# A Quasi-Oppositional Heap-Based Optimization Technique for Power Flow Analysis by Considering Large Scale Photovoltaic Generator

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## Abstract

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## 1. Introduction

- A novel QOHBO technique is proposed to overcome the disadvantage of conventional power flow analysis.
- The impact of large-scale PVG on LF analysis is studied.
- The proposed method is able to provide multiple solutions that can be utilized for voltage stability analysis.
- The efficiency of the proposed technique is tested by applying it to ill-conditioned systems.
- Robustness of the algorithm is verified under maximum loadability limits and high R/X ratios.

## 2. Problem Formulation

#### 2.1. Power Flow Problem

#### 2.2. Power Flow Embedded with PVGs

## 3. Solution Methodology Using QOHBO

#### 3.1. Basic Heap-Based Optimization Technique

#### 3.2. Quasi-Oppositional Based Learning

## 4. LF Using QOHBO Technique

Algorithm 1 [28]. Main body of QOHBO technique |

$\begin{array}{l}for\hspace{0.17em}iter\leftarrow \hspace{0.17em}1\hspace{0.17em}to\hspace{0.17em}Max-iter\hspace{0.17em}do\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Calculate\hspace{0.17em}gamma(\gamma )\hspace{0.17em}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Calculate\hspace{0.17em}\mathrm{selection}\mathrm{of}\mathrm{proportion}{p}_{1}\hspace{0.17em}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Calculate\hspace{0.17em}\mathrm{selection}\mathrm{of}\mathrm{proportion}{p}_{2}\hspace{0.17em}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}I\leftarrow \hspace{0.17em}N\hspace{0.17em}to\hspace{0.17em}2\hspace{0.17em}do\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i\leftarrow \hspace{0.17em}heap[I].value\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}b\hspace{0.17em}i\leftarrow \hspace{0.17em}heap[parent(I)].value\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\%\hspace{0.17em}index\hspace{0.17em}of\hspace{0.17em}parent\hspace{0.17em}of\hspace{0.17em}I\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}c\hspace{0.17em}i\leftarrow \hspace{0.17em}heap[colleague(I)].value\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\%\hspace{0.17em}index\hspace{0.17em}of\hspace{0.17em}random\hspace{0.17em}co-worker\hspace{0.17em}of\hspace{0.17em}I\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\overrightarrow{B}\leftarrow \hspace{0.17em}{\overrightarrow{y}}_{bi}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\%\hspace{0.17em}position\hspace{0.17em}vector\hspace{0.17em}of\hspace{0.17em}parent\hspace{0.17em}of\hspace{0.17em}I\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\overrightarrow{S}\leftarrow \hspace{0.17em}{\overrightarrow{y}}_{ci}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\%\hspace{0.17em}position\hspace{0.17em}vector\hspace{0.17em}of\hspace{0.17em}random\hspace{0.17em}co-worker\hspace{0.17em}of\hspace{0.17em}I\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}k\leftarrow \hspace{0.17em}1\hspace{0.17em}to\hspace{0.17em}D\hspace{0.17em}do\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}p\leftarrow \hspace{0.17em}rand()\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{y}_{temp}^{k}\leftarrow \hspace{0.17em}update\hspace{0.17em}{y}_{temp}^{k}(iter)\hspace{0.17em}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}end\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}l\leftarrow \hspace{0.17em}1\hspace{0.17em}to\hspace{0.17em}D\hspace{0.17em}do\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{B}_{l}={y}_{l}^{\mathrm{min}}+{y}_{l}^{\mathrm{max}}/2\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}OY(I,l)={y}_{l}^{\mathrm{min}}+{y}_{l}^{\mathrm{max}}-{y}_{l}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}QOY(I,l)={B}_{l}+{r}_{1}\times ({B}_{l}-OY(I,l))\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}end\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}if\hspace{0.17em}f(QOY(I)f({\overrightarrow{y}}_{temp}))\hspace{0.17em}then\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\overrightarrow{y}}_{temp}\leftarrow \hspace{0.17em}QOY(I)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}end\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}if\hspace{0.17em}f({\overrightarrow{y}}_{temp})f({\overrightarrow{y}}_{i}(iter))\hspace{0.17em}then\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\overrightarrow{y}}_{i}^{k}(iter+1)\leftarrow \hspace{0.17em}{\overrightarrow{y}}_{i}(iter)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}end\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Heapify-up(I)\\ end\\ return\hspace{0.17em}{y}_{heap[1].value}\end{array}$ |

