# Development of a MATLAB-GAMS Framework for Solving the Problem Regarding the Optimal Location and Sizing of PV Sources in Distribution Networks

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## Abstract

**:**

## 1. Introduction

#### 1.1. General Context

#### 1.2. Literature Review

#### 1.3. Motivation, Contributions, and Scope

- i.
- A complete description of the mathematical formulation representing the problem of the siting and sizing of PV generators in distribution networks while considering the deactivation of maximum power point tracking.
- ii.
- A new optimization methodology based on the interconnection of MATLAB and GAMS that allows finding the best locally optimal solution to the problem under study.
- iii.
- A new master–slave methodology to solve the mathematical model representing the problem under study. In the master stage, the MATLAB software is used as a tool to develop the discrete version of the sine–cosine algorithm, with the aim to determine the locations of the PV generators. Then, in the slave stage, GAMS is used to solve the MINLP model that represents the studied problem, thus yielding the objective function value and the necessary nominal power to be generated by the PV systems.

#### 1.4. Document Structure

## 2. Mathematical Formulation

#### 2.1. Formulating Objective Function

#### 2.2. Set of Constraints

#### 2.3. Model Interpretation

## 3. Proposed Hybrid Optimization Approach

#### 3.1. Master Stage: DSCA

_{2}emissions, and voltage profiles, among others. The authors of [46] used the SCA in order to optimize the reactive power generated in power systems integrating a distributed generator, employing the IEEE 14-node test system. In [47], an automatic generation control was implemented, using the SCA to determine the gains of a PID controller to be applied in hydrothermal generators.

#### 3.2. Initial Population

#### 3.3. Evolution Criteria

#### 3.4. Updating the Individuals

Algorithm 1: Sine–cosine algorithm to solve the problem regarding the siting and sizing of PV generation sources [49]. |

#### 3.5. Slave Stage: GAMS

#### 3.6. Interface Connection

## 4. Test Systems

#### 4.1. First Test Feeder: IEEE 33-Node

#### 4.2. Second Test Feeder: IEEE 69-Node

#### 4.3. Additional Parametric Information

## 5. Numerical Results and Simulations

#### 5.1. Numerical Results for IEEE 33-Node Test System

#### 5.2. Comparison with Metaheuristics—IEEE 33-Node Test System

#### 5.3. Numerical Results for IEEE 69-Node Test System

#### 5.4. Comparison with Metaheuristics—IEEE 69-Node Test System

#### 5.5. Convergence Analysis

## 6. Conclusions and Future Works

- For the proposed simulation scenario, the master–slave methodology, when compared to the base case, allows for reductions of approximately USD/year 1,103,324.12 and USD/year 1,157,492.3 in the IEEE 33- and 69-node test systems, respectively. These results constitute the highest reductions when solving the problem regarding the siting and sizing of PV generators, with values of 29.82% for the 33-node system and 29.85% for the 69-node grid. This means that the MATLAB-GAMS interface achieves a high-quality solution for the problem addressed in this research paper.
- The proposed solution methodology succeeds in finding a high-quality solution to the problem, outperforming the solutions reported by the MIC in the specialized literature. The literature reports that the MIC solution ensures that the optimal global point is found via the interior-point method combined with the branch-and-cut optimization approach. However, the presence of binary variables increases the complexity of the problem, making it difficult to find the global optimum due to the approximations involved in the MIC approach. The above demonstrates that the proposed methodology is the best at solving problems that involve binary and continuous variables, as it separates the location problem from the sizing problem.
- As seen in the convergence curve, the proposed methodology is independent of the number of nodes in the system under study. Due to the high complexity of the MINLP model, as the number of nodes in the system increases, the solution space becomes larger, and the chances of getting stuck in a locally optimal solution increase. However, it takes the proposed methodology about 22 and 26 iterations to find a high-quality solution for the IEEE 33- and 69-bus test systems, respectively. The above leads to the conclusion that the MATLAB-GAMS interface is the best choice for solving the problem regarding the location and sizing of PV generators in distribution power grids.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Representation of the information exchange in the connection of the MATLAB-GAMS interfaces.

