# Analysis of Electric Energy Consumption Profiles Using a Machine Learning Approach: A Paraguayan Case Study

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## Abstract

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## 1. Introduction

- Weekly time series data, where the consumption of each feeder was aggregated on a weekly basis;
- Monthly time series data, where the consumption of each feeder was aggregated on a monthly basis;
- Statistical data set—a set of statistical features was calculated from the raw data;
- Seasonal and daily load curve feature data set—a set of features based on the daily load curve and seasonal consumption variations was computed.

- Analysis and comparison of the performance of different clustering algorithms using real electricity consumption data collected from a Paraguayan electricity provider.
- Study of the suitability of four different data processing strategies.
- Evaluation of the influence of distance metrics and linkage criteria for this particular case study.

## 2. Related Works

## 3. Materials and Methods

#### 3.1. Data

#### 3.2. Data Preprocessing

#### 3.3. Data Sets and Features

- Features from 1 to 5: The relative average power in each time period over the entire time series given by$${P}_{i}^{R}=\frac{{P}_{i}}{\widehat{P}}\phantom{\rule{1.em}{0ex}}\mathrm{for}\phantom{\rule{1.em}{0ex}}i=1,\dots ,5$$
- Feature 6: Mean relative standard deviation over the entire time series given by$$\widehat{\sigma}=\frac{1}{5}\sum _{i=1}^{5}\frac{{\sigma}_{i}}{{P}_{i}}$$
- Feature 7: A seasonal score given by$$\mathcal{S}=\sum _{i=1}^{5}\frac{|{P}_{i}^{W}-{P}_{i}^{S}|}{{P}_{i}}$$
- Feature 8: A weekend vs. weekday difference score given by$$\mathcal{W}=\sum _{i=1}^{5}\frac{|{P}_{i}^{WD}-{P}_{i}^{WE}|}{{P}_{i}}$$

#### 3.4. Distance Measurements

#### 3.5. Clustering Techniques

#### 3.5.1. K-Means

#### 3.5.2. Hierarchical Clustering

#### 3.5.3. K-Spectral Centroid

#### 3.6. Cluster Validity Indices

#### 3.7. Workflow

## 4. Results

- Comparison of the different clustering techniques studied to identify the best models according to the cluster validity index measures.
- Analysis of the consumption data of the best model found.l.

#### 4.1. Model Comparison

#### 4.2. Analysis of Selected Model

## 5. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## List of Symbols

Symbol | Description |

$x,X,y,Y$ | Time series |

${X}^{*}$ | Box–Cox transformation of time series |

${S}_{{x}^{*}}$ | Seasonal component of time series |

${T}_{{x}^{*}}$ | Trend component of time series |

${R}_{{x}^{*}}$ | Remainder component of time series |

$\mathcal{H}$ | Historical records where DOW is the same as the one on the missing record |

$\mu $ | Mean |

$\sigma $ | Standard deviation |

$\mathcal{S}$ | Skewness |

$\mathcal{K}$ | Kurtosis |

$\delta {\mathit{Z}}_{0}$ | Initial separation vector |

$\delta {\mathit{Z}}_{t}$ | Separation vector |

$\lambda $ | Maximum Lyapunov exponent |

$\mathcal{T}$ | Period |

$\mathcal{P}$ | Periodicity |

${\mathcal{P}}_{xx}\left(\omega \right)$ | Power spectral density |

$\mathcal{E}$ | Energy |

P | Mean electricity demand |

$\widehat{P}$ | Mean daily demand over a complete time series |

${P}^{S}$ | Mean summer demand |

${P}^{W}$ | Mean winter demand |

${P}^{WD}$ | Mean weekday demand |

${P}^{WE}$ | Mean weekend demand |

${P}^{R}$ | Relative average power |

$\widehat{\sigma}$ | Mean relative standard deviation |

$\mathcal{S}$ | Seasonal score |

$\mathcal{W}$ | Weekend vs. weekday difference score |

$\mathcal{O}$ | Set of objects |

${n}_{o}$ | Size of a set of objects |

C | Cluster |

c | Centroid of a cluster |

E | Sum of squared distances between objects and their centroid in all clusters |

${d}_{e}$ | Euclidean distance |

${d}_{DTW}$ | Dynamic time warping distance |

M | Cost matrix for DTW |

$wp$ | Optimal warping path |

${m}_{wp}$ | Cost function for DTW |

$\alpha $ | Average distance of a sample with respect to the others in the same cluster |

