# An Improved Equilibrium Optimizer for Solving Optimal Power Flow Problem

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## Abstract

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## 1. Introduction

- An improvement in the efficiency of powerful metaheuristic method named EO is investigated to solve optimal power flow problems in the power system.
- An improved equilibrium optimizer algorithm called IEO, combined with a chaotic equilibrium pool, nonlinear dynamic generation mechanism and golden sine algorithm, is developed to enhance the ability of the original EO algorithm to handle complex optimization objectives. The performance of the IEO algorithm is evaluated on 16 benchmark test functions, the Wilcoxon rank sum test and well-known CEC2014 test functions.
- The proposed IEO algorithm is applied to solve optimal power flow problems in the standard IEEE 30-bus test system. The performance of the proposed IEO algorithm is investigated in terms of fuel cost, active power transmission loss, and voltage deviation improvement. The results are compared with those of other improved algorithms and metaheuristic algorithms in the literature.

## 2. OPF Problem Formulation

#### 2.1. Objective Function

#### 2.1.1. Fuel Cost (FC)

#### 2.1.2. Active Power Transmission Loss (APL)

_{ij}) is the difference of the voltage phase angle between buses i and j.

#### 2.1.3. Voltage Deviation (VD)

#### 2.2. Contraints

#### 2.2.1. Equality Constraints

#### 2.2.2. Inequality Constraints

## 3. Improved Equilibrium Optimizer

#### 3.1. Equilibrium Optimizer (EO)

_{eq}is the concentration at the equilibrium state, and G is the mass generation rate of the control volume.

#### 3.2. Improved Equilibrium Optimizer (IEO)

#### 3.2.1. Chaotic Equilibrium Pool Leading Strategy

#### 3.2.2. Nonlinear Dynamic Generation Mechanism

#### 3.2.3. Golden Sine Position Update Strategy

#### 3.2.4. Detailed Steps for the Improved Equilibrium Optimizer

#### 3.2.5. Analysis of Time Complexity of Improved Algorithm

## 4. The Simulation Results

#### 4.1. Comparative Analysis of Algorithm Performance

#### 4.2. Convergence Analysis

#### 4.3. Comparison with Other Improved EO Algorithms

#### 4.4. Wilcoxon Rank Sum Test

#### 4.5. Experimental Analysis of CEC2014 Test Function

## 5. Application to Solve the OPF Optimization Problem

#### 5.1. Case 1: Fuel Cost Minimization (FC)

#### 5.2. Case 2: Minimization of Active Power Transmission Loss (APL)

#### 5.3. Case 3: Voltage Deviation (VD)

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Partial convergence curves of the proposed IEO and other metaheuristic algorithms on the benchmark test functions: (

**a**) Convergence curve of F1; (

**b**) Convergence curve of F2; (

**c**) Convergence curve of F3; (

**d**) Convergence curve of F7; (

**e**) Convergence curve of F11; (

**f**) Convergence curve of F12; (

**g**) Convergence curve of F13; (

**h**) Convergence curve of F14.

Fun No. | Name | Dim | Range | Optimal Value |
---|---|---|---|---|

F1 | Sphere | 30/100/500 | [−100, 100] | 0 |

F2 | shwefel.2.22 | 30/100/500 | [−10, 10] | 0 |

F3 | Schwefel.1.2 | 30/100/500 | [−100, 100] | 0 |

F4 | Schwefel.2.21 | 30/100/500 | [−100, 100] | 0 |

F5 | Step | 30/100/500 | [−100, 100] | 0 |

F6 | Quartic | 30/100/500 | [−1.28, 1.28] | 0 |

F7 | Schwefel2.26 | 30/100/500 | [−500, 500] | −418.9829 × dim |

F8 | Rastrigin | 30/100/500 | [−5.12, 5.12] | 0 |

F9 | Ackley | 30/100/500 | [−32, 32] | 0 |

F10 | Criewank | 30/100/500 | [−600, 600] | 0 |

F11 | Penalized 1 | 30/100/500 | [−50, 50] | 0 |

F12 | Apline | 30/100/500 | [−10, 10] | 0 |

F13 | Kowalik | 4 | [−5, 5] | 0.0003 |

F14 | Sheke_1 | 4 | [0, 10] | −10.1532 |

F15 | Sheke_2 | 4 | [0, 10] | −10.4028 |

F16 | Shekel_3 | 4 | [0, 10] | −10.5363 |

Algorithm | Parameter |
---|---|

SCA [30] | M = 2; |

ChOA [31] | f_{max} = 2.5, f_{min} = 0; |

SSA [32] | / |

MA [33] | G = 0.8, gdamp = 1, a_{1} = 1, a_{2} = 1.5, a_{3} = 1.5, dance = 5; |

PSO [34] | v_{max} = 6, v_{min} = −6, w_{max} = 0.9, w_{min} = 0.6, c_{1} = c_{2} = 2; |

GWO [35] | a_{max} = 2, a_{min} = 0; |

WOA [36] | a_{max} = 2, a_{min} = 0, b = 1; |

EO [23] | a_{1} = 2, a_{2} = 1, GP = 0.5; |

IEO | a_{1} = 2, a_{2} = 1; |

Algorithm | Algorithm Description |
---|---|

SCA | Sine–Cosine Algorithm |

ChOA | Chimp Optimization Algorithm |

SSA | Salp Swarm Algorithm |

MA | Mayfly Algorithm |

PSO | particle swarm optimization |

GWO | Gray Wolf Optimization Algorithm |

WOA | Whale Optimization Algorithm |

AOA | Arithmetic Optimization Algorithm |

EO | Equilibrium Optimizer |

m-EO | Modified Equilibrium Optimizer |

AEO | Adaptive Equilibrium Optimizer |

OB-L-EO | Opposition-Based Laplacian Equilibrium Optimizer |

HEO | High Equilibrium Optimizer |

MPA | Marine Predator Algorithm |

SMA | Slime Mould Algorithm |

MSCA | Modified Sine–Cosine Algorithm |

GOA | Grasshopper Optimization Algorithm |

MVO | Multi-Verse Optimization |

IPSO | Improved Particle Swarm Optimization |

HHO | Harris Hawks Optimization |

BHBO | Black-Hole-Based Optimization |

MOHS | Multi-Objective Harmony Search Algorithm |

DE | Differential Evolution |

FEA | Faster Evolutionary Algorithm |

MOJA | Multi-Objective Jaye Algorithm |

TLBO | Teaching–Learning-Based Optimization |

MOBSA | Multi-Objective Backtracking Search Algorithm |

Fun No. | Index | SCA | ChOA | SSA | MA | PSO | GWO | WOA | EO | IEO |
---|---|---|---|---|---|---|---|---|---|---|

F1 | Mean | 2.46 × 10^{1} | 4.71 × 10^{−6} | 2.23 × 10^{−7} | 1.67 × 10^{−5} | 2.66 | 2.65 × 10^{−27} | 3.36 × 10^{−72} | 4.09 × 10^{−41} | 0 |

Std | 6.76 × 10^{1} | 7.25 × 10^{−6} | 4.00 × 10^{−7} | 4.73 × 10^{−5} | 1.21 | 4.47 × 10^{−27} | 1.81 × 10^{−71} | 7.73 × 10^{−41} | 0 | |

F2 | Mean | 1.83 × 10^{−2} | 3.28 × 10^{−5} | 2.06 | 6.85 × 10^{−4} | 4.44 | 1.23 × 10^{−16} | 4.82 × 10^{−52} | 6.04 × 10^{−24} | 0 |

Std | 2.25 × 10^{−2} | 4.14 × 10^{−5} | 1.54 | 1.73 × 10^{−3} | 1.25 | 9.51 × 10^{−17} | 1.24 × 10^{−51} | 7.21 × 10^{−24} | 0 | |

F3 | Mean | 1.02 × 10^{4} | 1.96 × 10^{2} | 1.45 × 10^{3} | 5.53 × 10^{3} | 2.00 × 10^{2} | 1.05 × 10^{−5} | 4.33 × 10^{4} | 8.54 × 10^{−9} | 0 |

Std | 6.28 × 10^{3} | 4.00 × 10^{2} | 8.12 × 10^{2} | 1.30 × 10^{2} | 6.03 × 10^{1} | 2.32 × 10^{−5} | 1.73 × 10^{4} | 3.64 × 10^{−8} | 0 | |

F4 | Mean | 3.54 × 10^{1} | 3.06 × 10^{−1} | 9.72 | 4.51 × 10^{1} | 2.02 | 7.53 × 10^{−7} | 4.36 × 10^{1} | 1.23 × 10^{−10} | 0 |

