# Optimal Power Dispatch of DGs in Radial and Mesh AC Grids: A Hybrid Solution Methodology between the Salps Swarm Algorithm and Successive Approximation Power Flow Method

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## Abstract

**:**

## 1. Introduction

#### 1.1. General Context

_{2}emissions and higher operating costs associated with its distribution [5,6,7]. Different solutions to these problems have been sought, including the development of new energy management technologies and strategies to increase electricity production around the world, as well as the development and application of new energy distribution technologies such as Distributed Generators (DGs) [8] and energy storage elements (e.g., batteries, capacitors, ultracapacitors, and superinductors) [9]. These solutions have lead us to reconsider the way conventional energy transport systems operate [10]. For the above reasons, network operators and researchers in this field have looked for energy alternatives other than fossil fuels and proposed the use of renewable energies and the integration of DGs into electric networks to meet energy demands while remaining environmentally friendly [11,12,13].

#### 1.2. State of the Art

#### 1.3. Scope and Main Contributions

- A new solution methodology for solving the AC OPF problem based on a master–slave strategy by considering the reduction of power loss as objective function and all sets of constraints that make up the operation of a AC grid under a distributed generation environmental.
- An OPF solution approach that solves different distribution network topologies (radial and meshed) and improves recent literature reports based on combinatorial optimization algorithms such as continuous genetic algorithm, Multi-Verse Optimizer, black hole optimization, particle swarm optimization, and ant lion optimization.
- The implementation of a global parameter-tuning optimization algorithm to guarantee the same conditions for each technique being employed in terms of solution quality, repeatability, and processing times.

#### 1.4. Structure of the Paper

## 2. Mathematical Formulation

#### 2.1. Objective Function

#### 2.2. Set of Constraints

## 3. Proposed Solution Methodology

#### 3.1. Master Stage: Salp Swarm Algorithm (SSA)

#### Generating the Initial Population

#### 3.2. Slave Stage

## 4. Optimization Algorithms Employed for Comparison and Parameters

## 5. Test Scenarios and Considerations

#### 5.1. Radial Test Systems

#### 5.1.1. 10-Node Radial Test System

#### 5.1.2. 33-Node Radial Test System

#### 5.1.3. 69-Node Radial Test System

#### 5.2. Mesh Test System

#### 10-Node Mesh Test System

## 6. Simulations and Results

^{®}(version 2021b) running on a laptop with an Intel

^{®}Core

^{TM}[email protected] 1.80 GHz processor, 4 GB of RAM, a 225-GB solid-state drive, and Windows 11. To evaluate the repeatability and standard deviation of each technique and guarantee the same conditions for all, the techniques were tuned and executed 100 times.

#### 6.1. Radial Test Systems

#### 6.1.1. 10-Node Radial Test System

#### 6.1.2. 33-Node Radial Test System

#### 6.1.3. 69-Node Radial Test System

#### 6.2. Mesh Test Systems

#### 10-Node Mesh Test System

## 7. Conclusions

- In the case of radial networks, the SSA proved to be superior in terms of minimum ${P}_{loss}$ reduction, as it outperformed the other optimization algorithms by an average percentage of $0.0433\%$, $0.0107\%$, and $0.0327\%$ in the 10-, 33- and 69-node radial test systems, respectively. It produced such good results in short processing times and with low standard deviations: an average processing time of $3.49$ s, $9.94$ s, and $43.52$ s in the 10-, 33- and 69-node radial test systems, respectively, and an average STD of $0.013\%$ at the three penetration levels of distributed generation ($20\%$, $40\%$ and $60\%$). This demonstrates the superiority and convergence capacity of the SSA, which is why we may conclude that it is the most suitable optimization algorithm to solve the OPF problem in radial networks of any size.
- In the case of mesh networks, the SSA also proved its superiority, as it provided the best solution in terms of minimum ${P}_{loss}$ reduction in every test scenario, with an average reduction of $64.5125\%$, outperforming the other algorithms by an average percentage of $0.034\%$. It produced such results in processing times of around $0.058$ s and with an average STD of $0.0369\%$. This demonstrates the superiority of the SSA in providing the best solution in terms of minimum ${P}_{loss}$ reduction in very short processing times. Thus, we may conclude that it is the most suitable optimization algorithm to solve the OPF problem in mesh networks.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 6.**Percentage of reduction in minimum power losses obtained by the SSA in the 10-node radial test system compared to that of the other methodologies.

