# A Fractional-Order Archimedean Spiral Moth–Flame Optimization Strategy to Solve Optimal Power Flows

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## Abstract

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## 1. Introduction

- Utilizing the strengths of FO-AMFO, a novel application of the fractional evolutionary approach for ORPD is proposed.
- The design scheme is successfully applied to ORPD problems in order to minimize power losses while meeting load demand and operational constraints.
- The algorithm’s performance is determined by statistical findings in the form of probability plots, learning curves, and histogram analysis, which demonstrate the algorithm’s stability, resilience, and consistency as a precise and efficient optimization mechanism.

## 2. Problem Formulation

_{g}and r

_{h}are the penalty multipliers for constraints.

_{i}ϵ [V, T, Q]. It comprises the IEEE standard bus system’s Qi (capacitor bank), the voltage control bus Vi, and the transformer tap setting Ti. The total objective function that takes into account the penalty function for breaking both constraints is

_{G}), and total tap changers (N

_{T}). The upper and lower, or permissible, limits of these control variables are represented by the highest and lowest values of Qi, Vi and Ti in the aforementioned equations. Equation (3) states that any deviation from these bounds will result in a fitness penalty.

#### 2.1. System Constraints

_{Gi}and P

_{Di}, respectively, whereas the reactive power generation and demand at the ith bus are represented by Q

_{Gi}and Q

_{Di}. Bij and Gij are the line’s conductance and susceptance.

#### 2.2. Tap Limits

#### 2.3. Voltage Generation Limits

#### 2.4. Reactors Limits

_{T}, the total number of generators si N and buses with compensators are represented by N

_{C}.

## 3. Methodology

_{FPOS}) is the global optimum and the moth-associated flame (LB

_{FPOS}) is the local best position. Every iteration updates each moth’s position based on its current position and velocity. The updated velocity is in line with the moth’s initial velocity as well as social behavior patterns and cognitive processes.

