A Very Short-Term Photovoltaic Power Forecasting Model Using Linear Discriminant Analysis Method and Deep Learning Based on Multivariate Weather Datasets ^†

Abstract

Photovoltaic (PV)-system-generated solar energy has inconsistent and variable properties, which makes controlling electric power distribution and preserving grid stability extremely difficult. A photovoltaic (PV) system’s performance is profoundly affected by the amount of sunlight that reaches the solar cell, the season of the year, the ambient temperature, and the humidity of the air. Every renewable energy technology, sadly, has its problems. As a result, the system is unable to function at its highest or best level. To combat the unstable and intermittent performance of solar power output, it is essential to achieve a precise PV system output power. This work introduces a new approach to enhancing accuracy and extending the time range of very short-term solar energy forecasting (15 min step ahead) by using multivariate time series inputs in different seasons. First, Linear Discriminant Analysis (LDA) is used to select the relevant factors from the mixed meteorological input data. Secondly, two very short-term deep learning prediction models, CNN and LSTM, are used to predict PV power for a shuffled and reduced database of weather inputs. Finally, the predicted outputs from the two models are combined using a classification strategy. The proposed method is applied to one year of real data collected from a solar power plant located in southern Algeria to demonstrate that this technique can improve the forecasting accuracy compared to other techniques, as determined through the use of statistical analysis involving normalized root mean square error (NRMSE), mean absolute error (MAE), mean bias error (MBE), and determination coefficient. (R²).

1. Introduction

Nowadays, solar energy has become a crucial element in electricity production systems worldwide. For the proper functioning and cost-effective integration of solar power, accurate photovoltaic power forecasting is essential [1]. Several studies have been conducted, and different methods have been suggested in the literature. There are three categories of solar energy forecasting methods: physical methods, statistical methods, and machine learning methods [2,3]. In [4], N. Zemouri et al. proposed a novel approach using multi-model statistical ensembles to predict short-term GHI using the classification strategy. In [2], Kejun Wang et al. proposed a hybrid deep learning model LSTM-CNN applied to photovoltaic power prediction. In [5], Elham M. Al-Ali and al presented a hybrid CNN-LSTM-Transformer Model to forecast solar energy production. The rest of this paper is structured as follows: the next section provides a brief description of the proposed approach. To evaluate the proposed approach, we choose the statistical criteria, which are presented in Section 3; the practical experimental results and comparisons for the evaluation of the proposed technique are described in Section 3. Finally, the concluding remarks of this study can be found in Section 4.

2. Methodology

Figure 1 illustrates the operational flow of the proposed approach for executing very short-term forecasting of PV output power using different stage techniques. The prediction system is trained using real weather input data (solar radiation, temperature, relative humidity, pressure and wind speed) to forecast solar PV generation. After the actual methodological data have been subjected to dimensionality reduction to acquire an ideal collection of past variables, this collection serves as an input component for every predictive model trained using deep learning methods. The fusion of these two models enables the ultimate prediction of photovoltaic solar energy output. All these steps are detailed in the following subsection.

2.1. Meteorological Weather Input Parameters

In the present work, different meteorological weather input parameters are used to forecast solar power, which are as follows: temperature (T), relative humidity (RH), wind speed (WS) and solar radiation (SR) because these are uncomplicated to measure. The process of choosing input variables is a significant aspect of time series prediction. Selecting the appropriate input variables is an important step for modeling time series effectively. When there is a set of independent variables, it is necessary to perform variable selection to exclude variables that have minimal impact on the forecast and ensure important variables are not overlooked. The selection of variables is influenced by factors such as data availability, quality, and correlation with the target variable [6]. In order to determine the connection between the meteorological variables collected and the output of a solar power plant, a statistical analysis was conducted. Pearson correlation was utilized as a metric to assess the linear correlation between the two variables, which will be explained in the following subsection.