**Figure 7.**Percentage of reduction in average power losses obtained by the SSA in the 10-node radial test system compared to that of other methodologies.

**Figure 8.**Percentage of standard deviation obtained by the SSA in the 10-node radial test system compared to that of the other methodologies.

**Figure 9.**Percentage of reduction in minimum power losses obtained by the SSA in the 33-node radial test system compared to that of the other methodologies.

**Figure 10.**Percentage of reduction in average power losses obtained by the SSA in the 33-node radial test system compared to that of the other methodologies.

**Figure 11.**Percentage of standard deviation obtained by the SSA in the 33-node radial test system compared to that of the other methodologies.

**Figure 12.**Percentage of reduction in minimum power losses obtained by the SSA in the 69-node radial test system compared to that of the other methodologies.

**Figure 13.**Percentage of reduction in average power losses obtained by the SSA in the 69-node radial test system compared to that of the other methodologies.

**Figure 14.**Percentage of standard deviation obtained by the SSA in the 69-node radial test system compared to that of the other methodologies.

**Figure 15.**Percentage of reduction in minimum power losses obtained by the SSA in the 10-node mesh test system compared to that of the other methodologies.

**Figure 16.**Percentage of reduction in average power losses obtained by the SSA in the 10-node mesh test system compared to that of the other methodologies.

**Figure 17.**Percentage of standard deviation obtained by the SSA in the 10-node mesh test system compared to that of the other methodologies.

**Table 1.**Commercial and sequential programming solution methods reported in literature for solving the power dispatch problem in AC grids.

Commercial Software | ||
---|---|---|

Method | Year | Reference |

Particle Swarm Optimization- DigSILENT | 2009 | [26] |

GAMS-DICOPT | 2012 | [25] |

Genetic Algorithm- DigSILENT | 2021 | [27] |

Sequential Programming | ||

Method | Year | Reference |

Bio-geography Optimization Algorithm | 2010 | [29] |

Artificial Bee Swarm Optimization Algorithm | 2012 | [28] |

Turbulent Crazy Particle Swarm Optimization | 2017 | [30] |

Continuous Genetic Algorithm | 2018 | [31] |

Particle Swarm Optimization | 2018 | [32] |

Ant Lion Optimizer | 2021 | [33] |

Black Hole | 2021 | [15] |

Multi-Verse Optimizer | 2022 | [14] |

Parameters | |||
---|---|---|---|

Method |
Number of Particles |
Maximum Iterations |
Non-Improvement Iterations |

SSA | 78 | 433 | 154 |

MVO | 80 | 432 | 300 |

PSO | 58 | 723 | 252 |

ALO | 62 | 992 | 725 |

BH | 83 | 667 | 340 |

CGA | 57 | 551 | 551 |

Node i | Node j | ${\mathit{R}}_{\mathit{i}\mathit{j}}$ [$\mathsf{\Omega}$] | ${\mathit{X}}_{\mathit{i}\mathit{j}}$ [$\mathsf{\Omega}$] | P [kW] | Q [kVAr] |
---|---|---|---|---|---|

1 | 2 | 0.1233 | 0.4127 | 1840 | 460 |

2 | 3 | 0.2467 | 0.6051 | 980 | 340 |

2 | 4 | 0.7469 | 1.2050 | 1790 | 446 |

4 | 5 | 0.6984 | 0.6084 | 1598 | 1840 |

2 | 6 | 1.9837 | 1.7276 | 1610 | 600 |

6 | 7 | 0.9057 | 0.7886 | 780 | 110 |

7 | 8 | 2.0552 | 1.1640 | 1150 | 60 |

7 | 9 | 4.7953 | 2.7160 | 980 | 130 |

3 | 10 | 5.3434 | 3.0264 | 1640 | 200 |

5 | 10 | 0.1426 | 0.4522 | - | - |

8 | 10 | 0.2018 | 0.5214 | - | - |

10-Node Radial Test System | |||||||
---|---|---|---|---|---|---|---|

Method |
Node/ Power [kW] | Power Losses |
Vworst [pu]/ Node | Imax [A] | |||

Minimum [kW]/ Reduction [%] |
Average [kW]/ Reduction [%] | Time [s] | STD [%] | ||||

