# Genetic-Convex Model for Dynamic Reactive Power Compensation in Distribution Networks Using D-STATCOMs

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## Abstract

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## 1. Introduction

- Formulation of the mixed-integer nonlinear programming (i.e., MINLP) problem of the optimal location and sizing of D-STATCOMs in distribution networks into a mixed-integer convex optimization model using the branch power flow representation of the grid, which generates a new conic optimization model.
- Implementation of a new optimization strategy called the genetic-convex optimizer that works in a master–slave connection, where the master stage is guided by a classical Chu and Beasley genetic algorithm (CBGA) to determine the location of the D-STATCOMs, and the slave stage is entrusted with the optimal sizing of the D-STATCOMs via second-order cone programming (SOCP).
- Inclusion in the optimization model of the curves associated with residential, industrial, and commercial loads under an economic multi-period operation environment for radial distribution networks.
- Comparison of the proposed master–slave approach with the GAMS optimization package, where the results show that the proposed approach achieves a better reduction of the total operating costs in the test system.

## 2. Mixed-Integer Nonlinear Programming Model

#### 2.1. Objective Function Formulation

#### 2.2. Set of Constraints

## 3. Solution Methodology

#### 3.1. Slave Stage

#### 3.2. Master Stage

## 4. Electrical Distribution Grid Under Study

## 5. Computational Implementation

#### 5.1. Optimization Results

- The best objective function value is reached with population sizes of 20 and 40, with an average processing time of about $3.1144$ h; here, nodes 14, 30, and 32 are selected, with nominal rates of $0.2896$ MVAr, $0.5593$ MVAr, and $0.1177$ MVAr, respectively, producing a total annual operating cost of US$/year $109455.96$. This solution allows annual cost reductions of $16.20\%$.
- The worst solution reached by using the hybrid CBGA-SOCP approach corresponding to the nodes 12, 14, and 30, with D-STATCOM capacities of $0.1920$ MVAr, $0.1488$ MVAr, and $0.6556$ MVAr, respectively, generating a final objective function value of US$/year $109498.91$; however, the difference with respect to the best objective function is only $42.94$ dollars per year of operation.
- Regarding processing times in Table 4, it can be observed that the required processing times increase as a function of population size in the initial population, since to start searching the solution space, the CBGA requires the evaluation of the initial population, which consumes additional processing time.

#### 5.2. Comparison with the GAMS Optimization Package

#### 5.3. Additional Operative Gains

## 6. Conclusions and Recommendations

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Flowchart of the Chu and Beasley genetic algorithm (CBGA)-second-order cone programming (SOCP) approach for the optimal sizing and placement of D-STATCOMs in electrical distribution grids.

**Figure 3.**Grid configuration for the IEEE 33-bus system, with different shaded areas representing different types of load.

**Figure 6.**Dynamic reactive power behavior during a typical operation day for the D-STATCOMs in electric distribution networks in the IEEE 33-bus test feeder.

Node i | Node j | ${\mathit{R}}_{\mathit{ij}}$ ($\mathbf{\Omega}$) | ${\mathit{X}}_{\mathit{ij}}$ ($\mathbf{\Omega}$) | ${\mathit{P}}_{\mathit{j}}$ (kW) | ${\mathit{Q}}_{\mathit{j}}$ (kvar) | Node i | Node j | ${\mathit{R}}_{\mathit{ij}}$ ($\mathbf{\Omega}$) | ${\mathit{X}}_{\mathit{ij}}$ ($\mathbf{\Omega}$) | ${\mathit{P}}_{\mathit{j}}$ (kW) | ${\mathit{Q}}_{\mathit{j}}$ (kvar) |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 0.0922 | 0.0477 | 100 | 60 | 17 | 18 | 0.7320 | 0.5740 | 90 | 40 |

