A Modified Rao-2 Algorithm for Optimal Power Flow Incorporating Renewable Energy Sources
Abstract
:1. Introduction
- The proposed MRao-2 technique is used to achieve the accurate values of control variables of the OPF problem without RES, with RES, and with RES under contingency state.
- The fuel cost is the main objective function in five scenarios for the two IEEE 30 -bus and 118-bus systems to test the validation of the proposed algorithm.
- To check the robustness of this modified algorithm, its results are compared with five recent algorithms—ASO, TFWO, MPA, Rao-1, and Rao-3—as well as the original Rao-2 which are the strong algorithms in solving the modern power system problems and they are used in many published papers in the last two years so far.
2. Problem Formulation
2.1. General Structure of OPF
2.2. Objective Functions
2.2.1. Quadratic Total Fuel Cost
2.2.2. Total Emission
2.2.3. Power Loss Function
2.2.4. Voltage Deviation (VD) Function (Voltage Profile Improvement)
2.3. Constraints
2.3.1. Equality Constraints
2.3.2. Inequality Constraints
2.3.3. Power Balance Considering RES
3. The Proposed Optimization Technique
3.1. Rao Algorithm
- Step 1: Randomly distribute the population within the vector ranges.
- Step 2: Determine the objective value for each variable.
- Step 3: Define the worst and best solutions depending on the objective function’s values.
- Step 4: Upgrade the solutions by (25).
- Step 5: If any of the updated values fall outside of the range, they should be returned.
- Step 6: Evaluate the value of each search agent’s objective function.
- Step 7: Increase the number of iterations of the new one
- Step 8: If the iteration has reached its end, return the best value so far. If not, go on to Step 3.
3.2. Modified Rao Algorithm
3.2.1. Quasi-Oppositional
3.2.2. Levy Flight
4. Simulation Results
4.1. Test Systems
4.2. Case 1: The OPF without RES for the IEEE 30-Bus System
4.3. Case 2: OPF Incorporating RES for the IEEE 30-Bus System
4.4. Case 3: OPF Incorporating RES under Contingency State for IEEE 30-Bus System
4.5. Case 4: OPF without RES for the IEEE 118-Bus System
4.6. Case 5: OPF Incorporating RES for the IEEE 118-Bus System
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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IEEE 30-Bus System | IEEE 118-Bus System | |||
---|---|---|---|---|
Variables | Lower limit | Upper limit | Lower limit | Upper limit |
Voltages for all generator bus | 0.95 p.u | 1.1 p.u | 0.94 p.u | 1.06 p.u |
Voltages for all load bus | 0.95 p.u | 1.05 p.u | 0.95 p.u | 1.05 p.u |
Tap setting | 0.9 p.u | 1.1 p.u | 0.9 p.u | 1.1 p.u |
Reactive power of capacitor banks | 0 | 0.05 p.u | 0 | 0.3 p.u |
