Hourly photovoltaic power prediction based on signal decomposition and deep learning

Accurate photovoltaic (PV) power prediction is important for the utilization of solar energy resources. However, PV power is non-stationary due to the variable influence of meteorological factors, which poses a challenge for accurate forecasting. In this paper, a hybrid method based on signal decomposition and a deep learning model is proposed. The hybrid model integrates complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) and the Informer model. The CEEMDAN algorithm is used to separate different modes from the photovoltaic power sequence, enhancing its predictability. The deep learning model, the Informer, is employed to capture the complex relationship between photovoltaic power data and its historical data as well as external meteorological factors, ultimately enabling multi-step forecasting of photovoltaic power data. In hourly PV power forecasting experiments using a public dataset, the model exhibits significant performance improvements when compared to benchmark models such as LSTM, GRU, and Transformer. Specifically, the RMSE reduces by 6.07%-34.74% and the MAE reduces by 7.07%-37.5%. The results demonstrate that the hybrid model exhibits accurate predictive performance in the task of hourly photovoltaic power forecasting.


Introduction
The extensive use of fossil fuels unavoidably leads to issues such as greenhouse gas emissions and environmental pollution.Consequently, countries worldwide are actively shifting towards renewable energy as a more sustainable and environmentally friendly energy solution.PV power is poised to play a crucial role in the energy transition due to its safety, abundance, and versatility as a renewable energy source.According to the International Energy Agency's World Energy Outlook (WEO) 2022 publication, solar PV has emerged as the third most prominent renewable energy technology, following hydropower and wind energy [1].However, due to the direct or indirect influence of meteorological factors such as irradiance, temperature, wind speed, and others, PV power generation has a high degree of uncertainty.This uncertainty can have a series of impacts on power system control, grid integration, and energy planning.Therefore, the development of more accurate photovoltaic power forecasting models has significant importance.
The primary methods for photovoltaic power prediction include physical models, statistical models, machine learning models, and hybrid models.Physical methods utilize the meteorological forecast results from numerical weather prediction (NWP) systems and calculate photovoltaic power values directly using mathematical equations.However, due to significant computational resource requirements and limitations such as the relatively low output frequency of mainstream NWP systems, physical methods have substantial constraints.Statistical models predict photovoltaic power based on the historical trends of photovoltaic power.Commonly used statistical models for photovoltaic power prediction include Autoregressive Moving Average [2] and Extreme Learning Machines.However, due to the simple structure, statistical models struggle to fit the photovoltaic power fluctuations caused by short-term external environmental changes.In recent years, data-driven machine learning methods have been widely applied in photovoltaic power prediction, including Support Vector Machines, Random Forests, and Neural Networks [3].However, the data generated from the operation of photovoltaic power plants is often characterized by large volumes, multiple sources, and heterogeneity.Shallow machine learning models struggle to extract useful information from big data.Deep learning models, as an important branch of machine learning, addressing these limitations by increasing the depth of network layers to extract deeper abstract features from raw data, have been widely applied in photovoltaic power prediction.Feng and Zhang [4] used CNN as the prediction model, with satellite cloud imagery as input, to achieve ultra-short-term prediction of photovoltaic power.As a variant of the Transformer model, the Informer [5] model introduces a ProbSparse self-attention mechanism, effectively reducing the model's computational complexity.Additionally, the proposed multi-step prediction method mitigates the issue of error accumulation during multi-step forecasting.The Informer model has demonstrated excellent performance in various application domains.
Although the performance of individual models continues to improve, performance limitations are inevitably present.Signal decomposition is considered to be an effective method for reducing the nonstationarity of time series data.The mainstream data decomposition algorithms include empirical mode decomposition (EMD) and ensemble empirical mode decomposition (EEMD).For example, Li et al. [6] used EMD and Extreme Learning Machines to predict PV power.However, the independent modes obtained through the EMD algorithm exhibit mode mixing phenomena, and the EEMD algorithm, by introducing Gaussian white noise, enhances decomposition stability and alleviates mode mixing issues encountered in EMD.However, it comes with a higher computational cost, and the introduced white noise can be challenging to ignore.CEEMDAN, as an improvement over the EEMD algorithm, effectively addresses the issue of white noise residue.Therefore, a hybrid model CEEMDAN-Informer is proposed, which consists of two parts, a data decomposition module and a time series prediction module.The special information about the hybrid model is shown in the next.

