Short-term Prediction of PV Power Based on Combined Modal Decomposition and NARX-LSTM-LightGBM

1. Introduction

Solar power generation is safe and reliable and will not be affected by the energy crisis and fuel market instability factors, photovoltaic power generation is one of the most important forms of solar power generation, among many renewable energy sources, photovoltaic because of its unique cleanliness, low cost, high efficiency and abundant reserves and attention, whether from the perspective of protecting the Earth's environment or from the perspective of the sustainable development of the Earth's resources, the future of photovoltaic power generation Installed capacity will see significant growth. With the advancement of science and technology in recent years, PV power generation is growing quickly and accounting for an increasing share of power generation. PV systems play a significant role, particularly in remote locations of big PV power plants and residential power systems in rural areas[1,2,3].

However, the output of PV power varies substantially depending on the strength of the sun. PV electricity output ceases when the sun sets and the solar panels are not illuminated. Due to the impact of cloudy and rainy days, PV power production might vary significantly even during the day. Therefore, it is essential to understand the laws of PV power generation to get to maximize the use of solar energy and actively advance PV power generation. Power dispatch and configuration under precise and reliable PV power forecast plays a crucial function for the power system since the PV grid connection has such a significant impact on the power system[4,5,6,7].

At present, there are several techniques for forecasting PV power, which are broadly categorized as conventional techniques and artificial intelligence-based algorithms. Physical prediction techniques and time series prediction techniques are the two main traditional methodologies. The early stages of PV power prediction have seen extensive use of the time series prediction method, but it has lower prediction accuracy[8,9]. Traditional machine learning models, such as Random Forest (RF) [10] and Support Vector Machine (SVM)[11,12], which have higher prediction accuracy, are primarily used in applications of artificial intelligence algorithms. Neural networks are gradually becoming more popular in the field of PV prediction as deep learning advances. Numerous studies and experiments on the prediction of PV power by academics have demonstrated that combined prediction models typically produce better prediction outcomes than single models. Additionally, it can help when a single prediction model method produces large prediction errors at specific points. It is important to choose a model with strong predictive performance, and LSTM is widely used in processing problems for longer time series, especially for solving large and more complex nonlinear deep neural network models LSTM is preferred [13,14,15]. The Back Propagation (BP) neural network technique with time-delayed inputs serves as the foundation for the NARX neural network. The NARX neural network may successfully relate complicated dynamic interactions and react more swiftly to past state information by enhancing the time-delayed feedback connection from output to input. NARX has been used to resolve non-linear series forecasting issues in several disciplines and is appropriate for time series forecasting [16,17]. Although deep learning models can learn quickly and predict better outcomes, their structure is typically complex and computationally time-consuming, which results in low efficiency. An established and popular integrated learning strategy is the Gradient Boosting Machine (GBM). Extreme Gradient Boosting (XGBoost) is a gradient boosting tool that is incredibly scalable, adaptable, and versatile. Extreme Gradient Boosting employs regularization to control the overfitting issue to efficiently utilize resources and get around the drawbacks of earlier gradient boosting algorithms [18]. In comparison to XGBoost, LightGBM can speed up model training, use less memory, allow parallelized learning, and analyze massive amounts of data without compromising accuracy [19,20,21]. The accuracy of the combined model prediction outputs can be increased by giving various models varied weights. Using different weights for each time point of various forecasting models, also known as variable weight combination forecasting [22] or weighting the forecasts of two models using the inverse of error method [23], can both improve accuracy, and the choice of weights is a crucial issue in these studies. All of the aforementioned literature provide direct predictions of the starting PV power, but because of the complicated nonlinear and stochastic nature of the meteorological elements affecting PV power, it is challenging to produce precise forecasts using the initial PV power.

To increase prediction accuracy, modal decomposition techniques and machine learning models can delve deeper into the latent hidden information in the data. Wavelet analysis and empirical mode decomposition are two frequently employed techniques. Both approaches have drawbacks, but they can dissect the original waveform and increase prediction accuracy. EMD has issues including endpoint and over-envelope effects, and wavelet analysis is less adaptive [24,25]. The purpose of EEMD [26] is to enhance EMD by reducing the impact of mode aliasing. CEEMDAN is created by enhancing the algorithmic method for EMD, which incorporates the benefits of both EEMD and EMD while also speeding up the decomposition process [27,28,29]. All of the aforementioned literature established the efficiency of modal decomposition in PV power prediction, but only a single machine learning model for PV power prediction is not able to completely exploit the data information leading to the limitation of prediction accuracy.