## 5. Results and Discussion

Five bus system: | IEEE 14-bus system: | ||

${a}_{1}=0.0875$ | ${b}_{1}=-3.7031$ | ${a}_{1}=0.1331$ | ${b}_{1}=-14.4053$ |

${a}_{2}=-0.00184$ | ${b}_{2}=1.0691$ | ${a}_{2}=5\times {10}^{-5}$ | ${b}_{2}=1.6904$ |

${a}_{3}=-0.1034$ | ${b}_{3}=2.6334$ | ${a}_{3}=0.1002$ | ${b}_{3}=2.5049$ |

## 6. Conclusions

- The proposed method has the ability to provide multiple solutions simultaneously. For instance, the proposed QOHBO has provided three solutions in a single run for the IEEE 14-bus system when compared to the single solution provided by the conventional NR method.
- The proposed method provides a solution to ill-conditioned systems where conventional techniques fail. For example, the proposed QOHBO has been able to provide a solution for ill-conditioned 13-bus system where conventional methods such as NRLF, FDLF, and GS failed to produce the solution.
- The proposed QOHBO method provides better performance under maximum loadability and higher R/X ratios conditions. This performance is further enhanced with the integration of PVG. For instance, the line resistance multiplier for the critical conditions provided by QOHBO with PVG has been increased by 1, i.e., 5.9, when compared to NRLF and QOHBO, without PVG, is 4.9.
- The computational efficiency of the proposed method is higher when compared to other techniques. For example, the time taken by the proposed QOHBO to provide solution to IEEE 14-bus system is 0.0129 s when compared to 0.0684 s provided by GA.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Appendix B

#### Appendix B.1. Opposite Vector

#### Appendix B.2. Quasi-Oppositional Vector

_{1}denotes a random number generated between (0,1).

