A Modified Whale Optimizer for Single- and Multi-Objective OPF Frameworks
Abstract
:1. Introduction
1.1. Motivation
1.2. Literature Review
1.3. Contribution
- This research offers a novel optimization method for dealing with the OPF problem in power systems.
- This technique ensures fast convergence and improves search capability by iteratively exploring the neighborhood of the best compromise solution in successive descents.
- Validation of the suggested approach has been employed for single-objective and multi-objective OPF optimizations.
- From the viewpoint of economic and technical benefits, the proposed multi-objective WOA achieves significant improvement in cost minimization and power loss reduction, along with enhancement of bus voltage profile.
- The robustness is demonstrated through statistical evaluations of the lowest, average, maximum, and standard deviation of the obtained results.
1.4. Paper Organization
2. Formulation of the OPF Problem
2.1. Problem Objectives
2.2. Constraints
3. Proposed Solution Methodology
3.1. Whale Optimization Algorithm
- Shrinking Encircling Prey
- Bubble-Net Attacking Method (Exploitation Phase)
- (a)
- The mechanism of shrinking encircling [57]
- (b)
- Updating of spiral position
- (c)
- Search for prey (exploration phase) [58].
3.2. Proposed Multi-Objective WOA
4. Simulation Results
4.1. Results for Single-Objective Cases
4.2. Multi-Objective OPF Simulation Results
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Nomenclature
OPF | optimal power flow | buses Ref. voltage of, 1.0 p.u. | |
generated real and active power at bus i | load buses number | ||
active and reactive power demand at bus i | L-index | voltage stability index | |
capacitive/inductive reactive power of the present VAR source connected at bus i | , | voltage phase angles of buses i and j | |
voltage magnitude at the end buses i and j | , | sub-matrices of Y-bus matrix | |
VD | voltage deviation | generators number | |
conductance of the transmission line between bus i and bus j | voltage limits at bus i | ||
number of buses | reactive power limits at bus i | ||
generated active power limits of bus i. | voltage-controlled buses number | ||
transformer k tapping variation limit | transmission lines number | ||
existing VAR sources total number | transmission line maximum power flow | ||
total number of on-load tap changing transformers | cost function value for non-dominated solution against criteria | ||
transmission line power flow | position of the selected search agent | ||
reactive power output of the VAR source at bus e. | maximum number of iterations | ||
whale leader position | cost function value of solution | ||
recent iteration | the Pareto solutions number | ||
the objective functions number | is the generator active power | ||
minimal Euclidian distances | ai, bi, ci | the generator cost coefficients | |
non-dominated cost function |