Type of RES | No. of Bus | Value (MW) |
---|---|---|
biomass | 12 | 18.2 |
wind | 31 | 156 |
solar | 54 | 264 |
hydro | 76 | 77 |
hydro | 116 | 286 |
Algorithms | Parameters Setting |
---|---|
Common settings |
|
ASO |
|
MPA | FADs = 0.2, P = 0.5, C = 0.05, e = 0.25 |
ASO | TFWO | MPA | Rao-1 | Rao-2 | Rao-3 | MRao-2 | |
---|---|---|---|---|---|---|---|
PG1 (MW) | 176.9732 | 177.2422 | 176.8351 | 179.8369 | 177.887 | 177.5088 | 176.3625 |
PG2 (MW) | 48.91006 | 48.64558 | 48.67575 | 49.49712 | 49.36524 | 48.76959 | 49.07412 |
PG5 (MW) | 21.30213 | 21.36803 | 21.45982 | 22.21521 | 21.54786 | 20.94155 | 21.24651 |
PG8 (MW) | 20.9253 | 21.35596 | 21.88621 | 18.77388 | 21.6564 | 21.53046 | 21.37135 |
PG11 (MW) | 12.45411 | 11.86476 | 11.54409 | 10.38348 | 10.00796 | 11.80911 | 12.21261 |
PG13 (MW) | 12.00137 | 12.00148 | 12.02418 | 12 | 12.04775 | 12.0039 | 12.10508 |
V1 (p.u.) | 1.080029 | 1.079984 | 1.082237 | 1.078675 | 1.084304 | 1.080698 | 1.083304 |
V2 (p.u.) | 1.080992 | 1.030047 | 1.083237 | 1.027824 | 1.094344 | 1.099999 | 1.092657 |
V5 (p.u.) | 1.028033 | 1.082677 | 1.031479 | 1.085737 | 1.031264 | 1.028262 | 1.029766 |
V8 (p.u.) | 1.034358 | 1.035451 | 1.037666 | 1.033404 | 1.037539 | 1.036411 | 1.037062 |
V11 (p.u.) | 1.006403 | 1.05873 | 1.065692 | 1.078167 | 1.077243 | 1.031184 | 1.059477 |
V13 (p.u.) | 1.036055 | 1.067677 | 1.029076 | 1.057387 | 1.034243 | 1.099824 | 1.046984 |
T11 (6–9) | 0.96365 | 0.972727 | 0.963545 | 1.014786 | 1.031208 | 0.916034 | 1.002334 |
T12 (6–10) | 1.014561 | 1.012358 | 1.05331 | 0.983026 | 0.9 | 1.099979 | 0.953247 |
T15 (4–12) | 1.056913 | 0.994481 | 0.989785 | 1.001 | 0.964418 | 1.031275 | 0.971027 |
T36 (28–27) | 0.99403 | 0.971573 | 1.000675 | 0.975213 | 0.980459 | 0.987251 | 0.971058 |
QC10 (MVAR) | 3.0526 | 4.9784 | 3.2035 | 0.5727 | 0.1362 | 0.00562 | 3.7024 |
QC12 (MVAR) | 3.5939 | 0.5594 | 4.639 | 0.7186 | 1.3228 | 0.00426 | 2.0306 |
QC15 (MVAR) | 2.5611 | 4.635 | 3.9502 | 5 | 4.9242 | 4.9567 | 2.2152 |
QC17 (MVAR) | 1.6444 | 3.7878 | 1.5066 | 3.3725 | 4.2338 | 0.0702 | 4.6995 |
QC20 (MVAR) | 1.9898 | 4.5001 | 4.8618 | 4.4774 | 3.1484 | 4.9871 | 3.859 |
QC21 (MVAR) | 3.4191 | 4.1061 | 3.5977 | 3.9993 | 0.2586 | 4.8557 | 4.8858 |
QC23 (MVAR) | 4.7618 | 0.00168 | 4.3476 | 0.818 | 3.2847 | 0.0451 | 3.9984 |
QC24 (MVAR) | 1.1282 | 2.1995 | 4.5618 | 4.9692 | 4.9243 | 4.9741 | 4.8289 |
QC29 (MVAR) | 1.5646 | 0.4415 | 3.5686 | 2.1977 | 4.9685 | 2.3785 | 1.6698 |
Fuel cost ($/h) | 801.0005 | 800.6477 | 800.5804 | 800.8944 | 800.6166 | 800.848 | 800.4412 |
Emission (ton/h) | 0.295736 | 0.296049 | 0.295644 | 0.297821 | 0.296313 | 0.296384 | 0.295152 |
