# Outage Estimation in Electric Power Distribution Systems Using a Neural Network Ensemble

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## Abstract

**:**

## 1. Introduction

## 2. Outage and Weather Data

## 3. Deep Neural Network Ensemble Approach

#### 3.1. Layout of the Deep Neural Network Ensemble

#### 3.2. Supervised Learning in Sub-Networks

#### 3.3. Partitioning Using Unsupervised Learning

## 4. Theoretical Framework

#### 4.1. Gaussian Distribution Assumption

**,**is obtained using the expressions provided earlier in (14) and (15). Let us define an assignment mapping $\U0001d4c0:\mathcal{X}\to \mathcal{K}$. For the sake of conciseness, given any sample ${x}_{i}\in \mathcal{N}$, let $\U0001d4c0\left({x}_{i}\right)$ be written as ${\U0001d4c0}_{i}$. The joint probability of the dataset $\mathcal{N}$ is given by,

#### 4.2. Markov Random Field Viewpoint

## 5. Results

#### 5.1. Performance Metrics

- (i)
- Mean Absolute Error:

- (ii)
- Mean Squared Error:

- (iii)
- Slope: This is the slope of the best linear fit that passes through the origin.

- (iv)
- Coefficient of Correlation:

#### 5.2. Weather-Related Outage Prediction

#### 5.3. Animal-Related Outage Prediction

**,**and the outages ${y}_{i-1}$ of the immediately preceding week. The DNNE sub-networks were of size $3\times 5\times 1$ and the parameters were kept at $\eta =0.2,\gamma =0.95,\xi =0$, and ${T}_{\infty}=10$. The initial weights ${w}^{\left(k\right)}$ were very small random quantities. A cardinality $\left|\mathcal{K}\right|=4$ was used. A maximum of 150 epochs was allowed.

## 6. Conclusions

## 7. Nomenclature

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Overall architecture of the proposed deep neural network ensemble (DNNE). Internal weights appear in blue fonts.

**Figure 3.**Evolution of $\mathrm{MSE}$ (right) and ${\beta}_{i}^{\left(k\right)}$ (left) with $T$ for city D. Each trajectory (right) corresponds to a sample in dataset $\mathcal{N}$, that appear in distinct colors for clarity.

**Figure 4.**Scatter plots of lightning ${L}_{i}$ vs. wind speed ${W}_{i}$ for each sample $i$ in $\mathcal{N}$, colored according to the Voronoi partition in $\mathcal{K}$ that contains $i$. Note that $\mathrm{log}\left(1+{L}_{i}\right)$ has been used in the x-axis. Each plot corresponds to a city (

**A**–

**D**).

**Figure 5.**Regression plots for training data. The points correspond the observed outages ${y}_{i}$ (x-axis) and estimated outages ${\tilde{y}}_{i}\text{}$(y-axis) for samples in the training dataset $\mathcal{N}$ using four different methods. The regression lines are also shown. A different color is used for each model (see legend on the top-left figure). Each plot corresponds to a city (

**A**–

**D**).

**Figure 6.**Regression plots for test data. The points correspond to the observed outages ${y}_{i}$ (x-axis) and estimated outages ${\tilde{y}}_{i}\text{}$(y-axis) for samples in the test dataset $\mathcal{N}$ using four different methods. The regression lines are also shown. A different color is used for each model (see legend on the top-left figure). Each plot corresponds to a city (

**A**–

**D**).

City A (Weather) | ||||||||
---|---|---|---|---|---|---|---|---|

Dataset | Training | Test | ||||||

Metric | MAE | MSE | S | R | MAE | MSE | S | R |

Neural Network | 0.6009 | 2.4879 | 0.4555 | 0.6761 | 0.6433 | 2.3370 | 0.1254 | 0.4335 |

ADAboost+ | 0.3691 | 1.8251 | 0.5694 | 0.7860 | 0.5578 | 2.0558 | 0.2083 | 0.6079 |

