Determining Optimal Power Flow Solutions Using New Adaptive Gaussian TLBO Method
Abstract
:1. Introduction
- A new adaptive Gaussian mutation is integrated into the conventional TLBOs to enhance its exploration and exploration capabilities.
- The proposed algorithm can resolve a wide range of OPF problems for IEEE 30-bus, 57-bus, and 118-bus standard power systems with better outcomes than the state-of-the-art metaheuristic algorithms.
- Many-objective OPF problems to reduce the loss, emission, fuel cost, and voltage deviation in 12 cases are considered and solved by the proposed method to confirm its performance and applicability for various kinds of OPF problems.
2. OPF Problem Formulation
2.1. Problem Constraints
2.1.1. Equality Constraints
2.1.2. Inequality Constraints of the OPF Problem
- Constraints relating to the generation units
- According to [14], the transformers’ taps can be adjusted as follows.
- As stated in [14], the generation of volt ampere reactive (VAR) compensating units are constrained as follows.
2.2. Constraint Handling
3. Proposed Algorithm
3.1. TLBO
3.1.1. Teaching Phase
3.1.2. Learning Phase
3.2. GTLBO Using Gaussian Mutation Strategy
3.2.1. Gaussian Distribution
3.2.2. Adaptive GTLBO
3.3. Solution Method
- Step 1: Initialize AGTLBO using the parameters from the under-study test system. The maximum number of iterations and population (Npop) for the 30-bus system have been chosen 500 and 25, respectively, 1000 and 40 for the 57-bus system, and 5000 and 60 for the 118-bus system.
- Step 2: Using Equations (21)–(24), generate the starting population or students depending on the number of populations for i = 1: Npop, the lowest and maximum values of control variables and , system constraints, and OPF constraint constraints using Equations (4)–(10).
- Step 3: Using the system and OPF problem constraints, determine the under-study objective function of the initially generated solutions based on (11) and (12) and save them as initial answers of the initial population.
- Step 4 (Adaptive Gaussian Teaching phase): Enhance the answers from the previous step based on (21), apply the system and OPF problem constraints, and calculate their under-study objective function based on (11) and (12). Then, during the subsequent iteration, choose the best answer (new student) from the current and prior solutions provided.
- Step 5: Compare the best student solution (new student) to the previous step’s greatest global solution (teacher). If the new student solution is superior to the best global (teacher) solution, the former takes precedence; otherwise, the old (teacher) solution is retained in the algorithm memory.
- Step 6 (Adaptive Gaussian Learning phase): Based on (24) and the system and optimal OPF problem constraints, enhance the previous step solutions and construct their under-study objective function based on (11) and (12). Then, during the subsequent iteration, choose the student’s answer from among the current and previously generated solutions.
- Step 7: Compare the student solution to the previous step’s teacher solution. If the student solution is superior to the teacher solution, the student solution takes precedence over the teacher solution; otherwise, the prior student solution is stored in the algorithm memory.
- Step 8: Has the algorithm termination requirement been met? Proceed to Step 4 and continue the optimization process if yes; if not, proceed to Step 3.
4. Numerical Results
4.1. Cases Definition
4.1.1. The Studied Cases for the IEEE 30-Bus System
- Minimizing the fuel cost (Case 1).
- Minimizing piecewise quadratic fuel cost functions (Case 2).
- Minimizing the emission (Case 3).
- Minimizing the actual power loss (Case 4).
- Minimizing the fuel cost considering valve point effect (VPE) (Case 5).
- Minimizing the fuel cost and actual power loss (Case 6).
- Minimizing the fuel cost and voltage deviation (Case 7).
- Minimizing the fuel cost, emissions, voltage deviation and losses (Case 8).
4.1.2. The Studied Cases for the IEEE 57-Bus System
- Minimizing the fuel cost (Case 9).
- Minimizing the fuel cost while improving voltage profile (Case 10).
- Minimizing the fuel cost, emissions, voltage deviation and losses (Case 11).
4.1.3. The Studied Case for the IEEE 118-Bus System
- Minimizing the fuel cost (Case 12).
4.2. EEE 30-Bus Test System
4.2.1. Case 1: Minimizing the Fuel Cost
4.2.2. Case 2: Minimizing the Total Fuel Cost Considering Multi-Fuel Sources
4.2.3. Case 3: Minimizing the Emission
4.2.4. Case 4: Minimizing the Real Power Loss
4.2.5. Case 5: Minimizing the Fuel Cost Considering Valve-Point Effects (VPE)
4.2.6. Case 6: Minimizing the Fuel Cost and Real Power Loss
4.2.7. Case 7: Minimizing the Fuel Cost and Voltage Deviation