Std | 1.24 × 10^{1} | 2.45 × 10^{−1} | 2.18 | 2.21 × 10^{−1} | 2.50 × 10^{−1} | 7.96 × 10^{−7} | 2.83 × 10^{1} | 1.34 × 10^{−10} | 0 | |

F5 | Mean | 1.43 × 10^{1} | 3.70 | 1.74 × 10^{−6} | 1.19 × 10^{−5} | 2.53 | 8.01 × 10^{−1} | 4.07 × 10^{−1} | 8.97 × 10^{−6} | 5.48 × 10^{−7} |

Std | 1.31 × 10^{1} | 4.38 × 10^{−1} | 2.18 × 10^{−6} | 2.22 × 10^{−5} | 1.00 | 3.76 × 10^{−1} | 2.14 × 10^{−1} | 6.48 × 10^{−6} | 1.12 × 10^{−6} | |

F6 | Mean | 7.04 × 10^{−2} | 2.73 × 10^{−3} | 1.72 × 10^{−1} | 2.54 × 10^{−2} | 1.60 × 10^{1} | 1.91 × 10^{−3} | 3.27 × 10^{−3} | 1.34 × 10^{−3} | 1.02 × 10^{−}^{5} |

Std | 7.33 × 10^{−2} | 1.97 × 10^{−3} | 8.20 × 10^{−2} | 8.93 × 10^{3} | 1.50 × 10^{1} | 1.02 × 10^{−3} | 4.12 × 10^{−3} | 8.36 × 10^{−4} | 1.26 × 10^{−}^{5} | |

F7 | Mean | −3.72 × 10^{3} | −5.71 × 10^{3} | −7.53 × 10^{3} | −9.87 × 10^{3} | −6.59 × 10^{−3} | −6.13 × 10^{3} | −1.06 × 10^{4} | −8.84 × 10^{3} | −1.24 × 10^{4} |

Std | 2.96 × 10^{2} | 6.18 × 10^{2} | 6.39 × 10^{2} | 4.99 × 10^{2} | 1.26 × 10^{3} | 8.72 × 10^{2} | 1.92 × 10^{3} | 7.10 × 10^{2} | 2.82 × 10^{2} | |

F8 | Mean | 4.65 × 10^{1} | 1.17 × 10^{1} | 5.69 × 10^{1} | 5.13 × 10^{1} | 1.68 × 10^{2} | 3.45 | 3.79 × 10^{−15} | 1.89 × 10^{−15} | 0 |

Std | 4.79 × 10^{1} | 1.19 × 10^{1} | 1.71 × 10^{1} | 2.06 × 10^{1} | 3.24 × 10^{1} | 3.51 | 1.44 × 10^{−14} | 1.03 × 10^{−14} | 0 | |

F9 | Mean | 1.69 × 10^{1} | 2.00 × 10^{1} | 2.85 | 1.73 | 2.57 | 1.04 × 10^{−13} | 4.44 × 10^{−15} | 8.59 × 10^{−15} | 8.88 × 10^{−16} |

Std | 7.33 | 3.15 × 10^{−2} | 8.67 × 10^{−1} | 5.35 × 10^{−1} | 5.33 × 10^{−1} | 1.58 × 10^{−14} | 2.29 × 10^{−15} | 2.30 × 10^{−15} | 0 | |

F10 | Mean | 1.08 | 2.05 × 10^{−2} | 1.93 × 10^{−2} | 2.49 × 10^{−2} | 1.24 × 10^{−1} | 9.72 × 10^{−4} | 1.32 × 10^{−2} | 3.29 × 10^{−4} | 0 |

Std | 5.57 × 10^{−1} | 3.45 × 10^{−2} | 1.38 × 10^{−2} | 2.98 × 10^{−2} | 4.65 × 10^{−2} | 3.83 × 10^{−3} | 4.25 × 10^{−2} | 1.80 × 10^{−3} | 0 | |

F11 | Mean | 1.44 × 10^{4} | 4.69 × 10^{−1} | 7.16 | 5.20 × 10^{−1} | 4.04 × 10^{−2} | 4.89 × 10^{−2} | 2.27 × 10^{−2} | 4.60 × 10^{−7} | 2.35 × 10^{−8} |

Std | 5.90 × 10^{4} | 1.82 × 10^{−1} | 3.72 | 7.08 × 10^{−1} | 2.44 × 10^{−2} | 2.56 × 10^{−2} | 1.45 × 10^{−2} | 3.58 × 10^{−7} | 2.65 × 10^{−8} | |

F12 | Mean | 1.54 | 4.06 × 10^{−3} | 4.07 | 2.19 × 10^{−4} | 6.01 | 7.29 × 10^{−4} | 2.16 × 10^{−39} | 2.73 × 10^{−24} | 0 |

Std | 3.19 | 1.24 × 10^{−2} | 2.22 | 4.02 × 10^{−4} | 2.90 | 7.62 × 10^{−4} | 1.18 × 10^{−38} | 4.45 × 10^{−24} | 0 | |

F13 | Mean | 9.79 × 10^{−4} | 9.79 × 10^{−4} | 1.33 × 10^{−3} | 3.08 × 10^{−3} | 8.97 × 10^{−4} | 9.73 × 10^{−3} | 6.49 × 10^{−4} | 2.35 × 10^{−3} | 3.56 × 10^{−4} |

Std | 3.24 × 10^{−4} | 3.24 × 10^{−4} | 5.16 × 10^{−4} | 6.91 × 10^{−3} | 1.30 × 10^{−4} | 1.31 × 10^{−2} | 4.45 × 10^{−4} | 6.11 × 10^{−3} | 8.45 × 10^{−5} | |

F14 | Mean | −2.6931 | −1.6733 | −7.8901 | −5.9683 | −8.0397 | −9.3935 | −8.2692 | −8.1255 | −10.1532 |

Std | 1.9152 | 1.6798 | 3.112 | 3.5485 | 2.739 | 2.0068 | 2.7214 | 2.77 | 5.79 × 10^{−15} | |

F15 | Mean | −4.0105 | −3.6312 | −9.1095 | −6.6698 | −8.9849 | −10.2252 | −8.2252 | −9.3623 | −10.4028 |

Std | 1.6975 | 1.9582 | 2.6852 | 3.8057 | 2.9115 | 0.97032 | 3.1739 | 2.4134 | 1.04 × 10^{−15} | |

F16 | Mean | −4.7637 | −4.4468 | −8.0623 | −7.8211 | −9.5646 | −9.9948 | −7.1163 | −9.682 | −10.5363 |

Std | 2.2925 | 1.6923 | 3.3784 | 3.6619 | 2.2428 | 2.0586 | 3.5352 | 2.2541 | 2.06 × 10^{−15} |

Fun No. | Dim | m-EO [37] | AEO [38] | OB-L-EO [39] | HEO [40] | IEO | |||||
---|---|---|---|---|---|---|---|---|---|---|---|

Mean | Std | Mean | Std | Mean | Std | Mean | Std | Mean | Std | ||

30 | 0 | 0 | 1.62 × 10^{−104} | 7.47 × 10^{−104} | 6.76 × 10^{−212} | 0 | 5.60 × 10^{−196} | 0 | 0 | 0 | |

F1 | 100 | 1.53 × 10^{−304} | 0 | 1.47 × 10^{−97} | 4.93 × 10^{−97} | 0 | 0 | - | - | 0 | 0 |

500 | 0 | 0 | 3.56 × 10^{−78} | 1.48 × 10^{−77} | - | - | - | - | 0 | 0 | |

30 | 3.93 × 10^{−167} | 0 | 1.38 × 10^{−56} | 3.65 × 10^{−56} | 1.93 × 10^{−108} | 8.00 × 10^{−108} | 0 | 2.75 × 10^{−94} | 0 | 0 | |

F2 | 100 | 3.09 × 10^{−161} | 1.58 × 10^{−160} | 1.99 × 10^{−52} | 3.36 × 10^{−52} | 0 | 0 | - | - | 0 | 0 |

500 | 1.36 × 10^{−160} | 5.90 × 10^{−160} | 3.70 × 10^{−43} | 9.08 × 10^{−43} | - | - | - | - | 0 | 0 | |

30 | 2.71 × 10^{−306} | 0 | 4.24 × 10^{−38} | 2.30 × 10^{−37} | 6.93 × 10^{−187} | 0 | 6.44 × 10^{−198} | 3.09 × 10^{−106} | 0 | 0 | |

