# A Hybrid Jaya–Powell’s Pattern Search Algorithm for Multi-Objective Optimal Power Flow Incorporating Distributed Generation

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## Abstract

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## 1. Introduction

- The main contribution of this paper is to implement hybridization of two algorithms (Jaya and Powell’s Pattern Search) in different manners and at different levels to find the best option for hybridization.
- Powell’s Pattern Search method has been incorporated into the Jaya algorithm in three different ways, resulting in three variants, namely, J-PPS1, J-PPS2 and J-PPS3.
- The proposed hybrid Jaya and Powell’s Pattern Search method utilizes the exploration property of the Jaya algorithm and the exploitation quality of Powell’s Pattern Search method.
- This paper handles the OPF problem considering DG with four objectives functions simultaneously, namely, minimization of fuel cost, emission, real power losses and voltage profile improvement by converting the multi-objective OPF into a single objective OPF.
- In addition to Dragonfly Algorithm (DA), Grey Wolf Optimization (GWO) and Classical Jaya algorithms, three versions of hybrid Jaya and PPS, J-PPS1, J-PPS2 and J–PPS3 for the OPF problem are developed, wherein the excellent search capability of the PPS method has been exploited for further improvement of the solution provided by Jaya algorithm.

## 2. Problem Formulation

#### 2.1. OPF Objective Functions

#### 2.1.1. Fuel Cost Minimization

#### 2.1.2. Emission Cost Minimization

#### 2.1.3. Real Power Losses Minimization

#### 2.1.4. Voltage Profile Improvement

#### 2.2. Constraints

#### 2.3. Combined Objective Function (COF)

#### 2.4. Incorporation of Constraints

## 3. Jaya Algorithm

**Maximized (or minimized) value of objective function M(z)**

#### 3.1. Powell’s Pattern Search (PPS)

#### 3.2. Proposed Hybrid Jaya–PPS Algorithm

- Initialize the population with control variables and set maximum iteration count IterJmax and the number of iterations IterJmax for the PPS method.
- Set iteration Iter = 0.
- Identify the worst and best solutions in the population on the basis of the extended objective function value Equation (11).
- Modify the solutions using the best and the worst solutions Equation (12).
- If the modified solution is found to be better than the previous one, move to step vi, otherwise jump to step vii.
- Replace the previous solution with the modified one and jump to step viii.
- Retain previous solution.
- Increase iteration number by 1, i.e., Iter = Iter + 1.
- If Iter < IterJmax, then move to step iii, else move to step x.
- Select the best solution found by the Jaya algorithm as the initial point for the PPS method and apply the PPS method for IterPmax iterations to attain a better solution.
- Stop. Optimal solution achieved.

## 4. OPF Results and Discussion

_{ECM}, W

_{RPLM}and W

_{TVDM}as 19, 22 and 21, respectively, as reported in [9]. The same procedure can be used for different systems also.

_{ECM}, W

_{RPLM}and W

_{TVDM}as 300, 30 and 600, respectively. IEEE 57-bus system is modified by inserting DG units [26]. The optimal locations of the type 1 DG units are bus nos. 35 and 36 with the capacities of 47.9067 MW and 47.2636 MW, respectively.

- Case 1: OPF no DG in IEEE 30-bus test system.
- Case 2: OPF with DG in IEEE 30-bus test system.
- Case 3: OPF no DG in IEEE 57-bus test system.
- Case 4: OPF with DG in IEEE 57-bus test system.
- Case 5: OPF no DG in IEEE 118-bus system.

#### 4.1. Case 1: OPF No DG in IEEE 30-Bus Test System

#### 4.2. Case 2: OPF with DG in IEEE 30-Bus Test System

#### 4.3. Case 3: OPF No DG in IEEE 57-Bus Test System

#### 4.4. Case 4: OPF with DG in IEEE 57-Bus Test System

#### 4.5. Case 5: OPF No DG in IEEE 118-Bus System

## 5. Statistical Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

W | dependent variable |

X | control variable |

M (W, X) | objective function |

g (W, X) & h (W, X) | equality and inequality constraints |

NB | number of buses |

NLB | number of load buses |

Ntl | number of transmission lines |

NGN | numbers of generators |

NC | number of VAR compensation units |

NTR | number of regulating transformers |

${P}_{di}$ and ${Q}_{di}$ | active and reactive load |

${P}_{gi}$ and ${Q}_{gi}$ | Active & reactive power generations |

${P}_{Loss}$ and ${Q}_{Loss}$ | real and reactive power loss |

${U}_{{g}_{k}}^{min}$ and ${U}_{{g}_{k}}^{max}$ | minimum & maximum voltage limits of kth generator bus |

