# An Effective Power Dispatch of Photovoltaic Generators in DC Networks via the Antlion Optimizer

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## Abstract

**:**

## 1. Introduction

#### 1.1. General Context

#### 1.2. Motivation

#### 1.3. State of the Art

#### 1.4. Scope and Main Contributions

- The implementation of a mathematical formulation that observes all technical and operating constraints that constitute the operation of DC grids under a scenario of PV distributed generation.
- A new master–slave methodology to solve the optimal power dispatch in DC grids which employs the ALO and a matrix hourly power flow method and yields the best results with regard to solution quality (best solution and standard deviation).
- The implementation of two DC grids (GCN and SN) that represent the average PV power generation and demand reported in Colombia.
- A methodology that allows identifying the optimization method that optimally balances solution quality and processing times for solving the problem of optimal power dispatch in DC grids.
- The identification of the current economic needs associated with operating GCN and SN in Colombia.

#### 1.5. Paper Structure

## 2. Mathematical Formulation

#### 2.1. Objective Functions

#### 2.2. Set of Constraints

#### 2.3. Fitness Function

## 3. Codification and Optimization Methodology

#### 3.1. Antlion Optimizer

#### 3.1.1. Generating the Initial Population

#### 3.1.2. Evaluating the Fitness Function and Selecting the Incumbent

#### 3.1.3. Algorithm Advancement Method

Algorithm 1: Iterative process proposed for the ALO. |

#### 3.2. Matrix Hourly Power Flow

Algorithm 2: Iterative process for matrix hourly power flow based on successive approximations. |