Power loss (MW) | 9.177889 | 9.083431 | 9.036827 | 9.312149 | 9.123883 | 9.163424 | 8.983817 |
Voltage deviation (p.u.) | 0.334805 | 0.749458 | 0.575301 | 0.707313 | 0.916652 | 0.469378 | 0.868108 |
Time (s) | 95.06342 | 104.1479 | 166.5552 | 101.91743 | 94.84023 | 101.83725 | 169.6059 |
Algorithm | Min | Max | Average |
---|---|---|---|
MRao-2 | 800.4412 | 800.553 | 800.4872 |
Rao-2 | 800.6166 | 800.7965 | 800.7118 |
Rao-1 | 800.8944 | 801.2647 | 800.9678 |
Rao-3 | 800.848 | 800.9628 | 800.9067 |
MPA | 800.5804 | 800.8416 | 800.6659 |
TFWO | 800.6477 | 803.8754 | 801.1159 |
ASO | 801.0005 | 801.4358 | 801.101 |
MGOA [11] | 800.4744 | NA | NA |
ABC [52] | 800.6600 | 800.8715 | 801.8674 |
Jaya [47] | 800.4794 | 800.4928 | 800.5306 |
ARCBBO [53] | 800.5159 | 800.6412 | 800.9262 |
MSA [54] | 800.5099 | NA | NA |
Hybrid SFLA SA [55] | 801.79 | NA | NA |
HHO [33] | 801.4228 | NA | NA |
HHODE [33] | 800.9959 | NA | NA |
DE [56] | 801.23 | 801.622 | 801.282 |
ASO | TFWO | MPA | Rao-1 | Rao-2 | Rao-3 | MRao-2 | |
---|---|---|---|---|---|---|---|
PG1 (MW) | 166.3103 | 167.0352 | 166.5015 | 167.7043 | 167.5941 | 167.2709 | 167.2508 |
PG2 (MW) | 45.88087 | 46.29256 | 46.03059 | 47.38566 | 45.71914 | 47.19775 | 46.42704 |
PG5 (MW) | 20.9709 | 20.64382 | 20.45905 | 20.90393 | 20.6118 | 20.69004 | 20.64984 |
PG8 (MW) | 15.69756 | 15.68206 | 15.17763 | 13.74229 | 15.13705 | 14.47554 | 15.27324 |
PG11 (MW) | 10.79724 | 10.00009 | 11.10394 | 10 | 10.53956 | 10.00813 | 10 |
PG13 (MW) | 12.00262 | 12 | 12.32851 | 12 | 12.03885 | 12.04529 | 12 |
V1 (p.u.) | 1.077967 | 1.081966 | 1.0788 | 1.077875 | 1.080078 | 1.080089 | 1.07852 |
V2 (p.u.) | 1.072375 | 1.006269 | 1.1 | 1.094477 | 1.051656 | 1.063304 | 1.1 |
V5 (p.u.) | 1.033146 | 1.057191 | 1.031388 | 1.076221 | 1.077262 | 1.0572 | 1.032235 |
V8 (p.u.) | 1.031798 | 1.036781 | 1.038591 | 1.038888 | 1.039808 | 1.040699 | 1.026295 |
V11 (p.u.) | 1.02098 | 1.099826 | 1.093579 | 1.049587 | 1.04657 | 1.078359 | 1.047772 |
V13 (p.u.) | 1.042508 | 1.022207 | 1.014105 | 1.01495 | 1.009777 | 1.022799 | 1.062827 |
T11(6–9) | 0.986874 | 0.989094 | 1.027745 | 0.99534 | 1.09646 | 0.991078 | 0.98482 |
T12(6–10) | 1.005378 | 1.1 | 0.957009 | 0.928192 | 0.908495 | 1.073045 | 0.977984 |
T15(4–12) | 0.975554 | 0.987607 | 0.981413 | 0.981491 | 0.971842 | 0.970377 | 0.981403 |
T36(28–27) | 1.001887 | 0.99311 | 0.99195 | 0.997822 | 1.017273 | 1.010252 | 1.001331 |
QC10 (MVAR) | 4.1156 | 4.7537 | 2.7651 | 4.7977 | 2.8059 | 4.9307 | 4.9494 |
QC12 (MVAR) | 2.8466 | 4.821 | 3.8682 | 3.6157 | 1.4147 | 0.0171 | 0 |
QC15 (MVAR) | 3.4126 | 4.8818 | 0.5251 | 4.3005 | 1.2958 | 3.8049 | 0.0184 |
QC17 (MVAR) | 2.9106 | 4.2942 | 4.9994 | 0.3354 | 4.6224 | 3.1239 | 4.8752 |