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Applied Algorithm | Utilized Cost Function | Main Outcomes | |
---|---|---|---|
SOF | MOF | ||
Particle Swarm Optimization (PSO) [7,8] | X | X | Wheeling cost minimization |
Gravitational Search Algorithm [9] | X | Less-memory utilized | |
Tabu Search (TS) [10] | X | Quadratic cost curve with sine components | |
Biogeography Based Optimizer [11] | X | Consider UPFC devices | |
Differential Evolution (DE) [12] | X | X | Three-constraint handling techniques |
Genetic Algorithms [13,14] | X | X | Prohibited zones, valve-point effect, multi-fuels, and emission. |
Teaching Learning Based Optimization (TLBO) [15] | X | Applied for small- and large-scale test systems | |
Cuckoo Search (CS) [16] | X | Consider the valve loading effect | |
Harris’ Hawk Optimization Algorithm (HHO) [17,18] | X | Consider renewable energy integration | |
Artificial Bee Colony (ABC) [19] | X | Fast convergence | |
Hunger Games Search (HGS) [20] | X | Consider renewable energy integration | |
Heap Optimization Algorithm (HOA ) [21] | X | Consider renewable energy integration with variable nature of load over 24 h | |
Salp Swarm Optimizer [22] | X | Voltage stability analysis | |
Jaya Algorithm [23,24] | X | X | Use the Fuzzy Set Theory for the best compromise solution |
Fuzzy-Based Grenade Explosion Method [25] | X | X | Handling multi-objectives effectively |
Colliding Bodies Optimization Algorithm [26] | X | Prohibited zones constraints | |
Search Optimization Algorithm [27] | X | X | Consider emission and non-smooth cost functions using |
Electromagnetism-Like Mechanism [28] | X | Applied for small-scale test systems | |
League Championship Algorithm [29] | X | Consider the valve point effect | |
Differential Search Algorithm [30] | X | Robust, and a superior solution | |
Quasi-Oppositional Teaching Learning [31] | X | X | Highly distributed non-dominant Pareto optimal solutions |
Multi-Objective Harmony Algorithm [32] | X | X | Best compromise solution extraction using fuzzy membership |
Multi-Objective Non-Dominated Sorting Opposition Gravitational Algorithm [33] | X | X | A double-objective function is considered |
Modified Shuffle Frog Leaping Algorithm [34] | X | X | Run time proportionally increases as the number of control variables increases |
Multi-Objective Bees Algorithm [35] | X | X | Utilize fuel cost, emission, and loss |
Hybrid Firefly–Bat Algorithm with Constraints-Prior Object-Fuzzy Sorting Strategy (HFBA-COFS) [36] | X | Triple objective function | |
Adaptive Parallel Seeker Optimization Algorithm (APSOA) [37] | X | Consider thyristor-controlled series compensators (TCSCs) | |
Adaptive Group Search Optimization [38] | X | X | Consider the N-1 security index |
Sine–Cosine Algorithm [39] | X | Modified algorithm | |
Multi-Objective Differential-Based Harmony Search Algorithm [40] | X | X | Effective initialization method |
Differential Search Algorithm [41] | X | X | Use weighting factor for multi-objective |