## 4. Test Systems, Generation and Demand Curves, and Additional Considerations

#### 4.1. PV Generation Curves

#### 4.2. Power Demand Curves

#### 4.3. Test Systems

#### 4.4. Grid-Connected Test Feeder

#### 4.5. Standalone Test Feeder

#### 4.6. Comparison Methods

## 5. Simulation Results and Discussions

#### 5.1. Standalone Test Feeder

#### 5.2. Grid-Connected Test Feeder

#### 5.3. Average Processing Times

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Hernandez, J.; Velasco, D.; Trujillo, C. Analysis of the effect of the implementation of photovoltaic systems like option of distributed generation in Colombia. Renew. Sustain. Energy Rev.
**2011**, 15, 2290–2298. [Google Scholar] [CrossRef] - Dong, J.; Feng, T.T.; Sun, H.X.; Cai, H.X.; Li, R.; Yang, Y. Clean distributed generation in China: Policy options and international experience. Renew. Sustain. Energy Rev.
**2016**, 57, 753–764. [Google Scholar] [CrossRef] - Boumaiza, A.; Sanfilippo, A.; Mohandes, N. Modeling multi-criteria decision analysis in residential PV adoption. Energy Strategy Rev.
**2022**, 39, 100789. [Google Scholar] [CrossRef] - Zhao, Z.Y.; Zhang, S.Y.; Hubbard, B.; Yao, X. The emergence of the solar photovoltaic power industry in China. Renew. Sustain. Energy Rev.
**2013**, 21, 229–236. [Google Scholar] [CrossRef] - Chang, G.W.; Chinh, N.C. Coyote optimization algorithm-based approach for strategic planning of photovoltaic distributed generation. IEEE Access
**2020**, 8, 36180–36190. [Google Scholar] [CrossRef] - Singh, G.K. Solar power generation by PV (photovoltaic) technology: A review. Energy
**2013**, 53, 1–13. [Google Scholar] [CrossRef] - Solargis Solar Resource Maps of Colombia. Available online: https://solargis.com/maps-and-gis-data/download/colombia (accessed on 22 November 2022).
- Rakhshani, E.; Rouzbehi, K.J.; Sánchez, A.; Tobar, A.C.; Pouresmaeil, E. Integration of large scale PV-based generation into power systems: A survey. Energies
**2019**, 12, 1425. [Google Scholar] [CrossRef][Green Version] - Díaz González, J.J. Viabilidad Regulatoria para Implementar Sistemas de Micro Redes Con Fuentes No Convencionales de Energía Renovable–FNCER–por Intercolombia SAESP. 2018. Available online: http://bibliotecavirtualoducal.uc.cl/vufind/Record/oai:localhost:10185-28863 (accessed on 24 January 2023).
- Moreno, C.; Milanes, C.B.; Arguello, W.; Fontalvo, A.; Alvarez, R.N. Challenges and perspectives of the use of photovoltaic solar energy in Colombia. Int. J. Electr. Comput. Eng.
**2022**, 12, 4521–4528. [Google Scholar] [CrossRef] - De Brito, M.A.G.; Galotto, L.; Sampaio, L.P.; e Melo, G.d.A.; Canesin, C.A. Evaluation of the main MPPT techniques for photovoltaic applications. IEEE Trans. Ind. Electron.
**2012**, 60, 1156–1167. [Google Scholar] [CrossRef] - Mishra, V.L.; Chauhan, Y.K.; Verma, K. A critical review on advanced reconfigured models and metaheuristics-based MPPT to address complex shadings of solar array. Energy Convers. Manag.
**2022**, 269, 116099. [Google Scholar] [CrossRef] - Grisales-Noreña, L.F.; Montoya, O.D.; Ramos-Paja, C.A. An energy management system for optimal operation of BSS in DC distributed generation environments based on a parallel PSO algorithm. J. Energy Storage
**2020**, 29, 101488. [Google Scholar] [CrossRef] - Li, C.; De Bosio, F.; Chen, F.; Chaudhary, S.K.; Vasquez, J.C.; Guerrero, J.M. Economic dispatch for operating cost minimization under real-time pricing in droop-controlled DC microgrid. IEEE J. Emerg. Sel. Top. Power Electron.
**2016**, 5, 587–595. [Google Scholar] [CrossRef][Green Version] - Younes, Z.; Alhamrouni, I.; Mekhilef, S.; Reyasudin, M. A memory-based gravitational search algorithm for solving economic dispatch problem in micro-grid. Ain Shams Eng. J.