QC20 (MVAR) | 2.3832 | 2.9394 | 4.6997 | 4.691 | 4.394 | 3.1954 | 4.8711 |
QC21 (MVAR) | 2.9478 | 5 | 0.3764 | 1.8647 | 3.3121 | 0 | 5 |
QC23 (MVAR) | 1.4159 | 1.9167 | 2.9807 | 1.0238 | 4.9937 | 5 | 5 |
QC24 (MVAR) | 2.6985 | 5 | 0.8889 | 3.9491 | 4.9191 | 4.7007 | 4.9522 |
QC29 (MVAR) | 2.7382 | 0.3091 | 2.0465 | 1.5414 | 4.2786 | 2.302 | 2.282 |
Fuel cost ($/h) | 729.9074 | 729.6002 | 729.6347 | 729.6406 | 729.5025 | 729.5657 | 729.3429 |
Emission (ton/h) | 0.287894 | 0.288436 | 0.288163 | 0.288493 | 0.289114 | 0.28828 | 0.288559 |
Power loss (MW) | 8.271074 | 8.264871 | 8.212681 | 8.341641 | 8.245904 | 8.293049 | 8.21248 |
Voltage deviation (p.u.) | 0.487935 | 0.587188 | 0.723005 | 0.74643 | 0.599303 | 0.554658 | 0.890863 |
Time (s) | 96.3905 | 101.40315 | 154.3002 | 93.26448 | 92.40164 | 95.84038 | 166.5166 |
ASO | TFWO | MPA | Rao-1 | Rao-2 | Rao-3 | MRao-2 | |
---|---|---|---|---|---|---|---|
PG1 (MW) | 169.1157 | 167.646 | 168.4079 | 166.1685 | 168.162 | 168.2884 | 167.6707 |
PG2 (MW) | 47.10122 | 46.38156 | 46.73768 | 46.29319 | 46.34349 | 46.35133 | 46.43635 |
PG5 (MW) | 20.77245 | 20.68947 | 20.8021 | 20.22988 | 20.61748 | 20.70681 | 20.82097 |
PG8 (MW) | 12.4257 | 15.29242 | 13.2599 | 17.28305 | 14.80192 | 14.45443 | 14.76838 |
PG11 (MW) | 10.9089 | 10 | 10.93876 | 10 | 10.09605 | 10.24194 | 10.22822 |
PG13 (MW) | 12.0136 | 12 | 12.00331 | 12.02535 | 12.00538 | 12.00845 | 12.05377 |
V1 (p.u.) | 1.071178 | 1.080797 | 1.082793 | 1.088881 | 1.082847 | 1.081143 | 1.08205 |
V2 (p.u.) | 1.091177 | 1.0925 | 1.033038 | 1.063013 | 1.084106 | 1.082092 | 1.083835 |
V5 (p.u.) | 1.018346 | 1.031645 | 1.086276 | 1.054294 | 1.031456 | 1.026707 | 1.032011 |
V8 (p.u.) | 1.023794 | 1.036661 | 1.037818 | 1.034209 | 1.033635 | 1.033571 | 1.03592 |
V11 (p.u.) | 1.038897 | 1.071995 | 1.083139 | 1.059527 | 1.029808 | 1.074468 | 1.029754 |
V13 (p.u.) | 1.019225 | 1.042984 | 1.047633 | 1.039893 | 1.046966 | 1.045277 | 1.04714 |
T11 (6–9) | 0.966049 | 0.965995 | 1.05369 | 0.973616 | 1.040438 | 1.078513 | 1.06788 |
T12 (6–10) | 0.970151 | 1.099986 | 0.939359 | 1.016485 | 0.906038 | 0.9 | 0.900135 |
T15 (4–12) | 0.96075 | 0.984042 | 0.957996 | 0.961999 | 0.984907 | 0.988115 | 0.980633 |
T36 (28–27) | 1.014058 | 1.006899 | 1.011424 | 1.01808 | 1.017749 | 1.010059 | 1.010374 |
QC10 (MVAR) | 3.3514 | 1.8078 | 1.993 | 4.6449 | 0.0643 | 0.1501 | 0.0246 |
QC12 (MVAR) | 2.4055 | 4.5298 | 2.4846 | 0.515 | 0.9183 | 3.2165 | 0.0275 |
QC15 (MVAR) | 3.0776 | 5 | 4.3023 | 4.5803 | 4.4913 | 4.9203 | 1.5989 |
QC17 (MVAR) | 3.3283 | 5 | 2.7311 | 4.2387 | 4.9964 | 4.7269 | 4.994 |
QC20 (MVAR) | 3.9248 | 0 | 2.5187 | 0.2183 | 1.9472 | 0.0149 | 4.8959 |