Without DGs | - | 223.4181 | - | - | - | 0.9–1.1 | 590 |

20% penetration | |||||||

SSA | 5/0.05 | 116.9218/47.6668 | 116.9237/47.6660 | 3.49 | 0.0025 | 0.9723/8 | 433.3321 |

9/1589.82 | |||||||

10/928.41 | |||||||

MVO | 5/0.05 | 116.9220/47.6667 | 116.9250/47.6654 | 3.75 | 0.0049 | 0.9723/8 | 433.3324 |

9/1589.82 | |||||||

10/928.41 | |||||||

PSO | 5/0 | 116.9218/47.6668 | 117.2119/47.5370 | 4.50 | 1.3279 | 0.9723/8 | 433.3321 |

9/1589.55 | |||||||

10/928.73 | |||||||

ALO | 5/0.51 | 116.9473/47.6554 | 117.9188/47.2206 | 6.66 | 0.7210 | 0.9723/8 | 433.3827 |

9/1586.68 | |||||||

10/929.96 | |||||||

BH | 5/96.28 | 117.9244/47.2181 | 121.5254/45.6063 | 3.35 | 1.7463 | 0.9729/8 | 433.5938 |

9/1696.06 | |||||||

10/720.92 | |||||||

CGA | 5/18.86 | 117.0415/47.6132 | 117.4801/47.4169 | 3.29 | 0.1733 | 0.9725/8 | 433.4102 |

9/1619.67 | |||||||

10/878.08 | |||||||

40% penetration | |||||||

SSA | 5/1620.63 | 80.7608/63.8522 | 80.7610/63.8521 | 3.47 | 0.0003 | 0.9751/8 | 322.2693 |

9/1970.64 | |||||||

10/1445.29 | |||||||

MVO | 5/1619.69 | 80.7608/63.8522 | 80.7619/63.8517 | 3.68 | 0.0009 | 0.9752/8 | 322.2694 |

9/1971.25 | |||||||

10/1445.62 | |||||||

PSO | 5/1620.68 | 80.7608/63.8522 | 80.9785/63.7547 | 4.25 | 0.9097 | 0.9751/8 | 322.2693 |

9/1970.20 | |||||||

10/1445.69 | |||||||

ALO | 5/1570.43 | 80.7922/63.8381 | 81.8538/63.3629 | 6.61 | 1.7971 | 0.9752/8 | 322.2936 |

9/1979.08 | |||||||

10/1486.52 | |||||||

BH | 5/1606.93 | 80.9765/63.7556 | 82.4371/63.1019 | 3.29 | 1.0840 | 0.9751/8 | 323.3491 |

9/1969.06 | |||||||

10/1435.96 | |||||||

CGA | 5/1642.03 | 80.7807/63.8433 | 81.0075/63.7417 | 3.30 | 0.1791 | 0.9751/8 | 322.3464 |

9/1959.77 | |||||||

10/1433.01 | |||||||

60% penetration | |||||||

SSA | 5/2992.59 | 72.1260/67.7170 | 72.1260/67.7170 | 3.51 | 4.23$\times {10}^{-11}$ | 0.9771/8 | 235.1409 |

9/2235.17 | |||||||

10/1804.13 | |||||||

MVO | 5/2992.61 | 72.1260/67.7170 | 72.1260/67.7170 | 3.88 | 9.38$\times {10}^{-7}$ | 0.9771/8 | 235.1382 |