F3 | 100 | 8.50 × 10^{−297} | 0 | 4.01 × 10^{−9} | 1.99 × 10^{−8} | 0 | 0 | - | - | 0 | 0 |

500 | 4.63 × 10^{−293} | 0 | 2.64 × 10^{2} | 7.78 × 10^{2} | - | - | - | - | 0 | 0 | |

30 | 2.31 × 10^{−159} | 1.17 × 10^{−158} | 1.94 × 10^{−46} | 9.14 × 10^{−46} | 4.73 × 10^{−103} | 1.60 × 10^{−102} | 1.22 × 10^{−96} | 2.02 × 10^{−95} | 0 | 0 | |

F4 | 100 | 2.83 × 10^{−157} | 9.04 × 10^{−157} | 4.53 × 10^{−42} | 1.42 × 10^{−41} | 0 | 0 | - | - | 1.52 × 10^{−}^{314} | 0 |

500 | 1.48 × 10^{−154} | 7.46 × 10^{−154} | 5.05 × 10^{−28} | 2.35 × 10^{−27} | - | - | - | - | 7.39 × 10^{−}^{307} | 0 | |

30 | 9.23 × 10^{−5} | 4.18 × 10^{−5} | 6.45 × 10^{−6} | 4.98 × 10^{−6} | 9.09 × 10^{−5} | 5.98 × 10^{−5} | 1.11 × 10^{−3} | 1.34 × 10^{−3} | 5.48 × 10^{−}^{7} | 1.12 × 10^{−}^{6} | |

F5 | 100 | 5.28 × 10^{−3} | 4.24 × 10^{−3} | 3.49 | 7.01 × 10^{−1} | 1.53 × 10^{−1} | 2.05 × 10^{−1} | - | - | 9.17 × 10^{−}^{4} | 4.50 × 10^{−}^{4} |

500 | 4.92 × 10^{−2} | 5.50 × 10^{−2} | 8.99 × 10^{1} | 2.30 × 10^{−3} | - | - | - | - | 3.46 × 10^{−}^{3} | 1.91 × 10^{−}^{3} | |

30 | 2.47 × 10^{−4} | 2.23 × 10^{−4} | 1.10 × 10^{−3} | 5.87 × 10^{−4} | 4.70 × 10^{−4} | 3.05 × 10^{−4} | 1.18 × 10^{−5} | 7.32 × 10^{−5} | 1.02 × 10^{−}^{5} | 1.26 × 10^{−}^{5} | |

F6 | 100 | 3.47 × 10^{−4} | 2.50 × 10^{−4} | 2.30 × 10^{−3} | 8.46 × 10^{−4} | 1.47 × 10^{−4} | 1.22 × 10^{−4} | - | - | 1.25 × 10^{−}^{4} | 1.12 × 10^{−}^{4} |

500 | 5.11 × 10^{−4} | 3.90 × 10^{−4} | 4.80 × 10^{−3} | 2.30 × 10^{−3} | - | - | - | - | 2.02 × 10^{−}^{4} | 1.66 × 10^{−}^{4} | |

30 | −1.22 × 10^{4} | 1.02 × 10^{3} | −8.91 × 10^{3} | 6.21 × 10^{2} | −9.06 × 10^{3} | 9.28 × 10^{2} | - | - | −1.24 × 10^{4} | 2.82 × 10^{2} | |

F7 | 100 | −4.19 × 10^{4} | 9.63 | −2.58 × 10^{4} | 1.34 × 10^{3} | −2.85 × 10^{4} | 2.09 × 10^{3} | - | - | −4.19 × 10^{4} | 1.63 |

500 | −2.09 × 10^{5} | 1.87 × 10^{2} | −7.62 × 10^{4} | 5.92 × 10^{3} | - | - | - | - | −1.96 × 10^{5} | 1.13 × 10^{2} | |

30 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

F8 | 100 | 0 | 0 | 0 | 0 | 0 | 0 | - | - | 0 | 0 |

500 | 0 | 0 | 0 | 0 | - | - | - | - | 0 | 0 | |

30 | 8.88 × 10^{−16} | 0 | 5.98 × 10^{−15} | 1.79 × 10^{−15} | 8.88 × 10^{−16} | 4.01 × 10^{−31} | 8.88 × 10^{−16} | 0 | 8.88 × 10^{−16} | 0 | |

F9 | 100 | 8.88 × 10^{−16} | 0 | 6.81 × 10^{−15} | 1.70 × 10^{−15} | 8.88 × 10^{−16} | 8.88 × 10^{−16} | - | - | 8.88 × 10^{−16} | 0 |

500 | 8.88 × 10^{−16} | 0 | 7.52 × 10^{−15} | 2.03 × 10^{−15} | - | - | - | - | 8.88 × 10^{−16} | 0 | |

30 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

F10 | 100 | 0 | 0 | 0 | 0 | 0 | 0 | - | - | 0 | 0 |

500 | 0 | 0 | 0 | 0 | - | - | - | - | 0 | 0 | |

30 | 6.26 × 10^{−6} | 3.92 × 10^{−6} | 5.22 × 10^{−7} | 5.00 × 10^{−7} | 6.29 × 10^{−6} | 4.35 × 10^{−}^{6} | 1.79 × 10^{−5} | 5.06 × 10^{−5} | 2.35 × 10^{−}^{8} | 2.65 × 10^{−}^{8} | |

F11 | 100 | 2.18 × 10^{−5} | 1.78 × 10^{−5} | 3.45 × 10^{−2} | 8.20 × 10^{−3} | 5.83 × 10^{−4} | 1.14 × 10^{−3} | - | - | 8.00 × 10^{−}^{3} | 3.54 × 10^{−}^{3} |

500 | 1.98 × 10^{−5} | 2.25 × 10^{−5} | 6.44 × 10^{−1} | 3.48 × 10^{−2} | - | - | - | - | 9.75 × 10^{−}^{2} | 1.75 × 10^{−2} | |

30 | 1.65 × 10^{−165} | 0 | - | - | - | - | - | - | 0 | 0 | |

F12 | 100 | 1.65 × 10^{−164} | 0 | - | - | - | - | - | - | 0 | 0 |

500 | 5.61 × 10^{−160} | 0 | - | - | - | - | - | - | 0 | 0 |

Fun No. | m-EO [37] | AEO [38] | OB-L-EO [39] | HEO [40] | IEO | |||||
---|---|---|---|---|---|---|---|---|---|---|

Mean | Std | Mean | Std | Mean | Std | Mean | Std | Mean | Std | |

F13 | 1.55 × 10^{−1} | 4.93 × 10^{−1} | 1.10 × 10^{−2} | 3.70 × 10^{−3} | 4.61 × 10^{−4} | 3.47 × 10^{−4} | 1.02 | 3.87 | 3.56 × 10^{−4} | 8.45 × 10^{−5} |

F14 | −9.33 | 2.23 | −8.45 | 2.44 | −1.02 × 10^{1} | 9.12 × 10^{−6} | - | - | −1.02 × 10^{1} | 5.79 × 10^{−15} |

F15 | −1.01 × 10^{1} | 1.35 | −9.47 | 2.13 | −1.04 × 10^{1} | 7.99 × 10^{−6} | - | - | −1.04 × 10^{1} | 1.04 × 10^{−15} |

F16 | −1.03 × 10^{1} | 9.87 × 10^{−1} | −9.82 | 1.87 | −1.02 × 10^{1} | 1.36 | - | - | −1.05 × 10^{1} | 2.06 × 10^{−15} |

Fun No. | SCA | ChOA | SSA | MA | PSO | GWO | WOS | EO |
---|---|---|---|---|---|---|---|---|

p Value R | p Value R | p Value R | p Value R | p Value R | p Value R | p Value R | p Value R | |

F1 | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20}+ | 3.31 × 10^{−20} + |

F2 | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + |

F3 | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + |

F4 | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + |

F5 | 7.07 × 10^{−18} + | 7.07 × 10^{−18} + | 1.11 × 10^{−2} + | 3.26 × 10^{−13} + | 7.07 × 10^{−18} + | 7.07 × 10^{−18} + | 7.07 × 10^{−18} + | 7.49 × 10^{−16} + |

F6 | 7.07 × 10^{−18} + | 8.19 × 10^{−12} + | 7.07 × 10^{−18} + | 7.07 × 10^{−18} + | 7.07 × 10^{−18} + | 1.01 × 10^{−17} + | 1.17 × 10^{−15} + | 1.43 × 10^{−16} + |