## 4. Results and Discussion

#### 4.1. Case 1: ORPD 13 Variables for a 30-Bus IEEE System

#### 4.2. Case 2: ORPD for the IEEE 57-Bus System with 25 Control Variables

#### 4.3. Comparative Analysis through Statistics

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Yu, X.; Yu, X.; Lu, Y.; Sheng, J. Economic and emission dispatch using ensemble multi-objective differential evolution algorithm. Sustainability
**2018**, 10, 418. [Google Scholar] [CrossRef] - Jiang, S.; Ji, Z.; Wang, Y. A novel gravitational acceleration enhanced particle swarm optimization algorithm for wind–thermal economic emission dispatch problem considering wind power availability. Int. J. Electr. Power Energy Syst.
**2015**, 73, 1035–1050. [Google Scholar] [CrossRef] - Jadoun, V.K.; Pandey, V.C.; Gupta, N.; Niazi, K.R.; Swarnkar, A. Integration of renewable energy sources in dynamic economic load dispatch problem using an improved fireworks algorithm. IET Renew. Power Gener.
**2018**, 12, 1004–1011. [Google Scholar] [CrossRef] - Santra, D.; Sarker, K.; Mukherjee, A.; Mondal, S. Combined economic emission and load dispatch using hybrid metaheuristics. Int. J. Hybrid Intell.
**2019**, 1, 211–238. [Google Scholar] [CrossRef] - Sinha, N.; Chakrabarti, R.; Chattopadhyay, P. Evolutionary programming techniques for economic load dispatch. IEEE Trans. Evol. Comput.
**2003**, 7, 83–94. [Google Scholar] [CrossRef] - Gopalakrishnan, R.; Krishnan, A. An efficient technique to solve combined economic and emission dispatch problem using modified Ant colony optimization. Sadhana
**2013**, 38, 545–556. [Google Scholar] [CrossRef] - Faris, H.; Aljarah, I.; Al-Betar, M.A.; Mirjalili, S. Grey wolf optimizer: A review of recent variants and applications. Neural Comput. Appl.
**2017**, 30, 413–435. [Google Scholar] [CrossRef] - Manton, J. Optimization algorithms exploiting unitary constraints. IEEE Trans. Signal Process.
**2002**, 50, 635–650. [Google Scholar] [CrossRef] - Hu, Y.; Liu, K.; Zhang, X.; Su, L.; Ngai, E.; Liu, M. Application of evolutionary computation for rule discovery in stock algorithmic trading: A literature review. Appl. Soft Comput.
**2015**, 36, 534–551. [Google Scholar] [CrossRef] - Askarzadeh, A. Solving electrical power system problems by harmony search: A review. Artif. Intell. Rev.
**2016**, 47, 217–251. [Google Scholar] [CrossRef] - Binitha, S.; Sathya, S.S. A survey of bio inspired optimization algorithms. Int. J. Soft Comput. Eng.
**2012**, 2, 137–151. [Google Scholar] - Zhang, J.; Wu, Y.; Guo, Y.; Wang, B.; Wang, H.; Liu, H. A hybrid harmony search algorithm with differential evolution for day-ahead scheduling problem of a microgrid with consideration of power flow constraints. Appl. Energy
**2016**, 183, 791–804. [Google Scholar] [CrossRef] - Naderi, E.; Azizivahed, A.; Narimani, H.; Fathi, M.; Narimani, M.R. A comprehensive study of practical economic dispatch problems by a new hybrid evolutionary algorithm. Appl. Soft Comput.
**2017**, 61, 1186–1206. [Google Scholar] [CrossRef] - Neto, J.X.V.; Reynoso-Meza, G.; Ruppel, T.H.; Mariani, V.C.; dos Santos Coelho, L. Solving non-smooth economic dispatch by a new combination of continuous GRASP algorithm and differential evolution. Int. J. Electr. Power Energy Syst.
**2017**, 84, 13–24. [Google Scholar] [CrossRef] - Khan, B.S.; Raja, M.A.Z.; Qamar, A.; Chaudhary, N.I. Design of moth flame optimization heuristics for integrated power plant system containing stochastic wind. Appl. Soft Comput.
**2021**, 104, 107193. [Google Scholar] [CrossRef] - Muhammad, Y.; Khan, R.; Ullah, F.; Rehman, A.U.; Aslam, M.S.; Raja, M.A.Z. Design of fractional swarming strategy for solution of optimal reactive power dispatch. Neural Comput. Appl.
**2019**, 32, 10501–10518. [Google Scholar] [CrossRef] - El Ela, A.A.; Abido, M.A.; Spea, S.R. Differential evolution algorithm for optimal reactive power dispatch. Electr. Power Syst. Res.
**2011**, 81, 458–464. [Google Scholar] [CrossRef] - Mahadevan, K.; Kannan, P. Comprehensive learning particle swarm optimization for reactive power dispatch. Appl. Soft Comput.
**2010**, 10, 641–652. [Google Scholar] [CrossRef] - Khazali, A.; Kalantar, M. Optimal reactive power dispatch based on harmony search algorithm. Int. J. Electr. Power Energy Syst.
**2011**, 33, 684–692. [Google Scholar] [CrossRef] - Ghasemi, M.; Ghavidel, S.; Ghanbarian, M.M.; Habibi, A. A new hybrid algorithm for optimal reactive power dispatch problem with discrete and continuous control variables. Appl. Soft Comput.
**2014**, 22, 126–140. [Google Scholar] [CrossRef] - Mandal, B.; Roy, P.K. Optimal reactive power dispatch using quasi-oppositional teaching learning based optimization. Int. J. Electr. Power Energy Syst.
**2013**, 53, 123–134. [Google Scholar] [CrossRef] - Sulaiman, M.H.; Mustaffa, Z.; Mohamed, M.R.; Aliman, O. Using the gray wolf optimizer for solving optimal reactive power dispatch problem. Appl. Soft Comput.