## References

- Acharjee, P.; Goswami, S.K. A decoupled power flow algorithm using particle swarm optimization technique. Energy Convers. Manag.
**2009**, 50, 2351–2360. [Google Scholar] [CrossRef] - Kumar, P.; Singh, A.K. Load Flow Analysis with Wind Farms. In Handbook of Distributed Generation; Bansal, R., Ed.; Springer International Publishing: Cham, Switzerland, 2017; pp. 149–170. ISBN 978-3-319-51342-3. [Google Scholar]
- Salomon, C.P.; Lambert-Torres, G.; Martins, H.G.; Ferreira, C.; Costa, C.I.A. Load flow computation via Particle Swarm Optimization. In Proceedings of the 2010 9th IEEE/IAS International Conference on Industry Applications—INDUSCON 2010, Sao Paulo, Brazil, 8–10 November 2010; IEEE: Piscataway, NJ, USA, 2010; pp. 1–6. [Google Scholar]
- Karimi, M.; Shahriari, A.; Aghamohammadi, M.R.; Marzooghi, H.; Terzija, V. Application of Newton-based load flow methods for determining steady-state condition of well and ill-conditioned power systems: A review. Int. J. Electr. Power Energy Syst.
**2019**, 113, 298–309. [Google Scholar] [CrossRef] - Wang, Y.; Wu, H.; Xu, H.; Li, Q.; Liu, S. A general fast power flow algorithm for transmission and distribution networks. IEEE Access
**2020**, 8, 23284–23293. [Google Scholar] [CrossRef] - Portelinha, R.K.; Durce, C.C.; Tortelli, O.L.; Lourenço, E.M. Fast-decoupled power flow method for integrated analysis of transmission and distribution systems. Electr. Power Syst. Res.
**2021**, 196, 107215. [Google Scholar] [CrossRef] - Costilla-Enriquez, N.; Weng, Y.; Zhang, B. Combining Newton-Raphson and Stochastic Gradient Descent for Power Flow Analysis. IEEE Trans. Power Syst.
**2021**, 36, 514–517. [Google Scholar] [CrossRef] - Montoya, O.D.; Gil-González, W. On the numerical analysis based on successive approximations for power flow problems in AC distribution systems. Electr. Power Syst. Res.
**2020**, 187, 106454. [Google Scholar] [CrossRef] - Milano, F. Implicit Continuous Newton Method for Power Flow Analysis. IEEE Trans. Power Syst.
**2019**, 34, 3309–3311. [Google Scholar] [CrossRef] - Coletta, G.; Vaccaro, A.; Villacci, D. Fast and reliable uncertain power flow analysis by affine arithmetic. Electr. Power Syst. Res.
**2019**, 175, 105860. [Google Scholar] [CrossRef] - Albadi, M. Power Flow Analysis. Comput. Model. Eng.
**2020**, 1–15. [Google Scholar] [CrossRef] [Green Version] - Tostado-Véliz, M.; Kamel, S.; Jurado, F. A Robust Power Flow Algorithm Based on Bulirsch-Stoer Method. IEEE Trans. Power Syst.
**2019**, 34, 3081–3089. [Google Scholar] [CrossRef] - Wong, K.P.; Li, A.; Law, T.M.Y. Advanced constrained genetic algorithm load flow method. IEE Proc. Gener. Transm. Distrib.
**1999**, 146, 609–616. [Google Scholar] [CrossRef] - Tostado, M.; Kamel, S.; Jurado, F. Developed Newton-Raphson based Predictor-Corrector load flow approach with high convergence rate. Int. J. Electr. Power Energy Syst.
**2019**, 105, 785–792. [Google Scholar] [CrossRef] - Tostado-Veliz, M.; Alharbi, T.; Alrumayh, O.; Kamel, S.; Jurado, F. A Novel Power Flow Solution Paradigm for Well and Ill-Conditioned Cases. IEEE Access
**2021**, 9, 112425–112438. [Google Scholar] [CrossRef] - Oh, H. A Unified and Efficient Approach to Power Flow Analysis. Energies
**2019**, 12, 2425. [Google Scholar] [CrossRef] [Green Version] - Tostado-Véliz, M.; Kamel, S.; Jurado, F. Robust and efficient approach based on Richardson extrapolation for solving badly initialised/ill-conditioned power-flow problems. IET Gener. Transm. Distrib.