$\beta $ | Average distance of the same sample with respect to those in the nearest cluster |

⟆ | Silhouette index for a sample |

$SIL$ | Average score of the Silhouette index (Silhouette index) |

$\delta $ | Average distance between the centroid of a considered cluster and the objects that conform it |

$\mathcal{D}$ | Distance between centroids of two clusters |

R | Similarity score between clusters |

$DB$ | Davies–Bouldin index |

B | Between-cluster dispersion matrix |

W | Within-cluster dispersion matrix |

$CH$ | Calinski–Harabasz index |

## Abbreviations

DNOs | Distribution network operators |

DWT | Discrete wavelet transform |

NN | Neural networks |

SVM | Support vector machine |

DTW | Dynamic time warping |

LD | Linear dichroism |

DOW | Day of the week |

MLE | Maximum Lyapunov exponent |

FFT | Fast Fourier transform |

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**Figure 1.**Combo bar chart representing the percentage and total numbers of outliers detected on each feeder.

**Figure 4.**The four data sets that were formed from the hourly electricity consumption records of the feeders.

**Figure 7.**Variation in the Silhouette, Calinski–Harabasz and Davies–Bouldin validation index scores with respect to the number of clusters considered, for the K-means and hierarchical algorithms, with the ward, complete, centroid and average criteria for the latter.

**Figure 8.**Relationship between the clusters determined by the K-means and hierarchical model with the ward criterion for K = 6.

**Figure 9.**Consumption profiles determined in the K-means based model, where (

**a**) belongs to the box plot of the mean daily consumption for each cluster and (

**b**) corresponds to the mean consumption depending on the summer and winter seasons, as well as weekdays and weekends.

Time Period | Interval |
---|---|

1 | 10:00 p.m.–04:00 a.m. |

2 | 05:00 a.m.–09:00 a.m. |

3 | 10:00 a.m.–01:00 p.m. |

4 | 02:00 p.m.–05:00 p.m. |

5 | 06:00 p.m.–09:00 p.m. |

Criterion | Formula | Description |
---|---|---|

Single | $D({C}_{i},{C}_{j})\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\underset{o\in {C}_{i},\phantom{\rule{0.222222em}{0ex}}{o}^{\prime}\in {C}_{j}}{min}d(o,{o}^{\prime})$ | Determined by the distance of the nearest objects between clusters ${C}_{i}$ and ${C}_{j}$. |

Complete | $D({C}_{i},{C}_{j})\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\underset{o\in {C}_{i},\phantom{\rule{0.222222em}{0ex}}{o}^{\prime}\in {C}_{j}}{max}d(o,{o}^{\prime})$ | Determined by the distance of the farthest objects between clusters ${C}_{i}$ and ${C}_{j}$. |

Average | $D({C}_{i},{C}_{j})\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\frac{1}{\left|{C}_{i}\right|}\frac{1}{\left|{C}_{j}\right|}{\displaystyle \sum _{o\in {C}_{i}}}{\displaystyle \sum _{{o}^{\prime}\in {C}_{j}}}d(o,{o}^{\prime})$ | Determined by the average distance between the objects of clusters ${C}_{i}$ and ${C}_{j}$. |

Centroid | $D({C}_{i},{C}_{j})\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}d({c}_{i},{c}_{j})$ | Determined by the distance between the centroids ${c}_{i}$ and ${c}_{j}$ corresponding to clusters ${C}_{i}$ and ${C}_{j}$, respectively. |

Ward | $D({C}_{i},{C}_{j})\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}{\displaystyle \sum _{o\in {C}_{i}\cup {C}_{j}}}d{(o,{c}_{i,j})}^{2}$ | Determined by sum of the squares of the distance between all objects in cluster ${C}_{i}$ and ${C}_{j}$, and ${c}_{i,j}$, centroid of the new cluster merged from ${C}_{i}$ and ${C}_{j}$. |

Data Set | Algorithm | Distance | Linkage Criterion | Conformed Model ID |
---|---|---|---|---|