${Q}_{{g}_{k}}^{min}\text{}\mathrm{and}{Q}_{{g}_{k}}^{max}$ | minimum and maximum limits of reactive power output of kth generator |

${P}_{{g}_{k}}^{min}\text{}\mathrm{and}{P}_{{g}_{k}}^{max}$ | minimum and maximum active power limits of kth generating unit |

${T}_{k}^{max}$ and ${T}_{k}^{min}$ | maximum and minimum tap setting of kth transformer |

${U}_{{L}_{k}}^{max}\text{}\mathrm{and}\text{}{U}_{{L}_{k}}^{min}$ | maximum and minimum voltage limit of kth load bus |

${S}_{{l}_{k}}^{max}$ | maximum MVA flow in kth transmission line |

${C}_{1}$, ${C}_{2}$, ${C}_{3}$ and ${C}_{4}$ | penalty factors corresponding to limit violations |

${A}_{i},{B}_{i}$, and ${C}_{i}$ | fuel cost coefficients of the ith generating unit |

${P}_{{g}_{1}}$ | slack bus generator’s active power output |

${\alpha}_{i},{\beta}_{i},{\gamma}_{i},{\xi}_{i},{\lambda}_{i}$ | emission coefficients of ith generating unit |

pop | Population size |

Itermax | Maximum No. of iterations |

IterJmax | Maximum No. of Jaya iterations |

IterPmax | Maximum No. of PPS iterations |

NFE | Number of function evaluations |

JFE | Number of Jaya Function Evaluations |

PSFE | Number of PPS Function Evaluation |

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IEEE-30 Bus System | |||

Algorithm | Population | Iterations | Total NFE = 6000 |

Dragonfly Algorithm | 30 | 200 | 30 × 200 |

GWO Algorithm | 30 | 200 | 30 × 200 |

Jaya Algorithm | 30 | 200 | 30 × 200 |

J-PPS1 | 30 | 200 | 30JFE × 50 + 30PSFE × 150 |

J-PPS2 | 30 | 200 | 30JFE × 100 + 30PSFE × 100 |

J-PPS3 | 30 | 200 | 30JFE × 150 + 30 PSFE × 50 |

IEEE-57 bus system and IEEE 118 Bus System | |||

Algorithm | Population | Iterations | Total NFE = 12,000 |

Dragonfly Algorithm | 40 | 300 | 40 × 300 |

GWO Algorithm | 40 | 300 | 40 × 300 |

Jaya Algorithm | 40 | 300 | 40 × 300 |

J-PPS1 | 40 | 300 | 40JFE × 75 + 40PSFE × 225 |

J-PPS2 | 40 | 300 | 40JFE × 150 + 40PSFE × 150 |

J-PPS3 | 40 | 300 | 40JFE × 225 + 40PSFE × 75 |

S. No. | Control Variable | DA | GWO | Jaya | J-PPS1 | J-PPS2 | J-PPS3 |
---|---|---|---|---|---|---|---|

Generator real power output | |||||||

1 | Pg2 | 0.52656 | 0.52553 | 0.5167 | 0.5259 | 0.5266 | 0.527 |

2 | Pg5 | 0.31146 | 0.31068 | 0.32214 | 0.3156 | 0.3165 | 0.3155 |