F7 | 7.07 × 10^{−18} + | 7.07 × 10^{−18} + | 7.07 × 10^{−18} + | 7.07 × 10^{−18} + | 7.07 × 10^{−18} + | 7.07 × 10^{−18} + | 2.44 × 10^{−11} + | 3.15 × 10^{−12} + |

F8 | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 6.50 × 10^{−3} + | 1.23 × 10^{−2} + |

F9 | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 2.85 × 10^{−20} + | 1.54 × 10^{−11} + | 1.03 × 10^{−22} + |

F10 | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 6.25 × 10^{−21} + | 1.80 × 10^{−3} + | 2.30 × 10^{−2} + |

F11 | 7.07 × 10^{−18} + | 3.21 × 10^{−17} + | 7.07 × 10^{−18} + | 7.07 × 10^{−18} + | 7.07 × 10^{−18} + | 7.07 × 10^{−18} + | 7.07 × 10^{−18} + | 1.60 × 10^{−16} + |

F12 | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + |

F13 | 3.23 × 10^{−17} + | 7.07 × 10^{−18} + | 2.07 × 10^{−17} + | 1.26 × 10^{−6} + | 1.21 × 10^{−17} + | 6.10 × 10^{−3} + | 8.59 × 10^{−12} + | 1.55 × 10^{−8} + |

F14 | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.75 × 10^{−7} + | 3.68 × 10^{−10} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 1.89 × 10^{−20} + |

F15 | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.69 × 10^{−6} + | 2.57 × 10^{−20} + | 1.76 × 10^{−6} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 4.38 × 10^{−21} + |

F16 | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 3.23 × 10^{−5} + | 1.32 × 10^{−20} + | 3.80 × 10^{−7} + | 3.31 × 10^{−20} + | 3.31 × 10^{−20} + | 1.76 × 10^{−20} + |

+/=/− | 16/0/0 | 16/0/0 | 16/0/0 | 16/0/0 | 16/0/0 | 16/0/0 | 16/0/0 | 16/0/0 |

Fun No. | Function type | Function Name | Optimal Value |
---|---|---|---|

CEC01 | Unimodal Function | Rotated High Conditioned Elliptic Function | 100 |

CEC02 | Rotated Bent Cigar Function | 200 | |

CEC03 | Rotated Discus Function | 300 | |

CEC04 | Multimodal Function | Shifted and Rotated Rosenbrock’s Function | 400 |

CEC05 | Shifted and Rotated Ackley’s Function | 500 | |

CEC06 | Shifted and Rotated Weierstrass Function | 600 | |

CEC07 | Shifted and Rotated Griewank’s Function | 700 | |

CEC08 | Shifted Rastrigin’s Function | 800 | |

CEC09 | Shifted and Rotated Rastrigin’s Function | 900 | |

CEC10 | Shifted Schwefel’s Function | 1000 | |

CEC11 | Shifted and Rotated Schwefel’s Function | 1100 | |

CEC12 | Shifted and Rotated Katsuura Function | 1200 | |

CEC13 | Shifted and Rotated HappyCat Function | 1300 | |

CEC14 | Shifted and Rotated HGBat Function | 1400 | |

CEC15 | Shifted and Rotated Expanded Griewank’s plus Rosenbrock’s Function | 1500 | |

CEC16 | Shifted and Rotated Expanded Scaffer’s F6 Function | 1600 | |

CEC17 | Hybrid Function | Hybrid Function 1 (N = 3) | 1700 |

CEC18 | Hybrid Function 2 (N = 3) | 1800 | |

CEC19 | Hybrid Function 3 (N = 4) | 1900 | |

CEC20 | Hybrid Function 4 (N = 4) | 2000 | |

CEC21 | Hybrid Function 5 (N = 5) | 2100 | |

CEC22 | Hybrid Function 6 (N = 5) | 2200 | |

CEC23 | Composition Function | Composition Function 1 (N = 5) | 2300 |

CEC24 | Composition Function 2 (N = 3) | 2400 | |

CEC25 | Composition Function 3 (N = 3) | 2500 | |

CEC26 | Composition Function 4 (N = 5) | 2600 | |

CEC27 | Composition Function 5 (N = 5) | 2700 | |

CEC28 | Composition Function 6 (N = 5) | 2800 | |

CEC29 | Composition Function 7 (N = 3) | 2900 | |

CEC30 | Composition Function 8 (N = 3) | 3000 |

Fun No. | Index | SCA | ChOA | SSA | GWO | WOA | AOA | PSO | EO | IEO |
---|---|---|---|---|---|---|---|---|---|---|

CEC01 | Mean | 4.193 × 10^{8} | 5.988 × 10^{8} | 2.376 × 10^{7} | 8.936 × 10^{7} | 8.553 × 10^{7} | 1.362 × 10^{9} | 7.423 × 10^{6} | 7.740 × 10^{6} | 5.473 × 10^{6} |

Std | 1.085 × 10^{8} | 1.080 × 10^{8} | 1.168 × 10^{7} | 6.043 × 10^{7} | 5.417 × 10^{7} | 3.235 × 10^{8} | 3.835 × 10^{6} | 3.691 × 10^{6} | 3.216 × 10^{6} | |

CEC02 | Mean | 2.651 × 10^{10} | 4.381 × 10^{10} | 1.302 × 10^{4} | 2.464 × 10^{9} | 3.427 × 10^{9} | 7.125 × 10^{10} | 4.245 × 10^{5} | 2.111 × 10^{4} | 2.862 × 10^{3} |

Std | 4.297 × 10^{9} | 6.867 × 10^{9} | 1.066 × 10^{4} | 2.541 × 10^{9} | 3.135 × 10^{9} | 1.117 × 10^{10} | 2.267 × 10^{5} | 1.195 × 10^{4} | 5.698 × 10^{3} | |

CEC03 | Mean | 5.876 × 10^{4} | 8.265 × 10^{4} | 7.365 × 10^{4} | 4.529 × 10^{4} | 4.585 × 10^{4} | 8.101 × 10^{4} | 1.267 × 10^{4} | 8.549 × 10^{3} | 3.657 × 10^{3} |

Std | 1.192 × 10^{4} | 8.024 × 10^{3} | 1.941 × 10^{4} | 1.216 × 10^{4} | 1.056 × 10^{4} | 8.235 × 10^{3} | 8.234 × 10^{4} | 4.531 × 10^{3} | 2.577 × 10^{3} | |

CEC04 | Mean | 2.547 × 10^{3} | 3.335 × 10^{3} | 5.315 × 10^{2} | 6.967 × 10^{2} | 6.616 × 10^{2} | 1.226 × 10^{4} | 5.168 × 10^{2} | 5.129 × 10^{2} | 5.074 × 10^{2} |

Std | 7.605 × 10^{2} | 1.430 × 10^{3} | 4.515 × 10^{1} | 1.216 × 10^{2} | 5.316 × 10^{1} | 3.527 × 10^{3} | 2.935 × 10^{1} | 3.320 × 10^{1} | 2.681 × 10^{1} | |

CEC05 | Mean | 5.211 × 10^{2} | 5.211 × 10^{2} | 5.201 × 10^{2} | 5.210 × 10^{2} | 5.211 × 10^{2} | 5.209 × 10^{2} | 5.209 × 10^{2} | 5.209 × 10^{2} | 5.207 × 10^{2} |

Std | 5.832 × 10^{−}^{2} | 5.332 × 10^{−}^{2} | 9.926 × 10^{−}^{2} | 5.761 × 10^{−}^{2} | 5.325 × 10^{−}^{2} | 6.787 × 10^{−2} | 9.917 × 10^{−2} | 1.260 × 10^{−}^{1} | 4.640 × 10^{−}^{2} | |

CEC06 | Mean | 6.365 × 10^{2} | 6.364 × 10^{2} | 6.232 × 10^{2} | 6.163 × 10^{2} | 6.157 × 10^{2} | 6.392 × 10^{2} | 6.207 × 10^{2} | 6.105 × 10^{2} | 6.098 × 10^{2} |

Std | 2.705 | 2.021 | 3.635 | 3.113 | 2.646 | 1.982 | 3.647 | 3.226 | 1.081 | |

CEC07 | Mean | 9.172 × 10^{2} | 1.178 × 10^{3} | 7.021 × 10^{2} | 7.226 × 10^{2} | 7.231 × 10^{2} | 1.352 × 10^{3} | 7.006 × 10^{2} | 7.001 × 10^{2} | 7.000 × 10^{2} |

