# An Operational Approach to Multi-Objective Optimization for Volt-VAr Control

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## Abstract

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## 1. Introduction

- A general framework for formulating multi-objective Volt-VAr Optimization problems is proposed. Such a model has not been proposed before. The manner in which previous work on VVC–MOO can be expressed in terms of this framework is shown.
- More importantly, the application of simple techniques for MOO on VVC problems is investigated, and the operational interpretation of each of these techniques is discussed. This discussion of operational interpretation is novel to this field and is missing from previous work on VVC–MOO. This discussion, especially when the operational interpretation is intuitive, may lead to the faster adoption of these techniques.

## 2. Materials and Methods

#### 2.1. General Multi-Objective Volt-VAr Optimization

#### 2.1.1. The General VVO Problem

#### 2.1.2. Example Target Functions

- Target voltage and Conservation Voltage Reduction (CVR): For nominal voltages ${v}_{k}^{nom}$, the deviation function is the sum of differences, under some norm function $\left|\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\right|$:$${f}_{V}\left(\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\right)=\sum _{k}\left|{V}_{k}-{V}_{k}^{nom}\right|.$$Specifically, for CVR ${V}_{k}^{nom}={V}_{min}$, where ${V}_{min}$ is a regulatory set minimum voltage.Alternatively, especially when the nominal values vary considerably, the voltage deviation can be normalized (see e.g., [31,32,37]), resulting in$${f}_{V}\left(\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\right)=\sum _{k}\left|\frac{{V}_{k}-{V}_{k}^{nom}}{{V}_{k}^{nom}}\right|.$$
- Feeder voltage deviation: This can be seen as a variant of target voltage. To express this function, a re-indexing is used, such that ${V}_{m,n}$ is the voltage at element n of feeder m. The index $n=0$ is used to represent the feeder head. The voltage deviation is defined as ${V}_{m,n}-{V}_{m,0}$. Typically (see e.g., [30]), the maximum deviation in each feeder is maximized:$${f}_{Vdiff}\left(\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\right)=\sum _{m}\underset{n}{max}\left[{V}_{m,n}-{V}_{m,0}\right].$$In the case of a radial network, Equation (8) can be expressed in terms of the power as follows (see [29,30]):$${f}_{Vdiff}\left(\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\right)=\sum _{m}\underset{n}{max}\left[\frac{{r}_{m,n}{P}_{m,n}+{x}_{m,n}{Q}_{m,n}}{{V}_{m,0}}\right],$$
- Losses: The total losses due to energy dissipation can be written in the most general case as$${f}_{L}\left(\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\right)=\sum _{j}{I}_{j}^{2}{r}_{j},$$$${f}_{L}\left(\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\right)=\sum _{m}\sum _{n}{r}_{m,n}\frac{{P}_{m,n}^{2}+{Q}_{m,n}^{2}}{{V}_{m,0}^{2}}.$$
- Root power factor: Typically, the power factor is important at the root of the network; that is, the point where the network is fed. Using the index 1 for that node, the power factor at the root of the network can be written as$${f}_{PF}\left(\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\right)=\frac{{P}_{1}}{\sqrt{{P}_{1}^{2}+{Q}_{1}^{2}}}.$$
- Root active power: The root active power is the total power the network consumes. It is simple to express it as$${f}_{p}\left(\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\right)={P}_{1}.$$
- Root reactive power: In a similar way, the root reactive power is$${f}_{q}\left(\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\right)={Q}_{1}.$$
- Cost of controls: In the most generic way, the cost of control can be expressed as$${f}_{C}\left(\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\right)=h(\overrightarrow{X},\overrightarrow{Y}),$$$${f}_{C}\left(\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\right)=\sum _{a}{X}_{a}.$$

#### 2.1.3. Time and Scenario Based Optimization

#### 2.1.4. The Multi-Objective VVO Problem

#### 2.2. Techniques for Multi-Objective Optimization

#### 2.2.1. The Weighted-Sum Technique and the Efficient Curve

#### 2.2.2. The E-Constraint Technique

## 3. Simulation Results

#### 3.1. The Test Feeder

#### 3.2. Applying the Weighted-Sum Technique for Active and Reactive Power Optimization

#### 3.3. Applying the E-Constraint Technique for Active and Reactive Power Optimization

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**An illustration of the proposed multi-objective optimization–Volt-VAr Control (MMO–VVC) approach. Data are presented in parallelograms and functions are presented in rectangles.

**Figure 2.**Efficient curve example. Each plot is a feasible solution; the efficient curve is shown as a dashed line.

**Figure 3.**Modified single-line diagram of the test feeder, based on [44].

**Figure 4.**Calculated active power (P, in $\mathrm{W}$) and reactive power (Q, in $\mathrm{VAr}$) for all control points in the test feeder, with highlighted extreme points.

**Figure 5.**Calculated active power (P, in $\mathrm{W}$) and reactive power (Q, in $\mathrm{VAr}$) on the test feeder, with the efficient curve highlighted and $\alpha $ values.

**Figure 6.**Calculated active power (P, in $\mathrm{W}$) and reactive power (Q, in $\mathrm{VAr}$) for all control points in the test feeder, with highlighted points of interest.

Impedance of Line Ending at Bus | |||
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Bus # | Load | R ($\mathrm{\Omega}$) | X ($\mathrm{\Omega}$) |

1 | Slack bus, $0.8$ $\mathrm{MVAr}$ | ||

2 | $63\mathrm{kVA},cos\phi =0.7$ | 1.35309 | 1.32349 |

3 | $1\mathrm{k}\mathrm{\Omega}$ | 1.17024 | 1.14464 |

4 | $200\mathrm{k}\mathrm{VA},cos\phi =0.7$ | 0.84111 | 0.82271 |

5 | $1\mathrm{k}\mathrm{\Omega}$ | 1.52348 | 1.02760 |

6 | $200\mathrm{k}\mathrm{VA},cos\phi =0.7$ | 2.55727 | 1.72490 |

7 | $1\mathrm{k}\mathrm{\Omega}$ | 1.08820 | 0.73400 |

8 | $100\mathrm{k}\mathrm{VA},cos\phi =0.7$ | 1.25143 | 0.84410 |

9 | $100\mathrm{k}\mathrm{VA},cos\phi =0.7$ | 2.01317 | 1.35790 |

10 | $1\mathrm{k}\mathrm{\Omega}$ | 1.68671 | 1.13770 |

11 | $200\mathrm{k}\mathrm{VA},cos\phi =0.7$ | 1.79553 | 1.21110 |

12 | $1\mathrm{k}\mathrm{\Omega}$ | 2.44845 | 1.65150 |

13 | $63\mathrm{k}\mathrm{VA},cos\phi =0.7$ | 2.01317 | 1.35790 |

14 | $1\mathrm{k}\mathrm{\Omega}$ | 2.23081 | 1.50470 |

15 | $200\mathrm{k}\mathrm{VA},cos\phi =0.7$ | 1.19702 | 0.80740 |

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Raz, D.; Beck, Y. An Operational Approach to Multi-Objective Optimization for Volt-VAr Control. *Energies* **2020**, *13*, 5871.
https://doi.org/10.3390/en13225871

**AMA Style**

Raz D, Beck Y. An Operational Approach to Multi-Objective Optimization for Volt-VAr Control. *Energies*. 2020; 13(22):5871.
https://doi.org/10.3390/en13225871

**Chicago/Turabian Style**

Raz, David, and Yuval Beck. 2020. "An Operational Approach to Multi-Objective Optimization for Volt-VAr Control" *Energies* 13, no. 22: 5871.
https://doi.org/10.3390/en13225871