Materials and methods
The hybrid forecasting model mainly consists of two components: (1) Data decomposition module includes a time series decomposition module for the photovoltaic power series and other data processing operations for meteorological vary and data; (2) Time series prediction module is used to establish the relationship between photovoltaic power data and historical meteorological factors and forecast photovoltaic power.The overall framework of the proposed model is illustrated in Figure 1.

CEEMDAN algorithm
CEEMDAN is a variant of the EEMD algorithm that enhances IMF spectral separation by introducing specific noise during the decomposition steps and calculating residues to obtain each mode.The following are the detailed steps of this algorithm.
We set () as the original signal,   () as the white Gaussian noise, and   () as the  − ℎ mode of original signal.  (•) is defined to generate  − ℎ mode that decomposed by EMD, and   () ̂ is defined as the  − ℎ mode generated by CEEMDAN.
Firstly, adding Gaussian white noise to the signal () results in new signal   () (  = 1, ⋯ , ).Assigning    as the  − ℎ mode of   () , the first IMF component of the CEEMDAN decomposition is obtained by averaging the N IMFs produced by N operations of EMD, which is showed in Equation (1).After that, we add paired positive and negative Gaussian white noise to  1 () to obtain the new signal  1 () +  1 (  ()).Then  2 () ̂ is calculated by Equation (2).
We repeat Steps 3 and 4 for the next  until the signal has no more than two extrema and cannot be further decomposed.The last residual () can be calculated by Equation (5).

Informer model
The informer adopts the same encoder-decoder architecture as the transformer and introduces three improvement methods to address issues associated with the Transformer model in time series forecasting, such as high time and space complexity and error accumulation during multi-step predictions [5].
Firstly, the Informer model proposes the ProbSparse Self-Attention Mechanism which reduces the computational complexity from ( 2 ) to ().The self-attention coefficients of the transformer are shown in Equation ( 6) [5].The authors of Informer found that the classic self-attention mechanism exhibits sparsity, a phenomenon of long-tail distribution in the self-attention feature maps.Therefore, by deleting useless queries during the attention calculation process, the amount of calculation can be reduced.
Besides, the Informer assumes that the distribution of more important queries should have a greater difference from the uniform distribution.Therefore, calculating the Kullback-Leibler (KL) divergence between the probability distribution of each query and the uniform distribution can be used to measure the importance of the query.Finally, by scaling and simplifying the calculation equation, Equation ( 7) [5] is obtained.An "importance" score for each query is obtained through Equation (7).
Then by selecting top  =     pairs to calculate  ̅̅̅ (  , ) , the calculation complexity of attention matrix is reduced to ().Finally, the ProbSparse Self-Attention is defined as Equation (8) [5].
where  ̅ is a sparse matrix.Secondly, the block input and output shapes of the traditional transformer are the same, and the complexity brought by J Blocks is ( 2 ) * .As a result, the input of the model cannot become too long, which limits the scalability of the model.The self-attention distillation operation is proposed in Informer.Specifically, a down-sampling operation is performed between each layer to highlight the main attention by halving the shape of the sequence, allowing the model to accept longer sequence inputs and reducing memory and time consumption.The distilling procedure forwards from the  − ℎ layer to the ( + 1) − ℎ layer as Equation (9) [5].
Lastly, the Informer designed a relatively simple decoder that can output multi-step prediction values at once.Furthermore, the generative prediction method reduces the risk of error accumulation and enhances prediction accuracy.