Std | 3.107 × 10^{1} | 8.536 × 10^{1} | 1.456 × 10^{−}^{2} | 1.897 × 10^{1} | 1.949 × 10^{1} | 1.113 × 10^{1} | 1.663 × 10^{−1} | 1.543 × 10^{−}^{2} | 1.384 × 10^{−}^{2} | |

CEC08 | Mean | 1.073 × 10^{3} | 1.067 × 10^{3} | 9.572 × 10^{2} | 8.878 × 10^{2} | 8.892 × 10^{2} | 1.151 × 10^{3} | 9.242 × 10^{2} | 8.614 × 10^{2} | 8.552 × 10^{2} |

Std | 2.538 × 10^{1} | 2.213 × 10^{1} | 4.215 × 10^{1} | 2.445 × 10^{1} | 2.088 × 10^{1} | 3.074 × 10^{1} | 2.338 × 10^{1} | 1.554 × 10^{1} | 1.213 × 10^{1} | |

CEC09 | Mean | 1.205 × 10^{3} | 1.187 × 10^{3} | 1.062 × 10^{3} | 1.005 × 10^{3} | 1.023 × 10^{3} | 1.219 × 10^{3} | 1.045 × 10^{3} | 9.883 × 10^{2} | 9.680 × 10^{2} |

Std | 2.349 × 10^{1} | 2.147 × 10^{1} | 4.244 × 10^{1} | 2.038 × 10^{1} | 3.805 × 10^{1} | 2.791 × 10^{1} | 2.769 × 10^{1} | 2.784 × 10^{1} | 1.960 × 10^{1} | |

CEC10 | Mean | 7.670 × 10^{3} | 7.801 × 10^{3} | 4.780 × 10^{3} | 3.595 × 10^{3} | 3.564 × 10^{3} | 7.274 × 10^{3} | 4.258 × 10^{3} | 2.873 × 10^{3} | 2.814 × 10^{3} |

Std | 4.632 × 10^{2} | 9.379 × 10^{2} | 7.847 × 10^{2} | 7.425 × 10^{2} | 5.144 × 10^{2} | 6.115 × 10^{2} | 4.678 × 10^{2} | 6.086 × 10^{2} | 4.535 × 10^{2} | |

CEC11 | Mean | 8.682 × 10^{3} | 8.937 × 10^{3} | 5.050 × 10^{3} | 4.265 × 10^{3} | 4.481 × 10^{3} | 7.717 × 10^{3} | 4.505 × 10^{3} | 5.093 × 10^{3} | 4.862 × 10^{3} |

Std | 3.155 × 10^{2} | 2.901 × 10^{2} | 7.846 × 10^{2} | 6.525 × 10^{2} | 1.219 × 10^{3} | 5.136 × 10^{2} | 5.789 × 10^{2} | 7.821 × 10^{2} | 8.284 × 10^{2} | |

CEC12 | Mean | 1.203 × 10^{3} | 1.203 × 10^{3} | 1.201 × 10^{3} | 1.202 × 10^{3} | 1.203 × 10^{3} | 1.202 × 10^{3} | 1.202 × 10^{3} | 1.202 × 10^{3} | 1.201 × 10^{3} |

Std | 3.934 × 10^{−}^{1} | 4.052 × 10^{−}^{1} | 4.315 × 10^{−}^{1} | 1.214 | 1.104 | 4.557 × 10^{−1} | 6.887 × 10^{−1} | 3.582 × 10^{−}^{1} | 3.486 × 10^{−}^{1} | |

CEC13 | Mean | 1.304 × 10^{3} | 1.304 × 10^{3} | 1.301 × 10^{3} | 1.301 × 10^{3} | 1.301 × 10^{3} | 1.307 × 10^{3} | 1.301 × 10^{3} | 1.300 × 10^{3} | 1.300 × 10^{3} |

Std | 2.691 × 10^{−}^{1} | 5.682 × 10^{−}^{1} | 1.538 × 10^{−}^{1} | 3.385 × 10^{−}^{1} | 5.589 × 10^{−}^{1} | 9.561 × 10^{−}^{1} | 1.316 × 10^{−1} | 9.216 × 10^{−}^{2} | 7.381 × 10^{−}^{2} | |

CEC14 | Mean | 1.474 × 10^{3} | 1.556 × 10^{3} | 1.401 × 10^{3} | 1.407 × 10^{3} | 1.407 × 10^{3} | 1.639 × 10^{3} | 1.401 × 10^{3} | 1.400 × 10^{3} | 1.400 × 10^{3} |

Std | 1.115 × 10^{1} | 3.481 × 10^{1} | 2.232 × 10^{−}^{1} | 8.419 | 9.241 | 3.676 × 10^{1} | 1.764 × 10^{−1} | 1.406 × 10^{−}^{1} | 1.226 × 10^{−}^{1} | |

CEC15 | Mean | 1.854 × 10^{4} | 1.258 × 10^{5} | 1.514 × 10^{3} | 1.799 × 10^{3} | 1.800 × 10^{3} | 2.964 × 10^{5} | 1.515 × 10^{3} | 1.509 × 10^{3} | 1.508 × 10^{3} |

Std | 1.139 × 10^{4} | 9.523 × 10^{4} | 4.705 | 5.662 × 10^{2} | 7.871 × 10^{2} | 1.126 × 10^{5} | 2.048 | 3.235 | 2.926 | |

CEC16 | Mean | 1.613 × 10^{3} | 1.613 × 10^{3} | 1.613 × 10^{3} | 1.612 × 10^{3} | 1.612 × 10^{3} | 1.613 × 10^{3} | 1.613 × 10^{3} | 1.612 × 10^{3} | 1.611 × 10^{3} |

Std | 2.372 × 10^{−}^{1} | 2.239 × 10^{−}^{1} | 7.115 × 10^{−}^{1} | 6.959 × 10^{−}^{1} | 7.963 × 10^{−}^{1} | 3.641 × 10^{−1} | 4.339 × 10^{−1} | 6.615 × 10^{−}^{1} | 1.373 × 10^{−}^{1} | |

CEC17 | Mean | 1.385 × 10^{7} | 3.290 × 10^{7} | 1.265 × 10^{6} | 2.747 × 10^{6} | 2.720 × 10^{6} | 8.805 × 10^{7} | 6.051 × 10^{5} | 8.126 × 10^{5} | 5.846 × 10^{5} |

Std | 7.717 × 10^{6} | 1.947 × 10^{7} | 8.693 × 10^{5} | 2.881 × 10^{6} | 2.547 × 10^{6} | 3.984 × 10^{7} | 5.482 × 10^{5} | 8.573 × 10^{5} | 4.147 × 10^{5} | |

CEC18 | Mean | 3.134 × 10^{8} | 9.120 × 10^{8} | 1.094 × 10^{4} | 2.202 × 10^{7} | 1.692 × 10^{7} | 2.339 × 10^{9} | 4.065 × 10^{3} | 4.932 × 10^{3} | 3.767 × 10^{3} |

Std | 1.335 × 10^{8} | 9.997 × 10^{8} | 7.508 × 10^{3} | 4.057 × 10^{7} | 2.724 × 10^{7} | 1.625 × 10^{9} | 2.656 × 10^{3} | 3.687 × 10^{3} | 2.333 × 10^{3} | |

CEC19 | Mean | 2.033 × 10^{3} | 2.163 × 10^{3} | 1.920 × 10^{3} | 1.950 × 10^{3} | 1.970 × 10^{3} | 2.250 × 10^{3} | 1.915 × 10^{3} | 1.912 × 10^{3} | 1.909 × 10^{3} |

Std | 3.652 × 10^{1} | 1.070 × 10^{2} | 1.780 × 10^{1} | 3.450 × 10^{1} | 4.450 × 10^{1} | 1.155 × 10^{2} | 3.201 | 1.511 × 10^{1} | 2.194 | |

CEC20 | Mean | 4.452 × 10^{4} | 1.084 × 10^{5} | 3.280 × 10^{4} | 3.057 × 10^{4} | 2.486 × 10^{4} | 2.205 × 10^{5} | 1.761 × 10^{4} | 1.460 × 10^{4} | 1.246 × 10^{4} |

Std | 2.328 × 10^{4} | 3.923 × 10^{4} | 1.675 × 10^{4} | 1.265 × 10^{4} | 6.620 × 10^{4} | 8.616 × 10^{4} | 6.945 × 10^{3} | 5.620 × 10^{3} | 4.175 × 10^{3} | |