**2015**, 32, 286–292. [Google Scholar] [CrossRef] - Mei, R.N.S.; Sulaiman, M.H.; Mustaffa, Z.; Daniyal, H. Optimal reactive power dispatch solution by loss minimization using moth-flame optimization technique. Appl. Soft Comput.
**2017**, 59, 210–222. [Google Scholar] [CrossRef] - Ghamisi, P.; Couceiro, M.S.; Benediktsson, J.A. A Novel Feature Selection Approach Based on FODPSO and SVM. IEEE Trans. Geosci. Remote Sens.
**2014**, 53, 2935–2947. [Google Scholar] [CrossRef] - Ates, A.; Alagoz, B.B.; Kavuran, G.; Yeroglu, C. Implementation of fractional order filters discretized by modified Fractional Order Darwinian Particle Swarm Optimization. Measurement
**2017**, 107, 153–164. [Google Scholar] [CrossRef] - Couceiro, M.S.; Rocha, R.P.; Ferreira, N.M.F.; Machado, J.A.T. Introducing the fractional-order Darwinian PSO. Signal Image Video Process.
**2012**, 6, 343–350. [Google Scholar] [CrossRef] - Shahri, E.S.A.; Alfi, A.; Machado, J.T. Fractional fixed-structure H∞ controller design using augmented Lagrangian particle swarm optimization with fractional order velocity. Appl. Soft Comput.
**2019**, 77, 688–695. [Google Scholar] [CrossRef] - Machado, J.T.; Kiryakova, V. The chronicles of fractional calculus. Fract. Calc. Appl. Anal.
**2017**, 20, 307–336. [Google Scholar] [CrossRef] - Ghamisi, P.; Couceiro, M.S.; Martins, F.M.L.; Benediktsson, J.A. Multilevel Image Segmentation Based on Fractional-Order Darwinian Particle Swarm Optimization. IEEE Trans. Geosci. Remote Sens.
**2013**, 52, 2382–2394. [Google Scholar] [CrossRef] - Zhu, Q.; Yuan, M.; Chen, Y.; Liu, Y.-L.; Chen, W.-D.; Wang, H.-R. Research and application on fractional-order Darwinian PSO based adaptive extended kalman filtering algorithm. IAES Int. J. Robot. Autom.
**2014**, 3, 245–251. [Google Scholar] [CrossRef] - Yokoya, N.; Ghamisi, P. Land-cover monitoring using time-series hyperspectral data via fractional-order darwinian particle swarm optimization segmentation. In Proceedings of the 2016 8th Workshop on Hyperspectral Image and Signal Processing: Evolution in Remote Sensing (WHISPERS), Los Angeles, CA, USA, 21–24 August 2016; IEEE: Piscataway, NJ, USA, 2016; pp. 1–5. [Google Scholar]
- Wang, Y.-Y.; Zhang, H.; Qiu, C.-H.; Xia, S.-R. A Novel Feature Selection Method Based on Extreme Learning Machine and Fractional-Order Darwinian PSO. Comput. Intell. Neurosci.
**2018**, 2018, 1–8. [Google Scholar] [CrossRef] [PubMed] - Paliwal, K.; Singh, S.; Gaba, P. Feature selection approach of hyperspectral image using GSA-FODPSO-SVM. In Proceedings of the 2017 International Conference on Computing, Communication and Automation (ICCCA), Greater Noida, India, 5–6 May 2017; IEEE: Piscataway, NJ, USA, 2017; pp. 1070–1075. [Google Scholar]
- Łegowski, A.; Niezabitowski, M. Robot path control based on PSO with fractional-order velocity. In Proceedings of the 2016 International Conference on Robotics and Automation Engineering (ICRAE), Jeju, Korea, 27–29 August 2016; IEEE: Piscataway, NJ, USA, 2016; pp. 21–25. [Google Scholar]
- Kuttomparambil Abdulkhader, H.; Jacob, J.; Mathew, A.T. Fractional-order lead-lag compensator-based multi-band power system stabiliser design using a hybrid dynamic GA-PSO algorithm. IET Gener. Transm. Distrib.
**2018**, 12, 3248–3260. [Google Scholar] [CrossRef] - Kosari, M.; Teshnehlab, M. Non-linear fractional-order chaotic systems identification with approximated fraction-al-order derivative based on a hybrid particle swarm optimization-genetic algorithm method. J. AI Data Min.
**2018**, 6, 365–373. [Google Scholar] - Mirjalili, S. Moth-flame optimization algorithm: A novel nature-inspired heuristic paradigm. Knowl. Based Syst.
**2015**, 89, 228–249. [Google Scholar] [CrossRef] - Brzeziński, D.W.; Ostalczyk, P. About accuracy increase of fractional order derivative and integral computations by applying the Grünwald–Letnikov formula. Commun. Nonlinear Sci. Numer. Simul.
**2016**, 40, 151–162. [Google Scholar] [CrossRef] - Dai, C.; Chen, W.; Zhu, Y.; Zhang, X. Seeker Optimization Algorithm for Optimal Reactive Power Dispatch. IEEE Trans. Power Syst.
**2009**, 24, 1218–1231. [Google Scholar] [CrossRef] - Villa-Acevedo, W.M.; López-Lezama, J.M.; Valencia-Velásquez, J.A. A Novel Constraint Handling Approach for the Optimal Reactive Power Dispatch Problem. Energies
**2018**, 11, 2352. [Google Scholar] [CrossRef] - Babu, R.; Raj, S.; Dey, B.; Bhattacharyya, B. Optimal reactive power planning using oppositional grey wolf optimization by considering bus vulnerability analysis. Energy Convers. Econ.
**2021**, 3, 38–49. [Google Scholar] [CrossRef] - Khan, N.H.; Wang, Y.; Tian, D.; Raja, M.A.Z.; Jamal, R.; Muhammad, Y. Design of Fractional Particle Swarm Optimization Gravitational Search Algorithm for Optimal Reactive Power Dispatch Problems. IEEE Access
**2020**, 8, 146785–146806. [Google Scholar] [CrossRef]