4.2.8. Case 8: Minimizing the Fuel Cost, Emission, Voltage Deviation and Losses
4.3. IEEE 57-Bus Test System
4.3.1. Case 9: Minimizing the Fuel Cost
4.3.2. Case 10: Minimizing the Fuel Cost While Improving Voltage Profile
4.3.3. Case 11: Minimizing the Fuel Cost, Emission, Voltage Deviation and Losses
4.4. Case 12: IEEE 118-Bus System
4.5. Discussion on Results
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Control Variables Settings | Limits | Case 1 | Case 2 | Case 3 | Case 4 | Case 5 | Case 6 | Case 7 | Case 8 | |
---|---|---|---|---|---|---|---|---|---|---|
Lower | Upper | |||||||||
PG1 (MW) 1 | 50 | 250 | 177.1160 | 139.9996 | 64.0640 | 51.4932 | 198.7677 | 102.6327 | 176.3251 | 122.2170 |
PG2 (MW) | 20 | 80 | 48.7445 | 54.9999 | 67.5593 | 79.9983 | 44.7547 | 55.5362 | 49.0930 | 52.5169 |
PG5 (MW) | 15 | 50 | 21.3225 | 24.1810 | 50.0000 | 49.9997 | 18.5617 | 38.1125 | 21.5587 | 31.4731 |
PG8 (MW) | 10 | 35 | 21.2933 | 34.9987 | 35.0000 | 34.9997 | 10.0000 | 35.0000 | 22.1467 | 35.0000 |
PG11 (MW) | 10 | 30 | 11.9442 | 18.3761 | 30.0000 | 29.9999 | 10.0010 | 30.0000 | 12.1467 | 26.7432 |
PG13 (MW) | 12 | 40 | 12.0017 | 17.5743 | 40.0000 | 39.9998 | 12.0012 | 26.6486 | 12.0016 | 21.0381 |
VG1 (p.u.) | 0.95 | 1.1 | 1.0835 | 1.0756 | 1.0628 | 1.0618 | 1.0810 | 1.0697 | 1.0427 | 1.0729 |
VG2 (p.u.) | 0.95 | 1.1 | 1.0604 | 1.0583 | 1.0562 | 1.0571 | 1.0577 | 1.0575 | 1.0230 | 1.0573 |
VG5 (p.u.) | 0.95 | 1.1 | 1.0334 | 1.0328 | 1.0375 | 1.0382 | 1.0302 | 1.0357 | 1.0151 | 1.0326 |
VG8 (p.u.) | 0.95 | 1.1 | 1.0383 | 1.0414 | 1.0439 | 1.0445 | 1.0368 | 1.0437 | 1.0065 | 1.0411 |
VG11 (p.u.) | 0.95 | 1.1 | 1.0923 | 1.0937 | 1.0806 | 1.0846 | 1.0968 | 1.0982 | 1.0595 | 1.0401 |
VG13 (p.u.) | 0.95 | 1.1 | 1.0502 | 1.0563 | 1.0542 | 1.0542 | 1.0606 | 1.0576 | 0.9879 | 1.0235 |
T6–9 (p.u.) | 0.9 | 1.1 | 1.0669 | 1.0886 | 1.0457 | 1.0237 | 1.0810 | 1.0347 | 1.0830 | 1.0999 |
T6–10 (p.u.) | 0.9 | 1.1 | 0.9242 | 0.9000 | 0.9403 | 0.9734 | 0.9295 | 0.9498 | 0.9000 | 0.9533 |
T4–12 (p.u.) | 0.9 | 1.1 | 0.9783 | 0.9853 | 0.9985 | 0.9975 | 0.9980 | 0.9911 | 0.9397 | 1.0322 |
T28–27 (p.u.) | 0.9 | 1.1 | 0.9737 | 0.9776 | 0.9760 | 0.9766 | 0.9770 | 0.9749 | 0.9708 | 1.0050 |
QC10 (MVAR) | 0.0 | 5.0 | 4.2404 | 3.6102 | 4.3278 | 4.5447 | 5.0000 | 0.7485 | 5.0000 | 3.5672 |
QC12 (MVAR) | 0.0 | 5.0 | 2.8095 | 0.0895 | 5.0000 | 4.5668 | 4.6104 | 0.3659 | 0.2788 | 0.3126 |
QC15 (MVAR) | 0.0 | 5.0 | 4.3275 | 4.8197 | 4.6281 | 4.6210 | 3.8474 | 4.4830 | 4.9929 | 3.8845 |
QC17 (MVAR) | 0.0 | 5.0 | 5.0000 | 5.0000 | 4.9898 | 4.9974 | 4.8606 | 5.0000 | 0.0050 | 4.9997 |
QC20 (MVAR) | 0.0 | 5.0 | 5.0000 | 4.0709 | 4.1275 | 4.2175 | 4.8219 | 4.2613 | 4.9988 | 4.9991 |
QC21 (MVAR) | 0.0 | 5.0 | 4.9972 | 4.9595 | 5.0000 | 4.9995 | 4.9761 | 5.0000 | 4.9407 | 5.0000 |
QC23 (MVAR) | 0.0 | 5.0 | 3.1538 | 2.9699 | 3.2253 | 3.2410 | 3.5700 | 3.2503 | 4.9995 | 4.3039 |
QC24 (MVAR) | 0.0 | 5.0 | 5.0000 | 5.0000 | 4.9983 | 4.9998 | 4.9973 | 5.0000 | 4.9993 | 5.0000 |
QC29 (MVAR) | 0.0 | 5.0 | 2.7696 | 2.6055 | 2.5183 | 2.5435 | 2.4718 | 2.5520 | 2.6307 | 2.6200 |
Cost ($/h) | - | - | 800.4811 | 646.4511 | 944.3385 | 967.6336 | 832.1624 | 858.9928 | 803.7385 | 830.1559 |
Emission (t/h) | - | - | 0.3662 | 0.2835 | 0.2048 | 0.2073 | 0.4379 | 0.2290 | 0.3639 | 0.2529 |
Power losses (MW) | - | - | 9.0222 | 6.7296 | 3.2233 | 3.0906 | 10.6863 | 4.5300 | 9.8718 | 5.5823 |
V.D. (p.u.) | - | - | 0.9109 | 0.9046 | 0.9017 | 0.9086 | 0.8592 | 0.9428 | 0.0947 | 0.2975 |
Algorithm | Emission (t/h) | Fuel Cost (USD/h) | V.D. (p.u.) | Power Losses (MW) |
---|---|---|---|---|
AGTLBO | 0.3662 | 800.481 | 0.9109 | 9.0222 |
TLBO | 0.3668 | 800.674 | 0.9120 | 9.0198 |
TS (Tabu Search) [5] | - | 802.290 | - | - |
ABC (Artificial Bee Colony) [12] | 0.3651 | 800.660 | 0.9209 | 9.0328 |
AGSO (Adaptive Group Search Optimization) [14] | 0.3703 | 801.750 | - | - |
MHBMO (Modified Honey Bee Mating Optimization) [15] | - | 801.985 | - | 9.4900 |
MSA (Moth Swarm Algorithm) [16] | 0.3665 | 800.510 | 0.9036 | 9.0345 |
FPA (Flower Pollination Algorithm) [16] | 0.3596 | 802.798 | 0.3679 | 9.5406 |
MFO (Moth-Flame Optimization) [16] | 0.3685 | 800.686 | 0.7577 | 9.1492 |
MPSO-SFLA (Modified Particle Swarm Optimization and Shuffle Frog Leaping Algorithm) [21] | - | 801.750 | - | 9.5400 |
GWO (Grey Wolf Optimizer) [25] | - | 801.410 | - | 9.3000 |
MICA-TLA (Modified Imperialist Competitive Algorithm and Teaching Learning Algorithm) [45] | - | 801.049 | - | 9.1895 |
DE (Differential Evolution) [49] | - | 802.390 | - | 9.4660 |
SKH (Stud Krill Herd Algorithm) [52] | 0.3662 | 800.514 | - | 9.0282 |
SFLA-SA (Shuffle Frog Leaping Algorithm and Simulated Annealing) [53] | - | 801.790 | - | - |
