A Modified Rao2 Algorithm for Optimal Power Flow Incorporating Renewable Energy Sources
Abstract
:1. Introduction
 The proposed MRao2 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 118bus 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, Rao1, and Rao3—as well as the original Rao2 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 $it=it+1$
 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. QuasiOppositional
3.2.2. Levy Flight
4. Simulation Results
4.1. Test Systems
4.2. Case 1: The OPF without RES for the IEEE 30Bus System
4.3. Case 2: OPF Incorporating RES for the IEEE 30Bus System
4.4. Case 3: OPF Incorporating RES under Contingency State for IEEE 30Bus System
4.5. Case 4: OPF without RES for the IEEE 118Bus System
4.6. Case 5: OPF Incorporating RES for the IEEE 118Bus 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 30Bus System  IEEE 118Bus 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  Rao1  Rao2  Rao3  MRao2  

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 

MRao2  800.4412  800.553  800.4872 
Rao2  800.6166  800.7965  800.7118 
Rao1  800.8944  801.2647  800.9678 
Rao3  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  Rao1  Rao2  Rao3  MRao2  

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  Rao1  Rao2  Rao3  MRao2  

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  MRao2  800.4412  800.4872  800.4769  800.553  0.038822 
Rao2  800.6166  800.7118  800.7135  800.7965  0.052478  
Rao1  800.8944  800.9678  800.9277  801.2647  0.111619  
Rao3  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  MRao2  729.3429  729.4065  729.4001  729.4615  0.042289 
Rao2  729.5025  729.5599  729.5596  729.6205  0.040197  
Rao1  729.6406  729.6845  729.6815  729.7441  0.031135  
Rao3  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  MRao2  730.583  730.6588  730.6266  730.8189  0.09241 
Rao2  730.6201  730.7573  730.747  730.9311  0.141165  
Rao1  731.0468  731.132  731.1247  731.2359  0.090685  
Rao3  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  Rao1  Rao2  Rao3  MRao2 