**2019**, 13, 3524–3533. [Google Scholar] [CrossRef] - Tostado-Véliz, M.; Kamel, S.; Jurado, F. Power flow solution of Ill-conditioned systems using current injection formulation: Analysis and a novel method. Int. J. Electr. Power Energy Syst.
**2021**, 127, 106669. [Google Scholar] [CrossRef] - Pires, R.; Mili, L.; Chagas, G. Robust complex-valued Levenberg-Marquardt algorithm as applied to power flow analysis. Int. J. Electr. Power Energy Syst.
**2019**, 113, 383–392. [Google Scholar] [CrossRef] - Wong, K.P.; Li, A.; Law, M.Y. Development of constrained-genetic-algorithm load-flow method. IEE Proc. Gener. Transm. Distrib.
**1997**, 144, 91. [Google Scholar] [CrossRef] - Acharjee, P.; Goswami, S.K. Robust load flow based on local search. Expert Syst. Appl.
**2008**, 35, 1400–1407. [Google Scholar] [CrossRef] - Acharjee, P.; Goswami, S.K. Expert algorithm based on adaptive particle swarm optimization for power flow analysis. Expert Syst. Appl.
**2009**, 36, 5151–5156. [Google Scholar] [CrossRef] - Acharjee, P.; Goswami, S.K. Chaotic particle swarm optimization based robust load flow. Int. J. Electr. Power Energy Syst.
**2010**, 32, 141–146. [Google Scholar] [CrossRef] - Gnanambal, K.; Marimuthu, N.S.; Babulal, C.K. A Hybrid Differential Evolution Algorithm to Solve Power Flow Problem in rectangular coordinate. J. Electr. Syst.
**2010**, 6, 395–406. [Google Scholar] [CrossRef] - Shahnawaz Ahmed, S.; Mohsin, M. Analytical Determination of the Control Parameters for a Large Photovoltaic Generator Embedded in a Grid System. IEEE Trans. Sustain. Energy
**2011**, 2, 122–130. [Google Scholar] [CrossRef] - Juarez, R.T.; Fuerte-Esquivel, C.R.; Espinosa-Juarez, E.; Sandoval, U. Steady-State Model of Grid-Connected Photovoltaic Generation for Power Flow Analysis. IEEE Trans. Power Syst.
**2018**, 33, 5727–5737. [Google Scholar] [CrossRef] - Wang, Y.-B.; Wu, C.-S.; Liao, H.; Xu, H.-H. Steady-state model and power flow analysis of grid-connected photovoltaic power system. In Proceedings of the 2008 IEEE International Conference on Industrial Technology, Chengdu, China, 21–24 April 2008; IEEE: Piscataway, NJ, USA, 2008; pp. 1–6. [Google Scholar]
- Askari, Q.; Saeed, M.; Younas, I. Heap-based optimizer inspired by corporate rank hierarchy for global optimization. Expert Syst. Appl.
**2020**, 161, 113702. [Google Scholar] [CrossRef] - Barisal, A.K.; Prusty, R.C. Large scale economic dispatch of power systems using oppositional invasive weed optimization. Appl. Soft Comput. J.
**2015**, 29, 122–137. [Google Scholar] [CrossRef] - Bhattacharjee, K.; Bhattacharya, A.; Dey, S.H.N. Oppositional Real Coded Chemical Reaction Optimization for different economic dispatch problems. Int. J. Electr. Power Energy Syst.
**2014**, 55, 378–391. [Google Scholar] [CrossRef] - Tiew On, T. Development of Hybrid Constrained Genetic Algorithm and Particle Swarm Optimisation Algorithm for Load Flow; The Hong Kong Polytechnic University: Hong Kong, China, 2007. [Google Scholar]
- Mallick, S.; Rajan, D.V.; Thakur, S.S.; Acharjee, P.; Ghoshal, S.P. Development of a new algorithm for power flow analysis. Int. J. Electr. Power Energy Syst.
**2011**, 33, 1479–1488. [Google Scholar] [CrossRef]

