# Risk-Based Capacitor Placement in Distribution Networks

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## Abstract

**:**

## 1. Introduction

## 2. An Overview of the Concepts of Risk and Uncertainty

#### 2.1. Technical Risks

#### 2.2. Voltage Violation Risk

#### 2.3. Risk of Voltage Instability

#### 2.4. Load Uncertainty Risk

## 3. Load Modeling

_{L}, P

_{M}, P

_{R}) and shows that the expected load is around P

_{M}but not less than P

_{L}and not more than P

_{R}. If the estimated power at a load point is P

_{0}with maximum error e

_{1}and e

_{2}, the fuzzy number parameters corresponding to that load point can be obtained using the following equations:

## 4. Voltage Risk Modeling

_{max}and V

_{min}, respectively.

## 5. Instability Risk Analysis

^{2}-4c, the greater the distance between the real and positive answers will be. As shown in Figure 5, by increasing the distance between these solutions, the stable solution of the problem moves further away from the critical point.

## 6. Problem Formulation

#### 6.1. Economic Objective Function

#### 6.2. Voltage Risk Objective Function

#### 6.3. Instability Risk Objective Function

## 7. Solution Methodology

#### 7.1. Multi-Objective Model of Capacitive Placement in the Presence of Uncertainty

#### 7.2. Coding the Problem of Fixed and Switching Capacitor Placement

#### 7.3. Computational Steps of the Algorithm

#### 7.3.1. Initial Population

#### 7.3.2. Crossover

#### 7.3.3. Mutation

#### 7.3.4. Evaluation of Objective Functions

#### 7.3.5. Non-Dominant Ranking

_{1;}${d}_{i}^{2}$ is the ratio of the area of the domain of this point to the area of the objective function f

_{2}. D is the sum of these two ratios expressed as an index of the general domain related to this point, which is called the swarm distance.

#### 7.3.6. Stop

#### 7.3.7. Deciding on the Choice of the Final Answer

## 8. Numerical Studies and Results

#### 8.1. The First Experiment

#### 8.2. The Second Experiment

#### 8.3. Comparing the Proposed Method with Other Capacitor Placement Methods Considering the Fixed Capacitor

## 9. Conclusions

- The behavior of the network in the presence of both definite and non-deterministic information in various issues of the electrical energy distribution system can be investigated.
- The relationship between the uncertainty type of consumption of subscribers or in other words, different tariffs, is an interesting topic that helps in perfecting the way of modeling load uncertainty.
- By installing measuring devices, the amount of uncertainty in load information is reduced. However, installing these devices will increase the cost of the system. Therefore, the economic objective function model can be modified by considering these costs.
- One of the sources of uncertainty in power system studies is distributed generations. The power generation of each of these sources can be simulated with the model presented in this research.
- In the study of restructuring the distribution system, forecasting market information is associated with uncertainty. Using triangular fuzzy numbers can be a good solution for modeling this information.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviation

${\mathsf{\mu}}_{P}$ | Load membership function |

${\tilde{\mathrm{V}}}_{\mathrm{k}}$ | Bus voltage k in fuzzy |

${S}_{{V}_{k}}$ | Bus voltage risk k |

${\tilde{I}}_{i,j}$ | Passing current between bus i and j in fuzzy |

$SI$ | Stability index |

${\mu}_{{V}_{s}}$ | The membership function of the sub transmission bus voltage |

${\mathsf{\mu}}_{{S}_{l}}$ | Apparent power membership function |

${\mathsf{\mu}}_{{V}_{L}}$ | Voltage membership function |

${S}_{S{I}_{i}}$ | Risk of instability |

${\stackrel{\u02c7}{f}}_{c}$ | Economic objective function |

${C}_{c}$ | The reactive power value of the fixed capacitor |

$Ic$ | Fixed capacitor price |

${C}_{s}$ | Switching capacitor |

$Is$ | The price of the switchable capacitor |

${C}_{T}$ | Energy cost |

$hp$ | Horizon year |

$intr$ | Annual interest rate |

$infr$ | Annual interest rate |

${S}_{v}$ | Voltage risk objective function |

${S}_{SI}$ | Instability risk objective function |

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**Figure 9.**Pareto response space in fixed and switching capacitors placement in the distribution network under study.