2 | 3 | 0.4930 | 0.2511 | 90 | 40 | 2 | 19 | 0.1640 | 0.1565 | 90 | 40 |

3 | 4 | 0.3660 | 0.1864 | 120 | 80 | 19 | 20 | 1.5042 | 1.3554 | 90 | 40 |

4 | 5 | 0.3811 | 0.1941 | 60 | 30 | 20 | 21 | 0.4095 | 0.4784 | 90 | 40 |

5 | 6 | 0.8190 | 0.7070 | 60 | 20 | 21 | 22 | 0.7089 | 0.9373 | 90 | 40 |

6 | 7 | 0.1872 | 0.6188 | 200 | 100 | 3 | 23 | 0.4512 | 0.3083 | 90 | 50 |

7 | 8 | 1.7114 | 1.2351 | 200 | 100 | 23 | 24 | 0.8980 | 0.7091 | 420 | 200 |

8 | 9 | 1.0300 | 0.7400 | 60 | 20 | 24 | 25 | 0.8960 | 0.7011 | 420 | 200 |

9 | 10 | 1.0400 | 0.7400 | 60 | 20 | 6 | 26 | 0.2030 | 0.1034 | 60 | 25 |

10 | 11 | 0.1966 | 0.0650 | 45 | 30 | 26 | 27 | 0.2842 | 0.1447 | 60 | 25 |

11 | 12 | 0.3744 | 0.1238 | 60 | 35 | 27 | 28 | 1.0590 | 0.9337 | 60 | 20 |

12 | 13 | 1.4680 | 1.1550 | 60 | 35 | 28 | 29 | 0.8042 | 0.7006 | 120 | 70 |

13 | 14 | 0.5416 | 0.7129 | 120 | 80 | 29 | 30 | 0.5075 | 0.2585 | 200 | 600 |

14 | 15 | 0.5910 | 0.5260 | 60 | 10 | 30 | 31 | 0.9744 | 0.9630 | 150 | 70 |

15 | 16 | 0.7463 | 0.5450 | 60 | 20 | 31 | 32 | 0.3105 | 0.3619 | 210 | 100 |

16 | 17 | 1.2890 | 1.7210 | 60 | 20 | 32 | 33 | 0.3410 | 0.5302 | 60 | 40 |

Hour (h) | Ind. (p.u) | Res. (p.u) | Com. (p.u) | Hour (h) | Ind. (p.u) | Res. (p.u) | Com. (p.u) |
---|---|---|---|---|---|---|---|

1 | 0.56 | 0.69 | 0.20 | 13 | 0.95 | 0.99 | 0.89 |

2 | 0.54 | 0.65 | 0.19 | 14 | 0.96 | 0.99 | 0.92 |

3 | 0.52 | 0.62 | 0.18 | 15 | 0.90 | 1.00 | 0.94 |

4 | 0.50 | 0.56 | 0.18 | 16 | 0.83 | 0.96 | 0.96 |

5 | 0.55 | 0.58 | 0.20 | 17 | 0.78 | 0.96 | 1.00 |

6 | 0.58 | 0.61 | 0.22 | 18 | 0.72 | 0.94 | 0.88 |

7 | 0.68 | 0.64 | 0.25 | 19 | 0.71 | 0.93 | 0.76 |

8 | 0.80 | 0.76 | 0.40 | 20 | 0.70 | 0.92 | 0.73 |

9 | 0.90 | 0.90 | 0.65 | 21 | 0.69 | 0.91 | 0.65 |

10 | 0.98 | 0.95 | 0.86 | 22 | 0.67 | 0.88 | 0.50 |

11 | 1.00 | 0.98 | 0.90 | 23 | 0.65 | 0.84 | 0.28 |

12 | 0.94 | 1.00 | 0.92 | 24 | 0.60 | 0.72 | 0.22 |

Par. | Value | Unit | Par. | Value | Unit |
---|---|---|---|---|---|

${C}_{\mathrm{kWh}}$ | 0.1390 | US$kWh | T | 365 | Days |

${\Delta}_{h}$ | 1.00 | h | $\alpha $ | 0.30 | US$/MVAr${}^{3}$ |

$\beta $ | −305.10 | US$/MVAr${}^{2}$ | $\gamma $ | 127380 | US$/MVAr |

${c}_{1}$ | 6/2190 | 1/days | ${c}_{2}$ | 10 | Years |

No. of Iterations | Location and Size Node (MVAr) | ${\mathit{A}}_{\mathbf{cost}}$ (US$/year) | Proc. Times (h) |
---|---|---|---|

Benchmark case | — | 130,613.90 | — |

20 | $\left\{14\right(0.2896),\phantom{\rule{0.277778em}{0ex}}30(0.5593),\phantom{\rule{0.277778em}{0ex}}32(0.1177\left)\right\}$ | 109,455.96 | 3.0697 |

40 | $\left\{14\right(0.2896),\phantom{\rule{0.277778em}{0ex}}30(0.5593),\phantom{\rule{0.277778em}{0ex}}32(0.1177\left)\right\}$ | 109,455.96 | 3.1592 |

60 | $\left\{12\right(0.1920),\phantom{\rule{0.277778em}{0ex}}14(0.1488),\phantom{\rule{0.277778em}{0ex}}30(0.6556\left)\right\}$ | 109,498.91 | 3.3368 |

80 | $\left\{14\right(0.2896),\phantom{\rule{0.277778em}{0ex}}30(0.5489),\phantom{\rule{0.277778em}{0ex}}31(0.1281\left)\right\}$ | 109,472.55 | 3.4647 |

100 | $\left\{11\right(0.1982),\phantom{\rule{0.277778em}{0ex}}14(0.1539),\phantom{\rule{0.277778em}{0ex}}30(0.6509\left)\right\}$ | 109,496.84 | 3.6171 |

Sol. No. | Location (Node) | ${\mathit{A}}_{\mathbf{cost}}$ (US$/year) | Sol. No. | Location (Node) | ${\mathit{A}}_{\mathbf{cost}}$ (US$/year) |
---|---|---|---|---|---|

1 | $\{14,\phantom{\rule{0.277778em}{0ex}}30,\phantom{\rule{0.277778em}{0ex}}32\}$ | 109,455.96 | 11 | $\{13,\phantom{\rule{0.277778em}{0ex}}16,\phantom{\rule{0.277778em}{0ex}}30\}$ | 109,530.77 |