QC21 (MVAR) | 4.4199 | 5 | 0.24 | 3.2204 | 5 | 5 | 4.9499 |
QC23 (MVAR) | 2.5112 | 0.702 | 1.7923 | 4.9999 | 4.9924 | 0 | 3.431 |
QC24 (MVAR) | 4.5875 | 5 | 2.9447 | 1.9501 | 4.8343 | 4.9989 | 5 |
QC29 (MVAR) | 2.7613 | 0 | 2.76 × 10−5 | 0 | 0.00259 | 0.0514 | 0.0739 |
Fuel cost ($/h) | 731.4898 | 730.6851 | 731.0095 | 731.0468 | 730.6201 | 730.688 | 730.583 |
Emission (ton/h) | 0.289802 | 0.288885 | 0.289341 | 0.287874 | 0.289345 | 0.289416 | 0.288841 |
Power loss (MW) | 8.949145 | 8.620926 | 8.755022 | 8.605339 | 8.63787 | 8.662901 | 8.589888 |
Voltage deviation (p.u.) | 0.595555 | 0.626741 | 0.73608 | 0.693286 | 0.695892 | 0.665941 | 0.665648 |
Time (s) | 94.1653 | 98.40031 | 153.3302 | 95.61418 | 98.2997 | 97.24576 | 164.5099 |
Case No. | Algorithm | Min | Average | Median | Max | STD |
---|---|---|---|---|---|---|
Case 1 | MRao-2 | 800.4412 | 800.4872 | 800.4769 | 800.553 | 0.038822 |
Rao-2 | 800.6166 | 800.7118 | 800.7135 | 800.7965 | 0.052478 | |
Rao-1 | 800.8944 | 800.9678 | 800.9277 | 801.2647 | 0.111619 | |
Rao-3 | 800.848 | 800.9067 | 800.9167 | 800.9628 | 0.0403 | |
MPA | 800.5804 | 800.6659 | 800.6347 | 800.8416 | 0.081797 | |
TFWO | 800.6477 | 801.1159 | 800.855 | 803.8754 | 0.975128 | |
ASO | 801.0005 | 801.101 | 801.0422 | 801.4358 | 0.152133 | |
Case 2 | MRao-2 | 729.3429 | 729.4065 | 729.4001 | 729.4615 | 0.042289 |
Rao-2 | 729.5025 | 729.5599 | 729.5596 | 729.6205 | 0.040197 | |
Rao-1 | 729.6406 | 729.6845 | 729.6815 | 729.7441 | 0.031135 | |
Rao-3 | 729.5657 | 729.5888 | 729.5903 | 729.6361 | 0.021254 | |
MPA | 729.6347 | 729.674 | 729.6771 | 729.7095 | 0.024818 | |
TFWO | 729.6002 | 730.1646 | 729.782 | 732.1079 | 0.813312 | |
ASO | 729.9074 | 730.3542 | 730.251 | 731.555 | 0.513813 | |
Case 3 | MRao-2 | 730.583 | 730.6588 | 730.6266 | 730.8189 | 0.09241 |
Rao-2 | 730.6201 | 730.7573 | 730.747 | 730.9311 | 0.141165 | |
Rao-1 | 731.0468 | 731.132 | 731.1247 | 731.2359 | 0.090685 | |
Rao-3 | 730.688 | 730.8235 | 730.8296 | 730.9879 | 0.113336 | |
MPA | 731.0095 | 731.136 | 731.14 | 731.2927 | 0.111756 | |
TFWO | 730.6851 | 730.9124 | 730.8677 | 731.3553 | 0.23962 | |
ASO | 731.4898 | 731.8588 | 731.6576 | 732.9515 | 0.546641 |
Variables | Value | Variables | Value | Variables | Value | Variables | Value | Variables | Value |
---|---|---|---|---|---|---|---|---|---|
PG1 (MW) | 1.984061 | PG62 (MW) | 0.04968 | PG113 (MW) | 86.1052 | VG59 (p.u.) | 0.94707 | VG111 (p.u.) | 1.05221 |
PG4 (MW) | 0.351443 | PG65 (MW) | 343.880 | PG116 (MW) | 4.20199 | VG61 (p.u.) | 1.02331 | VG112 (p.u.) | 1.04184 |
PG6 (MW) | 1.344034 | PG66 (MW) | 340.152 | VG1 (p.u.) | 0.94 | VG62 (p.u.) | 0.95129 | VG113 (p.u.) | 1.02649 |
