# A Machine Learning Model Ensemble for Mixed Power Load Forecasting across Multiple Time Horizons

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}coefficient values ranging from 0.99 to 0.79 for prediction horizons ranging from 15 min to 24 h ahead, respectively. The method is compared to several state-of-the-art machine-learning approaches, as well as a different ensemble method, producing highly competitive results in terms of prediction accuracy.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Machine-Learning Methods Short Description

#### 2.1.1. Linear Regression

#### 2.1.2. Sparse Coding

#### 2.1.3. Support Vector Regression

#### 2.1.4. Neural Networks

#### 2.1.5. Random Forests

#### 2.2. Machine-Learning Model Ensemble

_{w}. At first, both ${\widehat{y}}_{1}$ and ${\widehat{y}}_{2}$ models appear ineffective as individual predictors of the $y$ time series. However, after closer inspection, ${\widehat{y}}_{2}$ performs better for the first half of $y$, while ${\widehat{y}}_{1}$ for the second half. By placing greater weight on the model with the best past prediction performance within the horizon h

_{w}, the proposed method is able to toggle towards the best available model for the current circumstance. The result is an overall superior prediction performance.

## 3. Case Study

#### 3.1. Problem and Data Description

#### 3.2. Data Preprocessing and Model Training

## 4. Results

^{2}), RMSE and MAE, considering them as representative and efficient criteria [106]. For comparative reasons, the table also contains the values of the indices for all submodels, as well as their percentage of ranking in the first place. This quantity, labeled as “Rank 1” in Table 2, denotes how many times each submodel scored the 1st rank among all submodels, i.e., achieved the lowest MAE.

## 5. Discussion

^{2}and RMSE. Moreover, this conclusion applies to all prediction time horizons. As the prediction horizon gets longer, the forecasting error increases, which is absolutely reasonable. The only exceptions are the R

^{2}and RMSE values obtained by the MLP model ensemble for 2 h prediction horizon, which slightly exceeds those of the proposed model. However, these differences cannot be considered significant as they are marginal, while on the other hand, the corresponding value of the MAE index clearly favors the proposed method. A result worth mentioning is the improvement of the multi-model performance over the current best sub-model that occurs in most cases while the horizon is getting longer. More specifically, the reduction of MAE that the proposed approach achieves over the best of the individual models ranges from 0.03411 to 0.3156. Such an improvement in performance could be partly explained by the occurrence of uncertainty in the load time series. As the prediction horizon is getting longer, the level of uncertainty is also increased, which is better addressed by the ensemble model than each individual submodel alone.

^{2}show that there is not just one model to prevail over the others in all cases. For the shorter prediction horizons and, more specifically, up to 3 h, LR and SR appear to achieve marginally smaller forecasting errors than their non-linear counterparts. Although the non-linearities are an intrinsic characteristic of mixed load [107], this behavior becomes more apparent as the prediction horizon is getting longer. As a result, models which are based on LR are able to provide robust results for very short-term forecasts. On the other hand, one major advantage of neural networks is their capability of modelling non-linear systems. An important observation is that neural networks appear to perform better for longer prediction horizons, and this can be attributed to the fact that, as the prediction horizon is getting longer, the non-linear properties of the load are becoming more dominant. Therefore, when predictions for longer horizons are required, MLP neural networks take the lead. However, the same does not apply to RBF networks. As stated above, in order for RBF networks to perform well, dense and suitable data are required. Consequently, their performance is reduced for 24-h prediction horizons, where the input information is poorer due to the resampling process. Although the remaining models of the pool, SVR and RF networks, present a moderate predictive capability, they contribute positively to the overall performance of the proposed model. This conclusion confirms our claim of the need to use multiple models in order to enhance the reliability of load predictions.

## 6. Conclusions and Prospects

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

AP | active power |

LR | linear regression |

ML | Machine-learning |

MAE | mean absolute error |

MLP | multi-layer perceptron |

NN | neural network |

RBF | radial basis function |

RF | random forests |

RES | renewable energy sources |

SBL | sparse Bayesian learning |

SR | sparse regression |

SVR | support vector regression |

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**Figure 1.**A typical fully connected multi-layer perceptron (MLP) neural network (NN) structure comprising of $N$ inputs, ${\mathit{x}}_{1},\dots ,{\mathit{x}}_{n}$, 2 hidden layers of $L$ neurons each, and one-dimensional output, $\widehat{\mathit{y}}$.