In response to the above questions, this paper proposes a combinatorial machine learning prediction model with multimodal decomposition, in which multimodal decomposition is applied to construct a high correlation feature matrix, and the combinatorial machine learning prediction model is dedicated to prediction accuracy improvement. At the data optimization level, the missing data are mean-filled and combined with the Pearson correlation coefficient method to screen out the weather feature variables with high correlation selection. Subsequently, the modal decomposition of PV power is performed by EMD, EEMD, and CEEMDAN, resulting in a series of subseries with different feature scales to increase feature diversity. At the innovative level, the combined prediction model embeds the NARX neural network into the LSTM neural network and LightGBM model to achieve multi-angle analysis, so that it gains the competence of extracting the hidden relationships of sequences and enhancing the parsing ability of PV data, while final prediction results are obtained by weighting the combination of the prediction results using the inverse of error method. Later, it is experimentally proved that the prediction accuracy of this model is significantly improved compared with the traditional model.

2. PV power prediction model framework

2.1. Method of the Optimal Weight Determination

Features like ambient temperature, PV inverter temperature, PV model temperature, solar irradiance, and wind speed have an impact on PV power generation, and various features contribute to and have varying degrees of impact on PV power output. To avoid the negative impacts of individual features on PV power prediction, the correlation coefficient of each feature variable regarding PV power is calculated using the Pearson correlation coefficient approach. With the following formula, the Pearson correlation coefficient evaluates the linear connection between two continuous variables:

$$P_{xy}=\frac{{\displaystyle \sum _{i=1}^n(x_i-\overline{x})(y_i-\overline{y})}}{\sqrt{{\displaystyle \sum _{i=1}^n{(x_i-\overline{x})}^2}{\displaystyle \sum _{i=1}^n{(y_i-\overline{y})}^2}}}$$

(1)

where $\overline{x}$ is the mean value of the characteristic variable $x$ ， $\overline{y}$ and is the mean value of the characteristic variable y.

2.2. Modal Decomposition

As PV power signals are non-stationary, it is imperative to develop a modal decomposition method without a parameter. EMD was recommended by NE. Huang et al. as a method for dissecting and examining nonlinear or non-stationary time series data. The advantage is that the signal is decomposed over the initial time scale of the data using adaptively created intrinsic modal functions rather than fixed basis functions. EEMD, which is based on the EMD method and adds normally distributed white noise to the original signal to make it consistently distributed across the frequency band at the interval of extreme points, successfully reduces the modal aliasing problem. CEEMADAN is an optimization algorithm based on EMD as well as the EEMD algorithm. The CEEMDAN algorithm enhances the speed of signal decomposition by adding a small amount of adaptive white noise to the EEMD algorithm. This fixes issues with the EEMD algorithm's incompleteness and reconstruction error after adding white noise. The implementation steps of decomposition are listed as follows[30]:

• Adding white noise $v^i(t)$ with a standard normal distribution to the raw signal. $s(t)$ ., The i-th signal is denoted as $s^i(t)=s(t)+v^i(t)$ ,i=1,2…, I. EMD decomposes the timing signal to obtain the corresponding subsequence $IM{F}_1^i$ and the residual error vector $r_1(t)$ :

  $$IM{F}_1=\frac{1}{I}{\displaystyle \sum _{i=1}^{I}IM{F}_1^i}$$

  (2)

  $$r_1(t)=s(t)-IM{F}_1$$

  (3)
• Adaptive white noise $v^i(t)$ is added to the error and i times of experiments are performed each time the results are decomposed using EMD $r_1^i(t)=x(t)+v^i(t)$ to obtain its first-order component $IM{F}_1$ . An error of the 2nd subsequence $r_2(t)$ removed from the 2nd subsequence $IM{F}_2$ for CEEMDAN decomposition:

  $$IM{F}_2=\frac{1}{I}{\displaystyle \sum _{i=1}^{I}IM{F}_1}$$

  (4)

  $$r_2(t)=s(t)-IM{F}_2$$

  (5)
• To acquire the components that satisfy the conditions and the corresponding errors, the aforementioned decomposition procedure is repeated. The repetition comes to an end if the error is a monotonic function and cannot be broken down by EMD. The original signal $s(t)$ can be expressed as:

  $$s(t)=IM{F}_i+r_n(t)$$

  (6)