3 | Pg8 | 0.35 | 0.35 | 0.3497 | 0.3496 | 0.3496 | 0.35 |

4 | Pg11 | 0.25774 | 0.26257 | 0.27264 | 0.2699 | 0.2692 | 0.2652 |

5 | Pg13 | 0.21671 | 0.21185 | 0.20712 | 0.2115 | 0.2091 | 0.2099 |

Generator voltage setting | |||||||

6 | Vg1 | 1.07429 | 1.07452 | 1.0728 | 1.0724 | 1.0735 | 1.0731 |

7 | Vg2 | 1.05972 | 1.06035 | 1.05906 | 1.0584 | 1.0597 | 1.0593 |

8 | Vg5 | 1.03127 | 1.03473 | 1.03371 | 1.0308 | 1.0332 | 1.0318 |

9 | Vg8 | 1.04147 | 1.0423 | 1.04155 | 1.0406 | 1.0409 | 1.0402 |

10 | Vg11 | 1.05456 | 1.05344 | 1.05018 | 1.0555 | 1.0431 | 1.0402 |

11 | Vg13 | 1.01607 | 1.01938 | 1.02735 | 1.0179 | 1.0206 | 1.0186 |

Transformer tap setting | |||||||

12 | T6-9 | 1.06778 | 1.08906 | 1.1 | 1.0991 | 1.0895 | 1.1 |

13 | T6-10 | 1.01404 | 0.9811 | 0.94836 | 0.9499 | 0.9586 | 0.9435 |

14 | T4-12 | 1.02163 | 1.01232 | 1.02587 | 1.0347 | 1.0304 | 1.0345 |

15 | T28-27 | 1.00183 | 1.00725 | 1.00342 | 1.0095 | 1.0024 | 1.0023 |

Shunt VAR source setting | |||||||

16 | Qc10 | 0.04965 | 0.0486 | 0 | 0.0104 | 0.0273 | 0.0225 |

17 | Qc12 | 0.00025 | 0.0009 | 0.00054 | 0.0498 | 0.0031 | 0.0477 |

18 | Qc15 | 0.03634 | 0.01863 | 0.04966 | 0.0344 | 0.0321 | 0.0474 |

19 | Qc17 | 0.04876 | 0.03188 | 0.05 | 0.0343 | 0.0441 | 0.05 |

20 | Qc20 | 0.0499 | 0.04829 | 0.04985 | 0.0478 | 0.0479 | 0.0481 |

21 | Qc21 | 0.05 | 0.05 | 0.04997 | 0.05 | 0.0499 | 0.0497 |

22 | Qc23 | 0.04898 | 0.04621 | 0.01739 | 0.0486 | 0.05 | 0.04 |

23 | Qc24 | 0.04978 | 0.05 | 0.04986 | 0.0497 | 0.0484 | 0.0482 |

24 | Qc29 | 0.02535 | 0.03226 | 0.03039 | 0.0344 | 0.0271 | 0.0274 |

COF | 965.3516 | 965.3025 | 965.2868 | 965.2159 | 965.1201 | 965.0228 | |

Fuel Cost | 829.3587 | 829.2395 | 831.5493 | 830.9938 | 830.8672 | 830.3088 | |

Emission | 0.2370 | 0.2373 | 0.2358 | 0.2355 | 0.2357 | 0.2363 | |

Real Power Loss (RPL) | 5.6859 | 5.6843 | 5.5780 | 5.6120 | 5.6175 | 5.6377 | |

Total Voltage Deviation (TVD) | 0.3046 | 0.3094 | 0.3114 | 0.2990 | 0.2948 | 0.2949 | |

L-Index | 0.1387 | 0.1389 | 0.1396 | 0.1392 | 0.1393 | 0.1388 | |

Pg1 | 122.8389 | 123.0213 | 122.1479 | 121.7620 | 121.9175 | 122.2777 |

Algorithm | Comb. Obj Fun (COF) | Fuel Cost ($/h) | Emission (ton/h) | RPL (MW) | TVD (pu) |
---|---|---|---|---|---|