Bus No. | Without Embedded PVG | With Embedded PVG | ||
---|---|---|---|---|

Voltage Magnitude (P.U.) | Voltage Angle (rad) | Voltage Magnitude (P.U.) | Voltage Angle (rad) | |

1 | 1.0400 | 0.0000 | 1.0400 | 0.0000 |

2 | 0.9614 | −0.1103 | 0.9621 | −0.0720 |

3 | 1.0200 | −0.0648 | 1.0200 | 0.0026 |

4 | 0.9203 | −0.1900 | 0.9203 | −0.1437 |

5 | 0.9683 | −0.1075 | 0.9692 | −0.0779 |

Bus No. | Without Embedded PVG | With Embedded PVG | ||
---|---|---|---|---|

Voltage Magnitude (P.U.) | Voltage Angle (rad) | Voltage Magnitude (P.U.) | Voltage Angle (rad) | |

1 | 1.0600 | 0.0000 | 1.0600 | 0.0000 |

2 | 1.0450 | −0.0870 | 1.0450 | −0.0620 |

3 | 1.0100 | −0.2224 | 1.0100 | −0.1818 |

4 | 1.0142 | −0.1790 | 1.0211 | −0.1267 |

5 | 1.0172 | −0.1530 | 1.0262 | −0.1000 |

6 | 1.0700 | −0.2516 | 1.0700 | −0.1033 |

7 | 1.0503 | −0.2313 | 1.0500 | −0.1553 |

8 | 1.0900 | −0.2313 | 1.0900 | −0.1553 |

9 | 1.0337 | −0.2589 | 1.0284 | −0.1706 |

10 | 1.0326 | −0.2625 | 1.0273 | −0.1635 |

11 | 1.0475 | −0.2591 | 1.0437 | −0.1356 |

12 | 1.0535 | −0.2665 | 1.0535 | −0.1223 |

13 | 1.0471 | −0.2672 | 1.0455 | −0.1271 |

14 | 1.0213 | −0.2804 | 1.0171 | −0.1695 |

Method | Test System | Load Multiplier | Solution |
---|---|---|---|

NRLF | IEEE 5-bus System | 1.91 | V(P.U.) = [1.0400 0.7914 0.9700 0.5566 0.7541] Angle (rad) = [0.0000 −0.5019 −0.5736 −0.8800 −0.4367] |

IEEE 14-Bus System | 3.6 | V(P.U.) = [1.0600 0.9950 0.9600 0.7436 0.7348 1.0200 0.8092 1.0400 0.7241 0.7386 0.8611 0.9364 0.8942 0.6964]; Angle (rad)= [0.0000 −0.5722 −1.3463 −1.0868 −0.9253 −1.5826 −1.4169 −1.4169 −1.5969 −1.6249 −1.6101 −1.6504 −1.6522 −1.7465]; | |

Proposed QOHBO | IEEE 5-bus System | 2 | V(P.U.)= [1.0400 0.8037 1.0200 0.5828 0.7606]; Angle (rad) = [0.0000 −0.5490 −0.6390 −0.9226 −0.4752]; |

IEEE 14-Bus System | 3.98 | V(P.U.) = [1.0600 1.0450 1.0100 0.7073 0.6798 1.0700 0.8002 1.0900 0.7044 0.7268 0.8784 0.9766 0.9269 0.6862]; Angle (rad) = [0.0000 −0.7270 −1.5901 −1.3247 −1.1363 −1.9355 −1.7197 −1.7197 −1.9291 −1.9654 −1.9573 −2.0046 −2.0052 −2.1044]; |

Solution Method | IEEE 5-Bus System | IEEE 14-Bus System |
---|---|---|

NRLF | 4.9 | 4.47 |

NRLF with Optimal multiplier [22] | - | 4.4225 |

Local search [22] | - | 4.4288 |

GA [22] | - | 4.4371 |

APSO [22] | - | 4.4371 |

Proposed QOHBO without PVG | 4.9 | 4.5 |

Proposed QOHBO with PVG | 5.9 | 4.9 |

Solution Method | IEEE 5-Bus System | IEEE 14-Bus System |
---|---|---|

NRLF | 0.012 | 0.043 |

NRLF with Optimal multiplier [22] | - | 0.0479 |

Local search [22] | 0.0428 | |

GA [22] | - | 0.0419 |

APSO [22] | - | 0.0419 |

Proposed QOHBO without PVG | 0.001 | 0.0418 |

Proposed QOHBO with PVG | 0.0001 | 0.0406 |

Bus No. | Proposed QOHBO-LF | NRLF [32] | FDLF [32] | GS [32] | ||||
---|---|---|---|---|---|---|---|---|

Voltage Magnitude (P.U.) | Voltage Angle (rad) | Voltage Magnitude (P.U.) | Voltage Angle (rad) | Voltage Magnitude (P.U.) | Voltage Angle (rad) | Voltage Magnitude (P.U.) | Voltage Angle (rad) | |

1 | 1.0000 | 0.0000 | NC | NC | NC | NC | NC | NC |

2 | 0.9745 | 0.0409 | ||||||

3 | 0.9426 | 0.0425 | ||||||

4 | 1.0630 | 0.1588 | ||||||

5 | 1.0442 | 0.0933 | ||||||

6 | 1.0672 | 0.1433 | ||||||

7 | 1.0177 | 0.2134 | ||||||

8 | 0.9430 | 0.2525 | ||||||

9 | 1.1000 | 0.1476 | ||||||

10 | 1.1000 | 0.1438 | ||||||

11 | 1.0000 | 0.0440 | ||||||

12 | 1.0370 | 0.1723 | ||||||

13 | 0.9693 | 0.0265 |

Bus No. | Solution 1 | Solution 2 | Solution 3 | |||
---|---|---|---|---|---|---|