**Figure 5.**Behavior of the objective function value for the test feeders: (

**a**) IEEE 33-node and (

**b**) IEEE 69-node.

Method/Algorithm | Acronym | Ref | Objective Function |
---|---|---|---|

Genetic Algorithm and Particle Swarm Optimizer | GA-PSO | [17] | Minimization of power losses |

Harmonic Search Algorithm | HSA | [38] | Minimizing power losses |

Mixed Integer Nonlinear Programming | MINLP | [39] | Minimizing real power losses |

Optimization based on Quasi-Oppositional Teaching | QOTLBO | [40] | Minimization of power losses, improving voltage profiles, and maximizing voltage stability |

Radial Basis Function Neural Network and Particle Swarm Optimization | RBFNN-PSO | [41] | Minimizing power losses |

Teaching Learning-Based Optimization | TLBO | [42] | Minimizing power losses, improving voltage profiles, and maximizing voltage stability |

Krill-Herd Algorithm | KHA | [43] | Minimizing power losses |

Search for Symbiotic Organisms | SOS | [18] | Minimizing active power losses |

Population-Based Incremental Learning and Particle Swarm Optimizer | PBIL-PSO | [19] | Minimizing power losses and square error in the voltage profiles |

Artificial Bee Colony Algorithm | ABC | [20] | Minimizing power distribution losses and enhancing voltage profiles |

Heuristic Algorithmic Approach | AHA | [21] | Minimizing power losses and improving voltage profiles |

Constructive Heuristic Vortex Search Algorithm | CHVSA | [22] | Minimizing active power losses |

Mixed-integer conic | MIC | [23] | Minimizing the costs of power purchasing and PV purchasing, investment, and operation |

Discrete-Continuous Vortex Search Algorithm | DCVSA | [24] | Minimizing the costs of power purchasing and PV purchasing, investment, and operation |

Generalized Normal Distribution Optimization | DCGNDO | [25] | Minimizing the costs of power purchasing and PV purchasing, investment, and operation |

Metaheuristic Optimizer based on a Modified Gradient | MGbMO | [26] | Minimizing the costs of power purchasing and PV purchasing, investment, and operation |

Discrete–Continuous version of the Crow Search Algorithm | DSCSA | [37] | Minimizing the costs of power purchasing and PV purchasing, investment, and operation |

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

${C}_{kWh}$ | 0.1390 | USD/kWh | ${t}_{e}$ | 2 | % |

${C}_{pv}$ | 1036.49 | USD/kWp | ${N}_{t}$ | 10 | años |

${C}_{O\&M}$ | 0.0019 | USD/kWh | ${P}_{i}^{pv,min}$ | 0 | kW |

$\Delta h$ | 1 | hour | ${P}_{i}^{pv,max}$ | 2400 | kW |

T | 365 | days | ${N}_{pv}^{ava}$ | 3 | - |

${t}_{a}$ | 10 | % | $\Delta V$ | $\pm 10$ | % |

Method | Location (Node) | Size (MW) | ${\mathit{A}}_{\mathit{c}\mathit{o}\mathit{s}\mathit{t}}$ (USD/Year) | Reduction (%) |
---|---|---|---|---|

Benchmark case | - | - | 3,700,455.38 | - |

MIC${}_{0}$ | [11, 16, 32] | [1.8400, 0.6799, 2.3083] | 2,603,465.00 | 29.64 |

MIC${}_{1/2}$ | [13, 24, 29] | [1.4116, 1.7145, 1.9471] | 2,597,283.00 | 29.81 |

MIC${}_{1}$ | [13, 24, 30] | [1.4392, 1.7745, 1.8596] | 2,597,139.00 | 29.82 |

DSCA-BONMIN | [14, 24, 30] | [1.3877, 1.8007, 1.8852] | 2,597,131.26 | 29.82 |

**Table 4.**Comparison of the results obtained by the metaheuristic optimizers and the proposed methodology.

Method | Location (Node) | Size (MW) | ${\mathit{A}}_{\mathit{c}\mathit{o}\mathit{s}\mathit{t}}$ (USD/Year) | Reduction (%) |
---|---|---|---|---|