Weekly time series | K-Means | Euclidean | - | week_k-means_euclid |

DTW | - | week_k-means_dtw | ||

Hierarchical | Euclidean | Single | week_hier_euclid_single | |

Complete | week_hier_euclid_complete | |||

Average | week_hier_euclid_average | |||

Centroid | week_hier_euclid_centroid | |||

Ward | week_hier_euclid_ward | |||

DTW | Single | week_hier_dtw_single | ||

Complete | week_hier_dtw_complete | |||

Average | week_hier_dtw_average | |||

Centroid | week_hier_dtw_centroid | |||

Ward | week_hier_dtw_ward | |||

K-Spectral Centroid | - | - | week_k-sc | |

Monthly time series | K-Means | Euclidean | - | month_k-means_euclid |

DTW | - | month_k-means_dtw | ||

Hierarchical | Euclidean | Single | month_hier_euclid_single | |

Complete | month_hier_euclid_complete | |||

Average | month_hier_euclid_average | |||

Centroid | month_hier_euclid_centroid | |||

Ward | month_hier_euclid_ward | |||

DTW | Single | month_hier_dtw_single | ||

Complete | month_hier_dtw_complete | |||

Average | month_hier_dtw_average | |||

Centroid | month_hier_dtw_centroid | |||

Ward | month_hier_dtw_ward | |||

K-Spectral Centroid | - | - | month_k-sc | |

Statistical Based | K-Means | Euclidean | - | stats_k-means |

Hierarchical | Euclidean | Single | stats_hier_single | |

Complete | stats_hier_complete | |||

Average | stats_hier_average | |||

Centroid | stats_hier_centroid | |||

Ward | stats_hier_ward | |||

Seasonal Based | K-Means | Euclidean | - | seas_k-means |

Hierarchical | Euclidean | Single | seas_hier_single | |

Complete | seas_hier_complete | |||

Average | seas_hier_average | |||

Centroid | seas_hier_centroid | |||

Ward | seas_hier_ward |

Rank | Model ID | Silhouette Score | Calinski–Harabasz Score | Davies–Bouldin Score | Clusters |
---|---|---|---|---|---|

1 | seas_k-means | 0.432 | 69.439 | 0.789 | 4 |

2 | seas_k-means | 0.428 | 78.807 | 0.730 | 6 |

3 | seas_hier_ward | 0.421 | 67.129 | 0.723 | 6 |

4 | seas_hier_complete | 0.415 | 74.284 | 0.735 | 7 |

5 | seas_hier_centroid | 0.403 | 42.509 | 0.562 | 4 |

6 | seas_hier_average | 0.402 | 58.848 | 0.618 | 7 |

7 | seas_hier_centroid | 0.400 | 62.466 | 0.610 | 8 |

8 | seas_hier_average | 0.397 | 55.111 | 0.696 | 5 |

9 | seas_hier_centroid | 0.397 | 56.494 | 0.680 | 6 |

10 | seas_hier_ward | 0.393 | 72.616 | 0.749 | 9 |

11 | seas_hier_complete | 0.391 | 70.890 | 0.868 | 9 |

12 | month_k-sc | 0.250 | 6.236 | 1.791 | 9 |

13 | week_k-means_dtw | 0.239 | 13.915 | 1.601 | 3 |

14 | week_hier_dtw_complete | 0.224 | 12.066 | 1.280 | 4 |

15 | week_hier_euclid_complete | 0.216 | 13.575 | 1.211 | 5 |

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**MDPI and ACS Style**

Morales, F.; García-Torres, M.; Velázquez, G.; Daumas-Ladouce, F.; Gardel-Sotomayor, P.E.; Gómez-Vela, F.; Divina, F.; Vázquez Noguera, J.L.; Sauer Ayala, C.; Pinto-Roa, D.P.;
et al. Analysis of Electric Energy Consumption Profiles Using a Machine Learning Approach: A Paraguayan Case Study. *Electronics* **2022**, *11*, 267.
https://doi.org/10.3390/electronics11020267

**AMA Style**

Morales F, García-Torres M, Velázquez G, Daumas-Ladouce F, Gardel-Sotomayor PE, Gómez-Vela F, Divina F, Vázquez Noguera JL, Sauer Ayala C, Pinto-Roa DP,
et al. Analysis of Electric Energy Consumption Profiles Using a Machine Learning Approach: A Paraguayan Case Study. *Electronics*. 2022; 11(2):267.
https://doi.org/10.3390/electronics11020267

**Chicago/Turabian Style**

Morales, Félix, Miguel García-Torres, Gustavo Velázquez, Federico Daumas-Ladouce, Pedro E. Gardel-Sotomayor, Francisco Gómez-Vela, Federico Divina, José Luis Vázquez Noguera, Carlos Sauer Ayala, Diego P. Pinto-Roa,
and et al. 2022. "Analysis of Electric Energy Consumption Profiles Using a Machine Learning Approach: A Paraguayan Case Study" *Electronics* 11, no. 2: 267.
https://doi.org/10.3390/electronics11020267