**2021**, 12, 1985–1994. [Google Scholar] [CrossRef] - Tan, Q.; Ding, Y.; Ye, Q.; Mei, S.; Zhang, Y.; Wei, Y. Optimization and evaluation of a dispatch model for an integrated wind-photovoltaic-thermal power system based on dynamic carbon emissions trading. Appl. Energy
**2019**, 253, 113598. [Google Scholar] [CrossRef] - Gao, S.; Jia, H.; Marnay, C. Techno-economic evaluation of mixed AC and DC power distribution network for integrating large-scale photovoltaic power generation. IEEE Access
**2019**, 7, 105019–105029. [Google Scholar] [CrossRef] - Lin, J.C.W.; Liu, Q.; Fournier-Viger, P.; Hong, T.P.; Voznak, M.; Zhan, J. A sanitization approach for hiding sensitive itemsets based on particle swarm optimization. Eng. Appl. Artif. Intell.
**2016**, 53, 1–18. [Google Scholar] [CrossRef] - Lin, J.C.W.; Yang, L.; Fournier-Viger, P.; Wu, J.M.T.; Hong, T.P.; Wang, L.S.L.; Zhan, J. Mining high-utility itemsets based on particle swarm optimization. Eng. Appl. Artif. Intell.
**2016**, 55, 320–330. [Google Scholar] [CrossRef] - Spea, S.R. Combined economic emission dispatch solution of an isolated renewable integrated micro-grid using crow search algorithm. In Proceedings of the 2019 21st International Middle East Power Systems Conference (MEPCON), Cairo, Egypt, 17–19 December 2019; pp. 47–52. [Google Scholar]
- Rosales-Muñoz, A.A.; Grisales-Noreña, L.F.; Montano, J.; Montoya, O.D.; Perea-Moreno, A.J. Application of the multiverse optimization method to solve the optimal power flow problem in direct current electrical networks. Sustainability
**2021**, 13, 8703. [Google Scholar] [CrossRef] - Velasquez, O.S.; Montoya Giraldo, O.D.; Garrido Arevalo, V.M.; Grisales Norena, L.F. Optimal power flow in direct-current power grids via black hole optimization. Adv. Electr. Electron. Eng.
**2019**, 17, 24–32. [Google Scholar] [CrossRef] - Rosales Muñoz, A.A.; Grisales-Noreña, L.F.; Montano, J.; Montoya, O.D.; Giral-Ramírez, D.A. Optimal Power Dispatch of Distributed Generators in Direct Current Networks Using a Master–Slave Methodology That Combines the Salp Swarm Algorithm and the Successive Approximation Method. Electronics
**2021**, 10, 2837. [Google Scholar] [CrossRef] - Garzon-Rivera, O.; Ocampo, J.; Grisales-Noreña, L.; Montoya, O.; Rojas-Montano, J. Optimal power flow in Direct Current Networks using the antlion optimizer. Stat. Optim. Inf. Comput.
**2020**, 8, 846–857. [Google Scholar] [CrossRef] - Grisales-Noreña, L.F.; Ocampo-Toro, J.A.; Rosales-Muñoz, A.A.; Cortes-Caicedo, B.; Montoya, O.D. An Energy Management System for PV Sources in Standalone and Connected DC Networks Considering Economic, Technical, and Environmental Indices. Sustainability
**2022**, 14, 16429. [Google Scholar] [CrossRef] - Grisales-Noreña, L.F.; Ramos-Paja, C.A.; Gonzalez-Montoya, D.; Alcalá, G.; Hernandez-Escobedo, Q. Energy management in PV based microgrids designed for the Universidad Nacional de Colombia. Sustainability
**2020**, 12, 1219. [Google Scholar] [CrossRef][Green Version] - Gbadamosi, S.L.; Nwulu, N.I. Optimal power dispatch and reliability analysis of hybrid CHP-PV-wind systems in farming applications. Sustainability
**2020**, 12, 8199. [Google Scholar] [CrossRef] - Luna, A.C.; Diaz, N.L.; Andrade, F.; Graells, M.; Guerrero, J.M.; Vasquez, J.C. Economic power dispatch of distributed generators in a grid-connected microgrid. In Proceedings of the 2015 9th International Conference on Power Electronics and ECCE Asia (ICPE-ECCE Asia), Seoul, Korea, 1–5 June 2015; pp. 1161–1168. [Google Scholar]
- Gil-González, W.; Montoya, O.D.; Holguín, E.; Garces, A.; Grisales-Noreña, L.F. Economic dispatch of energy storage systems in dc microgrids employing a semidefinite programming model. J. Energy Storage
