# A Scatter Search Heuristic for the Optimal Location, Sizing and Contract Pricing of Distributed Generation in Electric Distribution Systems

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

#### 2.1. Decision-Making Problem of the Distribution Company

#### 2.2. Decision-Making Problem of the Distributed Generation Owner

#### 2.3. Bilevel Modeling Framework

#### 2.4. Upper Level Optimization Problem

#### 2.5. Lower Level Optimization Problem

#### 2.6. Illustrative Example

## 3. Solution Approach

#### 3.1. Scatter Search General Structure

Algorithm 1. Scatter search for DG optimization | |

1 | $P=\varnothing $ |

2 | While $\left|P\right|\le n$ do |

3 | $S=DiverseGeneration()$ |

4 | If $S\notin P$ then |

5 | $P{\displaystyle \cup}\{S\}$ |

6 | End-if |

7 | End-While |

8 | Evaluate the solutions in $P$ and sort them by non-increasing profit (Equation (1)) |

9 | Build $RefSet$ with the best $b/2$ solutions of $P$, $RefSet=\{{S}_{1},\dots {S}_{b/2}\}$ |

10 | Add to $RefSet$ the most diverse $b/2$ solutions in $P$ with respect to those already in $RefSet$ |

11 | Sort $RefSet$ by non-increasing profit (Equation (1)) |

12 | $new=true$ |

13 | While (new) do |

14 | $new=false$ |

15 | For all $Sand{S}_{}^{\prime}\in RefSet$ do |

16 | If ($Sand{S}_{}^{\prime}$ has not been combined before) then |

17 | $\overline{S}=Combine(S,{S}^{\prime})$ |

18 | $\overline{S}=Improve(S)$ |

19 | $new=Update(RefSet,\overline{S})$ |

20 | End-if |

21 | End-For all |

22 | End-While |

23 | Return ${S}^{\ast}=argma{x}_{S\in RefSet}\{f(S)\}$ |

#### 3.2. Scatter Search Components

#### 3.2.1. Solution Representation and Objective Function Evaluation

#### 3.2.2. Distance Measure

#### 3.2.3. Diversification Generator Method

#### 3.2.4. Solution Combination Method

#### 3.2.5. Improvement Method

#### 3.2.6. Reference Set Update Method

## 4. Tests and Results

#### 4.1. Diversification Generator Method Comparison

#### 4.2. Comparison against Other Metaheuristics

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

Bus | P (MW) | Q (Mvar) | Bus | P (MW) | Q (Mvar) |
---|---|---|---|---|---|

1 | 0 | 0 | 18 | 0 | 0 |

2 | 0.1555 | 0.0820 | 19 | 0.0113 | 0.0057 |

3 | 0.1555 | 0.0820 | 20 | 0.0424 | 0.0198 |

4 | 0.0452 | 0.0226 | 21 | 0 | 0 |

5 | 0.0452 | 0.0226 | 22 | 0.1385 | 0.0707 |

6 | 0 | 0 | 23 | 2.5438 | 1.2719 |

7 | 0 | 0 | 24 | 0.5031 | 0.2544 |

8 | 0 | 0 | 25 | 0.0057 | 0.0028 |

9 | 0.0141 | 0.0057 | 26 | 0.9836 | 0.5992 |

10 | 0.0961 | 0.0480 | 27 | 0.0254 | 0.0141 |

11 | 0.1385 | 0.0678 | 28 | 0.3448 | 0.1781 |

12 | 0.4777 | 0.2459 | 29 | 2.4421 | 1.8598 |

13 | 0.0311 | 0.0141 | 30 | 0.0791 | 0.0396 |

14 | 0.1131 | 0.0565 | 31 | 0.2657 | 0.1752 |

15 | 0.3816 | 0.1979 | 32 | 0.1922 | 0.0961 |

16 | 0.2742 | 0.1215 | 33 | 0.0791 | 0.0396 |

17 | 0.0113 | 0.0057 | 34 | 0.4042 | 0.3024 |

Line | R (Ω) | X (Ω) | Line | R (Ω) | X (Ω) |
---|---|---|---|---|---|

1–2 | 0.0026 | 0.0025 | 17–18 | 0.0078 | 0.0064 |

2–3 | 0.0018 | 0.0013 | 18–20 | 0.0004 | 0.0003 |

3–4 | 0.0170 | 0.0138 | 20–21 | 0.0053 | 0.0038 |

