# Optimal Design of PV Systems in Electrical Distribution Networks by Minimizing the Annual Equivalent Operative Costs through the Discrete-Continuous Vortex Search Algorithm

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## Abstract

**:**

## 1. Introduction

- The generalization of the proposed master–slave optimization algorithm to accurately locate and size the PV sources in electrical distribution networks with AC or DC operating technologies, which were not previously reported in the current literature.
- The improvement of the current literature reports for the IEEE 33- and 69-bus systems with the classical Chu and Beasley genetic algorithm.

## 2. Mathematical Formulation

#### 2.1. Formulation of the Objective Function

#### 2.2. Set of Constraints

#### 2.3. Model Interpretation

## 3. Methodology Proposed

#### 3.1. Slave Stage: Matricial Backward/Forward Power Flow Method

- ${A}_{kl}=1,$ if the current through the line l leaves the node k;
- ${A}_{kl}=-1,$ if the current through the line l arrives the node k;
- ${A}_{kl}=0,$ if the line l is not connected to the node k.

**Remark**

**1.**

#### 3.2. Master Stage: DCVSA

Algorithm 1: Solution of the multiperiod power flow problem using the matricial backward/forward power flow formulation to determine the fitness function of the studied optimization problem |

#### 3.2.1. Initial Solution

#### 3.2.2. Candidate Solutions

#### 3.2.3. Updating of the Current Solution

#### 3.2.4. Radius Reduction

## 4. Test Systems

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

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

#### 4.3. Demand and Generation Curves

## 5. Numerical Results and Simulations

- i.
- Application of the DCVSA developed and its comparisons with existing methodologies into the IEEE 33- and IEEE 69-node test systems with their AC versions.
- ii.
- The minimization of the total annual operating cost using the proposed master–slave methodology for the DC versions of the IEEE 33- and IEEE 69-bus systems.

#### 5.1. Case 1: Results in the AC IEEE 33-Bus System

#### 5.2. Case 1: Results in the AC IEEE 69-Nodes System

#### 5.3. Case 2: Results in the DC IEEE 33-Bus System

#### 5.4. Case 2: Results in the DC IEEE 69-Node System

## 6. Conclusions and Future Works

- ✓
- The reduction from the base case reached by DCVSA was 27.04%, and 27.15% for the test systems in their AC version; in their DC versions, the reductions were 26.94% and 27.03%, respectively.
- ✓
- The proposed methodology obtained the lower standard deviation values when solving the PV units’ location and sizing problem for the IEEE 33- and IEEE 69-node test systems in their AC versions, with the values of US$/year 1154.08 and US$/year 2666.46, respectively. These values were considerably lower than the comparative DCCBGA, which confirmed the effectiveness and robustness of the proposed DCVSA to solve the studied problem ensuring that at each evaluation, the final objective function value will produce a small variation. In the case of the DC grids, these values were US$/year 1652.82 and US$/year 2710.94.
- ✓
- Regarding the voltage profiles of both systems in their AC version, it was observed that, during the period of maximum PV energy injection, i.e., hour 14, the voltage at some nodes is above the voltage at the substation node, with the magnitudes 1.0291 pu and 1.0419 pu, respectively, while the minimum voltage values found during the time period of maximum power demand and minimum PV energy injection, i.e., in hours 20 and 21, had values of 0.9038 pu and 0.90919 pu, respectively. The same behavior was experienced in the DC grid equivalents. The most significant characteristic of these results is that it recorded evidence that the voltage regulation bounds assigned in $\pm 10\%$ of the nominal voltage were always fulfilled by the solutions reached by the DCVSA.
- ✓
- The proposed solution methodology is independent of the number of nodes of the AC or DC network under study; however, in the number of nodes of the grid increase, the solution space size increases as well; this implies that the total processing times required to identify the optimal solution will also increase; however, these increments can be from a few minutes to hours, which is not a critical aspect in distribution system planning projects where the solution quality assumes the greatest importance instead of the total processing times.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Electrical configuration of the test feeders: (