**Figure 3.**IEEE 30-bus system [16].

**Figure 7.**IEEE 57-bus system [20].

**Figure 10.**Convergence characteristic curves of the proposed FO-AMFO algorithm for fractional orders of 0.1 to 0.9: (

**a**) fractional order δ = 0.1, (

**b**) fractional order δ = 0.2, (

**c**) fractional order δ = 0.3, (

**d**) fractional order δ = 0.4, (

**e**) fractional order δ = 0.5, (

**f**) fractional order δ = 0.6, (

**g**) fractional order δ = 0.7, (

**h**) fractional order δ = 0.8, (

**i**) fractional order δ = 0.9.

**Figure 12.**Statistical performance in terms of a histogram, a box plot and independent runs. (

**a**) Histogram analysis. (

**b**) Box plot for fitness. (

**c**) Fitness comparison.

Functions | GSA [37] | PSO [37] | MFO [37] | FO-AMFO |
---|---|---|---|---|

${F}_{1}(x)={\displaystyle \sum _{i=1}^{n}}{x}_{i}^{2}$ | 608.2432 | 1.321140 | 0.0001170 | 0.0099567 |

${F}_{2}(x)={\displaystyle \sum _{i=1}^{n}}\left|{x}_{i}\right|+{\displaystyle \prod _{i=1}^{n}}\left|{x}_{i}\right|$ | 22.7534 | 7.715640 | 0.0006390 | 0.0000077 |

${F}_{3}(x)={\displaystyle \sum _{i=1}^{n}}{\left({\displaystyle \sum _{j-1}^{i}}{x}_{j}\right)}^{2}$ | 135760.7 | 736.3190 | 696.73090 | 3.761800 |

${F}_{4}(x)=\underset{i}{\mathrm{m}\mathrm{a}\mathrm{x}}\left\{\left|{x}_{i}\right|,1\u2a7di\u2a7dn\right\}$ | 78.7854 | 12.97654 | 70.686460 | 24.662000 |

${F}_{5}(x)={\displaystyle \sum _{i=1}^{n-1}}\left[100{\left({x}_{i+1}-{x}_{i}^{2}\right)}^{2}+{\left({x}_{i}-1\right)}^{2}\right]$ | 741.005 | 77360.83 | 139.14870 | 97.876000 |

${F}_{6}(x)={\displaystyle \sum _{i=1}^{n}}{\left(\left[{x}_{i}+0.5\right]\right)}^{2}$ | 3080.67 | 286.6585 | 0.0001130 | 0.00000408 |

${F}_{7}(x)=({\displaystyle \sum _{i=1}^{n}}i{x}_{i}^{4}+\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{o}\mathrm{m}(0,1))$ | 0.11256 | 1.037426 | 0.0911550 | 0.0279000 |

${F}_{8}(x)={\displaystyle \sum _{i=1}^{n}}-{x}_{i}\mathrm{s}\mathrm{i}\mathrm{n}\left(\sqrt{\left|{x}_{i}\right|}\right)$ | −2352.36 | −3572.000 | −8496.780 | −3065.6000 |

${F}_{9}(x)={\displaystyle \sum _{i=1}^{n}}\left[{x}_{i}^{2}-10\mathrm{c}\mathrm{o}\mathrm{s}\left(2\pi {x}_{i}\right)+10\right]$ | 31.00015 | 124.1962 | 84.600090 | 19.996000 |

${F}_{10}(x)=\frac{1}{4000}{\displaystyle \sum _{i=1}^{n}}{x}_{i}^{2}-{\displaystyle \prod _{i=1}^{n}}\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{{x}_{i}}{\sqrt{i}}\right)+1$ | 0.048679 | 0.021654 | 0.0190800 | 0.296200 |