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 

P_{G1} (MW)  3.0911  P_{G62} (MW)  6.37571  P_{G113} (MW)  7.09147  V_{G59} (p.u.)  1.036427  V_{G111} (p.u.)  1.03604 
P_{G4} (MW)  7.265278  P_{G65} (MW)  298.824  P_{G116} (MW)  0  V_{G61} (p.u.)  0.956257  V_{G112} (p.u.)  0.97545 
P_{G6} (MW)  79.79656  P_{G66} (MW)  284.31  V_{G1} (p.u.)  1.01134  V_{G62} (p.u.)  1.054308  V_{G113} (p.u.)  1.04051 
P_{G8} (MW)  0.20039  P_{G69} (MW)  400.029  V_{G4} (p.u.)  1.03191  V_{G65} (p.u.)  0.974536  V_{G116} (p.u.)  0.97347 
P_{G10} (MW)  332.3392  P_{G70} (MW)  7.71916  V_{G6} (p.u.)  1.02025  V_{G66} (p.u.)  1.00127  T_{8} (8–5)  1.09856 
P_{G12} (MW)  73.21883  P_{G72} (MW)  0.27496  V_{G8} (p.u.)  0.94257  V_{G69} (p.u.)  1.020322  T_{32} (25–26)  0.90257 
P_{G15} (MW)  6.500848  P_{G73} (MW)  6.45688  V_{G10} (p.u.)  0.94115  V_{G70} (p.u.)  0.979919  T_{36} (17–30)  0.9 
P_{G18} (MW)  5.979357  P_{G74} (MW)  5.52285  V_{G12} (p.u.)  0.97438  V_{G72} (p.u.)  1.030921  T_{51} (37–38)  0.90084 
P_{G19} (MW)  1.346065  P_{G76} (MW)  2.53255  V_{G15} (p.u.)  1.05303  V_{G73} (p.u.)  1.005468  T_{93} (59–63)  1.09369 
P_{G24} (MW)  1.582226  P_{G77} (MW)  0.22016  V_{G18} (p.u.)  1.04361  V_{G74} (p.u.)  1.020992  T_{95} (61–64)  0.96853 
P_{G25} (MW)  159.4184  P_{G80} (MW)  363.967  V_{G19} (p.u.)  0.98928  V_{G76} (p.u.)  1.019236  T_{102} (65–66)  0.93325 
P_{G26} (MW)  215.2208  P_{G85} (MW)  0.07633  V_{G24} (p.u.)  0.96523  V_{G77} (p.u.)  1.034034  T_{107} (68–69)  0.91076 
P_{G27} (MW)  0.110372  P_{G87} (MW)  4.60698  V_{G25} (p.u.)  1.04943  V_{G80} (p.u.)  1.036756  T_{127} (80–81)  0.9 
P_{G31} (MW)  1.978596  P_{G89} (MW)  434.560  V_{G26} (p.u.)  1.00787  V_{G85} (p.u.)  1.049239  Q_{C34} (MVAR)  0.20760 
P_{G32} (MW)  2.320654  P_{G90} (MW)  0.12767  V_{G27} (p.u.)  0.97993  V_{G87} (p.u.)  1.059658  Q_{C44} (MVAR)  0.00941 
P_{G34} (MW)  0.120587  P_{G91} (MW)  17.5246  V_{G31} (p.u.)  0.98609  V_{G89} (p.u.)  1.010104  Q_{C45} (MVAR)  0.26098 
P_{G36} (MW)  5.158081  P_{G92} (MW)  1.45348  V_{G32} (p.u.)  0.94097  V_{G90} (p.u.)  0.963199  Q_{C46} (MVAR)  0.07600 
P_{G40} (MW)  32.66632  P_{G99} (MW)  4.11549  V_{G34} (p.u.)  0.94553  V_{G91} (p.u.)  1.042132  Q_{C48} (MVAR)  0.24244 
P_{G42} (MW)  3.238611  P_{G100} (MW)  206.177  V_{G36} (p.u.)  1.00105  V_{G92} (p.u.)  1.040622  Q_{C74} (MVAR)  0.2194 
P_{G46} (MW)  4.574268  P_{G103} (MW)  35.1167  V_{G40} (p.u.)  0.97223  V_{G99} (p.u.)  0.943471  Q_{C79} (MVAR)  5.6 × 10^{−5} 
P_{G49} (MW)  161.9397  P_{G104} (MW)  1.01542  V_{G42} (p.u.)  0.97289  V_{G100} (p.u.)  0.951227  Q_{C82} (MVAR)  0.00622 
P_{G54} (MW)  28.01801  P_{G105} (MW)  15.1016  V_{G46} (p.u.)  0.98664  V_{G103} (p.u.)  1.004165  Q_{C83} (MVAR)  0.27248 
P_{G55} (MW)  0.389909  P_{G107} (MW)  0  V_{G49} (p.u.)  1.00712  V_{G104} (p.u.)  0.987921  Q_{C105} (MVAR)  0.03938 
P_{G56} (MW)  12.21136  P_{G110} (MW)  18.5235  V_{G54} (p.u.)  1.04523  V_{G105} (p.u.)  0.943759  Q_{C107} (MVAR)  0.27336 
P_{G59} (MW)  108.0116  P_{G111} (MW)  35.5095  V_{G55} (p.u.)  1.05831  V_{G107} (p.u.)  0.960449  Q_{C110} (MVAR)  0.23428 
P_{G61} (MW)  122.726  P_{G112} (MW)  2.19141  V_{G56} (p.u.)  0.96064  V_{G110} (p.u.)  1.00083  
Fuel cost ($/h)  100738.5  
Power loss (MW)  88.04623  
Voltage deviation (p.u.)  0.778536 
Algorithm  ASO  TFWO  MPA  Rao1  Rao2  Rao3  MRao2 

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.; DomínguezGarcía, J.L. A Modified Rao2 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, DomínguezGarcía JL. A Modified Rao2 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 DomínguezGarcía. 2021. "A Modified Rao2 Algorithm for Optimal Power Flow Incorporating Renewable Energy Sources" Mathematics 9, no. 13: 1532. https://doi.org/10.3390/math9131532