Fruit Fly Optimization Algorithm (FFOA) [42] | X | X | Use weighting factor for multi-objective |
Moth Swarm Algorithm (MSA) [43] | X | X | Use Pareto front for multi-objective |
Crow Search Algorithm [44,45] | X | X | Economic and environmental objectives |
Sine Cosine Optimizer [46] | |||
Differential Evolution [47,48,49] | X | X | Pareto front for MOF |
Bus Number | Active Power Generation (MW) | Generation Cost Coefficients | |||
---|---|---|---|---|---|
Min. | Max. | a | b | c | |
1 | 50 | 200 | 0.0 | 2.0 | 0.00375 |
2 | 20 | 800 | 0.0 | 1.75 | 0.0175 |
5 | 15 | 50 | 0.0 | 1.0 | 0.0625 |
8 | 10 | 35 | 0.0 | 3.25 | 0.00834 |
11 | 10 | 30 | 0.0 | 3.0 | 0.025 |
13 | 12 | 40 | 0.0 | 3.0 | 0.025 |
OPF Formulation Case | Quadratic Fuel Cost | Active Power Loss | Voltage Deviation | Voltage Stability Index | |
---|---|---|---|---|---|
Mono-objective | Case 1 | X | |||
Case 2 | X | ||||
Case 3 | X | ||||
Case 4 | X | ||||
Bi-objective | Case 5 | X | X | ||
Case 6 | X | X | |||
Case 7 | X | X | |||
Four-objective | Case 8 | X | X | X | X |
Case | Case 1 | Case 2 | Case 3 | Case 4 | |
---|---|---|---|---|---|
Cont. Variable | |||||
Pg1 (MW) | 175.15 | 51.42 | 176.02 | 147.7839 | |
Pg2 (MW) | 46.19 | 80.00 | 47.17 | 44.9271 | |
Pg5 (MW) | 21.69 | 50.00 | 21.66 | 28.4973 | |
Pg8 (MW) | 23.97 | 35.00 | 22.17 | 20.0099 | |
Pg11 (MW) | 11.37 | 30.00 | 13.84 | 29.7263 | |
Pg13 (MW) | 14.53 | 40.00 | 12.13 | 19.8447 | |
Vg1 (p.u.) | 1.06 | 1.10 | 1.042 | 1.0979 | |
Vg2 (p.u.) | 1.03 | 1.10 | 1.026 | 1.082 | |
Vg5 (p.u.) | 1.01 | 1.08 | 1.013 | 1.0295 | |
Vg8 (p.u.) | 1.00 | 1.09 | 1.003 | 1.0484 | |
Vg11 (p.u.) | 1.08 | 1.10 | 1.065 | 1.0661 | |
Vg13 (p.u.) | 1.00 | 1.06 | 0.990 | 1.0793 | |
Tap6–9 | 1.05 | 1.09 | 1.084 | 0.9 | |
Tap6–10 | 0.95 | 1.01 | 0.904 | 1.0973 | |
Tap4–12 | 0.92 | 1.08 | 0.940 | 0.9033 | |
Tap28–27 | 0.94 | 1.04 | 0.967 | 0.9 | |
Qc10 (MVAR) | 1.92 | 3.24 | 4.124 | 4.9316 | |
Qc12 (MVAR) | 2.44 | 1.10 | 0.532 | 3.4673 | |
Qc15 (MVAR) | 3.77 | 4.74 | 2.991 | 0 | |
Qc17 (MVAR) | 2.93 | 4.66 | 1.830 | 2.6801 | |
Qc20 (MVAR) | 1.54 | 4.43 | 4.969 | 3.734 | |
Qc21 (MVAR) | 2.09 | 4.54 | 4.677 | 0 | |
Qc23 (MVAR) | 5.00 | 5.00 | 4.928 | 0 | |
Qc24 (MVAR) | 0.00 | 4.65 | 4.952 | 0 | |
Qc29 (MVAR) | 4.68 | 5.00 | 2.215 | 0 |
Case | Case 1 | Case 2 | Case 3 | Case 4 | |
---|---|---|---|---|---|
Fitness Fun. | |||||
Ploss (MW) | 8.82 | 2.88 | 9.7153 | 8.4723 | |
VD (p.u.) | 0.56 | 2.00 | 0.1053 | 1.0417 | |
L-index | 0.14 | 0.12 | 0.1371 | 0.1175 | |
Fuel cost (USD/h) | 800.26 | 967.14 | 803.3082 | 814.8988 |
Case-1 | WOA | CSADHS [60] | MICA–TLA [61] | Case-2 | WOA | ABC [62] | EGA–DQLF [13] |
800.26 USD/h | 801.59 USD/h | 801.05 USD/h | 2.88 MW | 3.11 MW | 3.2 MW | ||
Case-3 | WOA | DE [63] | BHBO [64] | Case-4 | WOA | ABC [62] | DE [63] |
0.1053 p.u | 0.1357 | 0.1262 | 0.1175 | 0.1379 | 0.1219 |
Case | Case 5 | Case 6 | Case 7 | Case 8 | |