9/2235.19 | |||||||

10/1804.14 | |||||||

PSO | 5/2992.59 | 72.1260/67.7170 | 72.1260/67.7170 | 2.39 | 1.22$\times {10}^{-10}$ | 0.9771/8 | 235.1409 |

9/2235.17 | |||||||

10/1804.13 | |||||||

ALO | 5/2993.04 | 72.1308/67.7149 | 72.7952/67.4175 | 6.70 | 1.6134 | 0.9770/8 | 236.2086 |

9/2219.08 | |||||||

10/1795.22 | |||||||

BH | 5/2941.39 | 72.1498/67.7064 | 73.1556/67.2562 | 3.78 | 1.1291 | 0.9773/8 | 236.3767 |

9/2267.73 | |||||||

10/1794.37 | |||||||

CGA | 5/3020.79 | 72.1345/67.7132 | 72.1848/67.6907 | 3.46 | 0.0610 | 0.9772/8 | 234.3459 |

9/2245.92 | |||||||

10/1783.47 |

33-Node Radial Test System | |||||||
---|---|---|---|---|---|---|---|

Method |
Node/ Power [kW] | Power Losses |
Vworst [p.u.]/ Node | Imax [A] | |||

Minimum [kW]/ Reduction [%] |
Average [kW]/ Reduction [%] | Time [s] | STD [%] | ||||

Without DGs | - | 210.9785 | - | - | - | 0.9–1.1 | 385 |

20% penetration | |||||||

SSA | 12/48.44 | 127.4984/39.5680 | 127.5044/39.5652 | 10.17 | 0.0077 | 0.9377/33 | 241.4931 |

15/396.14 | |||||||

31/340.61 | |||||||

MVO | 12/44.88 | 127.4984/39.5680 | 127.4994/39.5676 | 11.18 | 0.0009 | 0.9377/33 | 241.4931 |

15/398.94 | |||||||

31/341.37 | |||||||

PSO | 12/45.68 | 127.4984/39.5680 | 127.8911/39.3819 | 11.97 | 0.5240 | 0.9377/33 | 241.4931 |

15/398.71 | |||||||

31/340.81 | |||||||

ALO | 12/55.13 | 127.5029/39.5659 | 127.6270/39.5071 | 17.44 | 0.0910 | 0.9376/33 | 241.4970 |

15/391.34 | |||||||

31/338.68 | |||||||

BH | 12/88.70 | 127.6257/39.5077 | 128.4504/39.1168 | 9.19 | 0.4042 | 0.9358/18 | 241.5142 |

15/333.88 | |||||||

31/362.48 | |||||||

CGA | 12/76.31 | 127.5192/39.5582 | 127.6041/39.5180 | 9.27 | 0.0439 | 0.9376/33 | 241.4996 |

15/370.19 | |||||||

31/338.64 | |||||||

40% penetration | |||||||

SSA | 12/409.59 | 90.3771/57.1629 | 90.3779/57.1625 | 9.68 | 0.0012 | 0.9594/33 | 176.5392 |

15/397.41 | |||||||

31/763.40 | |||||||

MVO | 12/409.59 | 90.3771/57.1629 | 90.3777/57.1626 | 10.73 | 0.0008 | 0.9594/33 | 176.5392 |

15/397.41 | |||||||

31/763.40 | |||||||

PSO | 12/410.02 | 90.3771/57.1629 | 90.7890/56.9677 | 11.47 | 1.1588 | 0.9594/33 | 176.5392 |

15/397.60 | |||||||

31/762.78 | |||||||

ALO | 12/429.24 | 90.3861/57.1586 | 90.5850/57.0644 | 17.30 | 0.2181 | 0.9591/33 | 176.5422 |

15/388.74 | |||||||

31/752.38 | |||||||

BH | 12/348.19 | 90.5000/57.1047 | 91.7172/56.5277 | 9.04 | 0.7770 | 0.9594/33 | 176.7536 |

15/455.18 | |||||||

31/764.43 | |||||||

CGA | 12/432.88 | 90.4019/57.1511 | 90.4811/57.1136 | 9.48 | 0.0535 | 0.9591/33 | 176.5933 |

15/384.37 | |||||||

31/752.48 | |||||||

60% penetration | |||||||

SSA | 12/596.31 | 85.7789/59.3423 | 85.7789/59.3423 | 9.97 | 8.65$\times {10}^{-11}$ | 0.9700/33 | 114.2656 |