**Figure 2.**A typical Gaussian-based radial basis function (RBF) neural network (NN) structure comprising of $N$ inputs, ${\mathit{x}}_{1},\dots ,{\mathit{x}}_{n}$, $L$ neurons in the hidden layer, and one-dimensional output, $\widehat{\mathit{y}}$.

**Figure 3.**A typical random forest architecture comprising of $N$ tree learners, ${H}_{1},{H}_{2},\dots ,{H}_{N}$. The prediction set for each learner is averaged to produce the final predictions.

**Figure 4.**Schematic for a two-model version of the proposed method, where y denotes the real load, ${\widehat{\mathit{y}}}_{i}$ the prediction of the i-th model, $\widehat{\mathit{y}}$ the weighted prediction and k the current timestep. The ensemble model recognizing the superiority of ${\widehat{\mathit{y}}}_{1}$ over ${\widehat{\mathit{y}}}_{2}$, within the rolling window adapts its weights accordingly, achieving highly accurate prediction for the next timestep k + 1.

**Figure 5.**Operation of the rolling median threshold outlier detection algorithm. The data points marked as outliers exceed the median value of the time window multiplied by a user-specified threshold factor.

**Figure 6.**Overview of the proposed model ensemble. Its application in mixed load forecasting comprises a series of steps, i.e. raw data acquisition, data preprocessing, collection of input variables, splitting of the dataset in a training and a testing subset, training of submodels, generation of the next AP forecast by each submodel, weighting of the individual predictions, and, lastly, calculation of the next AP final forecast.

**Figure 7.**Scatterplots of actual versus predicted mixed load for (

**a**) 15-min, (

**b**) 1-h, (

**c**) 2-h, (

**d**) 3-h, (

**e**) 6-h, and (

**f**) 24-h ahead prediction. The predicted values residing on the diagonal line are identical to the actual values. Each mark refers to a data point and shows the deviation of its predicted value from its actual value.

**Figure 8.**Pie charts depicting the ranking of the submodels included in the proposed model ensemble for (

**a**) 15-min, (

**b**) 1-h, (

**c**) 2-h, (

**d**) 3-h, (

**e**) 6-h, and (

**f**) 24-h ahead prediction. Each pie chart refers to a ranking position and shows the percentage that each submodel was ranked in that position. Each submodel is represented by a different color and pattern.

**Figure 9.**Results for a randomly selected 12-h window for (

**a**) 15-min, (

**b**) 1-h, (

**c**) 2-h, (

**d**) 3-h, (

**e**) 6-h, and (

**f**) 24-h ahead predictions. Subgraphs labeled 1 depict actual and predicted value results, whereas subgraphs labeled 2 depict the best submodel performance results.

**Table 1.**Description of training variables of the forecasting models for the different prediction horizons examined in the case study. Each row of the table refers to the different groups of input variables, whereas the last row refers to the output variable.

Prediction Horizon | 15 min $\left(\mathit{t}+1\right)$ | 1 h $\left(\mathit{t}+4\right)$ | 2 h $\left(\mathit{t}+4\right)$ | 3 h $\left(\mathit{t}+12\right)$ | 6 h $\left(\mathit{t}+24\right)$ | 24 h $\left(\mathit{t}+96\right)$ |
---|---|---|---|---|---|---|

Current and past AP measures | $\begin{array}{c}{\mathit{p}}^{\left(t-i\right)},\\ i=0,95,671\end{array}$ | $\begin{array}{c}{\mathit{p}}^{\left(t-i\right)},\\ i=0,4,92,668\end{array}$ | $\begin{array}{c}{\mathit{p}}^{\left(t-i\right)},\\ i=0,8,88,664\end{array}$ | $\begin{array}{c}{\mathit{p}}^{\left(t-i\right)},\\ i=0,12,84,660\end{array}$ | $\begin{array}{c}{\mathit{p}}^{\left(t-i\right)},\\ i=0,24,72,648\end{array}$ | $\begin{array}{c}{\mathit{p}}^{\left(t-i\right)},\\ i=0,96,576\end{array}$ |