DNNE | 0.2611 | 0.7044 | 0.8406 | 0.9225 | 0.2802 | 0.6679 | 0.6400 | 0.8715 |

DNNE-H | 0.2078 | 0.6390 | 0.8504 | 0.9294 | 0.2516 | 0.7266 | 0.6046 | 0.8674 |

City B (Weather) | ||||||||

Dataset | Training | Test | ||||||

Metric | MAE | MSE | S | R | MAE | MSE | S | R |

Neural Network | 0.6973 | 4.3621 | 0.0871 | 0.2958 | 0.8778 | 5.0176 | 0.1012 | 0.4130 |

ADAboost+ | 0.3095 | 2.6210 | 0.3662 | 0.6947 | 0.4340 | 3.0395 | 0.3825 | 0.7173 |

DNNE | 0.1815 | 0.2603 | 0.9414 | 0.9724 | 0.3856 | 2.9584 | 0.3743 | 0.7382 |

DNNE-H | 0.1722 | 0.3762 | 0.9217 | 0.9610 | 0.4516 | 2.3559 | 0.5390 | 0.7867 |

City C (Weather) | ||||||||

Dataset | Training | Test | ||||||

Metric | MAE | MSE | S | R | MAE | MSE | S | R |

Neural Network | 1.3913 | 12.9613 | 0.3016 | 0.5494 | 2.4418 | 37.1506 | 0.1909 | 0.4231 |

ADAboost+ | 0.7070 | 8.8922 | 0.4409 | 0.7448 | 1.4621 | 22.9910 | 0.3530 | 0.7928 |

DNNE | 0.4493 | 3.0957 | 0.8276 | 0.9138 | 1.0227 | 9.6359 | 0.8172 | 0.8871 |

DNNE-H | 0.4475 | 2.8011 | 0.8402 | 0.9217 | 0.9915 | 9.4382 | 0.8080 | 0.8882 |

City D (Weather) | ||||||||

Dataset | Training | Test | ||||||

Metric | MAE | MSE | S | R | MAE | MSE | S | R |

Neural Network | 2.8051 | 35.9343 | 0.3312 | 0.5756 | 3.2873 | 76.7436 | 0.2314 | 0.4003 |

ADAboost+ | 1.4615 | 18.4254 | 0.5755 | 0.8263 | 2.4409 | 48.0201 | 0.3271 | 0.7963 |

DNNE | 0.9677 | 8.1498 | 0.8289 | 0.9225 | 1.4358 | 18.6595 | 0.7524 | 0.8883 |

DNNE-H | 0.0944 | 7.6422 | 0.8556 | 0.9262 | 1.1823 | 9.2845 | 0.8330 | 0.9473 |

City A (Animal) | ||||||||
---|---|---|---|---|---|---|---|---|

Dataset | Training | Test | ||||||

Metric | MAE | MSE | S | R | MAE | MSE | S | R |

Neural Network | 2.1843 | 9.8982 | 0.3021 | 0.5593 | 5.1144 | 1.8683 | 0.2867 | 0.3885 |

ADAboost+ | 0.8592 | 1.6513 | 0.8219 | 0.9518 | 0.6566 | 1.3111 | 0.5847 | 0.8325 |

DNNE | 0.8246 | 1.5039 | 0.8576 | 0.9467 | 0.5576 | 0.9390 | 0.6624 | 0.8859 |

City B (Animal) | ||||||||

Dataset | Training | Test | ||||||

Metric | MAE | MSE | S | R | MAE | MSE | S | R |

Neural Network | 3.1250 | 15.8521 | 0.4680 | 0.6857 | 3.7601 | 19.7885 | 0.4075 | 0.4263 |

ADAboost+ | 1.3293 | 3.5305 | 0.8170 | 0.9407 | 0.9759 | 2.4946 | 0.6102 | 0.8594 |

DNNE | 1.2829 | 3.1367 | 0.8423 | 0.9468 | 0.9125 | 2.2260 | 0.6530 | 0.8739 |

City C (Animal) | ||||||||

Dataset | Training | Test | ||||||

Metric | MAE | MSE | S | R | MAE | MSE | S | R |

Neural Network | 6.8084 | 84.4520 | 0.5896 | 0.7729 | 5.7810 | 54.3965 | 0.7059 | 0.7552 |

ADAboost+ | 2.8002 | 22.0193 | 0.8337 | 0.9485 | 2.6823 | 20.9659 | 0.7065 | 0.9589 |