15/397.74 | |||||||

31/980.32 | |||||||

MVO | 12/596.31 | 85.7789/59.3423 | 85.7789/59.3423 | 10.68 | 6.11$\times {10}^{-7}$ | 0.9700/33 | 144.2656 |

15/397.76 | |||||||

31/980.31 | |||||||

PSO | 12/596.32 | 85.7789/59.3423 | 85.7789/59.3423 | 6.63 | 8.00$\times {10}^{-6}$ | 0.9700/33 | 144.2657 |

15/397.74 | |||||||

31/980.32 | |||||||

ALO | 12/604.99 | 85.7813/59.3412 | 86.0098/59.2329 | 18.03 | 0.3471 | 0.9699/33 | 144.6453 |

15/388.35 | |||||||

31/976.24 | |||||||

BH | 12/598.86 | 85.8045/59.3302 | 86.3709/59.0618 | 9.80 | 0.6068 | 0.9694/33 | 146.3655 |

15/380.11 | |||||||

31/968.85 | |||||||

CGA | 12/594.56 | 85.7803/59.3417 | 85.7999/59.3324 | 10.07 | 0.0168 | 0.9699/33 | 144.7778 |

15/395.17 | |||||||

31/978.16 |

69-Node Radial Test System | |||||||
---|---|---|---|---|---|---|---|

Method | Node/Power [kW] | Power Losses | Vworst [pu]/Node | Imax [A] | |||

Minimum [kW]/Reduction [%] | Average [kW]/Reduction [%] | Time [s] | STD [%] | ||||