Dataset description
The data for the study was collected from the DKA Solar Centre (DKASC) [7], located in the Alice Springs Desert Knowledge Precinct in Australia (-23.7624, 133.8754).The solar array is fixedly installed on the ground.Its material is monocrystalline silicon.The area of the array is 27.4 m 2 and the rated power under standard test conditions is 5.236 KW.Table 1 displays the detailed statistical information of the features in the dataset.The dataset covers the period from July 22, 2013, to June 1, 2016, with a resolution of 1 hour, which contains a total of 25, 082 samples.The dataset is divided into training, validation, and test sets in a ratio of 7:1:2.The training set is used to train the model parameters, the validation set is used to choose the best model parameters, and the test set is used to evaluate the model's performance.Besides, missing values in the raw data are filled by using forward imputation, and normalization is applied to each feature.

Evaluation metrics
To comprehensively and objectively evaluate the predictive performance of the hybrid model, three mainstream metrics are used, namely mean absolute error (), root mean squared error (), and Pearson correlation coefficient

Results & discussion
In this work, LSTM, GRU, and Transformer models are selected as the backbone to demonstrate the utilization of the Informer model, and all of them form hybrid models with the CEEMDAN algorithm to validate the effectiveness of the proposed CEEMDAN-Informer method.In addition, experiments for 1-step, 3-step, and 6-step ahead were also conducted to analyze the multi-step forecasting capability of the proposed method.Table 2 lists the names of hyperparameters and their optimal values for the 1-hour ahead forecasts.In this study, all experiments were conducted by using PyTorch 1.9 on an RTX 3080Ti GPU within the Ubuntu 20.0 environment.Figure 2 displays the curves of model predictions and actual values with 1 step ahead for three consecutive days randomly sampled from the test dataset.From the graph, it can be observed that during the rapid increase phase of photovoltaic power, all models can make accurate predictions.However, at the extreme points of photovoltaic power, where the trend undergoes significant changes, there is a notable discrepancy between the predicted values and the actual values.The predictive results of the CEEMDAN-Informer hybrid model proposed in this study are closer to the actual values.The performance of the models at lead times of 1-step, 3-step, and 6-step ahead are shown in Table 3.Without using the CEEMDAN algorithm, the prediction performance of the Transformer and Informer model based on the self-attention mechanism is superior to the LSTM and GRU models.Specifically, in the case of a 1-step ahead, the Informer model shows a reduction of 29.6% in MAE and a reduction of 17.4% in RMSE compared to the LSTM model.Additionally, the  between predicted and actual values increased by 1.23%, effectively demonstrating the predictive performance of the Informer mode.Furthermore, it can be observed that the predictive errors of all models combined with the CEEMDAN algorithm are significantly reduced.For example, in the case of predicting 1-step, the CEEMDAN-Informer hybrid model proposed in this study achieves MAE and RMSE values of 0.105 KW and 0.201 KW, which represent a reduction of 19.8% and 28.4%, and the R-value reaches 0.993, demonstrating an increase of 0.8% relative to the Informer model.This effectively demonstrates the significant role of the CEEMDAN algorithm in reducing data non-stationarity and improving the predictive accuracy of the model.Finally, it can be observed that under all prediction conditions, the CEEMDAN-Informer method achieves the best performance.

Conclusions
Accurate PV power prediction is important for the utilization of solar energy resources.This paper introduces a hybrid prediction method composed of CEEMDAN and Informer.Specifically, the CEEMDAN algorithm, as a signal decomposition technique, decomposes photovoltaic power sequence data to reduce its non-linearity and non-stationarity, while the Informer model is implemented for feature extraction and photovoltaic power prediction.Furthermore, to assess the effectiveness and performance

Figure 2 .
Figure 2. The actual values and 1-step ahead predictions for three days randomly sampled.

Table 1 .
Detailed statistical information on the features in the dataset.

Table 2 .
Hyperparameter settings for the 1-step ahead prediction models.

Table 3 .
Forecasting performance of different models.