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

**b**) IEEE 69-bus system.

**Figure 4.**Percentage of reduction of the total grid operative costs in the IEEE 33-bus system in its AC version.

**Figure 5.**Impact of the PV inclusion in the IEEE 33-bus system: (

**a**) power injections in the slack source, and (

**b**) maximum current performance.

**Figure 6.**Voltage behavior during the day for the IEEE 33-bus system: (

**a**) maximum voltage magnitude, and (

**b**) minimum voltage magnitude.

**Figure 7.**Percentage of reduction of the total grid operative costs in the IEEE 69-bus system in its AC version.

**Figure 8.**Impact of the PV inclusion in the IEEE 69-bus system: (

**a**) power injections in the slack source, and (

**b**) maximum current performance.

**Figure 9.**Voltage behavior during the day for the IEEE 69-bus system: (

**a**) maximum voltage magnitude and (

**b**) minimum voltage magnitude.

**Figure 10.**Impact of the PV inclusion in the DC version of the IEEE 33-bus system: (

**a**) power injections in the slack source, and (

**b**) maximum current performance.

**Figure 11.**Voltage behavior during the day for the IEEE 33-bus system in its DC version: (

**a**) maximum voltage magnitude and (

**b**) minimum voltage magnitude.

**Figure 12.**Impact of the PV inclusion in the DC version of the IEEE 69-bus system: (

**a**) power injections in the slack source and (

**b**) maximum current performance.

**Figure 13.**Voltage behavior during the day for the IEEE 69-bus system in its DC version: (

**a**) maximum voltage magnitude, and (

**b**) minimum voltage magnitude.

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

${C}_{kWh}$ | 0.1390 | US$/kWh | T | 365 | días |

${t}_{a}$ | 10 | % | ${N}_{t}$ | 20 | años |

$\Delta h$ | 1 | h | ${t}_{e}$ | 2 | % |

${C}_{pv}$ | 1036.49 | US$/kWp | ${C}_{0\&M}$ | 0.0019 | US$/kWh |

${N}_{pv}^{ava}$ | 3 | - | $\Delta V$ | $\pm 10$ | % |

${s}_{k}^{pv,min}$ | 0 | kW | ${s}_{k}^{pv,max}$ | 2400 | kW |

${\alpha}_{1}$ | $100\times {10}^{4}$ | US$/V | ${\alpha}_{2}$ | $100\times {10}^{4}$ | US$/V |

${\alpha}_{3}$ | $100\times {10}^{4}$ | US$/W | ${\alpha}_{4}$ | $100\times {10}^{4}$ | US$/A |

Method | Site and Size (Node, MVAr) | ${\mathit{A}}_{\mathbf{cost}}$ (US$/year) | ${\mathit{f}}_{1}$ (US$/year) | ${\mathit{f}}_{2}$ (US$/year) |
---|---|---|---|---|

Bench. Case | - | 3,700,455.38 | 3,700,455.38 | 0 |

BONMIN | $\left\{17\left(1.35393\right),18\left(0.21051\right),33\left(2.14515\right)\right\}$ | 2,701,824.14 | 2,233,247.50 | 468,576.64 |

DCCBGA | $\left\{11\left(0.76046\right),15\left(0.96897\right),30\left(1.90598\right)\right\}$ | 2,699,932.29 | 2,240,724.98 | 459,207.31 |

DCVSA | $\left\{11\left(0.76061\right),14\left(1.08518\right),31\left(1.80295\right)\right\}$ | 2,699,761.71 | 2,238,872.09 | 460,889.62 |

**Table 3.**Numerical performance comparison between the DCVSA and DCCBGA in the IEEE 33-bus system after 100 consecutive evaluations.