Without DGs | - | 242.1523 | - | - | - | 0.9–1.1 | 400 |

20% penetration | |||||||

SSA | 26/0 | 133.5626/44.8435 | 133.6548/44.8055 | 44.62 | 0.1034 | 0.9397/64 | 252.6391 |

61/580.52 | |||||||

66/246.05 | |||||||

MVO | 26/0.01 | 133.5632/44.8433 | 133.5687/44.8410 | 44.84 | 0.0033 | 0.9385/69 | 252.5817 |

61/583.13 | |||||||

66/243.43 | |||||||

PSO | 26/0 | 133.5626/44.8435 | 134.1547/44.5990 | 57.16 | 1.5020 | 0.9385/69 | 252.5817 |

61/580.16 | |||||||

66/246.41 | |||||||

ALO | 26/0 | 133.6333/44.8143 | 134.6068/44.4123 | 76.89 | 0.5786 | 0.9390/69 | 252.6323 |

61/546.38 | |||||||

66/279.62 | |||||||

BH | 26/9.55 | 133.9468/44.6849 | 137.8053/43.0915 | 38.64 | 1.4990 | 0.9378/69 | 252.6825 |

61/595.61 | |||||||

66/220.52 | |||||||

CGA | 26/4.08 | 133.6923/44.7900 | 134.2007/44.5800 | 43.18 | 0.1652 | 0.9381/69 | 252.5921 |

61/595.66 | |||||||

66/226.83 | |||||||

40% penetration | |||||||

SSA | 26/152.93 | 86.4573/64.2963 | 86.4593/64.2955 | 42.06 | 0.0036 | 0.9634/69 | 183.5728 |

61/1254.04 | |||||||

66/246.17 | |||||||

MVO | 26/152.51 | 86.4574/64.2963 | 86.4585/64.2958 | 45.11 | 0.0017 | 0.9638/69 | 183.5712 |

61/1253.71 | |||||||

66/246.91 | |||||||

PSO | 26/152.72 | 86.4574/64.2963 | 86.6493/64.2170 | 56.62 | 0.6638 | 0.9638/69 | 183.5711 |

61/1252.84 | |||||||

66/247.57 | |||||||

ALO | 26/152.77 | 86.4817/64.2862 | 87.0658/64.0450 | 81.02 | 0.6258 | 0.9639/69 | 183.6309 |

61/1243.67 | |||||||

66/255.96 | |||||||

BH | 26/208.65 | 86.9818/64.0797 | 90.4786/62.6357 | 45.23 | 1.9240 | 0.9632/69 | 183.9434 |

61/1110.03 | |||||||

66/330.27 | |||||||

CGA | 26/144.73 | 86.4671/64.2923 | 86.6006/64.2371 | 37.99 | 0.0974 | 0.9638/69 | 183.5754 |

61/1274.50 | |||||||

66/233.87 | |||||||

60% penetration | |||||||

SSA | 26/382.16 | 76.9578/68.2193 | 76.9578/68.2193 | 43.89 | 5.42$\times {10}^{-9}$ | 0.9784/69 | 134.0925 |

61/1641.63 | |||||||

66/246.24 | |||||||

MVO | 26/382.16 | 76.9578/68.2193 | 76.9578/68.2193 | 44.49 | 1.31$\times {10}^{-6}$ | 0.9784/69 | 134.0951 |

61/1641.63 | |||||||

66/246.21 | |||||||

PSO | 26/382.17 | 76.9578/68.2193 | 76.9578/68.2193 | 55.59 | 1.46$\times {10}^{-8}$ | 0.9784/69 | 134.0926 |

61/1641.64 | |||||||

66/246.23 | |||||||

ALO | 26/386.59 | 76.9593/68.2186 | 77.3907/68.0405 | 86.72 | 0.7409 | 0.9785/69 | 133.6689 |

61/1637.61 | |||||||

66/251.20 | |||||||

BH | 26/358.03 | 76.9986/68.2024 | 79.0719/67.3462 | 43.35 | 1.8238 | 0.9778/69 | 136.5195 |

61/1653.47 | |||||||

66/227.85 | |||||||

CGA | 26/382.31 | 76.9593/68.2186 | 76.9859/68.2077 | 38.06 | 0.0237 | 0.9784/69 | 134.4437 |

61/1629.83 | |||||||

66/253.45 |

10-Node Mesh Test System | |||||||
---|---|---|---|---|---|---|---|

Method |
Node/ Power [kW] | Power Losses |
Vworst [pu]/ Node | Imax [A] | |||

Minimum [kW]/ Reduction [%] |
Average [kW]/ Reduction [%] | Time [s] | STD [%] | ||||