Voltage Magnitude (P.U.) | Voltage Angle (rad) | Voltage Magnitude (P.U.) | Voltage Angle (rad) | Voltage Magnitude (P.U.) | Voltage Angle (rad) | |

1 | 1.0600 | 0.0000 | 1.0600 | 0.0000 | 1.0600 | 0.0000 |

2 | 1.0450 | −0.2598 | 1.0450 | −0.0870 | 1.0450 | −2.4130 |

3 | 1.0100 | −0.5856 | 1.0100 | −0.2224 | 1.0100 | −2.5765 |

4 | 0.5607 | −0.4967 | 1.0142 | −0.1790 | 0.8014 | −2.4474 |

5 | 0.4906 | −0.3076 | 1.0172 | −0.1530 | 0.6721 | −2.3380 |

6 | 1.0700 | −3.2109 | 1.0700 | −0.2516 | 1.0700 | −2.5282 |

7 | 0.5750 | −1.4133 | 1.0503 | −0.2313 | 0.9537 | −2.5039 |

8 | 1.0900 | −1.4133 | 1.0900 | −0.2313 | 1.0900 | −2.5039 |

9 | 0.4426 | −1.9847 | 1.0337 | −0.2589 | 0.9402 | −2.5299 |

10 | 0.4554 | −2.3885 | 1.0326 | −0.2625 | 0.9548 | −2.5344 |

11 | 0.7016 | −2.9596 | 1.0475 | −0.2591 | 1.0075 | −2.5326 |

12 | 0.9857 | −3.2113 | 1.0535 | −0.2665 | 1.0465 | −2.5439 |

13 | 0.9215 | −3.1658 | 1.0471 | −0.2672 | 1.0331 | −2.5429 |

14 | 0.5129 | −2.7932 | 1.0213 | −0.2804 | 0.9613 | −2.5556 |

Solution Method | 5-Bus System | IEEE 14-Bus System | Ill-Conditioned 13-Bus System |
---|---|---|---|

NRLF | 0.0282 s | 0.0522 s | - |

FDLF [1] | 0.0518 s | 0.0776 s | - |

NRLF with Optimal multiplier [22] | 0.0365 s | 0.0632 s | - |

Local search [22] | 0.0360 s | 0.585 s | - |

GA [22] | 0.0372 s | 0.0684 s | - |

PSO method [1] | 0.0318 s | 0.0603 s | - |

APSO [22] | 0.0370 s | 0.0598 s | - |

PSO with update [1] | 0.0498 s | 0.0601 s | - |

Proposed QOHBO | 0.0105 s | 0.0129 s | 0.0101 s |

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## Share and Cite

**MDPI and ACS Style**

Basetti, V.; Rangarajan, S.S.; Kumar Shiva, C.; Verma, S.; Collins, R.E.; Senjyu, T.
A Quasi-Oppositional Heap-Based Optimization Technique for Power Flow Analysis by Considering Large Scale Photovoltaic Generator. *Energies* **2021**, *14*, 5382.
https://doi.org/10.3390/en14175382

**AMA Style**

Basetti V, Rangarajan SS, Kumar Shiva C, Verma S, Collins RE, Senjyu T.
A Quasi-Oppositional Heap-Based Optimization Technique for Power Flow Analysis by Considering Large Scale Photovoltaic Generator. *Energies*. 2021; 14(17):5382.
https://doi.org/10.3390/en14175382

**Chicago/Turabian Style**

Basetti, Vedik, Shriram S. Rangarajan, Chandan Kumar Shiva, Sumit Verma, Randolph E. Collins, and Tomonobu Senjyu.
2021. "A Quasi-Oppositional Heap-Based Optimization Technique for Power Flow Analysis by Considering Large Scale Photovoltaic Generator" *Energies* 14, no. 17: 5382.
https://doi.org/10.3390/en14175382