**Table 2.**Description of IEEE 30 standard systems [16].

Characteristics | IEEE 30 Bus 13 Variables |
---|---|

Total Buses | 30 |

Load Buses | 24 |

Total Generators | 6 |

Total Transformers | 4 |

Total Capacitors | 9 |

Total Reactors | 0 |

Total Branches | 41 |

Control Variables | Reported Results Proposed | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

D-E [17] | G-A [18] | (PSO) [19] | HS-A [19] | MICA-(IWO) [20] | IC-A [20] | I-WO [21] | G-WO [22] | (MFO) [23] | FO- (AMFO) | |

V_{g-1} | 1.095 | 1.072 | 1.031 | 1.072 | 1.079 | 1.078 | 1.069 | 1.100 | 1.100 | 1.028 |

V_{g-2} | 1.085 | 1.063 | 1.011 | 1.062 | 1.070 | 1.069 | 1.060 | 1.096 | 1.094 | 1.021 |

V_{g-5} | 1.062 | 1.037 | 1.022 | 1.039 | 1.048 | 1.069 | 1.036 | 1.080 | 1.075 | 0.985 |

V_{g-8} | 1.065 | 1.044 | 1.003 | 1.042 | 1.048 | 1.047 | 1.038 | 1.080 | 1.771 | 0.990 |

V_{g-11} | 1.026 | 1.013 | 0.974 | 1.031 | 1.075 | 1.034 | 1.029 | 1.093 | 1.086 | 1.002 |

V_{g-13} | 1.014 | 1.089 | 0.998 | 1.068 | 1.070 | 1.071 | 1.055 | 1.100 | 1.100 | 1.028 |

T_{h6-9} | 1.017 | 1.022 | 0.970 | 1.010 | 1.030 | 1.080 | 1.050 | 1.040 | 1.041 | 1.054 |

T_{h6-10} | 0.979 | 0.991 | 1.020 | 1 | 0.990 | 0.950 | 0.960 | 0.950 | 0.950 | 1.045 |

T_{h4-12} | 0.977 | 0.996 | 1.010 | 0.990 | 1 | 1 | 0.970 | 0.950 | 0.955 | 0.922 |

T_{h27-28} | 1.008 | 0.97 | 0.990 | 0.970 | 0.980 | 0.970 | 0.970 | 0.950 | 0.957 | 0.938 |

Q-c_{3} | 20.223 | 5.350 | 17 | 34 | −7 | −6 | 8 | 12 | 7.103 | 30 |

Q-c_{10} | 9.584 | 36 | 13 | 12 | 23 | 36 | 35 | 30 | 30.796 | 29.870 |

Q-c_{24} | 13.029 | 12.417 | 23 | 10 | 12 | 11 | 11 | 8 | 9.898 | 30 |

P_{L} | 4.88 | 4.87 | 5.88 | 5.10 | 4.84 | 4.84 | 4.92 | 4.61 | 4.60 | 4.219 |

**Table 4.**Description of the IEEE 57-bus standard system [20].

Characteristics | IEEE 57-Bus 25 Variables |
---|---|

Total Buses | 57 |

Load Buses | 45 |

Total Generators | 7 |

Total Transformers | 17 |

Total Capacitors | 3 |

Total Reactors | 15 |

Total Branches | 80 |

Control Variables | Base Case [16] | FO- DPSO [16] | SOA [39] | MFO [15] | IWO [20] | CLPS [40] | GWO [41] | FPSO-GSA [42] | FO-AMFO |
---|---|---|---|---|---|---|---|---|---|

