# Performance Analysis of Statistical, Machine Learning and Deep Learning Models in Long-Term Forecasting of Solar Power Production

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## Abstract

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## 1. Introduction

- Temporal aspect: A time series model considers the temporal aspect of the data, specifically the order in which the data points occur, while functional forecasting modeling is more suited to modeling physical systems, which do not have a specific temporal aspect.
- Complexity: Time series models can be more complex than functional models because they need to consider patterns and trends in the data, which may not be captured by a simpler functional model.
- Feature Engineering: Time series ML models often rely on feature engineering, which is the process of creating new features from the raw data, to improve the prediction performance. Functional forecasting modeling does not require feature engineering.

- Data Assumptions: Time series models assume that the data are generated by a stationary process and make assumptions about the underlying distribution of the data, while Bayesian learning models use Bayes’ theorem to update the probability of a hypothesis as more evidence becomes available.
- Predictive model: Time series models are specifically designed to predict future values of a time series based on historical data, whereas Bayesian learning models can be used for a wide range of applications, including time series forecasting, but they may not be as well-suited to the task as a dedicated time series forecasting model.
- Modeling techniques: Time series machine learning models use techniques such as ARIMA, LSTM, and Prophet to improve the prediction performance, whereas Bayesian learning models use techniques such as Markov Chain Monte Carlo (MCMC) and variational inference to estimate the parameters of the model.

- This study aims to provide a comprehensive comparison of popular forecasting models for long-term solar power generation forecasting, an area where there has been limited research.
- The study seeks to understand the relationship between the forecasting model’s input variables and forecasting accuracy.
- The study investigates how the performance of different models changes as the prediction horizon changes.
- The study compares the performance of hybrid and ensemble models to that of single models.
- The study assesses the performance of statistical, ML, DL, and ensemble forecasting models when limited input variables and datasets are available.

## 2. Limitation of the Existing Empirical-Based Forecasting Model

#### 2.1. The Sunshine-Based Model

_{sc}and n are referred to as solar constant (1367 W/m

^{2}) and the number of days starting from 1 January, respectively.

#### 2.2. The Cloud-Based Model

## 3. Materials and Methods

#### 3.1. Data Collection

#### 3.2. Data Pre-Processing

#### 3.3. Data Visualization

#### 3.4. Test-Train Split

#### 3.5. Building a Model

#### 3.6. Model Evaluation

_{i}corresponds to the actual value and Y

_{j}is the forecast value.

## 4. Building a Model

#### 4.1. Univariate Models

#### 4.1.1. Statistical Model (ARIMA)

- p: corresponds to the quantity of lag observation in a model.
- d: corresponds to the number of times the observed value is different from its lagged value.
- q: corresponds to the order of lagged prediction error.

- Seasonality versus non-seasonality

_{t}is time series, p, d, and q are non-seasonal ARIMA parameters, and e

_{t}is the white noise.

_{t−1}

- Stationarity of the data

- Determining p, d, and q parameters

#### 4.1.2. The Machine Learning Model (SVR)

- Terminologies
- 1.
- Hyperplane

- 2.
- Kernel

- 3.
- Support Vectors

- 4.
- Boundary lines

- Importing libraries and training dataset

- Selection of Kernel

- Determining the SVR parameters

- Correlation matrix and predictions

#### 4.1.3. Deep Learning Model (LSTM)

- Gates
- 1.
- Forget gate

_{t}) and previous layer output (y

_{t−1}), and to apply the sigmoid activation function (σ) on the available information. The activation function results in a value of forget gate (f

_{t}) into a binary value (either 0 or 1) [43]. The value of f

_{t}being 1 means remember every piece of information, and the value of f

_{t}being 0 means forget every piece of information. Forget gate (f

_{t}) is calculated as

_{f}is the weight matrix between forget gate and input gate where information is either stored or forgotten, and ${\mathsf{\beta}}_{\mathrm{f}}$ is the gate layer’s bias term.

- 2.
- Input gate

_{t−1}) and current input (X

_{t}) are sent to the sigmoid activation function ($\mathsf{\sigma}$) to determine which values will be updated. Then, to govern the network, the same two inputs are fed into the tanh activation function. Finally, the tanh output (C

_{t}) is multiplied by the sigmoid output (I

_{t}) to determine which information is critical for updating the cell state [1]. Input gate (I

_{t}) is evaluated as,

_{t}= σ(X

_{t}× U

_{i}+ H

_{t−1}× W

_{i})

_{t}= I

_{t}× tanh (C

_{t})

_{i}is the weight matrix of the sigmoid operator between the input and output gate and U

_{i}is the weight matrix of the input.