Base Case | 1336.64501 | 902.00457 | 0.22232 | 5.84233 | 1.16014 |

DA | 965.35164 | 829.35878 | 0.23705 | 5.68593 | 0.30469 |

GWO | 965.30257 | 829.23953 | 0.23731 | 5.68435 | 0.30945 |

Jaya | 965.28681 | 831.54930 | 0.23582 | 5.57800 | 0.31147 |

J-PPS1 | 965.2159 | 830.9938 | 0.2355 | 5.6120 | 0.2990 |

J-PPS2 | 965.1201 | 830.8672 | 0.2357 | 5.6175 | 0.2948 |

J-PPS3 | 965.0228 | 830.3088 | 0.2363 | 5.6377 | 0.2949 |

MSA [11] | 965.2905 | 830.639 | 0.25258 | 5.6219 | 0.29385 |

MPSO [11] | 986.0063 | 833.6807 | 0.25251 | 6.5245 | 0.18991 |

MDE [11] | 973.6116 | 829.0942 | 0.2575 | 6.0569 | 0.30347 |

MFO [11] | 965.8077 | 830.9135 | 0.25231 | 5.5971 | 0.33164 |

FPA [11] | 971.9076 | 835.3699 | 0.24781 | 5.5153 | 0.49969 |

MSA [6] | * | 838.9233 | 0.2116 | 5.6149 | 0.1535 |

ABC [6] | * | 835.5230 | 0.2076 | 5.3948 | 0.1380 |

CSA [6] | * | 834.5125 | 0.2099 | 5.4250 | 0.1373 |

GWO [6] | * | 851.0491 | 0.2057 | 4.8925 | 0.2015 |

BSOA [6] | * | 830.7115 | 0.2251 | 5.7446 | 0.1836 |

MJAYA [6] | * | 833.3410 | 0.2064 | 5.1779 | 0.1196 |

MOEA/D-SF [12] | - | 883.322 | 0.21867 | 4.4527 | 0.1322 |

MOMICA [10] | - | 830.1884 | 0.2523 | 5.5851 | 0.2978 |

MOICA [10] | - | 831.2251 | 0.267 | 6.0223 | 0.4046 |

MNSGA-II [10] | - | 834.5616 | 0.2527 | 5.6606 | 0.4308 |

BB-MOPSO [10] | - | 833.0345 | 0.2479 | 5.6504 | 0.3945 |

NKEA [10] | - | 834.6433 | 0.2491 | 5.8935 | 0.4448 |

FKH [9] | - | 828.3271 | 0.2549 | 5.3828 | 0.4925 |

KH [9] | - | 827.7054 | 0.2526 | 5.4977 | 0.4930 |

FA [9] | - | 829.5778 | 0.2527 | 5.5104 | 0.5661 |

S. No. | Control Variable | DA | GWO | Jaya | J-PPS1 | J-PPS2 | J-PPS3 |
---|---|---|---|---|---|---|---|

Generator Real power output | |||||||

1 | Pg2 | 0.51902 | 0.51557 | 0.51579 | 0.516 | 0.5161 | 0.5162 |

2 | Pg5 | 0.31184 | 0.31164 | 0.31143 | 0.3119 | 0.3116 | 0.311 |

3 | Pg8 | 0.3500 | 0.34999 | 0.3500 | 0.35 | 0.35 | 0.35 |

4 | Pg11 | 0.25881 | 0.2574 | 0.26254 | 0.2596 | 0.2619 | 0.261 |

5 | Pg13 | 0.20565 | 0.2019 | 0.20564 | 0.2063 | 0.2039 | 0.2056 |

Generator voltage setting | |||||||

6 | Vg1 | 1.07105 | 1.07422 | 1.06371 | 1.0709 | 1.0732 | 1.0724 |

7 | Vg2 | 1.05748 | 1.06016 | 1.04926 | 1.0579 | 1.0594 | 1.0595 |

8 | Vg5 | 1.03158 | 1.03316 | 1.02234 | 1.0297 | 1.0329 | 1.0325 |

9 | Vg8 | 1.04001 | 1.04209 | 1.03217 | 1.0381 | 1.0434 | 1.0423 |

10 | Vg11 | 1.09934 | 1.04159 | 1.04782 | 1.0459 | 1.0379 | 1.0416 |

11 | Vg13 | 1.02435 | 1.01592 | 1.02994 | 1.023 | 1.0143 | 1.0164 |

Transformer tap setting | |||||||

12 | T6-9 | 1.01015 | 1.09902 | 1.09958 | 1.0981 | 1.0997 | 1.0998 |

13 | T6-10 | 1.1 | 0.92446 | 0.92521 | 0.958 | 0.9565 | 0.9585 |

14 | T4-12 | 1.0343 | 1.02314 | 1.03347 | 1.0398 | 1.0182 | 1.0241 |

15 | T28-27 | 1.0054 | 1.02139 | 1.00165 | 1.0028 | 1.013 | 1.0089 |

Shunt VAR source setting | |||||||

16 | Qc10 | 0.00001 | 0.00136 | 0.00394 | 0.0436 | 0.0498 | 0.0494 |

17 | Qc12 | 0.01415 | 0.04996 | 0.00005 | 0.0422 | 0.0344 | 0.0251 |

18 | Qc15 | 0.04982 | 0.03682 | 0.04071 | 0.0457 | 0.0385 | 0.0454 |

19 | Qc17 | 0.03936 | 0.05 | 0.04992 | 0.0484 | 0.05 | 0.0457 |

20 | Qc20 | 0.02121 | 0.00011 | 0.04959 | 0.0481 | 0.05 | 0.0484 |

21 | Qc21 | 0.04912 | 0.05 | 0.04867 | 0.0492 | 0.05 | 0.05 |

22 | Qc23 | 0.03649 | 0.05 | 0.03892 | 0.0386 | 0.04 | 0.0397 |

23 | Qc24 | 0.05 | 0.05 | 0.04854 | 0.0482 | 0.05 | 0.05 |

24 | Qc29 | 0.01492 | 0.05 | 0.02444 | 0.012 | 0.0235 | 0.0211 |

COF | 938.5816 | 938.4980 | 938.3787 | 937.6646 | 937.3837 | 937.3486 | |

Fuel Cost | 811.9476 | 811.2105 | 812.3347 | 811.9609 | 811.8993 | 811.8635 | |

Emission Cost | 0.2328 | 0.2340 | 0.2327 | 0.2329 | 0.2330 | 0.2329 | |

Real Power Loss | 5.2318 | 5.2836 | 5.2871 | 5.2381 | 5.2171 | 5.2214 | |

Total Voltage Deviation | 0.3385 | 0.3142 | 0.2525 | 0.2875 | 0.2990 | 0.2946 | |