V-_{G (1)} | 1.040 | 1.04 | 1.060 | 1.060 | 1.06 | 1.054 | 1.060 | 1.100 | 1.100 |

V-_{G (2)} | 1.011 | 1.029 | 1.058 | 1.058 | 1.059 | 1.052 | 1.056 | 1.098 | 1.0991 |

V-_{G (3)} | 0.985 | 1.009 | 1.043 | 1.046 | 1.047 | 1.033 | 1.037 | 1.087 | 1.0889 |

V-_{G (6)} | 0.980 | 0.977 | 1.035 | 1.042 | 1.038 | 1.031 | 1.020 | 1.080 | 1.0832 |

V-_{G (8)} | 1.050 | 0.985 | 1.054 | 1.060 | 1.059 | 1.049 | 1.044 | 1.100 | 1.1000 |

V-_{G (9)} | 0.980 | 0.967 | 1.036 | 1.042 | 1.027 | 1.030 | 1.029 | 1.084 | 1.0848 |

V-_{G (12)} | 1.015 | 0.908 | 1.033 | 1.037 | 1.037 | 1.034 | 1.031 | 0.960 | 1.0808 |

T-_{TS (4-18)} | 0.970 | 0.9 | 1 | 0.950 | 1.05 | 0.990 | 0.984 | 1.007 | 0.900 |

T-_{TS (4-18)} | 0.978 | 0.920 | 0.96 | 1.007 | 1.0 | 0.980 | 0.932 | 1.084 | 0.900 |

T-_{TS (21-20)} | 1.043 | 1.026 | 1.01 | 1.006 | 1.07 | 0.990 | 0.957 | 0.995 | 1.0032 |

T-_{TS (24-26)} | 1.043 | 1.007 | 1.01 | 1.007 | 1.02 | 1.010 | 0.996 | 0.990 | 0.9879 |

T-_{TS (7-29)} | 0.967 | 0.907 | 0.97 | 0.975 | 0.97 | 0.990 | 0.963 | 0.996 | 0.9000 |

T-_{TS (34-32)} | 0.965 | 0.987 | 0.97 | 0.972 | 0.99 | 0.930 | 0.981 | 1.007 | 0.9802 |

T-_{TS (11-41)} | 0.955 | 0.901 | 0.9 | 0.900 | 0.9 | 0.910 | 1.062 | 0.990 | 0.9000 |

T-_{TS (15-45)} | 0.955 | 0.9 | 0.97 | 0.9718 | 0.96 | 0.9700 | 0.9755 | 0.9906 | 0.9000 |

T-_{TS (14-46)} | 0.900 | 0.9 | 0.95 | 0.953 | 0.95 | 0.9500 | 0.9639 | 1.0028 | 0.9000 |

T-_{TS (10-51)} | 0.930 | 0.916 | 0.96 | 0.9673 | 0.98 | 0.9800 | 0.9723 | 0.9900 | 0.9108 |

T-_{TS (13-49)} | 0.895 | 0.9 | 0.92 | 0.9278 | 0.93 | 0.9500 | 0.9248 | 1.0027 | 0.9000 |

T-_{TS (11-43)} | 0.958 | 0.9 | 0.96 | 0.964 | 0.99 | 0.9500 | 0.9554 | 1.0844 | 0.9000 |

T-_{TS (40-56)} | 0.958 | 0.998 | 1 | 0.9998 | 1.01 | 1.000 | 1.1000 | 1.0023 | 1.1024 |

T-_{TS (39-57)} | 0.980 | 0.994 | 0.96 | 0.9606 | 1.04 | 0.9600 | 0.9976 | 0.9900 | 0.9844 |

T-_{TS (955)} | 0.940 | 0.9 | 0.97 | 0.9789 | 0.96 | 0.9700 | 0.9845 | 1.0951 | 0.900 |

Q_{SC (18)} | 0 | 4 | 9.984 | 9.9968 | 0.0442 | 0.0988 | 1.8917 | 4.9846 | 1.1581 |

Q_{SC (25)} | 0 | 15 | 5.904 | 5.9000 | 0.0443 | 0.0542 | 5.2489 | 4.9992 | 4.9028 |

Q_{SC (53)} | 0 | 11.67 | 6.288 | 6.300 | 0.0615 | 0.0628 | 5.1513 | 4.3653 | 4.7725 |

P_{loss} (MW) | 27.86 | 26.68 | 24.26 | 24.252 | 24.593 | 24.89 | 24.752 | 22.918 | 21.6210 |

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**MDPI and ACS Style**

Wadood, A.; Ahmed, E.; Rhee, S.B.; Sattar Khan, B.
A Fractional-Order Archimedean Spiral Moth–Flame Optimization Strategy to Solve Optimal Power Flows. *Fractal Fract.* **2024**, *8*, 225.
https://doi.org/10.3390/fractalfract8040225

**AMA Style**

Wadood A, Ahmed E, Rhee SB, Sattar Khan B.
A Fractional-Order Archimedean Spiral Moth–Flame Optimization Strategy to Solve Optimal Power Flows. *Fractal and Fractional*. 2024; 8(4):225.
https://doi.org/10.3390/fractalfract8040225

**Chicago/Turabian Style**

Wadood, Abdul, Ejaz Ahmed, Sang Bong Rhee, and Babar Sattar Khan.
2024. "A Fractional-Order Archimedean Spiral Moth–Flame Optimization Strategy to Solve Optimal Power Flows" *Fractal and Fractional* 8, no. 4: 225.
https://doi.org/10.3390/fractalfract8040225