# Data Analytics for Admittance Matrix Estimation of Poorly Monitored Distribution Grids

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## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Exact Power-Flow Equations

#### 2.2. Extended DC Power Flow Approximation

## 3. Solution Methodology

#### 3.1. Regression Model

#### 3.2. Proposed Solution

Algorithm 1 Construction of the matrices of (8). |

Step 1: Compute the Voltage Drop Vector |

${\left(\right)}^{\left(\right)}={\left(\right)}_{\begin{array}{c}\Delta \left(\right)open="|"\; close="|">{V}_{01}\end{array}}^{\Delta \left(\right)open="|"\; close="|">{V}_{02}}\vdots \\ \Delta \left(\right)open="|"\; close="|">{V}_{0i}$ |

where: |

(1) $i=1,2,\dots ,{N}_{bus}-1$; |

(2) $t=1,2,\dots ,\mathrm{number}\phantom{\rule{4.pt}{0ex}}\mathrm{of}\phantom{\rule{4.pt}{0ex}}\mathrm{periods}$; |

Step 2: Define the R-X Vector |

(1) R = ${\left[\begin{array}{ccccccccccc}{R}_{11}& \cdots & {R}_{1i}& {R}_{22}& \cdots & {R}_{2i}& {R}_{33}& \cdots & {R}_{3i}& \cdots & {R}_{ii}\end{array}\right]}^{\top}$ |

(2) X = ${\left[\begin{array}{ccccccccccc}{X}_{11}& \cdots & {X}_{1i}& {X}_{22}& \cdots & {X}_{2i}& {X}_{33}& \cdots & {X}_{3i}& \cdots & {X}_{ii}\end{array}\right]}^{\top}$ |

where: $i=1,2,\dots ,{N}_{bus}-1$; |

Step 3: Compute the Measurements Matrix |

$\left[\begin{array}{cc}\Delta P/\left|V\right|& \Delta Q/\left|V\right|\end{array}\right]$ = $\left[\begin{array}{cc}{\left[\Delta {P}_{i}/\left|{V}_{i}\right|\right]}_{1}& {\left[\Delta {Q}_{i}/\left|{V}_{i}\right|\right]}_{1}\\ {\left[\Delta {P}_{i}/\left|{V}_{i}\right|\right]}_{2}& {\left[\Delta {Q}_{i}/\left|{V}_{i}\right|\right]}_{2}\\ \vdots & \vdots \\ {\left[\Delta {P}_{i}/\left|{V}_{i}\right|\right]}_{t}& {\left[\Delta {Q}_{i}/\left|{V}_{i}\right|\right]}_{t}\end{array}\right]$ |

where: |

(1) $i=1,2,\dots ,{N}_{bus}-1$; |

(2) Element’s position on sub-matrices $[\Delta {P}_{i}/|{V}_{i}{\left|\right]}_{t}$ and $[\Delta {Q}_{i}/|{V}_{i}{\left|\right]}_{t}$ must be coherent with |

the R-X vector defined in Step 2, maintaining the same linear structure of (7). |

#### 3.3. Feeder Laterals and Non-Metered Buses

## 4. Data and Results

#### 4.1. Data Set Generation

#### 4.2. Results

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Feeder lateral example where i and k correspond to standard network load buses and j corresponds to a node to which feeder laterals are connected to supply consumers $1,2,\dots ,n$.

**Figure 3.**Sensitivity Analysis: (

**a**) Parameter error evolution with total active power losses. (

**b**) Matrix condition number evolution with the number of network buses.

**Figure 4.**Sub-grid topology decomposition. Reference nodes of each sub-grid are circled in red. Sub-grids (

**a**–

**c**) are connected to the grey circles of sub-grid (

**d**) whose reference node is bus 1.

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**MDPI and ACS Style**

Leal, P.C.; Ferreira, D.M.V.P.; Carvalho, P.M.S.
Data Analytics for Admittance Matrix Estimation of Poorly Monitored Distribution Grids. *Energies* **2022**, *15*, 8961.
https://doi.org/10.3390/en15238961

**AMA Style**

Leal PC, Ferreira DMVP, Carvalho PMS.
Data Analytics for Admittance Matrix Estimation of Poorly Monitored Distribution Grids. *Energies*. 2022; 15(23):8961.
https://doi.org/10.3390/en15238961

**Chicago/Turabian Style**

Leal, Pedro C., Diogo M. V. P. Ferreira, and Pedro M. S. Carvalho.
2022. "Data Analytics for Admittance Matrix Estimation of Poorly Monitored Distribution Grids" *Energies* 15, no. 23: 8961.
https://doi.org/10.3390/en15238961