---|---|---|---|---|---|
Cont. Variable | |||||
Pg1 (MW) | 104.27 | 177.1531 | 77.2559 | 180.45 | |
Pg2 (MW) | 55.26 | 46.2841 | 78.5356 | 51.25 | |
Pg5 (MW) | 41.20 | 23.0439 | 42.8331 | 17.105 | |
Pg8 (MW) | 32.31 | 20.5680 | 31.7522 | 11.065 | |
Pg11 (MW) | 27.68 | 12.4723 | 20.6790 | 14.743 | |
Pg13 (MW) | 27.16 | 13.2523 | 36.5573 | 19.337 | |
Vg1 (p.u.) | 1.10 | 1.0569 | 1.0996 | 1.0663 | |
Vg2 (p.u.) | 1.09 | 1.0351 | 1.0885 | 1.0492 | |
Vg5 (p.u.) | 1.06 | 1.0046 | 1.0630 | 0.9937 | |
Vg8 (p.u.) | 1.07 | 1.0029 | 1.0581 | 1.0299 | |
Vg11 (p.u.) | 1.07 | 1.0185 | 1.0195 | 1.1000 | |
Vg13 (p.u.) | 1.04 | 1.0404 | 1.0282 | 1.0036 | |
Tap6–9 | 1.02 | 0.9906 | 1.0726 | 1.0579 | |
Tap6–10 | 1.05 | 0.9718 | 1.0475 | 1.1000 | |
Tap4–12 | 1.04 | 0.9986 | 1.0165 | 1.0083 | |
Tap28–27 | 0.98 | 0.9617 | 0.9981 | 0.9057 | |
Qc10 (MVAR) | 1.99 | 2.6224 | 4.6249 | 3.1676 | |
Qc12 (MVAR) | 3.43 | 2.7815 | 3.6859 | 5.000 | |
Qc15 (MVAR) | 2.29 | 2.3091 | 2.5605 | 4.999 | |
Qc17 (MVAR) | 2.60 | 3.3258 | 3.1081 | 4.689 | |
Qc20 (MVAR) | 4.03 | 4.6138 | 4.0475 | 5.000 | |
Qc21 (MVAR) | 4.70 | 4.2020 | 4.1126 | 5.000 | |
Qc23 (MVAR) | 2.30 | 2.5341 | 4.7037 | 4.899 | |
Qc24 (MVAR) | 4.23 | 4.1295 | 3.1570 | 2.5508 | |
Qc29 (MVAR) | 2.84 | 2.3603 | 2.1575 | 4.2789 |
Case | Case 5 | Case 6 | Case 7 | Case 8 | |
---|---|---|---|---|---|
Fittness | |||||
VD (p.u.) | 0.88 | 0.1741 | 0.4694 | 0.4985 | |
L-index | 0.13 | 0.1368 | 0.1328 | 0.1253 | |
Fuel cost (USD/h) | 862.53 | 802.5395 | 907.1819 | 805.0121 | |
Ploss (MW) | 4.47 | 9.4857 | 4.2134 | 9.199 |
Case No. | Min | Max | Standard Deviation | Average |
---|---|---|---|---|
Case 1 | 800.26 | 800.47 | 0.067411 | 800.3034 |
Case 2 | 2.88 | 2.889821 | 0.003713 | 2.88473 |
Case 3 | 0.1053 | 0.106769 | 0.000527 | 0.105965 |
Case 4 | 0.1175 | 0.117975 | 0.000153 | 0.117753 |
Algorithm | PSO | SSA | SCA | CSA | MPA | NSWOA | |
---|---|---|---|---|---|---|---|
Fitness Fun. | |||||||
VD (p.u.) | 0.4905 | 0.4880 | 0.5280 | 0.4186 | 0.5791 | 0.4985 | |
L-index | 0.1212 | 0.1229 | 0.1291 | 0.1256 | 0.1193 | 0.1253 | |
Fuel cost (USD/h) | 803.4415 | 803.0555 | 814.198 | 804.457 | 802.8512 | 805.0121 | |
Ploss(MW) | 9.5773 | 9.9199 | 9.9970 | 9.6696 | 9.2256 | 9.1620 |
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El-Dabah, M.; Ebrahim, M.A.; El-Sehiemy, R.A.; Alaas, Z.; Ramadan, M.M. A Modified Whale Optimizer for Single- and Multi-Objective OPF Frameworks. Energies 2022, 15, 2378. https://doi.org/10.3390/en15072378
El-Dabah M, Ebrahim MA, El-Sehiemy RA, Alaas Z, Ramadan MM. A Modified Whale Optimizer for Single- and Multi-Objective OPF Frameworks. Energies. 2022; 15(7):2378. https://doi.org/10.3390/en15072378
Chicago/Turabian StyleEl-Dabah, Mahmoud, Mohamed A. Ebrahim, Ragab A. El-Sehiemy, Z. Alaas, and M. M. Ramadan. 2022. "A Modified Whale Optimizer for Single- and Multi-Objective OPF Frameworks" Energies 15, no. 7: 2378. https://doi.org/10.3390/en15072378