2 | $\{14,\phantom{\rule{0.277778em}{0ex}}30,\phantom{\rule{0.277778em}{0ex}}31\}$ | 109,472.54 | 12 | $\{8,\phantom{\rule{0.277778em}{0ex}}14,\phantom{\rule{0.277778em}{0ex}}30\}$ | 109,713.95 |

3 | $\{11,\phantom{\rule{0.277778em}{0ex}}14,\phantom{\rule{0.277778em}{0ex}}30\}$ | 109,496.84 | 13 | $\{13,\phantom{\rule{0.277778em}{0ex}}29,\phantom{\rule{0.277778em}{0ex}}30\}$ | 109,720.81 |

4 | $\{12,\phantom{\rule{0.277778em}{0ex}}14,\phantom{\rule{0.277778em}{0ex}}30\}$ | 109,498.91 | 14 | $\{12,\phantom{\rule{0.277778em}{0ex}}30,\phantom{\rule{0.277778em}{0ex}}31\}$ | 109,770.17 |

5 | $\{12,\phantom{\rule{0.277778em}{0ex}}16,\phantom{\rule{0.277778em}{0ex}}30\}$ | 109,501.60 | 15 | $\{13,\phantom{\rule{0.277778em}{0ex}}28,\phantom{\rule{0.277778em}{0ex}}30\}$ | 109,905.94 |

6 | $\{10,\phantom{\rule{0.277778em}{0ex}}14,\phantom{\rule{0.277778em}{0ex}}30\}$ | 109,504.58 | 16 | $\{10,\phantom{\rule{0.277778em}{0ex}}12,\phantom{\rule{0.277778em}{0ex}}30\}$ | 109,908.50 |

7 | $\{13,\phantom{\rule{0.277778em}{0ex}}30,\phantom{\rule{0.277778em}{0ex}}31\}$ | 109,511.63 | 17 | $\{10,\phantom{\rule{0.277778em}{0ex}}30,\phantom{\rule{0.277778em}{0ex}}31\}$ | 109,955.44 |

8 | $\{13,\phantom{\rule{0.277778em}{0ex}}30,\phantom{\rule{0.277778em}{0ex}}33\}$ | 109,513.62 | 18 | $\{9,\phantom{\rule{0.277778em}{0ex}}12,\phantom{\rule{0.277778em}{0ex}}30\}$ | 109,960.30 |

9 | $\{13,\phantom{\rule{0.277778em}{0ex}}17,\phantom{\rule{0.277778em}{0ex}}30\}$ | 109,515.72 | 19 | $\{13,\phantom{\rule{0.277778em}{0ex}}29,\phantom{\rule{0.277778em}{0ex}}31\}$ | 109,961.47 |

10 | $\{10,\phantom{\rule{0.277778em}{0ex}}16,\phantom{\rule{0.277778em}{0ex}}30\}$ | 109,525.51 | 20 | $\{8,\phantom{\rule{0.277778em}{0ex}}12,\phantom{\rule{0.277778em}{0ex}}30\}$ | 110,069.41 |

**Table 6.**Numerical results obtained after implementing mixed-integer nonlinear programming (MINLP) model in GAMS.

Solver | Location and Size Node (MVAr) | ${\mathit{A}}_{\mathbf{cost}}$ (US$/year) |
---|---|---|

Benchmark case | — | 130,613.90 |

BONMIN | $\left\{8\right(0.2980),\phantom{\rule{0.277778em}{0ex}}25(0.0920),\phantom{\rule{0.277778em}{0ex}}30(0.5127\left)\right\}$ | 109,560.85 |

COUENNE | $\left\{13\right(0.1850),\phantom{\rule{0.277778em}{0ex}}16(0.0825),\phantom{\rule{0.277778em}{0ex}}32(0.4478\left)\right\}$ | 109,791.14 |

Genetic-Convex | $\left\{14\right(0.2896),\phantom{\rule{0.277778em}{0ex}}30(0.5593),\phantom{\rule{0.277778em}{0ex}}32(0.1177\left)\right\}$ | 109,455.96 |

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## Share and Cite

**MDPI and ACS Style**

Montoya, O.D.; Chamorro, H.R.; Alvarado-Barrios, L.; Gil-González, W.; Orozco-Henao, C.
Genetic-Convex Model for Dynamic Reactive Power Compensation in Distribution Networks Using D-STATCOMs. *Appl. Sci.* **2021**, *11*, 3353.
https://doi.org/10.3390/app11083353

**AMA Style**

Montoya OD, Chamorro HR, Alvarado-Barrios L, Gil-González W, Orozco-Henao C.
Genetic-Convex Model for Dynamic Reactive Power Compensation in Distribution Networks Using D-STATCOMs. *Applied Sciences*. 2021; 11(8):3353.
https://doi.org/10.3390/app11083353

**Chicago/Turabian Style**

Montoya, Oscar Danilo, Harold R. Chamorro, Lazaro Alvarado-Barrios, Walter Gil-González, and César Orozco-Henao.
2021. "Genetic-Convex Model for Dynamic Reactive Power Compensation in Distribution Networks Using D-STATCOMs" *Applied Sciences* 11, no. 8: 3353.
https://doi.org/10.3390/app11083353