ARCBBO (Adaptive Real Coded Biogeography-Based Optimization) [51] | 0.3663 | 800.516 | 0.8867 | 9.0255 |
MGBICA (Modified Gaussian Bare-Bones Imperialist Competitive Algorithm) [54] | 0.3296 | 801.141 | - | - |
IEP (Improved Evolutionary Programming) [50] | - | 802.460 | - | - |
PSOGSA (Particle Swarm Optimization and Gravitational Search Algorithm) [55] | - | 800.499 | 0.1267 | 9.0339 |
EP (Evolutionary Programming) [56] | - | 803.570 | - | - |
Algorithm | Emission (t/h) | Fuel Cost (USD/h) | V.D. (p.u.) | Power Losses (MW) |
---|---|---|---|---|
AGTLBO | 0.2835 | 646.4511 | 0.9046 | 6.7296 |
TLBO | 0.2835 | 647.1344 | 0.8927 | 6.8409 |
LTLBO (Lévy Teaching–Learning-Based Optimization) [28] | 0.2835 | 647.4315 | 0.8896 | 6.9347 |
MDE (Modified Differential Evolution) [49] | - | 647.8460 | - | 7.095 |
MFO (Moth-Flame Optimization) [16] | 0.28336 | 649.2727 | 0.47024 | 7.2293 |
IEP (Improved Evolutionary Programming) [50] | - | 649.3120 | - | - |
MICA-TLA (Modified Imperialist Competitive Algorithm and Teaching Learning Algorithm) [45] | - | 647.1002 | - | 6.8945 |
FPA (Flower Pollination Algorithm) [16] | 0.28083 | 651.3768 | 0.31259 | 7.2355 |
SSO (Social Spider Optimization) [37] | - | 663.3518 | - | - |
GABC (Gbest Guided Artificial Bee Colony Algorithm) [57] | - | 647.0300 | 0.8010 | 6.8160 |
MSA (Moth Swarm Algorithm) [16] | 0.28352 | 646.8364 | 0.84479 | 6.8001 |
MPSO-SFLA (Modified Particle Swarm Optimization and Shuffle Frog Leaping Algorithm) [21] | - | 647.5500 | - | - |
Algorithm | Emission (t/h) | Fuel Cost (USD/h) | V.D. (p.u.) | Power Losses (MW) |
---|---|---|---|---|
AGTLBO | 0.2048 | 944.3385 | 0.9017 | 3.2233 |
TLBO | 0.2048 | 944.5446 | 0.7979 | 3.2944 |
ARCBBO (Adaptive Real Coded Biogeography-Based Optimization) [51] | 0.2048 | 945.1597 | 0.8647 | 3.2624 |
ABC (Artificial Bee Colony) [12] | 0.2048 | 944.4391 | 0.8463 | 3.2470 |
GBICA (Gaussian Bare-Bones Imperialist Competitive Algorithm) [54] | 0.2049 | 944.6516 | - | - |
FPA (Flower Pollination Algorithm) [16] | 0.2052 | 948.9490 | 0.4276 | 4.4920 |
AGSO (Adaptive Group Search Optimization) [14] | 0.2059 | 953.6290 | - | - |
DSA (Differential Search Algorithm) [22] | 0.2058 | 944.4086 | - | 3.2437 |
MFO (Moth-Flame Optimization) [16] | 0.2049 | 945.4553 | 0.7097 | 3.4295 |
MSA (Moth Swarm Algorithm) [16] | 0.2048 | 944.5003 | 0.8739 | 3.2358 |
MSFLA (Modified Shuffle Frog Leaping Algorithm) [53] | 0.2056 | - | - | - |
MPSO-SFLA (Modified Particle Swarm Optimization and Shuffle Frog Leaping Algorithm) [21] | 0.2052 | - | - | - |
Algorithm | Emission (t/h) | Fuel Cost (USD/h) | V.D. (p.u.) | Power Losses (MW) |
---|---|---|---|---|
AGTLBO | 0.2073 | 967.6336 | 0.9086 | 3.0906 |
TLBO | 0.2072 | 967.7195 | 0.8821 | 3.1104 |
ABC (Artificial Bee Colony) [12] | 0.2073 | 967.6810 | 0.9008 | 3.1078 |
FPA (Flower Pollination Algorithm) [16] | 0.2076 | 967.1138 | 0.3893 | 3.5661 |
MFO (Moth-Flame Optimization) [16] | 0.2073 | 967.6785 | 0.9156 | 3.1111 |
MSA (Moth Swarm Algorithm) [16] | 0.2073 | 967.6636 | 0.8887 | 3.1005 |
DSA (Differential Search Algorithm) [22] | 0.2083 | 967.6493 | - | 3.0945 |
GWO (Grey Wolf Optimizer) [25] | - | 968.3800 | - | 3.4100 |
Jaya [32] | - | 967.6827 | 0.1272 | 3.1035 |
ARCBBO (Adaptive Real Coded Biogeography-Based Optimization) [51] | 0.2073 | 967.6605 | 0.8913 | 3.1009 |
EEA (Efficient Evolutionary Algorithm) [58] | - | 952.3785 | - | 3.2823 |
EGA (Enhanced Genetic Algorithm) [58] | - | 967.9300 | - | 3.2440 |
EGA-DQLF (Enhanced Genetic Algorithm to Decoupled Quadratic Load Flow) [59] | - | 967.8600 | 0.1218 | 3.2008 |
ALC-PSO (Particle Swarm Optimization with an Aging Leader and Challengers) [60] | - | 967.7683 | - | 3.1700 |
Algorithm | Emission (t/h) | Fuel Cost (USD/h) | V.D. (p.u.) | Power Losses (MW) |
---|---|---|---|---|
AGTLBO | 0.4379 | 832.1624 | 0.8592 | 10.6863 |
TLBO | 0.4382 | 832.5325 | 0.8654 | 10.6870 |
PSO (Particle Swarm Optimization) [11] | - | 832.6871 | - | - |
SP-DE (Differential Evolution Integrated Self-Adaptive Penalty) [61] | 0.43651 | 832.4813 | 0.7504 | 10.6762 |
Algorithm | Emission (t/h) | Fuel Cost (USD/h) | V.D. (p.u.) | Power Losses (MW) |
---|---|---|---|---|
AGTLBO | 0.2290 | 858.9928 | 0.9428 | 4.5300 |
TLBO | 0.2307 | 859.0191 | 0.9152 | 4.5687 |
MSA (Moth Swarm Algorithm) [16] | 0.2289 | 859.1915 | 0.92852 | 4.5404 |
SF-DE (Differential Evolution Integrated Superiority of Feasible Solutions) [61] | 0.2289 | 859.1458 | 0.92731 | 4.5245 |
Algorithm | Emission (t/h) | Fuel Cost (USD/h) | V.D. (p.u.) | Power Losses (MW) |
---|---|---|---|---|
AGTLBO | 0.3639 | 803.7385 | 0.0947 | 9.8718 |
TLBO | 0.3565 | 804.2391 | 0.1056 | 9.9565 |
MPSO (Modified Particle Swarm Optimization) [16] | 0.3636 | 803.9787 | 0.1202 | 9.9242 |