- 3.
- Output gate

_{t−1}) and current input (x

_{t}) are both sent to the sigmoid activation function, much as the input gate. After going through the sigmoid function, the output is multiplied by the output of the hyperbolic function (tanh) to yield the current hidden state (H

_{t}). The final outputs are the current state (C

_{t}) and the present hidden state (H

_{t}). The following equation governs the output gate [44].

_{t}= σ(X

_{t}× U

_{o}+ H

_{t−1}× W

_{o})

_{t}= O

_{t}× tanh (C

_{t})

_{o}is the weight matrix of the output gate and U

_{i}is the weight matrix of the input.

- Execution steps

- Network Architecture

#### 4.2. Multivariate Models

#### 4.2.1. Multivariate-LSTM

#### 4.2.2. Stacked LSTMs

#### 4.2.3. Bi-Directional LSTM

#### 4.2.4. Encoder-Decoder LSTM

#### 4.2.5. Stacked GRU

#### 4.3. Ensemble Model

#### 4.3.1. Random Forrest (RF)

_{1}, N

_{2}, N

_{3}and N

_{4}features.

#### 4.3.2. ARIMA-LSTM

## 5. Models’ Performance Comparison and Evaluation

#### 5.1. Comparison between Univariate Models (ARIMA, SVR, LSTM)

#### 5.2. Comparison between Different Multivariate Models

#### 5.3. Comparison of Different Ensemble Models

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 15.**Comparison between the ARIMA, SVR, and LSTM model for solar power forecasting. Row 1, 2, 3 and 4 represents comparison between models for next 1-day, 3-days, 5-days and 15-days prediction respectively. Column 1, 2, 3 represents ARIMA, SVR and LSTM model respectively (red line = actual, blue line = Predicted).

**Figure 18.**Performance of the Stacked LSTM model for predicting 15 days-ahead solar power generation (red line = actual value, blue line = predicted value).

**Figure 20.**Peformance of the RF model for predicting next 15 days-ahead solar power generation (red line = actual value, blue line = predicted value).

Model | t = 1 Day | t = 3 Days | t = 5 Days | t = 15 Days |
---|---|---|---|---|

ARIMA | 0.42 | 0.8 | 1.36 | 2.24 |

SVR | 0.07 | 0.11 | 0.16 | 0.34 |

u-LSTM | 0.08 | 0.12 | 0.19 | 0.28 |

m-LSTM | 0.06 | 0.10 | 0.12 | 0.18 |

s-LSTM | 0.05 | 0.08 | 0.11 | 0.16 |

GRU | 0.07 | 0.10 | 0.13 | 0.20 |

s-GRU | 0.06 | 0.10 | 0.12 | 0.17 |

ED-LSTM | 1.97 | 2.01 | 2.12 | 2.13 |

b-LSTM | 0.09 | 0.13 | 0.15 | 0.19 |

ARIMA-LSTM | 0.12 | 0.17 | 0.20 | 0.25 |

RF | 0.03 | 0.07 | 0.10 | 0.13 |

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## Share and Cite

**MDPI and ACS Style**

Sedai, A.; Dhakal, R.; Gautam, S.; Dhamala, A.; Bilbao, A.; Wang, Q.; Wigington, A.; Pol, S.
Performance Analysis of Statistical, Machine Learning and Deep Learning Models in Long-Term Forecasting of Solar Power Production. *Forecasting* **2023**, *5*, 256-284.
https://doi.org/10.3390/forecast5010014

**AMA Style**

Sedai A, Dhakal R, Gautam S, Dhamala A, Bilbao A, Wang Q, Wigington A, Pol S.
Performance Analysis of Statistical, Machine Learning and Deep Learning Models in Long-Term Forecasting of Solar Power Production. *Forecasting*. 2023; 5(1):256-284.
https://doi.org/10.3390/forecast5010014

**Chicago/Turabian Style**

Sedai, Ashish, Rabin Dhakal, Shishir Gautam, Anibesh Dhamala, Argenis Bilbao, Qin Wang, Adam Wigington, and Suhas Pol.
2023. "Performance Analysis of Statistical, Machine Learning and Deep Learning Models in Long-Term Forecasting of Solar Power Production" *Forecasting* 5, no. 1: 256-284.
https://doi.org/10.3390/forecast5010014