4–5 | 0.0004 | 0.0003 | 20–22 | 0.0071 | 0.0071 |

4–6 | 0.0036 | 0.0033 | 21–23 | 0.0007 | 0.0004 |

6–7 | 0.0010 | 0.0009 | 22–25 | 0.0004 | 0.0003 |

7–8 | 0.0076 | 0.0057 | 22–24 | 0.0007 | 0.0005 |

8–9 | 0.0003 | 0.0003 | 24–26 | 0.0105 | 0.0065 |

9–10 | 0.0105 | 0.0074 | 24–27 | 0.0037 | 0.0037 |

9–11 | 0.0190 | 0.0172 | 26–28 | 0.0004 | 0.0003 |

10–12 | 0.0086 | 0.0065 | 28–30 | 0.0005 | 0.0004 |

12–15 | 0.0095 | 0.0065 | 30–33 | 0.0037 | 0.0034 |

11–14 | 0.0037 | 0.0037 | 28–31 | 0.0070 | 0.0052 |

11–13 | 0.0077 | 0.0064 | 27–29 | 0.0014 | 0.0013 |

13–16 | 0.0017 | 0.0011 | 29–32 | 0.0038 | 0.0035 |

16–17 | 0.0038 | 0.0037 | 32–34 | 0.0047 | 0.0034 |

17–19 | 0.0103 | 0.0103 | - | - | - |

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**Figure 2.**(

**a**) 5-bus distribution system; (

**b**) impact of distributed generation (DG) location in voltage profile; and (

**c**) impact of DG location in power losses.

**Figure 5.**34-bus distribution system (Source [20]).

SS Variant | DG Owner Profit | Average Running Time (s) | Best Profit | ||
---|---|---|---|---|---|

Solution 1 | Solution 2 | Solution 3 | |||

SS-Sist | 165,036 | 165,036 | 165,036 | 422 | 165,036 |

SS-Rand | 165,952 | 166,303 | 166,731 | 407 | 166,731 |

SS-SistRand | 165,339 | 165,484 | 165,119 | 607 | 165,484 |

SS Variant | DG Unit (Bus, Price($/MWh), Size (MW)) | DG Owner Profit | ||
---|---|---|---|---|

1 | 2 | 3 | ||

SS-Sist | (24, 77.0, 1.5) | (29, 77.0, 1.5) | (30, 77.0, 1.5) | 165,036 |

SS-Rand | (27, 77.0, 1.5) | (29, 77.0, 1.5) | (30, 77.0, 1.5) | 166,731 |

SS-SistRand | (24, 77.0, 1.5) | (27, 77.0, 1.5) | (31, 77.0, 1.5) | 165,484 |

**Table 3.**Results for SS, memetic algorithm (MA) and genetic algorithm (GA) on the 34-bus distribution system.

Method | DG Owner Profit | Average Running Time (s) | Best Profit | ||
---|---|---|---|---|---|

Solution 1 | Solution 2 | Solution 3 | |||

SS-Rand | 165,952 | 166,303 | 166,731 | 407 | 166,731 |

SS-Sist | 165,036 | 165,036 | 165,036 | 422 | 165,036 |

MA | 156,128 | 157,601 | 155,423 | 302,400 | 157,601 |

GA | 25,100 | 21,500 | 35,000 | 1200 | 35,000 |

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**MDPI and ACS Style**

Pérez Posada, A.F.; Villegas, J.G.; López-Lezama, J.M.
A Scatter Search Heuristic for the Optimal Location, Sizing and Contract Pricing of Distributed Generation in Electric Distribution Systems. *Energies* **2017**, *10*, 1449.
https://doi.org/10.3390/en10101449

**AMA Style**

Pérez Posada AF, Villegas JG, López-Lezama JM.
A Scatter Search Heuristic for the Optimal Location, Sizing and Contract Pricing of Distributed Generation in Electric Distribution Systems. *Energies*. 2017; 10(10):1449.
https://doi.org/10.3390/en10101449

**Chicago/Turabian Style**

Pérez Posada, Andrés Felipe, Juan G. Villegas, and Jesús M. López-Lezama.
2017. "A Scatter Search Heuristic for the Optimal Location, Sizing and Contract Pricing of Distributed Generation in Electric Distribution Systems" *Energies* 10, no. 10: 1449.
https://doi.org/10.3390/en10101449