Without DGs | - | 190.3237 | - | - | - | 0.9–1.1 | 590 |

20% penetration | |||||||

SSA | 5/0 | 104.7510/44.9617 | 104.7707/44.9513 | 4.16 | 0.0446 | 0.9793/8 | 433.0907 |

9/1039.33 | |||||||

10/1472.33 | |||||||

MVO | 5/0 | 104.75110/44.9616 | 104.7540/44.9601 | 4.09 | 0.0021 | 0.9793/8 | 433.0907 |

9/1039.96 | |||||||

10/1471.71 | |||||||

PSO | 5/0.02 | 104.7511/44.9616 | 105.3226/44.6613 | 4.72 | 1.8071 | 0.9793/8 | 433.0907 |

9/1038.24 | |||||||

10/1473.40 | |||||||

ALO | 5/32.05 | 104.7986/44.9367 | 105.0366/44.8116 | 6.53 | 0.1796 | 0.9793/8 | 433.1153 |

9/1012.16 | |||||||

10/1466.94 | |||||||

BH | 5/1.87 | 104.9699/44.8467 | 105.9958/44.3076 | 3.48 | 0.5380 | 0.9793/8 | 433.4899 |

9/1037.61 | |||||||

10/1463.23 | |||||||

CGA | 5/18.12 | 104.8075/44.9320 | 105.0660/44.7962 | 3.40 | 0.1174 | 0.9793/8 | 433.1163 |

9/1087.18 | |||||||

10/1405.83 | |||||||

40% penetration | |||||||

SSA | 5/587.06 | 58.4855/69.2705 | 58.5107/69.2573 | 3.94 | 0.0580 | 0.9838/7 | 321.8763 |

9/1222.72 | |||||||

10/3213.55 | |||||||

MVO | 5/586.03 | 58.4855/69.2705 | 58.4882/69.2691 | 3.81 | 0.0058 | 0.9838/7 | 321.8764 |

9/1224.24 | |||||||

10/3213.06 | |||||||

PSO | 5/611.96 | 58.4859/69.2703 | 64.6277/66.0433 | 4.38 | 24.2119 | 0.9838/7 | 321.8764 |

9/1227.34 | |||||||

10/3184.03 | |||||||

ALO | 5/526.13 | 58.4985/69.2637 | 58.6598/69.1789 | 6.37 | 0.2907 | 0.9838/7 | 321.9142 |

9/1215.08 | |||||||

10/3281.26 | |||||||

BH | 5/1253.67 | 58.6297/69.1947 | 60.1293/68.4068 | 3.45 | 1.2234 | 0.9838/7 | 321.8891 |

9/1241.74 | |||||||

10/2527.77 | |||||||

CGA | 5/813.33 | 58.5195/69.2526 | 58.6400/69.1894 | 3.44 | 0.1372 | 0.9838/7 | 321.8762 |

9/1215.85 | |||||||

10/2994.17 | |||||||

60% penetration | |||||||

SSA | 5/2447.21 | 39.3867/79.3054 | 39.3886/79.3044 | 3.91 | 0.0081 | 0.9874/6 | 211.8432 |

9/1395.92 | |||||||

10/3691.86 | |||||||

MVO | 5/2440.87 | 39.3867/79.3054 | 39.3874/79.3050 | 3.88 | 0.0018 | 0.9874/6 | 211.8432 |

9/1396.49 | |||||||

10/3697.63 | |||||||

PSO | 5/2448.40 | 39.3867/79.3054 | 40.7435/78.5925 | 4.31 | 10.3116 | 0.9874/6 | 211.8432 |

9/1396.09 | |||||||

10/3690.50 | |||||||

ALO | 5/2445.34 | 39.3976/79.2997 | 39.6632/79.1601 | 6.56 | 0.6903 | 0.9874/6 | 212.0355 |

9/1399.98 | |||||||

10/3685.25 | |||||||

BH | 5/3065.27 | 39.5207/79.2350 | 40.6407/78.6465 | 3.40 | 1.5767 | 0.9873/6 | 212.0846 |

9/1368.25 | |||||||

10/3096.05 | |||||||

CGA | 5/2378.87 | 39.3908/79.3033 | 39.4689/79.2622 | 3.56 | 0.1044 | 0.9874/6 | 211.8915 |

9/1399.13 | |||||||

10/3755.89 |

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## Share and Cite

**MDPI and ACS Style**

Rosales-Muñoz, A.A.; Montano, J.; Grisales-Noreña, L.F.; Montoya, O.D.; Andrade, F.
Optimal Power Dispatch of DGs in Radial and Mesh AC Grids: A Hybrid Solution Methodology between the Salps Swarm Algorithm and Successive Approximation Power Flow Method. *Sustainability* **2022**, *14*, 13408.
https://doi.org/10.3390/su142013408

**AMA Style**

Rosales-Muñoz AA, Montano J, Grisales-Noreña LF, Montoya OD, Andrade F.
Optimal Power Dispatch of DGs in Radial and Mesh AC Grids: A Hybrid Solution Methodology between the Salps Swarm Algorithm and Successive Approximation Power Flow Method. *Sustainability*. 2022; 14(20):13408.
https://doi.org/10.3390/su142013408

**Chicago/Turabian Style**

Rosales-Muñoz, Andrés Alfonso, Jhon Montano, Luis Fernando Grisales-Noreña, Oscar Danilo Montoya, and Fabio Andrade.
2022. "Optimal Power Dispatch of DGs in Radial and Mesh AC Grids: A Hybrid Solution Methodology between the Salps Swarm Algorithm and Successive Approximation Power Flow Method" *Sustainability* 14, no. 20: 13408.
https://doi.org/10.3390/su142013408