Pg1 | 119.0998 | 120.0336 | 119.1471 | 119.2581 | 119.2671 | 119.2414 | |

L-Index (LI) | 0.1046 | 0.1017 | 0.1036 | 0.1027 | 0.1017 | 0.1019 |

S. No | Control Variable | DA | GWO | Jaya | J-PPS1 | J-PPS2 | J-PPS3 |
---|---|---|---|---|---|---|---|

Generator active power output | |||||||

1 | Pg2 | 0.9998 | 1 | 1 | 0.9997 | 0.9994 | 0.998 |

2 | Pg3 | 0.52822 | 0.63533 | 0.57092 | 0.6064 | 0.6052 | 0.6058 |

3 | Pg6 | 0.9934 | 0.92366 | 0.87967 | 1 | 0.9631 | 0.9196 |

4 | Pg8 | 3.15449 | 3.13702 | 3.21403 | 3.1116 | 3.1263 | 3.1868 |

5 | Pg9 | 0.99979 | 1 | 0.99992 | 0.9998 | 0.9994 | 0.9964 |

6 | Pg12 | 4.09988 | 4.0985 | 4.09921 | 4.0848 | 4.1 | 4.1 |

Generator voltage setting | |||||||

7 | Vg1 | 1.03896 | 1.04785 | 1.0333 | 1.0248 | 1.0321 | 1.0292 |

8 | Vg2 | 0.95129 | 1.09823 | 1.09987 | 1.1 | 1.0873 | 1.0761 |

9 | Vg3 | 1.0799 | 0.97546 | 1.08977 | 0.95 | 1.1 | 1.0919 |

10 | Vg6 | 0.95 | 1.02 | 0.97047 | 1.0289 | 1.0209 | 1.0343 |

11 | Vg8 | 0.99115 | 0.99545 | 1.00747 | 1.011 | 1.0133 | 1.0118 |

12 | Vg9 | 0.95157 | 1.03031 | 0.97345 | 1.0273 | 1.0485 | 1.0125 |

13 | Vg12 | 1.00525 | 1.01681 | 1.01609 | 1.0214 | 1.0109 | 1.0211 |

Transformer tap setting | |||||||

14 | T4-18 | 1.06804 | 1.09919 | 0.98026 | 0.9344 | 0.9205 | 1.0968 |

15 | T4-18 | 0.90157 | 0.9 | 0.91699 | 1.0891 | 1.0094 | 0.9272 |

16 | T21-20 | 1.00286 | 0.97166 | 1.09774 | 0.9698 | 0.9791 | 0.9821 |

17 | T24-25 | 0.95085 | 1.03327 | 1.09303 | 0.9958 | 1.0922 | 1.0199 |

18 | T24-25 | 1.0109 | 1.06638 | 0.90024 | 1.0045 | 0.9006 | 0.975 |

19 | T24-26 | 1.04559 | 1.03573 | 1.0347 | 1.0213 | 1.0562 | 1.0157 |

20 | T7-29 | 0.92573 | 0.95287 | 0.94193 | 0.9534 | 0.9446 | 0.9336 |

21 | T34-32 | 0.92662 | 0.93717 | 0.93982 | 0.9444 | 0.916 | 0.9304 |

22 | T11-41 | 0.90366 | 0.9 | 0.90013 | 0.9073 | 0.9 | 0.9116 |

23 | T15-45 | 0.9482 | 0.96314 | 0.94642 | 0.9546 | 0.9364 | 0.9581 |

24 | T14-46 | 0.94622 | 0.96437 | 0.97134 | 0.9641 | 0.976 | 0.977 |

25 | T10-51 | 0.98181 | 1.02617 | 0.99404 | 0.9977 | 0.9789 | 0.9893 |

26 | T13-49 | 0.92296 | 0.9 | 0.93097 | 0.9086 | 0.912 | 0.9 |

27 | T11-43 | 0.91315 | 0.9141 | 0.92554 | 0.9278 | 0.9504 | 0.95 |

28 | T40-56 | 1.09814 | 1.03063 | 1.06585 | 1.0017 | 0.9851 | 0.9818 |

29 | T39-57 | 0.90081 | 0.95428 | 0.91838 | 0.9324 | 0.9307 | 0.9135 |

30 | T9-55 | 0.97945 | 0.94635 | 0.99248 | 0.9699 | 0.9735 | 1.0015 |

Shunt VAR source setting | |||||||

31 | Qc18 | 0.05841 | 0.0326 | 0.00118 | 0.1531 | 0 | 0.0857 |

32 | Qc25 | 0.08446 | 0.19243 | 0.12834 | 0.1706 | 0.0792 | 0.1308 |

33 | Qc53 | 0.14752 | 0.10041 | 0.14808 | 0.1577 | 0.0795 | 0.1171 |

COF | 43,887.4176 | 43,864.8418 | 43,833.6421 | 43,825.8807 | 43,793.8820 | 43,788.6319 | |

Fuel Cost | 42,584.4552 | 42,587.9655 | 42,547.0948 | 42,575.9726 | 42,580.0946 | 42,564.4608 | |

Emission Cost | 1.3577 | 1.3447 | 1.3708 | 1.3336 | 1.3433 | 1.3566 | |

Real Power Loss | 13.6065 | 13.2727 | 12.772 | 12.5408 | 12.5242 | 12.5079 | |

Voltage Deviation | 0.8124 | 0.7921 | 0.8202 | 0.7893 | 0.7251 | 0.7365 | |

Pg1 | 186.8485 | 184.6217 | 187.1970 | 183.1103 | 183.9874 | 182.6473 | |

L-Index | 0.2638 | 0.2429 | 0.2512 | 0.2418 | 0.2669 | 0.2501 |

Algorithm | COF | FCost ($/h) | Emission (ton/h) | PLoss (MW) | TVD (pu) |
---|---|---|---|---|---|