Average AP measures | $\frac{{\displaystyle \mathbf{\sum}_{n=0}^{3}{\mathit{p}}^{\left(t-n\right)}}}{4}$ | $\frac{{\displaystyle \mathbf{\sum}_{n=0}^{3}{\mathit{p}}^{\left(t-n\right)}}}{4}$ | $\frac{{\displaystyle \mathbf{\sum}_{n=0}^{7}{\mathit{p}}^{\left(t-n\right)}}}{8}$ | $\frac{{\displaystyle \mathbf{\sum}_{n=0}^{11}{\mathit{p}}^{\left(t-n\right)}}}{12}$ | $\frac{{\displaystyle \mathbf{\sum}_{n=0}^{23}{\mathit{p}}^{\left(t-n\right)}}}{24}$ | $\frac{{\displaystyle \mathbf{\sum}_{n=0}^{95}{\mathit{p}}^{\left(t-n\right)}}}{96}$ |

Difference AP measures | $\begin{array}{c}{\mathit{p}}^{\left(t\right)}-{\mathit{p}}^{\left(t-i\right)},\\ i=1\end{array}$ | $\begin{array}{c}{\mathit{p}}^{\left(t\right)}-{\mathit{p}}^{\left(t-i\right)},\\ i=4\end{array}$ | $\begin{array}{c}{\mathit{p}}^{\left(t\right)}-{\mathit{p}}^{\left(t-i\right)},\\ i=8\end{array}$ | $\begin{array}{c}{\mathit{p}}^{\left(t\right)}-{\mathit{p}}^{\left(t-i\right)},\\ i=12\end{array}$ | $\begin{array}{c}{\mathit{p}}^{\left(t\right)}-{\mathit{p}}^{\left(t-i\right)},\\ i=24\end{array}$ | $\begin{array}{c}{\mathit{p}}^{\left(t\right)}-{\mathit{p}}^{\left(t-i\right)},\\ i=96\end{array}$ |

Weather measures | $\begin{array}{c}{w}^{\left(t+i\right)},\\ i=0\end{array}$ | $\begin{array}{c}{w}^{\left(t+i\right)},\\ i=4\end{array}$ | $\begin{array}{c}{w}^{\left(t+i\right)},\\ i=4,8\end{array}$ | $\begin{array}{c}{w}^{\left(t+i\right)},\\ i=4,8,12\end{array}$ | $\begin{array}{c}{w}^{\left(t+i\right)},\\ i=16,20,24\end{array}$ | $\begin{array}{c}{w}^{\left(t+i\right)},\\ i=88,92,96\end{array}$ |

Future AP forecasts (output variable) | ${\widehat{\mathit{p}}}^{(t+1)}$ | ${\widehat{\mathit{p}}}^{(t+4)}$ | ${\widehat{\mathit{p}}}^{(t+8)}$ | ${\widehat{\mathit{p}}}^{(t+12)}$ | ${\widehat{\mathit{p}}}^{(t+24)}$ | ${\widehat{\mathit{p}}}^{(t+96)}$ |

**Table 2.**Performance of the proposed multi-model scheme, the MLP model ensemble of [76], and individual machine-learning models for each prediction horizon. The values of ΜAΕ, RMSE, and R

^{2}, achieved by each model, are presented, as well as the percentage that each submodel achieved the lowest MAE among all submodels.