2.3. NARX Neural Network

The NARX neural network incorporates two time-delay structures from the signals at the input and output to describe the model of a nonlinear discrete system. The parametric formulas for the NARX neural network are as follows:

$$y(n)=f[x(n);\overline{u}(n);\overline{y}(n-1)]$$

(7)

$$y(n)=f[x(n),\dots x(n-d_E);\stackrel{\Lambda }{x}(n),\dots ,\stackrel{\Lambda }{x}(n-d_y+1)]$$

(8)

where $x(n)$ and $y(n)$ denote the input and output values of the NARX neural network at the discrete moment n, respectively, $d_E$ and $d_y$ greater than or equal to 0 are the maximum delay order of the input and output, respectively.

The NARX neural network's feedback loop and delay mechanism improve the ability to recall previous time-series data in order to better mine the non-linear sequence relationship of time-series data. The precise construction of the NARX neural network is shown in Figure 1.

NARX neural network has two layers of feedforward networks, with a linear transfer function in the output layer and a hidden layer having a sigmoid function $\sigma (x)$ , calculated as:

$$\sigma (x)=\frac{1}{1+\exp (-\sigma )}$$

(9)

The network has a time delay structure to store sequential prior values of u(n) and y(n), and the output y(n) is fed back to the input of the network. The input vectors are inserted through two-time delay structures of the input and output signals that demonstrate over-jump connections in the time-expanded network, increasing the capacity of gradient descent to propagate back with shorter paths. This increases the NARX neural network's capacity for historical data analysis[31].

2.4. Long Short-Term Memory

The issue of gradient disappearance produced by Recurrent Neural Network (RNN) structures can be resolved using LSTM neural networks. Through its special gate control and memory system, the LSTM neural network can fully use time series data. the structure of the LSTM neural network is shown in Figure 2.

The LSTM neural network mitigates the gradient explosion and gradient disappearance problem by employing three gating structures: input gates $I_t$ , forgetting gates $f_t$ , and output gates $O_t$ , which are responsible for managing the interaction between memory units implemented by tanh functions, Sigmoid functions, and matrix multiplication. In addition, the forgetting gate removes irrelevant states that mislead the prediction process and retains only the important information to be forwarded to the hidden layer. The value of the forgetting gate ranges from 0 to 1, where a higher value means that the information is the most important to select for retention, and a result of 0 requires complete discard[32].

Here, contrary to the input gate, the output gate checks its effect on the state of other memory cells. The LSTM gates, hidden outputs, and cell states are given as follows[33]:

$$I_t=\sigma (W_1{x}_t+U_i{h}_{t-1}+b_I)$$

(10)

$$f_t=\sigma (W_f{x}_t+U_f{h}_{t-1}+b_f)$$

(11)

$$O_t=\sigma (W_o{x}_t+U_o{h}_{t-1}+b_0)$$

(12)

$$\tilde{C}=f_t\cdot C_{t-1}$$

(13)

$$h_t=O_t\cdot \tanh (C_t)$$

(14)

where $x_t$ and $c_t$ denote the input and storage units at time t, respectively. $b$ , $W$ , and $U$ are the deviation, cycle weight, and input weight of each gate, respectively. $h_{t-1}$ is the hidden layer of each gate $x$ in moment t-1. The flow of the LSTM neural network operation is shown in Figure. First, $h_{t-1}$ , $C_{t-1}$ , and $x_t$ pass the input information to the LSTM neural network cell. Each gate of the LSTM neural network interacts with the input to generate a logic function-based function that, after $f_t$ , constructs a new cell state $c_t$ , $x_t$ and $h_{t-1}$ move to the forgetting gate, using 0 and 1 to quantify the importance of the input information to be used to decide whether the input information is stored or not. Then, the forgetting gate will update the cell state with the new important information. Finally, the remaining state values are calculated by the hidden layer of the LSTM neural network.

