0.23 | 0.23 | 0.23 | 0.23 | 0.23 | 0.23 |
(MW) | 4.54 | 4.39 | 4.54 | 4.80 | 4.58 | 4.04 |
VD (p.u.) | 0.93 | 0.88 | 0.95 | 1.01 | 0.90 | 1.78 |
L index | 0.14 | 0.14 | 0.14 | 0.14 | 0.14 | 0.12 |
Total cost | 1040.81 | 1044.05 | 1041.22 | 1055.72 | 1041.67 | 1018.45 |
Solutions | MSA | MDE | MPSO | FPA | MFO | HIC-GWA |
---|---|---|---|---|---|---|
Fuel cost ($/h) | 803.31 | 803.21 | 803.98 | 803.66 | 803.79 | 802.45 |
Emission (t/h) | 0.36 | 0.36 | 0.36 | 0.37 | 0.36 | 0.36 |
(MW) | 9.72 | 9.60 | 9.92 | 9.93 | 9.87 | 9.95 |
VD (p.u.) | 0.11 | 0.13 | 0.12 | 0.14 | 0.11 | 0.10 |
L index | 0.15 | 0.15 | 0.15 | 0.15 | 0.15 | 0.13 |
Total cost | 814.15 | 815.86 | 816.00 | 817.32 | 814.35 | 812.05 |
Solutions | MSA | MDE | MPSO | FPA | MFO | HIC-GWA |
---|---|---|---|---|---|---|
Fuel cost ($/h) | 801.22 | 802.10 | 801.70 | 801.15 | 801.67 | 797.80 |
Emission (t/h) | 0.36 | 0.35 | 0.36 | 0.37 | 0.34 | 0.37 |
(MW) | 8.98 | 9.06 | 9.20 | 9.32 | 8.56 | 8.75 |
VD (p.u.) | 0.93 | 0.89 | 0.83 | 0.88 | 0.84 | 1.98 |
L index | 0.14 | 0.14 | 0.14 | 0.14 | 0.14 | 0.11 |
Total cost | 814.94 | 815.84 | 815.44 | 814.91 | 815.43 | 808.38 |
Solutions | MSA | MDE | MPSO | FPA | MFO | HIC-GWA |
---|---|---|---|---|---|---|
Fuel cost ($/h) | 804.48 | 806.67 | 807.65 | 805.54 | 804.56 | 802.00 |
Emission (t/h) | 0.36 | 0.37 | 0.36 | 0.36 | 0.36 | 0.36 |
(MW) | 9.95 | 10.72 | 10.76 | 10.18 | 9.95 | 9.67 |
VD (p.u.) | 0.92 | 0.57 | 0.43 | 0.45 | 0.91 | 1.97 |
L index | 0.14 | 0.14 | 0.14 | 0.14 | 0.14 | 0.11 |
Total cost | 832.32 | 834.63 | 835.75 | 833.84 | 832.43 | 823.06 |
Solutions | MSA | MDE | MPSO | FPA | MFO | HIC-GWA |
---|---|---|---|---|---|---|
Fuel cost ($/h) | 830.6 | 829.1 | 833.7 | 835.4 | 830.9 | 817.6 |
Emission (t/h) | 0.3 | 0.3 | 0.3 | 0.2 | 0.3 | 0.3 |
(MW) | 5.6 | 6.1 | 6.5 | 5.5 | 5.6 | 5.3 |
VD (p.u.) | 0.3 | 0.3 | 0.2 | 0.5 | 0.3 | 0.2 |
L index | 1.5 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 |
Total cost | 965.3 | 973.6 | 986.0 | 971.9 | 965.8 | 944.0 |
Solutions | MSA | MDE | MPSO | FPA | MFO | HIC-GWA |
---|---|---|---|---|---|---|
Fuel cost ($/h) | 129640.72 | 130444.57 | 132039.21 | 129688.72 | 129708.08 | 129633.70 |
(MW) | 73.26 | 71.64 | 112.85 | 74.32 | 74.71 | 76.80 |
VD (p.u.) | 3.07 | 1.31 | 1.15 | 2.54 | 2.38 | 3.13 |
L index | 0.06 | 0.07 | 0.07 | 0.06 | 0.06 | 0.06 |
Fuel cost ($/h) | 112,545.51 |
Wind cost ($/h) | 5340.42 |
PV cost ($/h) | 4211.38 |
(MW) | 76.64 |
VD (p.u.) | 3.13 |
L index | 0.06 |
Parameters | 30 Bus Power System | Parameters | 118 Bus Power System | ||
---|---|---|---|---|---|
Cost ($/h) | Deviation (%) | Cost ($/h) | Deviation (%) | ||
Normal Solution | 798.20 | 0.0 | Normal Solution | 129,633.70 | 0.0 |
= 15 + 5 | 797.38 | +0.1017 | = 200 + 30 | 129,631.93 | +0.00137 |
= 15 − 5 | 799.07 | −0.1102 | = 200 − 30 | 129,636.79 | −0.00238 |
= 5 + 2 | 797.33 | +0.1082 | = 40 + 10 | 129,632.44 | +0.00098 |
= 5 − 2 | 797.00 | +0.1491 | = 40 − 10 | 129,630.66 | +0.00235 |
= 5 + 2 | 797.12 | +0.1341 | = 10 + 3 | 129,631.77 | +0.00149 |
= 5 − 2 | 798.98 | −0.0984 | = 10 − 3 | 129,634.92 | −0.00094 |
All (up) | 797.06 | +0.1420 | All (up) | 129,630.94 | +0.00213 |
All (Down) | 799.08 | −0.1110 | All (Down) | 129,645.84 | −0.00936 |
Algorithm | Best Cost ($/h) | Worst Cost ($/h) | Average Cost ($/h) |
---|---|---|---|
MSA | 646.84 | 648.03 | 646.86 |
MPSO | 646.73 | 656.23 | 649.86 |
MDE | 650.28 | 653.40 | 651.26 |
MFO | 649.27 | 650.62 | 649.89 |
FPA | 651.38 | 654.33 | 652.96 |
LTLBO | 647.43 | 647.86 | 647.47 |
ABC | 649.09 | 659.77 | 654.08 |
GABC | 647.03 | 647.12 | 647.08 |
HIC-GWA | 645.85 | 647.03 | 645.87 |
Algorithm | Time (s) |
---|---|
MICA-TLA | 30.74 |
LTLBO | 22.78 |
HMPSO-SFLA | 19.06 |
MPSO | 16.05 |
MDE | 15.63 |
MSA | 14.91 |
FPA | 14.79 |
HIC-GWA | 14.34 |
MFO | 14.33 |
© 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/).
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Ben Hmida, J.; Javad Morshed, M.; Lee, J.; Chambers, T. Hybrid Imperialist Competitive and Grey Wolf Algorithm to Solve Multiobjective Optimal Power Flow with Wind and Solar Units. Energies 2018, 11, 2891. https://doi.org/10.3390/en11112891
Ben Hmida J, Javad Morshed M, Lee J, Chambers T. Hybrid Imperialist Competitive and Grey Wolf Algorithm to Solve Multiobjective Optimal Power Flow with Wind and Solar Units. Energies. 2018; 11(11):2891. https://doi.org/10.3390/en11112891
Chicago/Turabian StyleBen Hmida, Jalel, Mohammad Javad Morshed, Jim Lee, and Terrence Chambers. 2018. "Hybrid Imperialist Competitive and Grey Wolf Algorithm to Solve Multiobjective Optimal Power Flow with Wind and Solar Units" Energies 11, no. 11: 2891. https://doi.org/10.3390/en11112891