MFO (Moth-Flame Optimization) [16] | 0.36355 | 803.7911 | 0.10563 | 9.8685 |
MOICA (Multi-Objective Imperialist Competitive Algorithm) [62] | - | 805.0345 | 0.1004 | - |
NKEA (Neighborhood Knowledge-based Evolutionary Algorithm) [62] | - | 804.9612 | 0.099 | - |
BB-MOPSO (Bare-Bones Modified Particle Swarm Optimization) [62] | - | 804.9639 | 0.1021 | - |
MNSGA-II (Modified Non-dominated Sorting GA-II) [62] | - | 805.0076 | 0.0989 | - |
MOMICA (Multi-Objective Modified Imperialist Competitive Algorithm) [62] | 0.3552 | 804.9611 | 0.0952 | 9.8212 |
Algorithm | Emission (t/h) | Fuel Cost (USD/h) | V.D. (p.u.) | Power Losses (MW) |
---|---|---|---|---|
AGTLBO | 0.2529 | 830.1559 | 0.2975 | 5.5823 |
TLBO | 0.2538 | 831.3186 | 0.2982 | 5.5847 |
MFO (Moth-Flame Optimization) [16] | 0.25231 | 830.9135 | 0.33164 | 5.5971 |
MSA (Moth Swarm Algorithm) [16] | 0.25258 | 830.639 | 0.29385 | 5.6219 |
MOICA (Multi-Objective Imperialist Competitive Algorithm) [62] | 0.267 | 831.2251 | 0.4046 | 6.0223 |
NKEA (Neighborhood Knowledge-based Evolutionary Algorithm) [62] | 0.2491 | 834.6433 | 0.4448 | 5.8935 |
BB-MOPSO (Bare-Bones Modified Particle Swarm Optimization) [62] | 0.2479 | 833.0345 | 0.3945 | 5.6504 |
MNSGA-II (Modified Non-dominated Sorting GA-II) [62] | 0.2527 | 834.5616 | 0.4308 | 5.6606 |
MOMICA (Multi-Objective Modified Imperialist Competitive Algorithm) [62] | 0.2523 | 830.1884 | 0.2978 | 5.5851 |
Control Variables Settings | Algorithms | |||||
---|---|---|---|---|---|---|
EADDE (Evolving Ant Direction Differential Evolution) [63] | GSA (Gravitational Search Algorithm) [64] | ABC (Artificial Bee Colony) [12] | MICA (Modified Imperialist Competitive Algorithm) [45] | LTLBO (Lévy Teaching–Learning-Based Optimization) [28] | AGTLBO | |
PG1 (MW) | 143.150 | 142.369 | 142.811 | 142.520 | 142.870 | 143.910 |
PG2 (MW) | 95.290 | 92.630 | 90.033 | 88.001 | 88.953 | 92.021 |
PG3 (MW) | 45.320 | 45.318 | 44.515 | 44.596 | 44.788 | 45.324 |
PG6(MW) | 73.600 | 72.355 | 74.200 | 72.086 | 72.577 | 70.000 |
PG8 (MW) | 464.850 | 464.743 | 454.848 | 460.325 | 460.906 | 458.585 |
PG9 (MW) | 83.440 | 84.999 | 96.885 | 98.354 | 94.179 | 96.025 |
PG12 (MW) | 361.240 | 363.951 | 362.772 | 360.179 | 361.687 | 360.000 |
VG1 (p.u.) | 1.050 | 1.059 | 1.042 | 1.033 | 1.038 | 1.060 |
VG2(p.u.) | 1.048 | 1.058 | 1.041 | 1.037 | 1.042 | 1.059 |
VG3 (p.u.) | 1.041 | 1.060 | 1.039 | 1.028 | 1.031 | 1.055 |
VG6 (p.u.) | 1.049 | 1.060 | 1.055 | 1.044 | 1.047 | 1.060 |
VG8 (p.u.) | 1.056 | 1.060 | 1.064 | 1.060 | 1.060 | 1.060 |
VG9 (p.u.) | 1.034 | 1.060 | 1.037 | 1.027 | 1.028 | 1.039 |
VG12 (p.u.) | 1.041 | 1.046 | 1.041 | 1.019 | 1.022 | 1.044 |
T4–18 (p.u.) | 1.051 | 0.900 | 0.938 | 0.961 | 0.901 | 1.012 |
T4–18 (p.u.) | 0.907 | 0.900 | 1.050 | 1.004 | 1.080 | 0.966 |
T21–20 (p.u.) | 1.038 | 0.909 | 0.975 | 1.021 | 1.019 | 1.011 |
T24–25 (p.u.) | 1.004 | 1.059 | 0.950 | 1.038 | 1.034 | 1.100 |
T24–25 (p.u.) | 0.962 | 0.999 | 1.013 | 0.972 | 1.025 | 0.970 |
T24–26 (p.u.) | 0.986 | 0.922 | 1.000 | 1.029 | 1.026 | 1.026 |
T7–29 (p.u.) | 0.984 | 0.932 | 1.013 | 0.983 | 0.984 | 0.990 |
T34–32 (p.u.) | 0.908 | 1.088 | 0.913 | 0.963 | 0.971 | 0.964 |
T11–41 (p.u.) | 0.923 | 1.039 | 0.900 | 0.933 | 0.900 | 0.906 |
T15–45 (p.u.) | 0.991 | 1.043 | 1.013 | 0.951 | 0.954 | 0.976 |
T14–46 (p.u.) | 0.982 | 1.025 | 0.988 | 0.935 | 0.940 | 0.966 |
T10–51 (p.u.) | 0.989 | 0.954 | 1.000 | 0.949 | 0.950 | 0.967 |
T13–49 (p.u.) | 0.966 | 0.929 | 0.963 | 0.904 | 0.912 | 0.933 |
T11–43 (p.u.) | 0.972 | 1.099 | 0.963 | 0.941 | 0.948 | 0.965 |
T40–56 (p.u.) | 0.997 | 0.969 | 0.963 | 1.017 | 0.993 | 0.990 |
T39–57 (p.u.) | 1.002 | 1.062 | 0.925 | 0.985 | 0.970 | 0.947 |
T9–55 (p.u.) | 1.045 | 1.094 | 0.988 | 0.974 | 0.972 | 0.996 |
QC18 (MVAR) | 9.030 | 15.243 | 16.000 | 18.660 | 17.590 | 8.950 |
QC25 (MVAR) | 8.170 | 14.403 | 15.000 | 13.620 | 17.410 | 18.110 |
QC53 (MVAR) | 20.130 | 15.102 | 14.000 | 14.310 | 15.080 | 15.410 |
Cost ($/h) | 41,713.620 | 41,695.872 | 41,693.959 | 41,683.048 | 41,679.545 | 41,678.310 |
Emission (t/h) | - | - | - | - | - | 1.901 |
V.D. (p.u.) | 1.098 | - | - | 1.449 | - | 1.618 |
Power losses (MW) | 16.090 | - | - | 15.266 | 15.159 | 15.064 |
Algorithm | Emission (t/h) | Fuel Cost (USD/h) | V.D. (p.u.) | Power Losses (MW) |
---|---|---|---|---|
AGTLBO | 1.901 | 41,678.310 | 1.618 | 15.064 |
TLBO | 1.928 | 41,679.891 | 1.651 | 15.153 |
ICBO (Improved Colliding Bodies Optimization) [11] | - | 41,697.330 | 1.317 | 15.547 |
MPSO (Modified Particle Swarm Optimization) [16] | 1.944 | 41,678.676 | 1.340 | 15.127 |
MFO (Moth-Flame Optimization) [16] | 2.004 | 41,686.412 | 1.294 | 15.611 |
DSA (Differential Search Algorithm) [22] | - | 41,686.820 | 1.083 | - |