PG8 (MW) | 10.24637 | PG69 (MW) | 415.633 | VG4 (p.u.) | 1.00609 | VG65 (p.u.) | 0.94352 | VG116 (p.u.) | 0.96645 |
PG10 (MW) | 376.1126 | PG70 (MW) | 4.69474 | VG6 (p.u.) | 1.00546 | VG66 (p.u.) | 1.01735 | T8 (8–5) | 0.91057 |
PG12 (MW) | 76.80127 | PG72 (MW) | 7.50899 | VG8 (p.u.) | 0.94545 | VG69 (p.u.) | 1.03288 | T32 (25–26) | 1.09451 |
PG15 (MW) | 1.528393 | PG73 (MW) | 10.3911 | VG10 (p.u.) | 0.94035 | VG70 (p.u.) | 0.97696 | T36 (17–30) | 1.09266 |
PG18 (MW) | 46.72157 | PG74 (MW) | 5.8267 | VG12 (p.u.) | 0.99321 | VG72 (p.u.) | 1.03499 | T51 (37–38) | 0.9 |
PG19 (MW) | 0.067021 | PG76 (MW) | 20.4488 | VG15 (p.u.) | 1.01157 | VG73 (p.u.) | 0.98931 | T93 (59–63) | 1.00128 |
PG24 (MW) | 2.611653 | PG77 (MW) | 5.51401 | VG18 (p.u.) | 0.96454 | VG74 (p.u.) | 0.99958 | T95 (61–64) | 1.03762 |
PG25 (MW) | 196.2022 | PG80 (MW) | 451.524 | VG19 (p.u.) | 1.04577 | VG76 (p.u.) | 0.99539 | T102 (65–66) | 0.95931 |
PG26 (MW) | 280.9463 | PG85 (MW) | 0 | VG24 (p.u.) | 0.99640 | VG77 (p.u.) | 0.97826 | T107 (68–69) | 0.95758 |
PG27 (MW) | 98.34095 | PG87 (MW) | 0.95058 | VG25 (p.u.) | 0.97686 | VG80 (p.u.) | 1.01302 | T127 (80–81) | 1.05250 |
PG31 (MW) | 0.751755 | PG89 (MW) | 483.822 | VG26 (p.u.) | 0.94265 | VG85 (p.u.) | 0.97493 | QC34 (MVAR) | 3.111 |
PG32 (MW) | 18.9298 | PG90 (MW) | 2.75380 | VG27 (p.u.) | 1.01799 | VG87 (p.u.) | 0.94034 | QC44 (MVAR) | 29.931 |
PG34 (MW) | 0.070676 | PG91 (MW) | 0 | VG31 (p.u.) | 1.03403 | VG89 (p.u.) | 1.03117 | QC45 (MVAR) | 29.497 |
PG36 (MW) | 4.986125 | PG92 (MW) | 1.17943 | VG32 (p.u.) | 0.99471 | VG90 (p.u.) | 1.02089 | QC46 (MVAR) | 28.169 |
PG40 (MW) | 1.913409 | PG99 (MW) | 16.5592 | VG34 (p.u.) | 1.00824 | VG91 (p.u.) | 1.04970 | QC48 (MVAR) | 0 |
PG42 (MW) | 1.394682 | PG100 (MW) | 200.116 | VG36 (p.u.) | 0.99818 | VG92 (p.u.) | 1.01643 | QC74 (MVAR) | 24.96 |
PG46 (MW) | 13.76368 | PG103 (MW) | 23.1161 | VG40 (p.u.) | 0.99148 | VG99 (p.u.) | 1.02329 | QC79 (MVAR) | 28.765 |
PG49 (MW) | 209.701 | PG104 (MW) | 99.6802 | VG42 (p.u.) | 1.02358 | VG100 (p.u.) | 0.99312 | QC82 (MVAR) | 27.479 |
PG54 (MW) | 48.30631 | PG105 (MW) | 0.13269 | VG46 (p.u.) | 0.98153 | VG103 (p.u.) | 1.05004 | QC83 (MVAR) | 24.519 |
PG55 (MW) | 26.09714 | PG107 (MW) | 0.19020 | VG49 (p.u.) | 1.00041 | VG104 (p.u.) | 1.05559 | QC105 (MVAR) | 27.063 |
PG56 (MW) | 80.54259 | PG110 (MW) | 0.42630 | VG54 (p.u.) | 0.99326 | VG105 (p.u.) | 0.96614 | QC107 (MVAR) | 6.934 |
PG59 (MW) | 128.7814 | PG111 (MW) | 33.7688 | VG55 (p.u.) | 1.05871 | VG107 (p.u.) | 0.94274 | QC110 (MVAR) | 29.781 |
PG61 (MW) | 146.4049 | PG112 (MW) | 5.1539 | VG56 (p.u.) | 1.05977 | VG110 (p.u.) | 1.00997 | ||
Fuel cost ($/h) | 131457.8 | ||||||||