**2019**, 21, 1–8. [Google Scholar] [CrossRef] - Grisales-Noreña, L.F.; Rosales-Muñoz, A.A.; Cortés-Caicedo, B.; Montoya, O.D.; Andrade, F. Optimal Operation of PV Sources in DC Grids for Improving Technical, Economical, and Environmental Conditions by Using Vortex Search Algorithm and a Matrix Hourly Power Flow. Mathematics
**2023**, 11, 93. [Google Scholar] [CrossRef] - Lin, C.W.; Hong, T.P.; Yang, K.T.; Wang, S.L. The GA-based algorithms for optimizing hiding sensitive itemsets through transaction deletion. Appl. Intell.
**2015**, 42, 210–230. [Google Scholar] [CrossRef] - Lin, J.C.W.; Zhang, Y.; Zhang, B.; Fournier-Viger, P.; Djenouri, Y. Hiding sensitive itemsets with multiple objective optimization. Soft Comput.
**2019**, 23, 12779–12797. [Google Scholar] [CrossRef] - NASA. NASA Prediction of Worldwide Energy Resources, Washington, DC, United States. Available online: https://power.larc.nasa.gov/ (accessed on 21 September 2022).
- XM SA ESP. Sinergox Database, Colombia. Available online: https://sinergox.xm.com.co/Paginas/Home.aspx (accessed on 21 September 2022).
- Instituto de Planificación y Promoción de Soluciones Energéticas para Zonas No Interconectadas. Informes Mensuales de Telimetría, Colombia. Available online: https://ipse.gov.co/cnm/informe-mensuales-telemetria/ (accessed on 21 September 2022).
- Zolghadr-Asli, B.; Bozorg-Haddad, O.; Chu, X. Crow search algorithm (CSA). In Advanced Optimization by Nature-Inspired Algorithms; Springer: Berlin/Heidelberg, Germany, 2018; pp. 143–149. [Google Scholar]
- Kennedy, J.; Eberhart, R. Particle swarm optimization. In Proceedings of the ICNN’95-International Conference on Neural Networks, Perth, WA, Australia, 27 November–1 December 1995; Volume 4, pp. 1942–1948. [Google Scholar]
- Mirjalili, S.; Mirjalili, S.M.; Hatamlou, A. Multi-verse optimizer: A nature-inspired algorithm for global optimization. Neural Comput. Appl.
**2016**, 27, 495–513. [Google Scholar] [CrossRef] - Mirjalili, S.; Gandomi, A.H.; Mirjalili, S.Z.; Saremi, S.; Faris, H.; Mirjalili, S.M. Salp Swarm Algorithm: A bio-inspired optimizer for engineering design problems. Adv. Eng. Softw.
**2017**, 114, 163–191. [Google Scholar] [CrossRef] - Doğan, B.; Ölmez, T. A new metaheuristic for numerical function optimization: Vortex Search algorithm. Inf. Sci.
**2015**, 293, 125–145. [Google Scholar] [CrossRef] - Mirjalili, S. The ant lion optimizer. Adv. Eng. Softw.
**2015**, 83, 80–98. [Google Scholar] [CrossRef] - Mirjalili, S.; Jangir, P.; Saremi, S. Multi-objective ant lion optimizer: A multi-objective optimization algorithm for solving engineering problems. Appl. Intell.
**2017**, 46, 79–95. [Google Scholar] [CrossRef] - Yang, X.S. Nature-Inspired Metaheuristic Algorithms; Luniver Press: Beckington, UK, 2010. [Google Scholar]
- Falaghi, H.; Ramezani, M.; Haghifam, M.R.; Milani, K.R. Optimal selection of conductors in radial distribution systems with time varying load. In Proceedings of the CIRED 2005-18th International Conference and Exhibition on Electricity Distribution, Turin, Italy, 6–9 June 2005; pp. 1–4. [Google Scholar]
- Grisales-Noreña, L.; Montoya-Giraldo, O.; Gil-González, W. Optimal Integration of Distributed Generators into DC Microgrids Using a Hybrid Methodology: Genetic and Vortex Search Algorithms. Arab. J. Sci. Eng.
**2022**, 47, 14657–14672. [Google Scholar] [CrossRef] - Abou El Ela, A.; El-Sehiemy, R.A.; Shaheen, A.; Shalaby, A. Application of the crow search algorithm for economic environmental dispatch. In Proceedings of the 2017 nineteenth international Middle East power systems conference (MEPCON), Cairo, Egypt, 19–21 December 2017; pp. 78–83. [Google Scholar]
- Verma, S.; Shiva, C.K. A novel salp swarm algorithm for expansion planning with security constraints. Iran. J. Sci. Technol. Trans. Electr. Eng.
**2020**, 44, 1335–1344. [Google Scholar] [CrossRef]