LTLBO (Lévy Teaching–Learning-Based Optimization) [28] | - | 41,679.545 | - | 15.159 |
MICA (Modified Imperialist Competitive Algorithm) [45] | - | 41,683.048 | 1.449 | 15.266 |
ARCBBO (Adaptive Real Coded Biogeography-Based Optimization) [51] | - | 41,686.000 | - | 15.377 |
EADDE (Evolving Ant Direction Differential Evolution) [63] | - | 41,713.620 | 1.098 | 16.090 |
Algorithm | Emission (t/h) | Fuel Cost (USD/h) | V.D. (p.u.) | Power Losses (MW) |
---|---|---|---|---|
AGTLBO | 1.92 | 41,707.97 | 0.71 | 15.79 |
TLBO | 2.00 | 41,715.20 | 0.71 | 16.17 |
MPSO (Modified Particle Swarm Optimization) [16] | 2.01 | 41,721.61 | 0.68 | 16.25 |
MFO (Moth-Flame Optimization) [16] | 2.01 | 41,718.87 | 0.68 | 16.22 |
MSA (Moth Swarm Algorithm) [16] | 1.96 | 41,714.98 | 0.68 | 15.92 |
EADDE (Evolving Ant Direction Differential Evolution) [63] | - | 42,051.44 | 0.79 | 19.32 |
MICA (Modified Imperialist Competitive Algorithm) [45] | - | 41,974.43 | 0.54 | 20.30 |
MICA-TLA (Modified Imperialist Competitive Algorithm and Teaching Learning Algorithm) [45] | - | 41,959.18 | 0.54 | 19.91 |
Control Variables Settings | Algorithms | |||
---|---|---|---|---|
EADDE (Evolving Ant Direction Differential Evolution) [63] | MICA (Modified Imperialist Competitive Algorithm) [45] | MICA-TLA (Modified Imperialist Competitive Algorithm and Teaching Learning Algorithm) [45] | AGTLBO | |
PG1 (MW) | 153.820 | 163.830 | 152.900 | 142.857 |
PG2 (MW) | 83.670 | 99.993 | 99.805 | 88.694 |
PG3 (MW) | 71.560 | 42.111 | 42.718 | 45.094 |
PG6(MW) | 54.790 | 29.671 | 37.046 | 71.816 |
PG8 (MW) | 506.750 | 467.689 | 474.497 | 459.858 |
PG9 (MW) | 79.930 | 100.000 | 77.787 | 96.653 |
PG12 (MW) | 319.600 | 367.810 | 385.955 | 361.617 |
VG1 (p.u.) | 1.0058 | 0.9923 | 0.9908 | 1.0224 |
VG2(p.u.) | 1.0026 | 0.9997 | 1.0020 | 1.0202 |
VG3 (p.u.) | 1.0108 | 1.0232 | 1.0191 | 1.0140 |
VG6 (p.u.) | 1.0370 | 0.9998 | 1.0000 | 1.0282 |
VG8 (p.u.) | 1.0619 | 1.0038 | 1.0024 | 1.0477 |
VG9 (p.u.) | 1.0248 | 1.0139 | 1.0165 | 1.0152 |
VG12 (p.u.) | 1.0208 | 1.0399 | 1.0421 | 1.0093 |
T4–18 (p.u.) | 0.9840 | 1.1000 | 0.9570 | 1.0141 |
T4–18 (p.u.) | 1.0167 | 0.9270 | 1.0083 | 0.9424 |
T21–20 (p.u.) | 1.0047 | 0.9754 | 0.9725 | 0.9949 |
T24–25 (p.u.) | 1.0205 | 1.0355 | 1.0700 | 0.9914 |
T24–25 (p.u.) | 1.0219 | 1.0857 | 1.0618 | 1.0112 |
T24–26 (p.u.) | 1.0081 | 0.9958 | 0.9957 | 1.0256 |
T7–29 (p.u.) | 1.0077 | 0.9955 | 0.9992 | 1.0016 |
T34–32 (p.u.) | 0.9276 | 0.9121 | 0.9194 | 0.9397 |
T11–41 (p.u.) | 0.9007 | 0.9000 | 0.9000 | 0.9000 |
T15–45 (p.u.) | 0.9332 | 0.9073 | 0.9118 | 0.9553 |
T14–46 (p.u.) | 0.9540 | 0.9924 | 0.9958 | 0.9566 |
T10–51 (p.u.) | 1.0014 | 1.0223 | 1.0259 | 0.9714 |
T13–49 (p.u.) | 0.9499 | 0.9000 | 0.9000 | 0.9275 |
T11–43 (p.u.) | 0.9893 | 0.9912 | 0.9595 | 0.9506 |
T40–56 (p.u.) | 0.9002 | 0.9648 | 1.0272 | 0.9934 |
T39–57 (p.u.) | 1.0252 | 0.9000 | 0.9000 | 0.9427 |
T9–55 (p.u.) | 1.0062 | 1.0162 | 1.0185 | 0.9975 |
QC18 (MVAR) | 19.6200 | 4.7000 | 0.0000 | 5.0604 |
QC25 (MVAR) | 19.1500 | 19.7700 | 20.7800 | 16.7240 |
QC53 (MVAR) | 11.0900 | 29.1100 | 30.0000 | 15.2132 |
Cost ($/h) | 42,051.44 | 41,974.4346 | 41,959.1774 | 41,707.9679 |
Emission (t/h) | - | - | - | 1.9194 |
V.D. (p.u.) | 0.7882 | 0.5416 | 0.539 | 15.7901 |
Power losses (MW) | 19.32 | 20.303 | 19.909 | 0.7099 |
Algorithm | Emission (t/h) | Fuel Cost (USD/h) | V.D. (p.u.) | Power Losses (MW) |
---|---|---|---|---|
AGTLBO | 1.4328 | 41,929.387 | 0.9526 | 13.2563 |
TLBO | 1.4357 | 41,932.013 | 0.9528 | 13.6420 |
MOMICA (Multi-Objective Modified Imperialist Competitive Algorithm) [62] | 1.496 | 41,983.059 | 0.7970 | 13.6969 |
MOICA (Multi-Objective Imperialist Competitive Algorithm) [62] | 1.7605 | 41,998.566 | 0.8748 | 13.3353 |
MNSGA-II (Modified Non-dominated Sorting GA-II) [62] | 1.4965 | 42,070.825 | 0.8896 | 14.4557 |
BB-MOPSO (Bare-Bones Modified Particle Swarm Optimization) [62] | 1.5336 | 41,994.019 | 1.0742 | 12.6090 |
NKEA (Neighborhood Knowledge-based Evolutionary Algorithm) [62] | 1.5174 | 42,065.996 | 1.0420 | 13.9764 |
Control Variables | AGTLBO | Control Variables | AGTLBO |
---|---|---|---|
PG1 (MW) | 149.2946 | T24–25 (p.u.) | 0.9861 |
PG3 (MW) | 51.8000 | T24–26 (p.u.) | 1.0194 |
PG2 (MW) | 100.0000 | T24–25 (p.u.) | 1.0999 |
PG6(MW) | 100.0000 | T7–29 (p.u.) | 1.0262 |
PG9 (MW) | 100.0000 | T11–41 (p.u.) | 1.0999 |
PG8 (MW) | 378.9279 | T34–32 (p.u.) | 0.9220 |
PG12 (MW) | 384.0338 | T15–45 (p.u.) | 0.9771 |
VG2(p.u.) | 1.0559 | T10–51 (p.u.) | 1.0018 |
VG1 (p.u.) | 1.0599 | T14–46 (p.u.) | 0.9731 |
VG3 (p.u.) | 1.0420 | T13–49 (p.u.) | 0.9265 |
VG6 (p.u.) | 1.0487 | T11–43 (p.u.) | 0.9330 |
VG9 (p.u.) | 1.0321 | T39–57 (p.u.) | 0.9085 |
VG8 (p.u.) | 1.0600 | T40–56 (p.u.) | 0.9700 |