DNNE | 2.2725 | 11.0242 | 0.9344 | 0.9736 | 2.1174 | 11.4240 | 0.8028 | 0.9425 |

City D (Animal) | ||||||||

Dataset | Training | Test | ||||||

Metric | MAE | MSE | S | R | MAE | MSE | S | R |

Neural Network | 8.6310 | 154.8318 | 0.5912 | 0.7800 | 6.2469 | 68.8767 | 0.9819 | 0.7399 |

ADAboost+ | 3.2699 | 33.8051 | 0.8637 | 0.9569 | 2.2847 | 12.5119 | 0.7104 | 0.9082 |

DNNE | 2.8531 | 19.7962 | 0.9320 | 0.9741 | 1.8747 | 7.5639 | 0.8686 | 0.9333 |

Symbol | Meaning |
---|---|

$\mathcal{K}$ | Set of partitions (or sub-networks) |

$D$ | Dimensionality of input space |

$H$ | Number of hidden neurons in each sub-network |

$\mathcal{X}$ | Input space |

$k$ | Index of a partition or sub-network |

${\mathcal{X}}^{\left(k\right)}$ | Partition with index $k$ |

${w}^{\left(k\right)}$ | Vector of all weights (and biases) of sub-network $k$ |

$\mathcal{N}$ | Dataset of samples (either training or test) |

${\mathcal{N}}^{\left(k\right)}$ | Subset of samples included in ${\mathcal{X}}^{\left(k\right)}$ |

${c}^{\left(k\right)}$ | Estimated centroid, i.e., mean of samples in ${\mathcal{N}}^{\left(k\right)}$ |

$i$ | Index of a sample |

${x}_{i}$ | $D\times 1$ input vector in any sample $i$ |

${y}_{i}$ | Real outage frequency in any sample $i$ |

${\tilde{y}}_{i}$ | Estimated outage frequency in any sample $i$ |

${\tilde{y}}_{i}^{\left(k\right)}$ | Output of sub-network $k$ for any sample $i$ |

$\eta $ | Supervised learning rate |

${\beta}_{i}^{\left(k\right)}$ | Weight applied to any sample $i$ in sub-network $k$ |

$\xi $ | (Optional) regularization parameter |

$R(\xb7)$ | (Optional) regularization function |

${f}_{NN}(\xb7)$ | Sub-network output as a function of input and weights |

$T$ | Boltzmann parameter |

${T}_{\infty}$ | Initial value of $T$ |

${\delta}_{i}^{\left(k\right)}$ | Error in the output of sub-network $k$ for sample $i$(normalized) |

$Z$ | Partition function |

$\gamma $ | Cooling rate |

$\Omega $ | Set of all weights and centroids in the deep neural network ensemble |

$MAE$ | Mean absolute error |

$MSE$ | Mean squared error |

$S$ | Slope of regression line |

$R$ | Correlation coefficient |

${L}_{i}$ | Total daily lightning strikes corresponding to sample $i$ |

${W}_{i}$ | Maximum daily wind gust speed corresponding to sample $i$ |

${F}_{i}$ | Level of fair-weather days in the week corresponding to sample $i$ |

${M}_{i}$ | Month type of the week corresponding to sample $i$ |

${y}_{i-1}$ | Level of outages in the previous week |

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## Share and Cite

**MDPI and ACS Style**

Das, S.; Kankanala, P.; Pahwa, A.
Outage Estimation in Electric Power Distribution Systems Using a Neural Network Ensemble. *Energies* **2021**, *14*, 4797.
https://doi.org/10.3390/en14164797

**AMA Style**

Das S, Kankanala P, Pahwa A.
Outage Estimation in Electric Power Distribution Systems Using a Neural Network Ensemble. *Energies*. 2021; 14(16):4797.
https://doi.org/10.3390/en14164797

**Chicago/Turabian Style**

Das, Sanjoy, Padmavathy Kankanala, and Anil Pahwa.
2021. "Outage Estimation in Electric Power Distribution Systems Using a Neural Network Ensemble" *Energies* 14, no. 16: 4797.
https://doi.org/10.3390/en14164797