**Figure 7.**Average reductions (

**a**) and standard deviation (

**b**) obtained by the optimization methods with respect to the ALO regarding the economic, technical, and environmental objective functions in the standalone system.

**Figure 8.**Power supplied by the distributed generators located in the standalone network for the three objective functions considered.

**Figure 9.**Average reductions (

**a**) and standard deviation (

**b**) obtained by the optimization methods with respect to the ALO regarding economic, technical, and environmental objective functions in the grid-connected network.

**Figure 10.**Power supplied by the distributed generators located in the grid-connected network for the three objective functions considered.

Method | Year | Objective Function | Considered | Not Considered | Reference |
---|---|---|---|---|---|

Genetic algorithm | 2016 | Minimization of operating costs | Variable power demand and generation Costs associated with renewable and conventional generators voltage profile limits Average solution analysis | Average processing time analysis Standard deviation analysis Comparison methods Maximum current line | [14] |

Non-dominant sorting genetic algorithm-II | 2019 | Minimization of operating costs and energy losses | Best solution analysis Comparison methods | Average solution analysis Average processing time analysis Standard deviation analysis Maximum current line | [17] |

Crow search algorithm | 2019 | Minimization of operating costs and $C{O}_{2}$ emissions | Best solution analysis Comparison methods | Average solution analysis Average processing time analysis Standard deviation analysis Maximum current line | [20] |

Black hole optimization | 2019 | Minimization of power losses | Best solution analysis Comparison methods | Average solution analysis Average processing time analysis Standard deviation analysis Maximum current line | [22] |

Parallel particle swarm optimizer | 2020 | Minimization of the energy purchasing costs | Best solution analysis Average processing time analysis Average solution analysis Standard deviation analysis Comparison methods voltage profile limits Maximum current line | [13] | |

Antlion optimizer | 2020 | Minimization of power losses | Best solution analysis Average processing time analysis Comparison methods | Average solution analysis Standard deviation analysis Maximum current line | [24] |

Salp swarm algorithm | 2021 | Minimization of power losses | Best solution anylisis Average processing time analysis Average solution analysis Standard deviation analysis Comparison methods Maximum current line | variable power generation and demand | [23] |

Memory-based gravitational search algorithm | 2021 | Minimization of operating costs | Best solution analysis Comparison methods | Average processing time analysis Average solution analysis Standard deviation analysis Maximum current line | [15] |

Multiverse optimizer | 2021 | Minimization of power losses | Best solution analysis Average solution analysis Standard deviation analysis Comparison methods | Average processing time analysis Maximum current line | [21] |

Salp swarm algorithm | 2022 | Minimization of energy losses, operating costs, and $C{O}_{2}$ emissions | Average solution analysis Average processing time analysis Standard deviation analysis Comparison methods Maximum current line | [30] |

Line l | Node i | Node j | ${\mathit{R}}_{\mathit{ij}}\phantom{\rule{0.222222em}{0ex}}\left(\mathbf{\Omega}\right)$ | ${\mathit{P}}_{\mathit{j}}\phantom{\rule{0.222222em}{0ex}}$(kW) | ${\mathit{I}}_{\mathit{l}}^{max}\phantom{\rule{0.222222em}{0ex}}$(A) |
---|---|---|---|---|---|