CEC21 | Mean | 4.066 × 10^{6} | 1.130 × 10^{7} | 4.137 × 10^{5} | 8.915 × 10^{5} | 1.334 × 10^{6} | 2.428 × 10^{7} | 3.351 × 10^{5} | 4.266 × 10^{5} | 3.056 × 10^{5} |

Std | 2.341 × 10^{6} | 3.245 × 10^{6} | 3.892 × 10^{5} | 2.086 × 10^{6} | 2.125 × 10^{6} | 2.126 × 10^{7} | 2.665 × 10^{5} | 4.378 × 10^{5} | 2.319 × 10^{5} | |

CEC22 | Mean | 3.260 × 10^{3} | 3.177 × 10^{3} | 2.794 × 10^{3} | 2.673 × 10^{3} | 2.61 × 10^{3} | 4.971 × 10^{3} | 3.045 × 10^{3} | 2.658 × 10^{3} | 2.588 × 10^{3} |

Std | 2.169 × 10^{2} | 2.647 × 10^{2} | 2.090 × 10^{2} | 2.054 × 10^{2} | 2.762 × 10^{2} | 2.105 × 10^{3} | 2.926 × 10^{2} | 2.045 × 10^{2} | 2.022 × 10^{2} | |

CEC23 | Mean | 2.716 × 10^{3} | 2.745 × 10^{3} | 2.639 × 10^{3} | 2.641 × 10^{3} | 2.644 × 10^{3} | 2.511 × 10^{3} | 2.614 × 10^{3} | 2.616 × 10^{3} | 2.615 × 10^{3} |

Std | 1.951 × 10^{1} | 4.447 × 10^{1} | 1.235 × 10^{1} | 1.272 × 10^{1} | 1.045 × 10^{1} | 6.233 × 10^{1} | 4.451 × 10^{−2} | 4.481 × 10^{−}^{4} | 7.714 × 10^{−}^{2} | |

CEC24 | Mean | 2.611 × 10^{3} | 2.600 × 10^{3} | 2.642 × 10^{3} | 2.600 × 10^{3} | 2.600 × 10^{3} | 2.600 × 10^{3} | 2.624 × 10^{3} | 2.600 × 10^{3} | 2.600 × 10^{3} |

Std | 1.953 × 10^{1} | 4.626 × 10^{−}^{2} | 7.915 | 1.347 × 10^{−}^{2} | 1.094 × 10^{−}^{2} | 8.515 × 10^{−2} | 6.836 | 4.430 × 10^{−}^{3} | 3.170 × 10^{−}^{3} | |

CEC25 | Mean | 2.741 × 10^{3} | 2.712 × 10^{3} | 2.719 × 10^{3} | 2.712 × 10^{3} | 2.713 × 10^{3} | 2.700 × 10^{3} | 2.175 × 10^{3} | 2.702 × 10^{3} | 2.700 × 10^{3} |

Std | 1.191 × 10^{1} | 1.369 × 10^{1} | 6.265 | 4.859 | 5.356 | 0 | 4.658 | 3.651 | 0 | |

CEC26 | Mean | 2.703 × 10^{3} | 2.796 × 10^{3} | 2.701 × 10^{3} | 2.738 × 10^{3} | 2.741 × 10^{3} | 2.782 × 10^{3} | 2.777 × 10^{3} | 2.737 × 10^{3} | 2.720 × 10^{3} |

Std | 5.079 × 10^{−}^{1} | 4.839 × 10^{1} | 1.473 × 10^{−}^{1} | 5.968 × 10^{1} | 4.956 × 10^{1} | 3.367 × 10^{1} | 4.289 × 10^{1} | 4.885 × 10^{1} | 4.784 × 10^{1} | |

CEC27 | Mean | 3.864 × 10^{3} | 3.923 × 10^{3} | 3.573 × 10^{3} | 3.377 × 10^{3} | 3.384 × 10^{3} | 3.832 × 10^{3} | 3.560 × 10^{3} | 3.384 × 10^{3} | 3.309 × 10^{3} |

Std | 2.728 × 10^{2} | 2.002 × 10^{2} | 2.062 × 10^{2} | 1.288 × 10^{2} | 1.498 × 10^{2} | 5.433 × 10^{2} | 2.390 × 10^{2} | 1.113 × 10^{2} | 1.030 × 10^{2} | |

CEC28 | Mean | 5.593 × 10^{3} | 5.745 × 10^{3} | 4.270 × 10^{3} | 4.190 × 10^{3} | 4.205 × 10^{3} | 4.970 × 10^{3} | 6.778 × 10^{3} | 3.846 × 10^{3} | 3.813 × 10^{3} |

Std | 4.547 × 10^{2} | 2.475 × 10^{2} | 4.646 × 10^{2} | 4.412 × 10^{2} | 3.744 × 10^{2} | 2.700 × 10^{3} | 8.031 × 10^{2} | 2.422 × 10^{2} | 1.592 × 10^{2} | |

CEC29 | Mean | 3.026 × 10^{7} | 5.153 × 10^{7} | 4.547 × 10^{6} | 3.430 × 10^{6} | 1.248 × 10^{7} | 3.318 × 10^{8} | 4.657 × 10^{3} | 2.44 × 10^{6} | 1.716 × 10^{6} |

Std | 1.490 × 10^{7} | 3.314 × 10^{7} | 7.902 × 10^{6} | 7.411 × 10^{6} | 8.591 × 10^{6} | 2.614 × 10^{8} | 1.894 × 10^{3} | 4.13 × 10^{6} | 3.348 × 10^{6} | |

CEC30 | Mean | 4.652 × 10^{5} | 8.103 × 10^{5} | 3.876 × 10^{4} | 8.482 × 10^{4} | 8.637 × 10^{4} | 5.306 × 10^{6} | 7.785 × 10^{3} | 8.451 × 10^{3} | 7.306 × 10^{3} |

Std | 1.775 × 10^{5} | 2.142 × 10^{5} | 2.146 × 10^{4} | 5.220 × 10^{4} | 5.851 × 10^{4} | 3.825 × 10^{6} | 3.215 × 10^{3} | 6.567 × 10^{3} | 2.764 × 10^{3} |

**Table 10.**The controlling parameters of the IEEE 30-bus system for cases 1 to 3 using the IEO based optimization method.

Parameters | Selective Objective | ||
---|---|---|---|

FC | APL | VD | |

P_{g}_{1} (MW) | 177.0567 | 51.2493 | 176.3369 |

P_{g}_{2} (MW) | 48.6972 | 80.0000 | 48.9664 |

P_{g}_{3} (MW) | 21.3043 | 50.0000 | 21.6539 |

P_{g}_{4} (MW) | 21.0814 | 35.0000 | 22.0081 |

P_{g}_{5} (MW) | 11.8842 | 30.0000 | 12.2721 |

P_{g}_{6} (MW) | 12.0000 | 40.0000 | 12.0000 |

V_{1} (p.u.) | 1.1000 | 1.1000 | 1.0391 |

V_{2} (p.u.) | 1.0879 | 1.0976 | 1.0227 |

V_{3} (p.u.) | 1.0617 | 1.0799 | 1.0156 |

V_{4} (p.u.) | 1.0694 | 1.0869 | 1.0051 |

V_{5} (p.u.) | 1.1000 | 1.1000 | 1.0245 |

V_{6} (p.u.) | 1.1000 | 1.1000 | 0.9951 |

T_{11(6–9)} | 1.0447 | 1.0546 | 1.0403 |

T_{12(6–10)} | 0.9000 | 0.9000 | 0.9000 |

T_{15(4–12)} | 0.9863 | 0.9841 | 0.9506 |

T_{36(28–27)} | 0.9657 | 0.9727 | 0.9691 |

Q_{C}_{1} (MVAR) | 5.0000 | 5.0000 | 4.8490 |

Q_{C}_{2} (MVAR) | 5.0000 | 5.0000 | 0.3485 |

Q_{C}_{3} (MVAR) | 5.0000 | 4.9996 | 4.9997 |

Q_{C}_{4} (MVAR) | 5.0000 | 5.0000 | 0.0048 |

Q_{C}_{5} (MVAR) | 5.0000 | 4.8218 | 5.0000 |

Q_{C6} (MVAR) | 5.0000 | 5.0000 | 4.9999 |

Q_{C}_{7} (MVAR) | 3.8491 | 3.6385 | 5.0000 |

Q_{C}_{8} (MVAR) | 5.0000 | 5.0000 | 4.9997 |

Q_{C}_{9} (MVAR) | 2.7434 | 2.5216 | 2.6104 |

Fuel Cost ($/h) | 799.0680 | 999.9988 | 803.6371 |

APL (MW) | 8.6245 | 2.8506 | 9.9343 |

VD | 1.8583 | 2.0489 | 0.0944 |

Parameter | Fuel Cost (FC) | ||||||||
---|---|---|---|---|---|---|---|---|---|

IEO | EO | MPA [13] | SMA [15] | MSCA [43] | GOA [44] | MVO [44] | IPSO [45] | HHO [46] | |

P_{g}_{1} (MW) | 177.0567 | 176.8497 | 177.032 | 176.2134 | 177.401 | - | - | 177.0431 | 176.97 |