Benchmark case | - | - | 3,700,455.38 | - |

DCVSA | [11, 14, 31] | [0.7606, 1.0852, 1.8030] | 2,699,761.71 | 27.0424 |

DCGNDO | [10, 16, 31] | [1.0083, 0.9137, 1.7257] | 2,699,671.76 | 27.0436 |

DCCSA | [10, 16, 31] | [1.0093, 0.9138, 1.7246] | 2,699,671.76 | 27.0449 |

DSCA-BONMIN | [14, 24, 30] | [1.3877, 1.8007, 1.8852] | 2,597,131.26 | 29.82 |

Method | Location (Node) | Size (MW) | ${\mathit{A}}_{\mathit{c}\mathit{o}\mathit{s}\mathit{t}}$ (USD/Year) | Reduction (%) |
---|---|---|---|---|

Benchmark case | - | - | 3,878,199.93 | - |

MIC${}_{0}$ | [23, 27, 46] | [2.3578, 0.0585, 2.4000] | 2,752,021.00 | 29.04 |

MIC${}_{1/2}$ | [17, 49, 61] | [1.0977, 1.7981, 2.4000] | 2,721,282.00 | 29.83 |

MIC${}_{1}$ | [17, 49, 61] | [1.0977, 1.7981, 2.4000] | 2,721,282.00 | 29.83 |

DSCA-BONMIN | [8, 17, 61] | [2.0714, 0.8300, 2.3999] | 2,720,707.63 | 29.85 |

**Table 6.**Comparison of the results obtained by the metaheuristic optimizers and the proposed methodology.

Method | Location (Node) | Size (MW) | ${\mathit{A}}_{\mathit{c}\mathit{o}\mathit{s}\mathit{t}}$ (USD/Year) | Reduction (%) |
---|---|---|---|---|

Benchmark case | - | - | 3,700,455.38 | - |

DCVSA | [16, 61, 63] | [0.2632, 2.2719, 2.2934] | 2,825,264.56 | 27.1502 |

DCGNDO | [21, 61, 64] | [0.4812, 2.4, 0.9259] | 2,824,923.38 | 27.1589 |

DCCSA | [21, 61, 64] | [0.4816, 2.4, 0.9254] | 2,824,923.05 | 27.1589 |

DSCA-BONMIN | [8, 17, 61] | [2.0714, 0.8300, 2.3999] | 2,720,707.63 | 29.85 |

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**MDPI and ACS Style**

Guzmán-Romero, D.S.; Cortés-Caicedo, B.; Montoya, O.D.
Development of a MATLAB-GAMS Framework for Solving the Problem Regarding the Optimal Location and Sizing of PV Sources in Distribution Networks. *Resources* **2023**, *12*, 35.
https://doi.org/10.3390/resources12030035

**AMA Style**

Guzmán-Romero DS, Cortés-Caicedo B, Montoya OD.
Development of a MATLAB-GAMS Framework for Solving the Problem Regarding the Optimal Location and Sizing of PV Sources in Distribution Networks. *Resources*. 2023; 12(3):35.
https://doi.org/10.3390/resources12030035

**Chicago/Turabian Style**

Guzmán-Romero, David Steveen, Brandon Cortés-Caicedo, and Oscar Danilo Montoya.
2023. "Development of a MATLAB-GAMS Framework for Solving the Problem Regarding the Optimal Location and Sizing of PV Sources in Distribution Networks" *Resources* 12, no. 3: 35.
https://doi.org/10.3390/resources12030035