1 | 1 | 2 | 0.0922 | 100 | 320 |

2 | 2 | 3 | 0.4930 | 90 | 280 |

3 | 3 | 4 | 0.3660 | 120 | 195 |

4 | 4 | 5 | 0.3811 | 60 | 195 |

5 | 5 | 6 | 0.8190 | 60 | 195 |

6 | 6 | 7 | 0.1872 | 200 | 95 |

7 | 7 | 8 | 1.7114 | 200 | 85 |

8 | 8 | 9 | 1.0300 | 60 | 70 |

9 | 9 | 10 | 1.0400 | 60 | 55 |

10 | 10 | 11 | 0.1966 | 45 | 55 |

11 | 11 | 12 | 0.3744 | 60 | 55 |

12 | 12 | 13 | 1.4680 | 60 | 40 |

13 | 13 | 14 | 0.5416 | 120 | 40 |

14 | 14 | 15 | 0.5910 | 60 | 25 |

15 | 15 | 16 | 0.7463 | 60 | 20 |

16 | 16 | 17 | 1.2890 | 60 | 20 |

17 | 17 | 18 | 0.7320 | 90 | 20 |

18 | 2 | 19 | 0.1640 | 90 | 30 |

19 | 19 | 20 | 1.5042 | 90 | 25 |

20 | 20 | 21 | 0.4095 | 90 | 20 |

21 | 21 | 22 | 0.7089 | 90 | 20 |

22 | 3 | 23 | 0.4512 | 90 | 85 |

23 | 23 | 24 | 0.8980 | 420 | 70 |

24 | 24 | 25 | 0.8900 | 420 | 40 |

25 | 6 | 26 | 0.2030 | 60 | 85 |

26 | 26 | 27 | 0.2842 | 60 | 85 |

27 | 27 | 28 | 1.0590 | 60 | 70 |

28 | 28 | 29 | 0.8042 | 120 | 70 |

29 | 29 | 30 | 0.5075 | 200 | 55 |

30 | 30 | 31 | 0.9744 | 150 | 40 |

31 | 31 | 32 | 0.3105 | 210 | 25 |

32 | 32 | 33 | 0.3410 | 60 | 20 |

Line l | Node i | Node j | ${\mathit{R}}_{\mathit{ij}}\phantom{\rule{0.222222em}{0ex}}\left(\mathbf{\Omega}\right)$ | ${\mathit{P}}_{\mathit{j}}\phantom{\rule{0.222222em}{0ex}}$(kW) | ${\mathit{I}}_{\mathit{l}}^{max}\phantom{\rule{0.222222em}{0ex}}$(A) |
---|---|---|---|---|---|