2.2. Correlation between Weather Inputs Data and Power

The process of choosing input variables is a significant aspect of time series prediction. Selecting the appropriate input variables is critical for modeling time series effectively. When there is a set of independent variables, it is necessary to perform variable selection to exclude variables that have minimal impact on the forecast and ensure important variables are not overlooked. The selection of variables is influenced by factors such as data availability, quality, and correlation with the target variable [7]. In order to determine the connection between the meteorological variables collected and the output of a solar power plant, a statistical analysis was conducted. Pearson correlation is utilized as a metric to assess the linear correlation between the two variables. It calculates the strength and direction of the relationship between the variables, ranging from −1 to +1. A value of +1 indicates a perfect positive linear relationship, 0 indicates no linear relationship, and −1 indicates a perfect negative linear relationship. Equation (1) represents the mathematical expression of the Pearson correlation coefficient.

$$R=\frac{\mathit{n}\sum \mathit{X}\mathit{Y}-\sum \mathit{X}\sum \mathit{Y}}{\sqrt{\left(\mathit{n}\sum {\mathit{X}}^2-{\left(\sum \mathit{X}\right)}^2\right).\left(\mathit{n}\sum {\mathit{Y}}^2-{\left(\sum \mathit{Y}\right)}^2\right)}}$$

(1)

Table 1. Performance comparison of solar power vs. meteorological data in terms of correlation coefficient (R) Bolded values are the best in the comparison.

Relationship	SP(kw)	T(C°)	SR(w/m2)	RH(%)	WS(m/s)	Pres(HPa)
SP(kw)	1.0000	0.9329	0.8520	−0.5015	0.1032	0.1711
T(C°)	0.9329	1.0000	0.8430	−0.6964	0.1220	0.0562
SR(w/m2)	0.852	0.8430	1.0000	−0.4595	0.0162	0.0702
RH(%)	−0.5015	−0.6964	−0.4595	1.0000	−0.2519	0.3060
WS(m/s)	0.1032	0.1220	0.01620	−0.2519	1.0000	−0.3088
Pres(HPa)	0.1711	0.0562	0.07020	0.3060	−0.3088	1.0000

shows the assessed effectiveness of the correlations between SP (solar power) and the different meteorological components, emphasizing the best results in bold. The impact of these relationships was evaluated using the correlation coefficient (R), which shows the degree to which the two variables are related.

2.3. Deep Learning Application Method to Forecast Power

The following section contains a brief introduction to the standard LSTM and 1D-CNN models that are used to forecast the power output.

2.3.1. LSTM-RNN Predictor Model

LSTM is a new kind of RNN, and it is substantially distinct compared to standard ANNs, which are often used to represent sequential data, such as time series or natural language. LSTMs, however, are able to recall long-term dependencies because they were specifically created to address the vanishing gradient problem [8]. LSTM comprises four basic interconnected layers (three gates and cell state): cell state is used to store information; the forget gate controls how much information is forgotten by the cell memory; the input gate is used to determine which inputs are utilized to update the memory of the cell; finally, the output gate defines which components of the cell memory need to be updated to change the LSTM cell’s hidden state. Figure 2a illustrates the LSTM’s fundamental unit architecture. The output cell ${\mathbf{c}}_{\mathbf{t}}$ and upward output $h_t$ are determined using the following equations:

$${\tilde{\mathit{c}}}_{\mathit{t}}=\mathbf{tanh}({\mathit{W}}_{\mathit{c}}{\mathit{x}}_{\mathit{t}}+{\mathit{U}}_{\mathit{c}}{\mathit{h}}_{\mathit{t}-\mathit{t}}+{\mathit{b}}_{\mathit{c}})$$

(2)

$${\mathit{c}}_{\mathit{t}}={\mathit{f}}_{\mathit{t}}⨂{\mathit{c}}_{\mathit{t}-1}+{\mathit{i}}_{\mathit{t}}⨂{\tilde{\mathit{c}}}_{\mathit{t}}$$

(3)

$${\mathit{h}}_{\mathit{t}}={\mathit{o}}_{\mathit{t}}⨂\mathbf{tanh}\left({\mathit{c}}_{\mathit{t}}\right)$$

(4)

where $\sigma \mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}$ represents sigmoid and hyperbolic tangent activation functions, respectively, and $⨂$ denotes the operation of matrix multiplication. In [8], a general introduction to the LSTM model is presented. The second deep learning prediction method used in our work is CNN, as mentioned in the following subsection.