Power loss (MW) | 96.68278 | ||||||||
Voltage deviation (p.u.) | 0.730363 |
Algorithm | ASO | TFWO | MPA | Rao-1 | Rao-2 | Rao-3 | MRao-2 |
---|---|---|---|---|---|---|---|
Fuel cost ($/h) | 133,610.8 | 132,132.2 | 131,942.6 | 131,817.9 | 131,490.7 | 131,793.1 | 131,457.8 |
Power loss (MW) | 61.83332 | 65.55476 | 71.94402 | 93.85931 | 95.46617 | 93.95222 | 96.68278 |
Voltage deviation (p.u.) | 0.658779 | 0.961026 | 1.152593 | 1.328297 | 0.998901 | 1.192274 | 0.730363 |
Time (s) | 800.709 | 809.028 | 1022.262 | 807.969 | 804.5724 | 806.71149 | 1160.264 |
Variables | Value | Variables | Value | Variables | Value | Variables | Value | Variables | Value |
---|---|---|---|---|---|---|---|---|---|
PG1 (MW) | 3.0911 | PG62 (MW) | 6.37571 | PG113 (MW) | 7.09147 | VG59 (p.u.) | 1.036427 | VG111 (p.u.) | 1.03604 |
PG4 (MW) | 7.265278 | PG65 (MW) | 298.824 | PG116 (MW) | 0 | VG61 (p.u.) | 0.956257 | VG112 (p.u.) | 0.97545 |
PG6 (MW) | 79.79656 | PG66 (MW) | 284.31 | VG1 (p.u.) | 1.01134 | VG62 (p.u.) | 1.054308 | VG113 (p.u.) | 1.04051 |
PG8 (MW) | 0.20039 | PG69 (MW) | 400.029 | VG4 (p.u.) | 1.03191 | VG65 (p.u.) | 0.974536 | VG116 (p.u.) | 0.97347 |
PG10 (MW) | 332.3392 | PG70 (MW) | 7.71916 | VG6 (p.u.) | 1.02025 | VG66 (p.u.) | 1.00127 | T8 (8–5) | 1.09856 |
PG12 (MW) | 73.21883 | PG72 (MW) | 0.27496 | VG8 (p.u.) | 0.94257 | VG69 (p.u.) | 1.020322 | T32 (25–26) | 0.90257 |
PG15 (MW) | 6.500848 | PG73 (MW) | 6.45688 | VG10 (p.u.) | 0.94115 | VG70 (p.u.) | 0.979919 | T36 (17–30) | 0.9 |
PG18 (MW) | 5.979357 | PG74 (MW) | 5.52285 | VG12 (p.u.) | 0.97438 | VG72 (p.u.) | 1.030921 | T51 (37–38) | 0.90084 |
PG19 (MW) | 1.346065 | PG76 (MW) | 2.53255 | VG15 (p.u.) | 1.05303 | VG73 (p.u.) | 1.005468 | T93 (59–63) | 1.09369 |
PG24 (MW) | 1.582226 | PG77 (MW) | 0.22016 | VG18 (p.u.) | 1.04361 | VG74 (p.u.) | 1.020992 | T95 (61–64) | 0.96853 |
PG25 (MW) | 159.4184 | PG80 (MW) | 363.967 | VG19 (p.u.) | 0.98928 | VG76 (p.u.) | 1.019236 | T102 (65–66) | 0.93325 |
PG26 (MW) | 215.2208 | PG85 (MW) | 0.07633 | VG24 (p.u.) | 0.96523 | VG77 (p.u.) | 1.034034 | T107 (68–69) | 0.91076 |
PG27 (MW) | 0.110372 | PG87 (MW) | 4.60698 | VG25 (p.u.) | 1.04943 | VG80 (p.u.) | 1.036756 | T127 (80–81) | 0.9 |
PG31 (MW) | 1.978596 | PG89 (MW) | 434.560 | VG26 (p.u.) | 1.00787 | VG85 (p.u.) | 1.049239 | QC34 (MVAR) | 0.20760 |
PG32 (MW) | 2.320654 | PG90 (MW) | 0.12767 | VG27 (p.u.) | 0.97993 | VG87 (p.u.) | 1.059658 | QC44 (MVAR) | 0.00941 |
PG34 (MW) | 0.120587 | PG91 (MW) | 17.5246 | VG31 (p.u.) | 0.98609 | VG89 (p.u.) | 1.010104 | QC45 (MVAR) | 0.26098 |
PG36 (MW) | 5.158081 | PG92 (MW) | 1.45348 | VG32 (p.u.) | 0.94097 | VG90 (p.u.) | 0.963199 | QC46 (MVAR) | 0.07600 |