VG12 (p.u.) | 1.0237 | T9–55 (p.u.) | 1.0140 |
T4–18 (p.u.) | 1.0298 | QC18 (MVAR) | 3.2131 |
T4–18 (p.u.) | 0.9801 | QC25 (MVAR) | 19.4187 |
T21–20 (p.u.) | 0.9831 | QC53 (MVAR) | 18.2685 |
Actual power output of generators (MW) | |||||||||
PG1~PG9 | 24.200 | 0.000 | 0.010 | 0.000 | 403.000 | 85.730 | 20.000 | 11.000 | 20.100 |
PG10~PG18 | 0.030 | 196.000 | 281.006 | 10.910 | 7.150 | 16.000 | 0.060 | 5.000 | 48.410 |
PG19~PG27 | 41.900 | 19.000 | 194.020 | 49.305 | 30.900 | 32.400 | 149.991 | 148.291 | 0.000 |
PG28~PG36 | 354.488 | 350.901 | 458.200 | 0.000 | 0.010 | 0.000 | 15.873 | 19.617 | 0.000 |
PG3~PG45 | 431.986 | 0.000 | 3.599 | 507.001 | 0.000 | 0.000 | 0.000 | 0.000 | 233.400 |
PG46~PG54 | 38.090 | 0.010 | 4.101 | 29.008 | 6.005 | 35.100 | 36.400 | 0.010 | 0.000 |
Voltage magnitude of generators (p.u.) | |||||||||
VG1~VG9 | 1.017 | 1.045 | 1.037 | 1.080 | 1.100 | 1.033 | 1.032 | 1.036 | 1.030 |
VG10~VG18 | 1.067 | 1.093 | 1.099 | 1.056 | 1.044 | 1.051 | 1.034 | 1.028 | 1.017 |
VG19~VG27 | 1.025 | 1.044 | 1.058 | 1.033 | 1.032 | 1.033 | 1.052 | 1.060 | 1.060 |
VG28~VG36 | 1.068 | 1.073 | 1.081 | 1.053 | 1.057 | 1.051 | 1.043 | 1.028 | 1.060 |
VG3~VG45 | 1.071 | 1.077 | 1.074 | 1.095 | 1.072 | 1.071 | 1.075 | 1.059 | 1.062 |
VG46~VG54 | 1.049 | 1.036 | 1.032 | 1.026 | 1.031 | 1.038 | 1.023 | 1.047 | 1.063 |
Transformers’ tap (p.u.) | |||||||||
T1~T9 | 1.040 | 1.050 | 0.971 | 0.977 | 1.000 | 1.010 | 0.976 | 0.970 | 0.980 |
VAR compensating units (MVAR) | |||||||||
QC1~QC9 | 29.900 | 0.000 | 0.000 | 2.000 | 19.669 | 6.000 | 9.000 | 27.991 | 29.900 |
QC10~QC14 | 29.9995 | 9.0001 | 29.9996 | 1.000 | 11.0050 | Cost (USD/h) | 129,543.56 | PLoss (MW) | 76.2098 |
Optimizer | Min | Mean | Max | Std. | Time (s) |
---|---|---|---|---|---|
AGTLBO | 129,543.6 | 129,551.9 | 129,562.4 | 8.5 | 737.2 |
CS-GWO (Crisscross Search Based Grey Wolf Optimizer) [65] | 129,544.0 | 129,558.9 | 129,568.8 | 10.7 | 4252.5 |
MSA (Moth Swarm Algorithm) [16] | 129,640.7 | - | - | - | - |
FPA (Flower Pollination Algorithm) [16] | 129,688.7 | - | - | - | - |
MFO (Moth-Flame Optimization) [16] | 129,708.1 | - | - | - | - |
PSOGSA (Particle Swarm Optimization and Gravitational Search Algorithm) [44] | 129,733.6 | - | - | - | - |
IABC (Improved Artificial Bee Colony Optimization) [66] | 129,862.0 | 129,895.0 | - | 40.8 | 4157.8 |
MCSA (Modified Crow Search Optimizer) [67] | 129,873.6 | - | - | - | - |
MRao-2 (Modified Rao-2 Algorithm) [68] | 131,457.8 | - | - | - | 1160.3 |
Rao-2 [68] | 131,490.7 | - | - | - | 804.6 |
Rao-3 [68] | 131,793.1 | - | - | - | 806.7 |
Rao-1 [68] | 131,817.9 | - | - | - | 808.0 |
SSO (Social Spider Optimization) [68] | 132,080.4 | - | - | - | - |
FHSA (Fuzzy Harmony Search Algorithm) [001] | 132,138.3 | 132,138.3 | 132,138.3 | 0.0 | - |
ICBO (Improved Colliding Bodies Optimization) [11] | 135,121.6 | - | - | - | - |
GWO (Grey Wolf Optimizer) [25] | 139,948.1 | 142,989.3 | 145,484.6 | 797.8 | 1766.2 |
EWOA (Effective Whale Optimization Algorithm) [69] | 140,175.8 | - | - | - | - |
Optimizer | Min | Mean | Max | Std. | Time (s) |
---|---|---|---|---|---|
Case 1 | |||||
AGTLBO | 800.4811 | 800.5316 | 800.5587 | 0.076 | 26.7 |
TLBO | 800.6735 | 800.8705 | 801.1324 | 0.542 | 26.6 |
ABC | 800.6802 | 800.9642 | 802.1506 | 1.819 | 25.1 |
BBO | 800.9527 | 802.3304 | 805.3919 | 3.994 | 28.9 |
PSO | 800.7016 | 801.4378 | 803.8826 | 2.075 | 27.0 |
Optimizer | Min | Mean | Max | Std. | Time (s) |
Case 2 | |||||
AGTLBO | 646.4511 | 646.6973 | 646.9005 | 0.094 | 26.8 |
TLBO | 647.1344 | 647.8260 | 648.4326 | 0.927 | 26.7 |
ABC | 647.4256 | 648.2013 | 648.8591 | 1.748 | 26.4 |
BBO | 647.7739 | 649.2835 | 651.7072 | 3.690 | 29.0 |
PSO | 647.6495 | 648.9424 | 650.6625 | 2.514 | 27.3 |
Optimizer | Min | Mean | Max | Std. | Time (s) |
Case 3 | |||||
AGTLBO | 0.20482 | 0.20483 | 0.20484 | 0.000 | 27.0 |
TLBO | 0.20484 | 0.20490 | 0.20495 | 0.006 | 27.0 |
ABC | 0.20484 | 0.20505 | 0.20557 | 0.009 | 26.2 |
BBO | 0.20490 | 0.20532 | 0.20603 | 0.010 | 29.1 |
PSO | 0.20484 | 0.20496 | 0.20551 | 0.009 | 27.5 |
Optimizer | Min | Mean | Max | Std. | Time (s) |
Case 4 | |||||
AGTLBO | 3.0906 | 3.0911 | 3.0920 | 0.000 | 27.5 |
TLBO | 3.1104 | 3.1133 | 3.1275 | 0.062 | 27.3 |
ABC | 3.1314 | 3.1682 | 3.2074 | 0.103 | 26.8 |
BBO | 3.1725 | 3.2490 | 3.3178 | 0.199 | 29.5 |
PSO | 3.1709 | 3.1770 | 3.2235 | 0.071 | 27.7 |
Optimizer | Min | Mean | Max | Std. | Time (s) |
Case 5 | |||||
AGTLBO | 832.1624 | 832.2364 | 832.3047 | 0.402 | 27.4 |
TLBO | 832.5325 | 832.8761 | 833.4422 | 0.953 | 27.3 |
ABC | 832.5860 | 833.0923 | 833.4986 | 1.046 | 26.9 |
BBO | 832.9234 | 834.7060 | 836.6011 | 2.814 | 29.4 |
PSO | 832.5966 | 832.9648 | 834.5746 | 1.983 | 28.0 |