2.5. LightGBM

The decision tree used as the foundation of the LightGBM base learner provides effective parallel training and offers the benefits of quicker training times, cheaper memory requirements, improved accuracy, and distributed assistance for quick processing of huge volumes of data. Histogram's decision tree algorithm. The leaf histogram is created by subtracting the speed-duplicated differences between the sibling and father node histograms. In order to calculate the information gain, the LightGBM algorithm employs Gradient-based One Side Sampling (GOSS), which not only reduces the number of samples used but also speeds up computation. Only the data with higher gradients are kept, while the data with smaller gradients are discarded. Exclusive Feature Bundling (EFB) makes several mutually exclusive features bound together, which can achieve the effect of reducing the data dimensionality[34]. The LightGBM algorithm effectively improves the operational efficiency through the leaf-wise growth strategy with depth limitation. It solves the problem that most GBDT tools use the inefficient level-wise decision tree growth strategy, which reduces the efficiency of machine learning by continuing to split and explore when the split gain is not high and achieves higher accuracy when the number of splits is the same[35].

2.6. Combined Forecasting Model and Process

The proposed model in this paper is a powerful combination of NARX, LSTM, and LightGBM models. The NARX-LSTM-LightGBM model merges the properties of each prediction model to obtain better results. In the proposed model NARX neural network is associated with the embedded memory that makes jump connections in the network. The NARX neural network is used in the paper to calculate the residual error. The residuals will reduce the dependence and sensitivity of the network structure on the time series. Let $E_n=[e_1,\dots ,e_n]$ be the error vector between the actual value $Y_t=[y_1,\dots ,y_n]$ and the predicted value ${\stackrel{\wedge }{Y}}_t=[\stackrel{\wedge }{y_1},\dots ,\stackrel{\wedge }{y_n}]$ of NARX. The residuals are calculated as follows:

$$E_n=Y_t-\stackrel{\wedge }{Y_t}=y_{i}t-{\displaystyle \sum _{i=1}^n{w}_i}{F}_{it}$$

(15)

where $F$ and $w$ denote the nonlinear mapping function of the NARX neural network and the corresponding weight values. For this purpose to fully utilize the hidden information of the time series, the residuals and predicted values from the preliminary PV power prediction from the NARX neural network are used to obtain new features as input. Finally, the LSTM and NARX neural networks are used to make predictions by the error inverse method with the data added to the new features as input to obtain the NARX-LSTM combined model results. The formula is as follows:

$$w_1=\frac{e_2}{e_1+e_2},w_2=\frac{e_1}{e_1+e_2}$$

(16)

$$f_P=w_1{f}_1+w_2{f}_2$$

(17)

where $e_1$ is the error of model 1, $e_2$ is the error of model 2; $w_1$ is the weight of model 1, $w_2$ is the weight of model 2; $f_1$ and $f_2$ are the predicted values of model 1 and model 2, respectively; $f_P$ is the weighted average of the error inverse method of the final combined model.

Since NARX-LSTM neural network is a combined model of deep learning and the LightGBM algorithm is a boosted tree machine learning model, with a low correlation between the model principles and LightGBM's better prediction performance, the two are combined. To obtain the final prediction results, the prediction results of the LightGBM algorithm and the combined NARX-LSTM algorithm are combined using the error inverse method. To better understand the proposed model, the combined prediction model is shown in Figure 3.

where $w_1$ and $w_2$ are the weights of the NARX neural network and the LSTM neural network, respectively; $w_3$ and $w_4$ are the weights of the combined NARX-LSTM model and the LightGBM algorithm, respectively. $f_{P1}$ is the weighted average of the error inverse method of the NARX neural network and the LSTM neural network; $f_{P2}$ is the weighted average of the error inverse method of the combined NARX-LSTM model and LightGBM.

To further enhance the exploration of the internal linkage of the historical time series, a prediction method based on the combined modal decomposition is proposed, and the flow chart is shown in Figure 4, which mainly consists of the following parts:

• After pre-processing the data, it only retains the data in the period of 5:00-20:00, analyzes the correlation of environmental features, selects the environmental variables with stronger correlation to be the features of the combined prediction model, and normalizes the features with higher correlation to improve the convergence speed and performance of the model.
• The EMD, EEMD, and CEEMDAN modal decomposition methods were selected to decompose the original PV power modalities, and the respective modal subseries were combined to construct the feature matrix for correlation analysis, and the subseries features with high correlation and environmental variables with strong correlation were selected to join the combined NARX-LSTM-LightGBM prediction model.
• Predictions are made by a combined modal decomposition NARX-LSTM-LightGBM model, and performance is evaluated.