Base Case | 53,828.14303 | 51,395.57064 | 2.76165 | 28.36589 | 1.25517 |

DA | 43,887.43700 | 42,584.46959 | 1.35770 | 13.60665 | 0.81243 |

GWO | 43,864.84184 | 42,587.97294 | 1.34478 | 13.27275 | 0.79209 |

Jaya | 43,833.62963 | 42,547.09273 | 1.37089 | 12.77199 | 0.82018 |

J-PPS1 | 43,825.8807 | 42,575.9726 | 1.33366 | 12.54089 | 0.78931 |

J-PPS2 | 43,793.88205 | 42,580.09468 | 1.34333 | 12.52422 | 0.72511 |

J-PPS3 | 43,788.63196 | 42,564.46087 | 1.35666 | 12.50792 | 0.73656 |

MOMICA [10] | - | 41,983.0585 | 1.496 | 13.6969 | 0.797 |

MOICA [10] | - | 41,998.5661 | 1.7605 | 13.3353 | 0.8748 |

MNSGA-II [10] | - | 42,070.82476 | 1.4965 | 14.4557 | 0.8896 |

BB-MOPSO [10] | - | 41,994.019127 | 1.5336 | 12.609 | 1.0742 |

NKEA [10] | - | 42,065.9964 | 1.5174 | 13.9764 | 1.042 |

MOEA/D-SF [12] | - | 42,648.69 | 1.3437 | 11.8862 | 0.6713 |

S. No | Control Variable | DA | GWO | Jaya | J-PPS1 | J-PPS2 | J-PPS3 |
---|---|---|---|---|---|---|---|

Generator active power output | |||||||

1 | Pg2 | 0.90988 | 0.9991 | 0.89863 | 0.9371 | 0.9368 | 0.9836 |

2 | Pg3 | 0.47575 | 0.49202 | 0.47457 | 0.4728 | 0.4807 | 0.468 |

3 | Pg6 | 0.59547 | 0.36955 | 0.46213 | 0.4847 | 0.3041 | 0.4147 |

4 | Pg8 | 3.33676 | 3.49929 | 3.47521 | 3.3894 | 3.5637 | 3.4798 |

5 | Pg9 | 0.99353 | 0.99999 | 0.99976 | 0.9991 | 0.9981 | 0.9999 |

6 | Pg12 | 3.94011 | 3.86025 | 3.94345 | 3.975 | 3.9554 | 3.8995 |

Generator voltage setting | |||||||

7 | Vg1 | 1.01833 | 1.0183 | 1.01557 | 1.01 | 1.0157 | 1.0166 |

8 | Vg2 | 1.1 | 1.09571 | 1.09894 | 1.0944 | 1.1 | 1.0623 |

9 | Vg3 | 1.06087 | 1.05416 | 1.06987 | 1.0323 | 1.0681 | 1.0749 |

10 | Vg6 | 0.95 | 1.08551 | 1.0955 | 1.0069 | 1.0442 | 1.0497 |

11 | Vg8 | 1.01291 | 1.01015 | 1.00346 | 0.9835 | 0.995 | 1.0026 |

12 | Vg9 | 1.01131 | 0.95038 | 0.98631 | 1.0073 | 1.0118 | 1.0272 |

13 | Vg12 | 1.01181 | 1.02021 | 1.00427 | 0.9893 | 1.0088 | 1.0031 |

Transformer tap setting | |||||||

14 | T4-18 | 0.90037 | 1.09424 | 0.9 | 0.9 | 1.1 | 1.0552 |