Method | Best (US$/year) | Mean (US$/year) | Worst (US$/year) | SD (US$/year) | Avg. Time (s) |
---|---|---|---|---|---|

BONMIN | 2,701,824.14 | 2,701,824.14 | 2,701,824.14 | 0 | 3.64 |

DCCBGA | 2,699,932.29 | 2,702,178.35 | 2,705,870.99 | 1221.67 | 5.30 |

DCVSA | 2,699,761.71 | 2,701,911.72 | 2,705,353.76 | 1154.08 | 170.23 |

Method | Site and Size (Node, MVAr) | ${\mathit{A}}_{\mathbf{cost}}$ (US$/year) | ${\mathit{f}}_{1}$ (US$/year) | ${\mathit{f}}_{2}$ (US$/year) |
---|---|---|---|---|

Bench. Case | - | 3,878,199.93 | 3,878,199.93 | 0 |

DCCBGA | $\left\{24\left(0.53255\right),61\left(1.89542\right),64\left(1.37716\right)\right\}$ | 2,825,783.32 | 2,345,138.38 | 480,644.95 |

DCVSA | $\left\{16\left(0.26321\right),61\left(2.27190\right),63\left(1.29335\right)\right\}$ | 2,825,261.56 | 2,341,670.47 | 483,591.08 |

**Table 5.**Numerical performance comparison between the DCVSA and DCCBGA in the IEEE 69-bus system after 100 consecutive evaluations.

Method | Best (US$/year) | Mean (US$/year) | Worst (US$/year) | SD (US$/year) | Avg. Time (s) |
---|---|---|---|---|---|

DCCBGA | 2,825,783.32 | 2,829,498.36 | 2,844,469.50 | 2827.18 | 22.36 |

DCVSA | 2,825,261.56 | 2,829,039.72 | 2,834,150.92 | 2666.56 | 887.64 |

Method | Site and Size (Node, MVAr) | ${\mathit{A}}_{\mathbf{cost}}$ (US$/year) | ${\mathit{f}}_{1}$ (US$/year) | ${\mathit{f}}_{2}$ (US$/year) |
---|---|---|---|---|

Bench. Case | - | 3,644,043.01 | 3,644,043.01 | 0 |

DCVSA | $\left\{9\left(0.58031\right),15\left(1.29137\right),31\left(1.71559\right)\right\}$ | 2,662,425.32 | 2,209,300.38 | 453,124.93 |

Method | Site and Size (Node, MVAr) | ${\mathit{A}}_{\mathit{cost}}$ (US$/year) | ${\mathit{f}}_{1}$ (US$/year) | ${\mathit{f}}_{2}$ (US$/year) |
---|---|---|---|---|

Bench. Case | - | 3,817,420.38 | 3,817,420.38 | 0 |

DCVSA | $\left\{23\left(0.77201\right),62\left(2.34027\right),63\left(0.61853\right)\right\}$ | 2,785,538.58 | 2,314,281.30 | 471,257.28 |

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## Share and Cite

**MDPI and ACS Style**

Cortés-Caicedo, B.; Molina-Martin, F.; Grisales-Noreña, L.F.; Montoya, O.D.; Hernández, J.C.
Optimal Design of PV Systems in Electrical Distribution Networks by Minimizing the Annual Equivalent Operative Costs through the Discrete-Continuous Vortex Search Algorithm. *Sensors* **2022**, *22*, 851.
https://doi.org/10.3390/s22030851

**AMA Style**

Cortés-Caicedo B, Molina-Martin F, Grisales-Noreña LF, Montoya OD, Hernández JC.
Optimal Design of PV Systems in Electrical Distribution Networks by Minimizing the Annual Equivalent Operative Costs through the Discrete-Continuous Vortex Search Algorithm. *Sensors*. 2022; 22(3):851.
https://doi.org/10.3390/s22030851

**Chicago/Turabian Style**

Cortés-Caicedo, Brandon, Federico Molina-Martin, Luis Fernando Grisales-Noreña, Oscar Danilo Montoya, and Jesus C. Hernández.
2022. "Optimal Design of PV Systems in Electrical Distribution Networks by Minimizing the Annual Equivalent Operative Costs through the Discrete-Continuous Vortex Search Algorithm" *Sensors* 22, no. 3: 851.
https://doi.org/10.3390/s22030851