1 | 1 | 2 | 0.0140 | 0 | 195 |

2 | 2 | 3 | 0.7463 | 0 | 145 |

3 | 3 | 4 | 0.4052 | 297.5 | 85 |

4 | 4 | 5 | 1.1524 | 0 | 70 |

5 | 5 | 6 | 0.5261 | 255 | 70 |

6 | 6 | 7 | 0.7127 | 0 | 55 |

7 | 7 | 8 | 1.6628 | 212.5 | 55 |

8 | 8 | 9 | 5.3434 | 0 | 20 |

9 | 9 | 10 | 2.1522 | 266.05 | 20 |

10 | 2 | 11 | 0.4052 | 85 | 70 |

11 | 11 | 12 | 1.1524 | 340 | 55 |

12 | 12 | 13 | 0.5261 | 297.5 | 40 |

13 | 13 | 14 | 1.2358 | 19125 | 25 |

14 | 14 | 15 | 2.8835 | 106.25 | 20 |

15 | 15 | 16 | 5.3434 | 255 | 20 |

16 | 3 | 17 | 1.2942 | 255 | 55 |

17 | 17 | 18 | 0.7027 | 127.5 | 40 |

18 | 18 | 19 | 3.3234 | 297.5 | 40 |

19 | 19 | 20 | 1.5172 | 340 | 20 |

20 | 20 | 21 | 0.7127 | 85 | 20 |

21 | 4 | 22 | 8.2528 | 106.25 | 20 |

22 | 5 | 23 | 9.1961 | 55.25 | 20 |

23 | 6 | 24 | 0.7463 | 69.7 | 20 |

24 | 8 | 25 | 2.0112 | 255 | 20 |

25 | 8 | 26 | 3.3234 | 63.75 | 20 |

26 | 26 | 27 | 0.5261 | 170 | 20 |

**Table 4.**Parameters used to calculate the economic, technical, and environmental objective functions.

Parameter | Value | Unit | Parameter | Value | Unit |
---|---|---|---|---|---|

${C}_{kWh}^{Urban}$ | 0.1302 | USD/kWh | $C{E}_{s}^{Urban}$ | 0.1644 | kg/kWh |

${C}_{kWh}^{Rural}$ | 0.2913 | USD/kWh | $C{E}_{s}^{Rural}$ | 0.2671 | kg/kWh |

${C}_{O\&M}^{pv}$ | 0.0019 | USD/kWh | $\mathrm{\Delta}V$ | ±10 | % |

Method | Optimization Parameter | Value |
---|---|---|

ALO | Number of particles | 95 |

Maximum iterations | 972 | |

Non-improving iterations | 292 | |

CSA | Number of particles | 177 |

Maximum iterations | 471 | |

Non-improving iterations | 295 | |

Awareness probability (${A}_{p}$) | 0.65826 | |

Flight length ($fl$) | 3.25058 | |

PSO | Number of particles | 159 |

Maximum iterations | 492 | |

Non-improving iterations | 229 | |

Maximum inertia (${W}_{max}$) | 0.99456 | |

Minimum inertia (${W}_{min}$) | 0.32458 | |

Cognitive component (${C}_{1}$) | 0.061368 | |

Social component (${C}_{2}$) | 1.5456 | |

MVO | Number of particles | 41 |

Maximum iterations | 1326 | |

Non-improving iterations | 188 | |

Wep-min | 0.68125 | |

Wep-max | 0.51768 | |

P parameter | 3 | |

SSA | Number of particles | 141 |

Maximum iterations | 1577 | |

Non-improving iterations | 547 | |

VSA | Number of particles | 163 |

Maximum iterations | 762 | |

Non-improving iterations | 762 | |

x parameter | 0.08 |

Standalone Test Feeder | |||
---|---|---|---|

Objectivefunction | ${E}_{loss}$ (kWh) | $Costs$ (USD) | $Emissions$ $\left(kgC{O}_{2}\right)$ |

Whitout PV-DGs | 489.3042 | 18,485.0507 | 16,951.2974 |

Average solution | |||

Objective function/Method | ${E}_{loss}$ (kWh) | $Costs$ (USD) | $Emissions$ $\left(kgC{O}_{2}\right)$ |