15 | T4-18 | 1.09998 | 0.9 | 1.0984 | 1.1 | 0.9078 | 0.9054 |

16 | T21-20 | 1.04432 | 0.98565 | 0.98642 | 0.9794 | 0.9887 | 0.9924 |

17 | T24-25 | 0.90006 | 1.1 | 1.034 | 1.0489 | 1.0928 | 1.0455 |

18 | T24-25 | 1.07936 | 0.9 | 0.97881 | 1.0794 | 1.0714 | 1.1 |

19 | T24-26 | 1.03695 | 1.00296 | 1.00993 | 0.9978 | 1.0712 | 1.0429 |

20 | T7-29 | 0.97201 | 0.97227 | 0.95166 | 0.9815 | 0.9461 | 0.9496 |

21 | T34-32 | 0.94356 | 1.00622 | 0.99596 | 0.986 | 0.989 | 0.9887 |

22 | T11-41 | 0.9784 | 0.96604 | 0.95685 | 0.9594 | 0.967 | 0.9653 |

23 | T15-45 | 0.97868 | 0.97901 | 0.98099 | 0.9703 | 0.9716 | 0.9806 |

24 | T14-46 | 0.97857 | 0.97699 | 0.96899 | 0.9481 | 0.9764 | 0.9876 |

25 | T10-51 | 0.98858 | 0.99304 | 0.98007 | 0.9786 | 0.9806 | 0.9783 |

26 | T13-49 | 0.92431 | 0.93811 | 0.93098 | 0.9007 | 0.9317 | 0.9271 |

27 | T11-43 | 0.99037 | 1.00372 | 0.98421 | 0.9637 | 0.9821 | 0.9837 |

28 | T40-56 | 0.91921 | 0.90084 | 0.93647 | 0.9182 | 0.9 | 0.9265 |

29 | T39-57 | 0.98075 | 0.99828 | 0.98771 | 0.9938 | 1.0123 | 0.979 |

30 | T9-55 | 0.95101 | 0.98913 | 0.98154 | 0.9159 | 0.9474 | 0.9383 |

Shunt VAR source setting | |||||||

31 | Qc18 | 0.18621 | 0.00004 | 0.12891 | 0.1062 | 0.0216 | 0.0212 |

32 | Qc25 | 0.06171 | 0.14674 | 0.11436 | 0.1869 | 0.169 | 0.1737 |

33 | Qc53 | 0.1296 | 0.19995 | 0.16624 | 0.1797 | 0.0752 | 0.065 |

COF | 39,200.1782 | 39,173.0979 | 39,162.8890 | 39,167.5961 | 39,165.9645 | 39,136.3249 | |

Fuel Cost | 38,120.8335 | 38,114.7354 | 38,105.9569 | 38,059.9136 | 38,048.2507 | 38,033.8329 | |

Emission Cost | 1.2751 | 1.3099 | 1.3218 | 1.3035 | 1.3612 | 1.3115 | |

Real Power Loss | 12.3189 | 13.1703 | 12.5706 | 12.8818 | 13.3724 | 12.9742 | |

Total Voltage Deviation | 0.5454 | 0.4504 | 0.4721 | 0.5469 | 0.5136 | 0.5329 | |

Pg1 | 142.7987 | 146.7800 | 142.8253 | 142.7015 | 145.1223 | 144.0540 | |

L-Index | 0.1393 | 0.1241 | 0.1290 | 0.12921 | 0.1252 | 0.1250 |

Algorithm | Fuel Cost ($/h) | TVD (pu) | PG69 (Slack Bus) | Power Loss | |
---|---|---|---|---|---|