Optimizer | Min | Mean | Max | Std. | Time (s) |
Case 9 | |||||
AGTLBO | 41,678.310 | 41,684.744 | 41,692.19 | 24.36 | 71.3 |
TLBO | 41,679.890 | 41,690.376 | 41,710.38 | 38.15 | 72.5 |
ABC | 41,681.704 | 41,695.058 | 41,712.74 | 45.90 | 83.8 |
BBO | 41,697.552 | 41,728.918 | 41,765.75 | 133.4 | 107.6 |
PSO | 41,683.951 | 41,697.120 | 41,714.54 | 75.62 | 88.9 |
Optimizer | Min | Mean | Max | Std. | Time (s) |
Case 12 | |||||
AGTLBO | 129,545.56 | 129,552.92 | 129,562.3 | 8.53 | 740.2 |
TLBO | 129,550.49 | 129,607.63 | 129,691.0 | 72.14 | 741.8 |
ABC | 129,677.83 | 129,732.46 | 130,896.6 | 446.8 | 1284 |
BBO | 129,734.06 | 129,985.25 | 132,268.8 | 995.2 | 1595 |
PSO | 129,703.48 | 129,867.01 | 131,573.7 | 610.7 | 1359 |
Variables | Limits | Case 1 | Case 2 | Case 3 | Case 4 | Case 5 | Case 6 | Case 7 | Case 8 | ||
---|---|---|---|---|---|---|---|---|---|---|---|
Min | Max | ||||||||||
Reactive power generation in the generators | |||||||||||
QG1 (MVar) | −20 | 200 | 13.0435 | 6.2865 | −1.0687 | −1.4183 | 6.4658 | 2.3913 | 6.1482 | 5.8087 | |
QG2 (MVar) | −20 | 10 | 9.9850 | 9.9980 | 9.9953 | 9.9870 | 9.9983 | 10.0000 | 9.6782 | 9.9999 | |
QG5 (MVar) | −15 | 80 | 30.0398 | 28.5295 | 24.1889 | 24.1153 | 29.3115 | 25.8902 | 49.8828 | 26.4629 | |
QG8 (MVar) | −15 | 60 | 29.7835 | 28.7604 | 29.1378 | 29.2384 | 29.6445 | 29.4048 | 41.5309 | 29.2938 | |
QG11 (MVar) | −10 | 50 | 27.9447 | 30.9132 | 21.1152 | 19.8527 | 32.3006 | 26.2968 | 30.4476 | 20.7654 | |
QG13 (MVar) | −15 | 60 | 0.2229 | 4.9166 | 4.1780 | 4.1746 | 8.1319 | 6.0366 | 14.5693 | 8.7635 | |
Voltage magnitude at all the buses | |||||||||||
V1 (p.u.) | 0.95 | 1.1 | 1.0835 | 1.0756 | 1.0628 | 1.0618 | 1.0810 | 1.0697 | 1.0427 | 1.0729 | |
V2 (p.u.) | 0.95 | 1.1 | 1.0604 | 1.0583 | 1.0562 | 1.0571 | 1.0577 | 1.0575 | 1.0230 | 1.0573 | |
V3 (p.u.) | 0.95 | 1.05 | 1.0500 | 1.0500 | 1.0500 | 1.0500 | 1.0500 | 1.0500 | 1.0081 | 1.0500 | |
V4 (p.u.) | 0.95 | 1.05 | 1.0422 | 1.0439 | 1.0468 | 1.0471 | 1.0431 | 1.0460 | 1.0001 | 1.0447 | |
V5 (p.u.) | 0.95 | 1.1 | 1.0334 | 1.0328 | 1.0375 | 1.0382 | 1.0302 | 1.0357 | 1.0151 | 1.0326 | |
V6 (p.u.) | 0.95 | 1.05 | 1.0394 | 1.0415 | 1.0439 | 1.0445 | 1.0391 | 1.0436 | 1.0034 | 1.0411 | |
V7 (p.u.) | 0.95 | 1.05 | 1.0290 | 1.0300 | 1.0332 | 1.0339 | 1.0275 | 1.0324 | 0.9999 | 1.0296 | |
V8 (p.u.) | 0.95 | 1.1 | 1.0383 | 1.0414 | 1.0439 | 1.0445 | 1.0368 | 1.0437 | 1.0065 | 1.0411 | |
V9 (p.u.) | 0.95 | 1.05 | 1.0393 | 1.0355 | 1.0416 | 1.0481 | 1.0357 | 1.0499 | 1.0000 | 1.0000 | |
V10 (p.u.) | 0.95 | 1.05 | 1.0467 | 1.0474 | 1.0453 | 1.0450 | 1.0437 | 1.0478 | 1.0084 | 1.0087 | |
V11 (p.u.) | 0.95 | 1.1 | 1.0923 | 1.0937 | 1.0806 | 1.0846 | 1.0968 | 1.0982 | 1.0595 | 1.0401 | |
V12 (p.u.) | 0.95 | 1.05 | 1.0500 | 1.0500 | 1.0500 | 1.0500 | 1.0500 | 1.0500 | 1.0087 | 1.0119 | |
V13 (p.u.) | 0.95 | 1.1 | 1.0502 | 1.0563 | 1.0542 | 1.0542 | 1.0606 | 1.0576 | 0.9879 | 1.0235 | |
V14 (p.u.) | 0.95 | 1.05 | 1.0401 | 1.0402 | 1.0402 | 1.0401 | 1.0396 | 1.0404 | 0.9996 | 1.0018 | |
V15 (p.u.) | 0.95 | 1.05 | 1.0399 | 1.0400 | 1.0393 | 1.0392 | 1.0389 | 1.0402 | 1.0005 | 1.0018 | |
V16 (p.u.) | 0.95 | 1.05 | 1.0426 | 1.0429 | 1.0415 | 1.0413 | 1.0412 | 1.0431 | 1.0005 | 1.0044 | |
V17 (p.u.) | 0.95 | 1.05 | 1.0433 | 1.0439 | 1.0420 | 1.0418 | 1.0407 | 1.0442 | 1.0008 | 1.0051 | |
V18 (p.u.) | 0.95 | 1.05 | 1.0337 | 1.0334 | 1.0321 | 1.0321 | 1.0319 | 1.0338 | 0.9944 | 0.9953 | |
V19 (p.u.) | 0.95 | 1.05 | 1.0332 | 1.0326 | 1.0311 | 1.0310 | 1.0309 | 1.0331 | 0.9940 | 0.9947 | |
V20 (p.u.) | 0.95 | 1.05 | 1.0383 | 1.0376 | 1.0359 | 1.0358 | 1.0357 | 1.0381 | 0.9994 | 1.0000 | |
V21 (p.u.) | 0.95 | 1.05 | 1.0381 | 1.0387 | 1.0367 | 1.0364 | 1.0352 | 1.0392 | 0.9997 | 1.0000 | |
V22 (p.u.) | 0.95 | 1.05 | 1.0387 | 1.0392 | 1.0373 | 1.0370 | 1.0358 | 1.0397 | 1.0004 | 1.0007 | |
V23 (p.u.) | 0.95 | 1.05 | 1.0365 | 1.0362 | 1.0355 | 1.0355 | 1.0352 | 1.0370 | 1.0000 | 0.9999 | |
V24 (p.u.) | 0.95 | 1.05 | 1.0320 | 1.0318 | 1.0306 | 1.0305 | 1.0294 | 1.0326 | 0.9945 | 0.9945 | |
V25 (p.u.) | 0.95 | 1.05 | 1.0366 | 1.0351 | 1.0344 | 1.0343 | 1.0330 | 1.0367 | 1.0000 | 1.0000 | |
V26 (p.u.) | 0.95 | 1.05 | 1.0193 | 1.0178 | 1.0170 | 1.0169 | 1.0156 | 1.0193 | 0.9820 | 0.9820 | |
V27 (p.u.) | 0.95 | 1.05 | 1.0479 | 1.0457 | 1.0453 | 1.0453 | 1.0436 | 1.0477 | 1.0122 | 1.0122 | |
V28 (p.u.) | 0.95 | 1.05 | 1.0353 | 1.0377 | 1.0399 | 1.0405 | 1.0348 | 1.0397 | 0.9999 | 1.0369 | |
V29 (p.u.) | 0.95 | 1.05 | 1.0365 | 1.0339 | 1.0333 | 1.0333 | 1.0314 | 1.0358 | 1.0000 | 1.0000 | |
V30 (p.u.) | 0.95 | 1.05 | 1.0220 | 1.0195 | 1.0190 | 1.0190 | 1.0171 | 1.0215 | 0.9851 | 0.9851 | |
The transmission power via i-th line to j-th (MW) | |||||||||||