Method | R^{2} | ΜAΕ | RMSE | Rank1 |
---|---|---|---|---|

15 min | ||||

Proposed | 0.98613 | 0.26120 | 0.4703 | - |

MLP ensemble | 0.9852 | 0.2760 | 0.4869 | - |

MLP | 0.98568 | 0.26936 | 0.4782 | 19.25% |

RBF | 0.98574 | 0.27095 | 0.4773 | 19.94% |

LR | 0.98562 | 0.26700 | 0.4793 | 10.49% |

SVR | 0.98541 | 0.26931 | 0.4829 | 14.23% |

RF | 0.98373 | 0.29531 | 0.5071 | 24.27% |

SR | 0.98561 | 0.26715 | 0.4795 | 11.82% |

1 h | ||||

Proposed | 0.93793 | 0.60224 | 0.9946 | - |

MLP ensemble | 0.9344 | 0.6330 | 1.0240 | - |

MLP | 0.91697 | 0.66794 | 1.1500 | 21.56% |

RBF | 0.93253 | 0.64235 | 1.0374 | 20.77% |

LR | 0.93168 | 0.64174 | 1.0438 | 8.91% |

SVR | 0.93008 | 0.65376 | 1.0562 | 10.70% |

RF | 0.92912 | 0.67311 | 1.0614 | 20.79% |

SR | 0.93045 | 0.64079 | 1.0532 | 17.27% |

2 h | ||||

Proposed | 0.88147 | 0.88279 | 1.3767 | - |

MLP ensemble | 0.8838 | 0.8965 | 1.3721 | - |

MLP | 0.84455 | 0.99479 | 1.5854 | 20.32% |

RBF | 0.87052 | 0.96255 | 1.4472 | 18.32% |

LR | 0.87233 | 0.93356 | 1.4377 | 11.20% |

SVR | 0.86949 | 0.93765 | 1.4537 | 11.38% |

RF | 0.86653 | 0.96596 | 1.4675 | 22.94% |

SR | 0.86953 | 0.93189 | 1.4534 | 15.92% |

3 h | ||||

Proposed | 0.84143 | 1.0599 | 1.5871 | - |

MLP ensemble | 0.8359 | 1.0859 | 1.6192 | - |

MLP | 0.78486 | 1.2504 | 1.8538 | 18.08% |

RBF | 0.82483 | 1.1367 | 1.6727 | 20.14% |

LR | 0.82241 | 1.1270 | 1.6843 | 8.74% |

SVR | 0.81914 | 1.1512 | 1.6997 | 11.82% |

RF | 0.81895 | 1.1391 | 1.7006 | 23.67% |

SR | 0.81893 | 1.1229 | 1.7007 | 17.54% |

6 h | ||||

Proposed | 0.83251 | 1.1144 | 1.6462 | - |

MLP ensemble | 0.8272 | 1.1951 | 1.6888 | - |

MLP | 0.83036 | 1.1758 | 1.6733 | 20.77% |

RBF | 0.80289 | 1.2848 | 1.8037 | 21.31% |

LR | 0.77800 | 1.3308 | 1.9141 | 10.34% |

SVR | 0.75400 | 1.4300 | 2.0150 | 16.42% |

RF | 0.81341 | 1.2119 | 1.7549 | 21.08% |

SR | 0.77373 | 1.3413 | 1.9325 | 10.08% |

24 h | ||||

Proposed | 0.78474 | 1.1835 | 1.8174 | - |

MLP ensemble | 0.7827 | 1.2372 | 1.8468 | - |

MLP | 0.78073 | 1.2313 | 1.8553 | 21.93% |

RBF | 0.73576 | 1.4119 | 2.0367 | 21.83% |

LR | 0.75712 | 1.3188 | 1.9526 | 11.16% |

SVR | 0.73669 | 1.3031 | 2.0331 | 16.82% |

RF | 0.76487 | 1.2694 | 1.9212 | 16.71% |

SR | 0.74761 | 1.3419 | 1.9905 | 11.56% |

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## Share and Cite

**MDPI and ACS Style**

Giamarelos, N.; Papadimitrakis, M.; Stogiannos, M.; Zois, E.N.; Livanos, N.-A.I.; Alexandridis, A.
A Machine Learning Model Ensemble for Mixed Power Load Forecasting across Multiple Time Horizons. *Sensors* **2023**, *23*, 5436.
https://doi.org/10.3390/s23125436

**AMA Style**

Giamarelos N, Papadimitrakis M, Stogiannos M, Zois EN, Livanos N-AI, Alexandridis A.
A Machine Learning Model Ensemble for Mixed Power Load Forecasting across Multiple Time Horizons. *Sensors*. 2023; 23(12):5436.
https://doi.org/10.3390/s23125436

**Chicago/Turabian Style**

Giamarelos, Nikolaos, Myron Papadimitrakis, Marios Stogiannos, Elias N. Zois, Nikolaos-Antonios I. Livanos, and Alex Alexandridis.
2023. "A Machine Learning Model Ensemble for Mixed Power Load Forecasting across Multiple Time Horizons" *Sensors* 23, no. 12: 5436.
https://doi.org/10.3390/s23125436