ALO | 359.6843 | 11,962.6688 | 10,930.9273 |

CSA | 369.1944 | 13,663.8328 | 12,534.4183 |

PSO | 362.0496 | 12,340.2908 | 11,267.5734 |

MVO | 360.0291 | 12,231.1691 | 11,131.5617 |

SSA | 359.8537 | 12,074.5543 | 11,039.5781 |

VSA | 359.8317 | 12,055.3410 | 11,016.6177 |

Percent average reduction (%) | |||

Objective function/Method | ${E}_{loss}$ | $Costs$ | $Emissions$ |

ALO | 26.4907 | 35.2846 | 35.5157 |

CSA | 24.5471 | 26.0817 | 26.0563 |

PSO | 26.0073 | 33.2418 | 33.5297 |

MVO | 26.4202 | 33.8321 | 34.3321 |

SSA | 26.4560 | 34.6794 | 34.8747 |

VSA | 26.4605 | 34.7833 | 35.0102 |

Standard deviation (%) | |||

Objective function/Method | ${E}_{loss}$ | $Costs$ | $Emissions$ |

ALO | 0.0010 | 0.0059 | 0.0032 |

CSA | 1.7548 | 2.3077 | 2.1093 |

PSO | 0.4095 | 1.7711 | 1.6491 |

MVO | 0.2356 | 2.4301 | 2.0192 |

SSA | 0.0230 | 0.4363 | 0.4329 |

VSA | 0.0212 | 0.3042 | 0.2886 |

Average processing time (s) | |||

Objective function/Method | ${E}_{loss}$ | $Costs$ | $Emissions$ |

ALO | 56.51 | 56.87 | 56.96 |

CSA | 6.54 | 6.76 | 6.74 |

PSO | 4.21 | 4.43 | 4.44 |

MVO | 2.02 | 1.80 | 1.90 |

SSA | 12.59 | 12.90 | 13.12 |

VSA | 7.69 | 7.84 | 7.78 |

**Table 7.**Simulations results obtained by the optimization algorithms in the grid-connected test feeder.

Grid-Connected Test Feeder | |||
---|---|---|---|

Objectivefunction | ${E}_{loss}$ (kWh) | $Costs$ (USD) | $Emissions$ $\left(kgC{O}_{2}\right)$ |

Whitout PV-DGs | 2186.2803 | 9776.3892 | 12345.1497 |

Average solution | |||

Objective function/Method | ${E}_{loss}$ (kWh) | $Costs$ (USD) | $Emissions$ $\left(kgC{O}_{2}\right)$ |

ALO | 1225.0193 | 7138.8122 | 8967.2586 |

CSA | 1270.1562 | 7407.9046 | 9328.7685 |

PSO | 1268.5973 | 7392.0432 | 9282.4081 |

MVO | 1231.2531 | 7298.7157 | 9187.9682 |

SSA | 1225.3323 | 7297.9712 | 9166.6746 |

VSA | 1225.2909 | 7249.3825 | 9108.9096 |

Percent average reduction (%) | |||

Objective function/Method | ${E}_{loss}$ | $Costs$ | $Emissions$ |

ALO | 43.9679 | 26.9791 | 27.3621 |

CSA | 41.9033 | 24.2266 | 24.4337 |

PSO | 41.9746 | 24.3888 | 24.8093 |

MVO | 43.6827 | 25.3434 | 25.5743 |

SSA | 43.9536 | 25.3511 | 25.7468 |

VSA | 43.9555 | 25.8481 | 26.2147 |

Standard deviation (%) | |||

Objective function/Method | ${E}_{loss}$ | $Costs$ | $Emissions$ |

ALO | 0.0046 | 0.0319 | 0.0296 |

CSA | 1.3806 | 1.8500 | 1.6987 |

PSO | 2.4065 | 2.2579 | 2.0891 |

MVO | 2.2694 | 1.2190 | 1.5868 |

SSA | 0.0131 | 0.7089 | 0.6306 |

VSA | 0.0108 | 0.5697 | 0.5676 |

Average processing time (s) | |||

Objective function/Method | ${E}_{loss}$ | $Costs$ | $Emissions$ |

ALO | 60.67 | 61.11 | 59.65 |

CSA | 36.37 | 36.45 | 36.87 |

PSO | 5.96 | 6.47 | 6.60 |

MVO | 2.45 | 2.47 | 2.48 |

SSA | 20.85 | 21.47 | 21.29 |

VSA | 9.93 | 10.37 | 10.45 |

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**MDPI and ACS Style**

Grisales-Noreña, L.F.; Rosales-Muñoz, A.A.; Montoya, O.D. An Effective Power Dispatch of Photovoltaic Generators in DC Networks via the Antlion Optimizer. *Energies* **2023**, *16*, 1350.
https://doi.org/10.3390/en16031350

**AMA Style**

Grisales-Noreña LF, Rosales-Muñoz AA, Montoya OD. An Effective Power Dispatch of Photovoltaic Generators in DC Networks via the Antlion Optimizer. *Energies*. 2023; 16(3):1350.
https://doi.org/10.3390/en16031350

**Chicago/Turabian Style**

Grisales-Noreña, Luis Fernando, Andrés Alfonso Rosales-Muñoz, and Oscar Danilo Montoya. 2023. "An Effective Power Dispatch of Photovoltaic Generators in DC Networks via the Antlion Optimizer" *Energies* 16, no. 3: 1350.
https://doi.org/10.3390/en16031350