2.3.2. CNN Predictor for 1D Signal

In 1989, deep ANNs, types of convolutional neural networks (CNNs), were first used to recognize zip codes [9]. At present, CNN methods have been created specifically to handle one-dimensional data, also called 1D-CNN. It has beeen very recently proposed for use in a number of applications and performed well, particularly for 1D signals, as in our case. The components that comprise a 1D-CNN model are as follows: convolution layers, pooling layers, and fully connected layers or dense layers, as shown in Figure 2b; the tasks of each layer are presented in [10,11,12], and the details of the mathematical equations are outlined in [13].

Figure 2. (a) LSTM unit architecture [14]. (b) 1D-CNNarchitecture.

Figure 2. (a) LSTM unit architecture [14]. (b) 1D-CNNarchitecture.

2.4. Supervised Dimensionality Reduction Based on the LDA Technique

Linear Discriminate Analysis (LDA) is one famous supervised learning technique used in statistics and another field to find a linear combination of features that characterizes or separates two or more classes of data [15]. It can be achieved through three principal steps: (1) calculate the separability between different classes called between-class variance (Sb), (2) calculate the distance between the mean and sample of each class, which is called the within-class variance (SW), and (3) construct the lower dimensional space, which maximizes between-class variance (Sb) and minimizes the within-class variance (SW), as mentioned in Figure 3.

2.5. Classification Aggregation Strategy

A combination strategy combines various techniques in order to enhance the performance of global forecasting. It presents a conscious decision to increase forecasting accuracy by using a practical approach; in the present work, two prediction models were combined using a classification strategy that is based on the hypothesis that several high-performance predictors may be associated with various regions of the input time series variable space [4] the best predictor is the one that minimizes error throughout the whole variable space wee used the RMSE error to suggested our proposed strategy.

3. Results and Discussion

The data that were used in this study are real data collected within 15 min during the period from 1 January 2019, to 31 December 2019, from a photovoltaic solar power plant with a capacity of 20 megawatts located in a city situated in East Algeria in North Africa. The meteorological data included are taken as the proposed model inputs. The collected dataset was divided into three subsets; this division was realized with the aim of obtaining acceptable and accurate forecasting data. The first subset includes a train set, containing 70%, and the second is regarded as a test set, containing 30% of the total data. All of these data are used to forecast solar power and to evaluate our approach proposed in this work; different criteria were used as follows: mean square error (NRMSE), mean absolute percentage error (MAPE), mean bias error (MBE), and coefficient of determination (R²) their equation is mentioned in [4]. The characters of the two methods of prediction used in this work are mentioned in

Table 2. Setting parameters of the prediction models.

Model	Parameters	Value
LSTM	Optimizer, mini batch size, number of hidden units, learning rate	ADAM, 80, 107, 0.0005
CNN	Convolutional layer, max-pooling layer, dropout layer	3, 5, 0.005
	initial learning rate, mini batch size	0.002, 2

.

The prediction accuracy of the proposed model is compared in

Table 3. Prediction accuracy of different proposed methods. The best results are highlighted in bold.

Methods	CNN	CNN-LDA	LSTM	LSTM-LDA	Class
NRMSE	0.1697	0.1480	0.0019	0.0016	0.0014
MAPE	43.6935	35.6423	17.7424	16.055	16.012
MBE	1.5077	1.0137	0.0580	0.0225	0.0183
R2	0.8701	0.8828	0.9234	0.9319	0.9337

, which shows that the used strategy based on classification aggregating performs significantly better than the other individual mode with LDA and without LDA in terms of R², with 0.9337, which indicates that the classification-based strategy outperforms the other ones by a large margin. Performance samples from the prediction of solar power with LSTM and CNN after using LDA reduction are compared with their combinations, as shown in Figure 4.

4. Conclusions

An aggregating deep learning technique with a dimensionality reduction framework called the Linear Discriminate Analysis (LDA) method is used to forecast solar power. Its combination based on a classification strategy is proposed in this paper. The suggested approach demonstrates experimental results in terms of forecasting tasks involving solar power, indicating that the combination strategy yields the best results compared with individual models. The good performance of the proposed models demonstrated here should make them more attractive for a large variety of forecasting problems.

Future Research

Under feasible forecasting conditions, the proposed approach has achieved good forecasting accuracy, and it can be further improved. As research work that can be carried out in the future, we can aim to propose other dimensionality reduction methods, such as PCA or MRMR, to add a block of decomposition to the signal of the weather time series using, for example, EMD, CEEMD or VMD, with the objective of improving the precision of the forecasts of the model of the proposed forecast.