P_{g}_{2} (MW) | 48.6972 | 48.9155 | 48.688 | 48.8501 | 48.632 | 48.0194 | 51.189 | 49.209 | 48.88 |

P_{g}_{3} (MW) | 21.3043 | 21.7782 | 21.305 | 21.5222 | 21.2376 | 20.9145 | 21.311 | 21.5135 | 21.42 |

P_{g}_{4} (MW) | 21.0814 | 18.8509 | 21.081 | 22.1311 | 20.8615 | 20.2342 | 21.173 | 22.648 | 22.02 |

P_{g}_{5} (MW) | 11.8842 | 13.8211 | 11.912 | 12.2063 | 11.9385 | 15.726 | 22.699 | 10.4146 | 12.29 |

P_{g}_{6} (MW) | 12.0000 | 12.1079 | 12.004 | 12.0000 | 12 | 13.5828 | 16.587 | 12 | 11.21 |

V_{1} (p.u.) | 1.1000 | 1.0998 | 1.100 | 1.0500 | 1.1 | 1.09356 | 1.0813 | 1.05 | - |

V_{2} (p.u.) | 1.0879 | 1.0796 | 1.088 | 1.0381 | 1.0867 | 1.040936 | 1.0689 | 1.0462 | - |

V_{3} (p.u.) | 1.0617 | 1.0345 | 1.062 | 1.0110 | 1.0604 | 0.969193 | 1.0406 | 1.0459 | - |

V_{4} (p.u.) | 1.0694 | 1.0452 | 1.069 | 1.0194 | 1.0923 | 0.987262 | 1.0442 | 1.0417 | - |

V_{5} (p.u.) | 1.1000 | 1.0996 | 1.100 | 1.1000 | 1.1 | 1.029317 | 1.0748 | 0.9523 | - |

V_{6} (p.u.) | 1.1000 | 1.0955 | 1.100 | 1.0999 | 1.1 | 1.001084 | 1.0111 | 1.05 | - |

T_{11(6–9)} | 1.0447 | 0.9940 | 1.045 | 0.9973 | 1.0439 | 1.066983 | 1.0525 | 1.01 | - |

T_{12(6–10)} | 0.9000 | 0.9136 | 0.900 | 0.9000 | 0.9144 | 1.084914 | 0.9602 | 0.98 | - |

T_{15(4–12)} | 0.9863 | 0.9619 | 0.987 | 1.0157 | 1.03 | 0.910429 | 0.9486 | 1.01 | - |

T_{36(28–27)} | 0.9657 | 0.9325 | 0.967 | 0.9403 | 0.9913 | 0.973125 | 0.9852 | 1.02 | - |

Q_{C}_{1} (MVAR) | 5.0000 | 4.6748 | 5.000 | 20.8943 | 0.0246 | 02.2169 | 2.4893 | 27.27 | - |

Q_{C}_{2} (MVAR) | 5.0000 | 0.3082 | 5.000 | - | 2.56 | 0.5252 | 1.3383 | - | - |

Q_{C}_{3} (MVAR) | 5.0000 | 0.0725 | 5.000 | - | 4.586 | 4.522 | 1.8017 | - | - |

Q_{C}_{4} (MVAR) | 5.0000 | 0.6951 | 5.000 | - | 2.4098 | 0.3904 | 0.1313 | - | - |

Q_{C}_{5} (MVAR) | 5.0000 | 4.6283 | 5.000 | - | 4.6378 | 2.5788 | 3.345 | - | - |

Q_{C6} (MVAR) | 5.0000 | 1.8789 | 5.000 | - | 0.3635 | 0.7132 | 0.482 | - | - |

Q_{C}_{7} (MVAR) | 3.8491 | 4.1374 | 3.661 | - | 3.1475 | 2.2812 | 0.9994 | - | - |

Q_{C}_{8} (MVAR) | 5.0000 | 0.1959 | 5.000 | 20.9865 | 4.8426 | 4.3131 | 3.2872 | 22.43 | - |

Q_{C}_{9} (MVAR) | 2.7434 | 3.6316 | 2.995 | - | 3.9411 | 1.1918 | 0.0411 | - | - |

Fuel Cost ($/h) | 799.0680 | 800.3361 | 799.072 | 802.5449 | 799.31 | 809.741 | 810.9011 | 801.97 | 801.829 |

APL (MW) | 8.6245 | 8.9235 | 8.622 | 9.5232 | 8.7327 | 10.09 | 7.68 | 13.39 | 9.387 |

VD | 1.8583 | 1.5860 | 1.852 | - | 1.4246 | 0.7165 | 0.3751 | - | - |

Parameter | Active Power Transmission Loss (APL) | ||||||||
---|---|---|---|---|---|---|---|---|---|

IEO | EO | MPA [13] | MSCA [43] | BHBO [47] | MOHS [48] | GWO [49] | DE [49] | FEA [50] | |

P_{g}_{1} (MW) | 51.2493 | 51.5220 | 51.250 | 52.08 | 67.3549 | 66.2759 | 51.81 | 51.82 | 59.3216 |

P_{g}_{2} (MW) | 80.0000 | 80.0000 | 80 | 79.28 | 72.8998 | 79.6413 | 80.00 | 79.99 | 74.8132 |

P_{g}_{3} (MW) | 50.0000 | 50.0000 | 50 | 50.00 | 48.1774 | 46.8835 | 50.00 | 49.99 | 49.8547 |

P_{g}_{4} (MW) | 35.0000 | 35.0000 | 35 | 35.00 | 33.3057 | 34.8880 | 35.00 | 35.00 | 34.9084 |

P_{g}_{5} (MW) | 30.0000 | 30.0000 | 30 | 30.00 | 27.6854 | 29.1213 | 30.00 | 29.98 | 28.1099 |

P_{g}_{6} (MW) | 40.0000 | 39.9702 | 40 | 39.97 | 37.4807 | 30.0558 | 40.00 | 40.00 | 39.7538 |

V_{1} (p.u.) | 1.1000 | 1.0520 | 1.100 | 1.10 | 1.0689 | 1.0774 | 1.1000 | 1.0288 | 1.0547 |

V_{2} (p.u.) | 1.0976 | 1.0467 | 1.098 | 1.07 | 1.0622 | 1.0638 | 1.0826 | 1.0537 | 1.0418 |

V_{3} (p.u.) | 1.0799 | 1.0254 | 1.080 | 1.08 | 1.0426 | 1.0365 | 1.0686 | 0.9782 | 1.0247 |

V_{4} (p.u.) | 1.0869 | 1.0334 | 1.087 | 1.10 | 1.0507 | 1.0497 | 0.9656 | 1.0056 | 1.0335 |

V_{5} (p.u.) | 1.1000 | 1.0995 | 1.100 | 1.10 | 1.0456 | 1.0955 | 1.0397 | 0.9518 | 1.0229 |

V_{6} (p.u.) | 1.1000 | 1.0997 | 1.100 | 1.10 | 1.0689 | 1.0979 | 1.0412 | 0.9642 | 1.0776 |