PG40 (MW) | 32.66632 | PG99 (MW) | 4.11549 | VG34 (p.u.) | 0.94553 | VG91 (p.u.) | 1.042132 | QC48 (MVAR) | 0.24244 |
PG42 (MW) | 3.238611 | PG100 (MW) | 206.177 | VG36 (p.u.) | 1.00105 | VG92 (p.u.) | 1.040622 | QC74 (MVAR) | 0.2194 |
PG46 (MW) | 4.574268 | PG103 (MW) | 35.1167 | VG40 (p.u.) | 0.97223 | VG99 (p.u.) | 0.943471 | QC79 (MVAR) | 5.6 × 10−5 |
PG49 (MW) | 161.9397 | PG104 (MW) | 1.01542 | VG42 (p.u.) | 0.97289 | VG100 (p.u.) | 0.951227 | QC82 (MVAR) | 0.00622 |
PG54 (MW) | 28.01801 | PG105 (MW) | 15.1016 | VG46 (p.u.) | 0.98664 | VG103 (p.u.) | 1.004165 | QC83 (MVAR) | 0.27248 |
PG55 (MW) | 0.389909 | PG107 (MW) | 0 | VG49 (p.u.) | 1.00712 | VG104 (p.u.) | 0.987921 | QC105 (MVAR) | 0.03938 |
PG56 (MW) | 12.21136 | PG110 (MW) | 18.5235 | VG54 (p.u.) | 1.04523 | VG105 (p.u.) | 0.943759 | QC107 (MVAR) | 0.27336 |
PG59 (MW) | 108.0116 | PG111 (MW) | 35.5095 | VG55 (p.u.) | 1.05831 | VG107 (p.u.) | 0.960449 | QC110 (MVAR) | 0.23428 |
PG61 (MW) | 122.726 | PG112 (MW) | 2.19141 | VG56 (p.u.) | 0.96064 | VG110 (p.u.) | 1.00083 | ||
Fuel cost ($/h) | 100738.5 | ||||||||
Power loss (MW) | 88.04623 | ||||||||
Voltage deviation (p.u.) | 0.778536 |
Algorithm | ASO | TFWO | MPA | Rao-1 | Rao-2 | Rao-3 | MRao-2 |
---|---|---|---|---|---|---|---|
Fuel cost ($/h) | 103,847.47 | 101,747.68 | 101,981.69 | 101,981.17 | 101,078.92 | 101,297.12 | 100,738.54 |
Power loss (MW) | 58.3333 | 85.062475 | 71.168974 | 89.992651 | 90.296046 | 91.006497 | 88.04623 |
Voltage deviation (p.u.) | 0.6645742 | 04387701 | 0.7712916 | 1.1227076 | 1.1595186 | 0.9923019 | 0.778536 |
Time (s) | 792.735 | 802.82732 | 1013.509 | 800.827 | 803.4047 | 798.4426 | 1136.06 |
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Hassan, M.H.; Kamel, S.; Selim, A.; Khurshaid, T.; Domínguez-García, J.L. A Modified Rao-2 Algorithm for Optimal Power Flow Incorporating Renewable Energy Sources. Mathematics 2021, 9, 1532. https://doi.org/10.3390/math9131532
Hassan MH, Kamel S, Selim A, Khurshaid T, Domínguez-García JL. A Modified Rao-2 Algorithm for Optimal Power Flow Incorporating Renewable Energy Sources. Mathematics. 2021; 9(13):1532. https://doi.org/10.3390/math9131532
Chicago/Turabian StyleHassan, Mohamed H., Salah Kamel, Ali Selim, Tahir Khurshaid, and José Luis Domínguez-García. 2021. "A Modified Rao-2 Algorithm for Optimal Power Flow Incorporating Renewable Energy Sources" Mathematics 9, no. 13: 1532. https://doi.org/10.3390/math9131532
APA StyleHassan, M. H., Kamel, S., Selim, A., Khurshaid, T., & Domínguez-García, J. L. (2021). A Modified Rao-2 Algorithm for Optimal Power Flow Incorporating Renewable Energy Sources. Mathematics, 9(13), 1532. https://doi.org/10.3390/math9131532