MW | MVAr | ||||

Base Case | 131,220.0208 | 1.4389 | 513.8101 | 132.8101 | 782.6073 |

J-PPS1 | 129,961.8924 | 1.4402 | 489.0344 | 113.4784 | 745.3196 |

J-PPS2 | 129,821.4309 | 1.5238 | 430.2158 | 118.5608 | 762.0786 |

J-PPS3 | 129,507.6123 | 1.3486 | 440.1366 | 109.6528 | 668.4798 |

Jaya | 130,165.8424 | 1.4991 | 482.2581 | 112.9269 | 740.0970 |

DA | 130,016.5235 | 1.4596 | 450.9608 | 119.1369 | 751.3072 |

GWO | 130,053.1453 | 1.4015 | 461.0356 | 108.2561 | 698.1435 |

IMFO [20] | 131,820.0000 | 1.5944 | 407.192 | 77.6522 | −910.020 |

Interior point [30] | 129,720.70 | N. A | N. A | N. A | N. A |

CC-ACOPF [31] | 129,662.0 | N. A | N. A | N. A | N. A |

NLP [32] | 129,700 | N. A | N. A | N. A | N. A |

QP [32] | 129,600 | N. A | N. A | N. A | N. A |

MIQP [32] | 129,600 | N. A | N. A | N. A | N. A |

ALC-PSO [33] | 129,546.0847 | N. A | N. A | N. A | N. A |

PSOGSA [34] | 129,733.58 | N. A | N. A | 73.21 | N. A |

GPU-PSO [35] | 129,627.03 | N. A | N. A | 76.984 | N. A |

Algorithm | Without DG | Incorporating DG | ||||||
---|---|---|---|---|---|---|---|---|

Best | Worst | Mean | Std. Deviation | Best | Worst | Mean | Std. Deviation | |

DA | 965.3516 | 966.4352 | 965.8734 | 0.02526 | 938.5816 | 939.1761 | 938.7554 | 0.02615 |

GWO | 965.3025 | 966.7339 | 965.7564 | 0.02151 | 938.4980 | 939.2469 | 938.8678 | 0.02387 |

Jaya | 965.2868 | 966.8154 | 965.8975 | 0.01983 | 938.3787 | 939.2543 | 938.9781 | 0.01955 |

Jaya–PPS1 | 965.2159 | 965.6587 | 965.4260 | 0.01851 | 937.6646 | 937.8942 | 937.7815 | 0.01829 |

Jaya–PPS2 | 965.1201 | 965.4089 | 965.2481 | 0.01809 | 937.3837 | 937.6582 | 937.4982 | 0.01785 |

Jaya–PPS3 | 965.0228 | 965.3261 | 965.2094 | 0.01132 | 937.3486 | 937.5803 | 937.4623 | 0.01105 |

Algorithm | Without DG | Incorporating DG | ||||||
---|---|---|---|---|---|---|---|---|

Best | Worst | Mean | Std. Deviation | Best | Worst | Mean | Std. Deviation | |

DA | 43,887.437 | 43,973.873 | 43,893.893 | 0.02988 | 39,200.178 | 39,218.879 | 39,207.656 | 0.02887 |

GWO | 43,864.841 | 43,896.887 | 43,871.698 | 0.02917 | 39,173.097 | 39,181.365 | 39,178.432 | 0.02828 |

Jaya | 43,833.629 | 43,845.953 | 43,839.894 | 0.02820 | 39,162.889 | 39,175.542 | 39,168.764 | 0.02812 |

Jaya–PPS1 | 43,825.880 | 43,839.720 | 43,833.542 | 0.02588 | 39,167.596 | 39,176.742 | 39,172.427 | 0.02609 |

Jaya–PPS2 | 43,793.882 | 43,804.659 | 43,800.752 | 0.02602 | 39,165.964 | 39,174.694 | 39,169.524 | 0.02531 |

Jaya–PPS3 | 43788.631 | 43797.462 | 43793.298 | 0.01299 | 39136.324 | 39140.437 | 39138.542 | 0.01297 |

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## Share and Cite

**MDPI and ACS Style**

Gupta, S.; Kumar, N.; Srivastava, L.; Malik, H.; Pliego Marugán, A.; García Márquez, F.P.
A Hybrid Jaya–Powell’s Pattern Search Algorithm for Multi-Objective Optimal Power Flow Incorporating Distributed Generation. *Energies* **2021**, *14*, 2831.
https://doi.org/10.3390/en14102831

**AMA Style**

Gupta S, Kumar N, Srivastava L, Malik H, Pliego Marugán A, García Márquez FP.
A Hybrid Jaya–Powell’s Pattern Search Algorithm for Multi-Objective Optimal Power Flow Incorporating Distributed Generation. *Energies*. 2021; 14(10):2831.
https://doi.org/10.3390/en14102831

**Chicago/Turabian Style**

Gupta, Saket, Narendra Kumar, Laxmi Srivastava, Hasmat Malik, Alberto Pliego Marugán, and Fausto Pedro García Márquez.
2021. "A Hybrid Jaya–Powell’s Pattern Search Algorithm for Multi-Objective Optimal Power Flow Incorporating Distributed Generation" *Energies* 14, no. 10: 2831.
https://doi.org/10.3390/en14102831