T_{11(6–9)} | 1.0546 | 0.9673 | 1.057 | 1.05 | 0.9907 | 0.9965 | 0.9625 | 0.9750 | 1.0125 |

T_{12(6–10)} | 0.9000 | 0.9063 | 0.900 | 0.95 | 0.9736 | 0.9124 | 0.9250 | 0.9000 | 0.9125 |

T_{15(4–12)} | 0.9841 | 0.9608 | 0.984 | 1.01 | 1.0144 | 0.9798 | 0.9500 | 0.9375 | 1.0125 |

T_{36(28–27)} | 0.9727 | 0.9356 | 0.973 | 0.99 | 0.9822 | 0.9499 | 0.9250 | 0.9250 | 1.0125 |

Q_{C}_{1} (MVAR) | 5.0000 | 4.8758 | 5.000 | 3.15 | 2.8915 | 0.0067 | 1.9463 | 4.9399 | 0.04 |

Q_{C}_{2} (MVAR) | 5.0000 | 4.9280 | 5.000 | 0.81 | 2.5199 | 0.0019 | 3.2861 | 4.9730 | 0.02 |

Q_{C}_{3} (MVAR) | 4.9996 | 3.5513 | 4.999 | 4.49 | 3.5486 | 0.0461 | 1.3935 | 4.9169 | 0.05 |

Q_{C}_{4} (MVAR) | 5.0000 | 5.0000 | 5.000 | 2.40 | 2.0410 | 0.0278 | 2.0929 | 4.9955 | 0.01 |

Q_{C}_{5} (MVAR) | 4.8218 | 3.3086 | 4.999 | 1.48 | 3.1853 | 0.0468 | 4.4512 | 4.3191 | 0.05 |

Q_{C6} (MVAR) | 5.0000 | 1.9816 | 5.000 | 4.64 | 2.7309 | 0.0417 | 4.9324 | 4.9790 | 0.00 |

Q_{C}_{7} (MVAR) | 3.6385 | 0.1478 | 3.713 | 3.17 | 3.1663 | 0.0050 | 5.0000 | 3.2145 | 0.02 |

Q_{C}_{8} (MVAR) | 5.0000 | 4.9375 | 5.000 | 4.69 | 3.3136 | 0.0208 | 4.8290 | 4.9843 | 0.05 |

Q_{C}_{9} (MVAR) | 2.5216 | 3.0386 | 2.540 | 1.80 | 1.8528 | 0.0267 | 1.1166 | 2.2067 | 0.02 |

Fuel Cost ($/h) | 999.9988 | 967.5659 | 999.845 | 965.648 | 932.8176 | 928.5099 | 968.38 | 968.23 | 952.3785 |

APL (MW) | 2.8506 | 3.0924 | 2.8513 | 2.9334 | 3.5035 | 3.5165 | 3.41 | 3.38 | 3.3541 |

VD | 2.0489 | 1.5225 | 2.048 | 1.5987 | 0.7993 | - | - | - | - |

Parameter | Voltage Deviation (VD) | ||||||||
---|---|---|---|---|---|---|---|---|---|

IEO | EO | MPA [13] | MSCA [43] | HHO [44] | BHBO [47] | MOJA [51] | TLBO [52] | MOBSA [53] | |

P_{g}_{1} (MW) | 176.3369 | 177.0716 | 175.172 | 112.585 | - | 172.0275 | 89.0808 | 176.7551 | - |

P_{g}_{2} (MW) | 48.9664 | 49.0630 | 48.703 | 79.76 | 56.607 | 48.0936 | 78.6206 | 48.8437 | 47.645 |

P_{g}_{3} (MW) | 21.6539 | 21.9624 | 21.515 | 22.25 | 23.951 | 21.8736 | 49.8306 | 21.5128 | 19.886 |

P_{g}_{4} (MW) | 22.0081 | 21.2153 | 22.328 | 25.09 | 14.786 | 20.8257 | 34.6289 | 21.9212 | 21.265 |

P_{g}_{5} (MW) | 12.2721 | 11.4725 | 12.300 | 29.95 | 17.467 | 20.8257 | 23.9941 | 12.2860 | 13.448 |

P_{g}_{6} (MW) | 12.0000 | 12.4194 | 13.184 | 20.85 | 19.229 | 15.2725 | 12.0077 | 12.0007 | 12.007 |

V_{1} (p.u.) | 1.0391 | 1.0404 | 1.035 | 1.01 | 1.0259 | 1.0338 | 1.0248 | 1.0429 | 1.0570 |

V_{2} (p.u.) | 1.0227 | 1.0227 | 1.019 | 0.99 | 1.0157 | 1.0170 | 1.0143 | 1.0259 | 1.0339 |

V_{3} (p.u.) | 1.0156 | 1.0076 | 1.010 | 1.02 | 1.0076 | 1.0116 | 1.0127 | 1.0148 | 1.0024 |

V_{4} (p.u.) | 1.0051 | 1.0038 | 1.010 | 1.05 | 1.0107 | 1.0027 | 1.0071 | 1.0071 | 1.0071 |

V_{5} (p.u.) | 1.0245 | 1.0530 | 1.062 | 1.05 | 1.0007 | 1.0435 | 1.0441 | 1.0253 | 0.9500 |

V_{6} (p.u.) | 0.9951 | 1.0094 | 0.997 | 0.99 | 1.0068 | 1.0141 | 1.0004 | 0.9876 | 1.0102 |

T_{11(6–9)} | 1.0403 | 1.0365 | 1.083 | 1.04 | 0.97984 | 1.0236 | 1.0646 | 1.0425 | 0.9663 |

T_{12(6–10)} | 0.9000 | 0.9050 | 0.909 | 0.95 | 0.96271 | 0.9250 | 0.9010 | 0.9000 | 0.9000 |

T_{15(4–12)} | 0.9506 | 0.9695 | 0.956 | 0.96 | 1.0038 | 0.9786 | 0.9574 | 0.9376 | 0.9941 |

T_{36(28–27)} | 0.9691 | 0.9473 | 0.969 | 0.95 | 0.96339 | 0.9633 | 0.9699 | 0.9702 | 0.9750 |

Q_{C}_{1} (MVAR) | 4.8490 | 3.4111 | 5.000 | 4.75 | 4.9891 | 2.9696 | 4.4080 | 0.0000 | 1.8465 |

Q_{C}_{2} (MVAR) | 0.3485 | 0.9139 | 0.855 | 4.13 | 4.6581 | 2.3947 | 0.0000 | 0.0000 | 4.9147 |

Q_{C}_{3} (MVAR) | 4.9997 | 4.7409 | 5.000 | 4.87 | 4.989 | 3.1905 | 4.8290 | 0.0000 | 5.0000 |

Q_{C}_{4} (MVAR) | 0.0048 | 0.3340 | 2.300 | 3.16 | 4.9891 | 3.0773 | 0.0773 | 0.0000 | 4.8485 |

Q_{C}_{5} (MVAR) | 5.0000 | 5.0000 | 5.000 | 4.93 | 3.4195 | 4.0279 | 4.9988 | 0.0000 | 5.0000 |

Q_{C6} (MVAR) | 4.9999 | 3.2971 | 5.000 | 4.91 | 4.9891 | 3.8901 | 4.8611 | 0.0000 | 5.0000 |

Q_{C}_{7} (MVAR) | 5.0000 | 2.3523 | 5.000 | 5.00 | 4.9891 | 3.7811 | 4.9784 | 0.0000 | 4.3718 |

Q_{C}_{8} (MVAR) | 4.9997 | 1.2541 | 5.000 | 4.93 | 3.4275 | 3.7777 | 4.9206 | 0.0000 | 5.0000 |

Q_{C}_{9} (MVAR) | 2.6104 | 0.6831 | 2.722 | 0.39 | 2.9353 | 2.3794 | 2.5858 | 0.0000 | 5.0000 |

Fuel Cost ($/h) | 803.6371 | 803.3426 | 803.9062 | 8.49.281 | 849.8061 | 804.5975 | 907.2475 | 803.7871 | 803.194 |

APL (MW) | 9.9343 | 9.9022 | 9.8005 | 7.0828 | 5.79 | 9.5778 | 4.7626 | 9.8641 | - |

VD | 0.0944 | 0.1323 | 0.0992 | 0.1030 | 0.1494 | 0.1262 | 0.0935 | 0.0945 | 0.1280 |

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## Share and Cite

**MDPI and ACS Style**

Lan, Z.; He, Q.; Jiao, H.; Yang, L.
An Improved Equilibrium Optimizer for Solving Optimal Power Flow Problem. *Sustainability* **2022**, *14*, 4992.
https://doi.org/10.3390/su14094992

**AMA Style**

Lan Z, He Q, Jiao H, Yang L.
An Improved Equilibrium Optimizer for Solving Optimal Power Flow Problem. *Sustainability*. 2022; 14(9):4992.
https://doi.org/10.3390/su14094992

**Chicago/Turabian Style**

Lan, Zhouxin, Qing He, Hongzan Jiao, and Liu Yang.
2022. "An Improved Equilibrium Optimizer for Solving Optimal Power Flow Problem" *Sustainability* 14, no. 9: 4992.
https://doi.org/10.3390/su14094992