**Figure 11.**Two-dimensional diagram of economic objective function and instability risk in different experiments.

**Figure 12.**Two-dimensional diagram of economic objective function and voltage risk in various experiments.

**Figure 15.**Comparison of reference [29] response with the Pareto Front obtained from fixed capacitor placement.

**Figure 16.**Comparison of reference [30] response with the Pareto Front obtained from fixed capacitor placement.

**Figure 17.**Comparison of reference [31] response with the Pareto Front obtained from fixed capacitor placement.

Bus No. | Base Load Level | Average Load Level | Maximum Load Level | ||
---|---|---|---|---|---|

Reactive Power (KVAr) | Active Power (KW) | Active Power (KW) | Active Power (KW) | Active Power (KW) | |

1 | 0 | 0 | 0 | 0 | 0 |

2 | 54 | 90 | 100 | 100 | 110 |

3 | 36 | 81 | 90 | 90 | 99 |

4 | 72 | 108 | 120 | 120 | 132 |

5 | 27 | 54 | 60 | 60 | 66 |

6 | 18 | 54 | 60 | 60 | 66 |

7 | 90 | 180 | 200 | 200 | 220 |

8 | 90 | 180 | 200 | 200 | 220 |

9 | 18 | 54 | 60 | 60 | 66 |

10 | 18 | 54 | 60 | 60 | 66 |

11 | 27 | 40.5 | 45 | 45 | 49.5 |

12 | 31.5 | 54 | 60 | 60 | 66 |

13 | 31.5 | 54 | 60 | 60 | 66 |

14 | 72 | 108 | 120 | 120 | 132 |

15 | 9 | 54 | 60 | 60 | 66 |

16 | 18 | 54 | 60 | 60 | 66 |

17 | 18 | 54 | 60 | 60 | 66 |

18 | 36 | 81 | 90 | 90 | 99 |

19 | 36 | 81 | 90 | 90 | 99 |

20 | 36 | 81 | 90 | 90 | 99 |

21 | 36 | 81 | 90 | 90 | 99 |

22 | 36 | 81 | 90 | 90 | 99 |

23 | 36 | 81 | 90 | 90 | 99 |

24 | 180 | 378 | 420 | 420 | 462 |

25 | 180 | 378 | 420 | 420 | 462 |

26 | 22.5 | 54 | 60 | 60 | 66 |

27 | 22.5 | 54 | 60 | 60 | 66 |

28 | 18 | 54 | 60 | 60 | 66 |

29 | 63 | 108 | 120 | 120 | 132 |

30 | 540 | 180 | 200 | 200 | 220 |

31 | 63 | 135 | 150 | 150 | 165 |

32 | 90 | 189 | 210 | 210 | 231 |

33 | 36 | 54 | 60 | 60 | 66 |

No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Standard Capacitance (KVAr) | 25 | 50 | 75 | 100 | 125 | 150 | 175 | 200 | 225 | 250 | 275 | 300 |

Response | Economic Situation | Voltage Risk | Instability Risk |
---|---|---|---|

Solution1 | Inappropriate | Suitable | Suitable |

Solution2 | Suitable | Suitable | Inappropriate |

Solution3 | Suitable | Inappropriate | Suitable |

Assumptions | Unit | Amount |
---|---|---|

Annual interest rates [23] | % | 9 |

Annual inflation rates [23] | % | 5.12 |

Permissible voltage limits | Per Unit | 0.93–1.05 |

Voltage stability limit | - | 0.71 |

Fixed capacitor cost factor (IC) | $/KVAr | 3 |

Switching capacitor cost factor (IS) [29] | $/KVAr | 3.2 |

Horizon year | - | 5 |

e1 (Maximum Error) | - | 0.1 |

e2 (Maximum Error) | - | 0.15 |

Experiment No. | Maximum Error e1 | Maximum Error e2 |
---|---|---|

1 | 0.1 | 0.15 |

2 | 0.15 | 0.20 |

3 | 0.20 | 0.25 |

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**MDPI and ACS Style**

Falaghi, H.; Ramezani, M.; Elyasi, H.; Farhadi, M.; Estebsari, A.
Risk-Based Capacitor Placement in Distribution Networks. *Electronics* **2022**, *11*, 3145.
https://doi.org/10.3390/electronics11193145

**AMA Style**

Falaghi H, Ramezani M, Elyasi H, Farhadi M, Estebsari A.
Risk-Based Capacitor Placement in Distribution Networks. *Electronics*. 2022; 11(19):3145.
https://doi.org/10.3390/electronics11193145

**Chicago/Turabian Style**

Falaghi, Hamid, Maryam Ramezani, Hasan Elyasi, Mahdi Farhadi, and Abouzar Estebsari.
2022. "Risk-Based Capacitor Placement in Distribution Networks" *Electronics* 11, no. 19: 3145.
https://doi.org/10.3390/electronics11193145