i-th | j-th | min | max | Case 1 | Case 2 | Case 3 | Case 4 | Case 5 | Case 6 | Case 7 | Case 8 |
1 | 2 | −130 | 130 | 115.303 | 89.8875 | 36.7666 | 26.3472 | 129.998 | 64.6271 | 114.884 | 78.3493 |
1 | 3 | −130 | 130 | 62.3312 | 50.2764 | 27.3180 | 25.1841 | 68.8903 | 38.0489 | 61.8038 | 44.0164 |
2 | 4 | −65 | 65 | 32.6647 | 27.4189 | 18.8148 | 19.8278 | 35.3420 | 21.8517 | 32.4656 | 24.1042 |
3 | 4 | −130 | 130 | 58.1391 | 46.7745 | 24.6186 | 22.5298 | 64.6006 | 35.0427 | 57.4753 | 40.7758 |
2 | 5 | −130 | 130 | 63.1242 | 57.9938 | 38.9298 | 39.2685 | 66.4483 | 47.1054 | 64.3868 | 52.0719 |
2 | 6 | −65 | 65 | 44.2342 | 36.4673 | 24.7146 | 25.4699 | 48.7312 | 28.8743 | 43.7856 | 31.9784 |
4 | 6 | −90 | 90 | 51.4320 | 40.3901 | 26.4317 | 25.3668 | 59.1888 | 31.3824 | 54.3112 | 35.0825 |
5 | 7 | −70 | 70 | 14.5951 | 15.9078 | 8.9353 | 8.6790 | 13.3175 | 12.4278 | 20.7297 | 14.2529 |
6 | 7 | −130 | 130 | 34.6959 | 36.8083 | 29.2347 | 28.9089 | 34.4332 | 33.2495 | 34.5826 | 35.1701 |
6 | 8 | −32 | 32 | 11.1907 | 1.0398 | 1.2698 | 1.2257 | 20.9236 | 1.0144 | 14.8467 | 0.9188 |
6 | 9 | −65 | 65 | 37.3850 | 41.5175 | 21.8485 | 14.3270 | 41.8665 | 21.5371 | 39.6149 | 26.9585 |
6 | 10 | −32 | 32 | 21.5345 | 26.5973 | 15.3814 | 9.5859 | 21.0635 | 13.9521 | 26.1334 | 19.9261 |
9 | 11 | −65 | 65 | 30.3903 | 35.9626 | 36.6858 | 35.9739 | 33.8135 | 39.8939 | 32.7811 | 33.8585 |
9 | 10 | −65 | 65 | 30.7155 | 33.6370 | 32.9659 | 33.1003 | 29.8433 | 35.3417 | 30.5355 | 35.6623 |
4 | 12 | −65 | 65 | 32.1972 | 26.3197 | 9.1445 | 9.1145 | 32.4045 | 17.8088 | 39.8056 | 21.6933 |
12 | 13 | −65 | 65 | 12.0038 | 18.2491 | 40.2176 | 40.2170 | 14.4968 | 27.3238 | 19.2733 | 22.7903 |
12 | 14 | −32 | 32 | 7.4644 | 7.4887 | 8.0749 | 8.0762 | 7.6036 | 7.5846 | 7.3937 | 7.4230 |
12 | 15 | −32 | 32 | 17.6596 | 17.7982 | 20.3299 | 20.3267 | 18.0452 | 18.2480 | 17.9536 | 17.5549 |
12 | 16 | −32 | 32 | 7.0459 | 7.0373 | 9.5598 | 9.5591 | 7.5158 | 7.2916 | 7.2073 | 6.6202 |
14 | 15 | −16 | 16 | 1.6797 | 1.7278 | 2.3560 | 2.3520 | 1.6852 | 1.8518 | 2.0094 | 1.6884 |
16 | 17 | −16 | 16 | 3.7079 | 3.7696 | 6.2836 | 6.2627 | 3.9549 | 4.0638 | 3.7891 | 3.2547 |
15 | 18 | −16 | 16 | 5.6989 | 5.6815 | 6.9444 | 6.9486 | 5.9704 | 5.7574 | 5.7253 | 5.4246 |
18 | 19 | −16 | 16 | 2.5879 | 2.5020 | 3.8208 | 3.8220 | 2.7700 | 2.6163 | 2.6637 | 2.2690 |
19 | 20 | −32 | 32 | 8.2377 | 8.1499 | 7.3147 | 7.3048 | 7.8839 | 8.1474 | 8.3012 | 8.4000 |
10 | 20 | −32 | 32 | 9.3365 | 9.3999 | 8.1634 | 8.1458 | 9.0765 | 9.3115 | 9.3222 | 9.6304 |
10 | 17 | −32 | 32 | 5.9364 | 5.9926 | 4.1399 | 4.0922 | 5.4267 | 5.8137 | 8.8603 | 6.3076 |
10 | 21 | −32 | 32 | 16.4266 | 16.7292 | 16.6950 | 16.6721 | 16.3154 | 16.9744 | 16.3198 | 16.7705 |
10 | 22 | −32 | 32 | 7.7913 | 7.9866 | 7.9727 | 7.9583 | 7.7181 | 8.1574 | 7.7232 | 8.0305 |
21 | 22 | −32 | 32 | 2.4705 | 2.2173 | 2.3852 | 2.4169 | 2.5910 | 2.2317 | 2.7244 | 2.5827 |
15 | 23 | −16 | 16 | 4.7732 | 4.9481 | 6.8064 | 6.7987 | 5.0315 | 5.4586 | 5.1982 | 5.0669 |
22 | 24 | −16 | 16 | 5.6352 | 6.1030 | 6.1871 | 6.1619 | 5.4637 | 6.6741 | 5.5657 | 6.4935 |
23 | 24 | −16 | 16 | 1.8013 | 1.9122 | 3.4405 | 3.4388 | 2.2318 | 2.2329 | 2.0005 | 2.0263 |
24 | 25 | −16 | 16 | 1.6791 | 1.0755 | 1.8803 | 1.8756 | 1.5255 | 1.3015 | 1.8040 | 1.4386 |
25 | 26 | −16 | 16 | 4.2593 | 4.2595 | 4.2596 | 4.2596 | 4.2598 | 4.2593 | 4.2648 | 4.2648 |
25 | 27 | −16 | 16 | 5.9595 | 5.3845 | 4.8998 | 4.9249 | 5.7419 | 5.1091 | 6.1251 | 5.6398 |
28 | 27 | −65 | 65 | 19.0340 | 18.4213 | 17.0131 | 17.0371 | 18.9522 | 17.6591 | 19.2448 | 18.4261 |
27 | 29 | −16 | 16 | 6.2047 | 6.1987 | 6.1962 | 6.1969 | 6.1956 | 6.1966 | 6.2080 | 6.2077 |
27 | 30 | −16 | 16 | 7.1177 | 7.1252 | 7.1290 | 7.1279 | 7.1316 | 7.1267 | 7.1360 | 7.1364 |
29 | 30 | −16 | 16 | 3.9652 | 3.9489 | 3.9403 | 3.9428 | 3.9360 | 3.9433 | 3.9565 | 3.9554 |
8 | 28 | −32 | 32 | 3.0987 | 4.5496 | 4.3463 | 4.3575 | 2.0975 | 4.4718 | 4.4075 | 4.6084 |
6 | 28 | −32 | 32 | 16.2347 | 13.9195 | 12.7138 | 12.7290 | 17.7043 | 13.2438 | 15.9782 | 13.8691 |
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Alanazi, A.; Alanazi, M.; Memon, Z.A.; Mosavi, A. Determining Optimal Power Flow Solutions Using New Adaptive Gaussian TLBO Method. Appl. Sci. 2022, 12, 7959. https://doi.org/10.3390/app12167959
Alanazi A, Alanazi M, Memon ZA, Mosavi A. Determining Optimal Power Flow Solutions Using New Adaptive Gaussian TLBO Method. Applied Sciences. 2022; 12(16):7959. https://doi.org/10.3390/app12167959
Chicago/Turabian StyleAlanazi, Abdulaziz, Mohana Alanazi, Zulfiqar Ali Memon, and Amir Mosavi. 2022. "Determining Optimal Power Flow Solutions Using New Adaptive Gaussian TLBO Method" Applied Sciences 12, no. 16: 7959. https://doi.org/10.3390/app12167959