3. Results and Discussions

3.1. Model performance evaluation indicators

The indicators used in this paper to predict the selected performance evaluation include the mean absolute error $e_{mae}$ , the mean absolute percentage error $e_{mape}$ , and the root mean square error $e_{rmse}$ , these percentage error measures are used because of their independent judgment and the efficiency of the judgment model. The formulas are as follows[36]:

$$e_{mae}=\frac{1}{n}{\displaystyle \sum _{i=1}^n|y_i-{\stackrel{\wedge }{y}}_i}|$$

(18)

$$e_{mape}=\frac{1}{n}{\displaystyle \sum _{i=1}^n\left|\frac{y_i-\stackrel{\wedge }{y_i}}{\overline{y}}\right|}$$

(19)

$$e_{rmse}=\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^n{(y_i-\stackrel{\wedge }{y_i})}^2}}$$

(20)

where $y_i$ and $\stackrel{\wedge }{y_i}$ are the actual and predicted values; $\overline{y}$ is the sample mean.

3.2. Data pre-processing

The performance of the proposed model was evaluated using real data sets, and the experimental study used measured data from Andre Agassi College, USA, to verify the generalization capability of the proposed model. The data set includes seven environmental characteristic variables: ambient temperature, PV inverter temperature, module temperature, irradiance, ambient humidity, wind speed, and wind direction.

The period of the dataset is January 1, and 2012-December 31, 2014, with a time interval of 15 minutes and a total of 96 sampling points a day. The time point of a day without PV power is eliminated to increase the efficiency of the model calculation, while the daily period of 5:00–20:00 with a time interval of 15 minutes and a total of 60 sampling points per day are kept. Missing data are filled by the mean fill method, and min-max normalization is done for the filled data, and the formula is as follows:

$$V_t=\frac{V_{t-1}+V_{t+1}}{2}$$

(21)

$$X_i=\frac{Z_i-Z_{\min }}{Z_{\max }-Z_{\min }}$$

(22)

where, $V_t$ is the missing value; $V_{t-1}$  and  are the values 15 minutes before and 15 minutes after the missing value, respectively. $X_i$ is the normalized value; $Z_i$ is the initial value; $Z_{\max }$ and $Z_{\min }$ are the maximum and minimum values, respectively.

3.3. PV power characteristics correlation analysis

The correlation of the characteristic variables has considerable significance to the accuracy of PV power prediction. When the correlation coefficient $\sigma \ge 0.7$ , the two variables can be considered as strongly correlated; when $0.4\le \sigma \le 0.7$ the two variables can be considered as moderately correlated; when $0.2\le \sigma <0.4$ the two variables can be considered as weakly correlated; when $\sigma <0.2$ the two variables can be considered as very weakly correlated; and when $\sigma <0$ the two variables can be considered as negatively correlated.

Table 1. Relative coefficients of power and individual characteristics.

Feature variables	Correlation factor	Correlation
Ambient temperature	0.42	Moderate
Inverter temperature	0.50	Moderate
Module temperature	0.69	Moderate
Irradiance	0.96	Strong
Relative humidity	-0.40	Negative
Wind speed	0.20	weak
Wind direction	-0.039	Negative

Table 1. Relative coefficients of power and individual characteristics.

Feature variables	Correlation factor	Correlation
Ambient temperature	0.42	Moderate
Inverter temperature	0.50	Moderate
Module temperature	0.69	Moderate
Irradiance	0.96	Strong
Relative humidity	-0.40	Negative
Wind speed	0.20	weak
Wind direction	-0.039	Negative

3.4. Combinatorial decomposition to build new features

This paper decomposes the original power using EMD, EEMD, and CEEMDAN decomposition methods to reduce the model's complexity, thoroughly excavate the intrinsic information, and obtain each subsequence to build the feature matrix. 14 subsequence components were obtained from EMD; 16 subsequence components were obtained from EEMD; and 15 subsequence components were obtained from CEEMDAN. A total of 45 subsequences. The decomposed results are illustrated in Figure 5, Figure 6 and Figure 7. The correlation analysis leaves the characteristic subseries with a higher correlation with PV power to achieve the complementarity of different decomposition methods and obtain the characteristic matrix because too many subseries will result in low computational efficiency of the combined model and the subseries with low correlation will affect the prediction accuracy of the model.

This paper selects the subseries with a correlation greater than 0.4 and combines them into a feature matrix. It is known from Table 2 that the correlation of IMF5 subseries of EMD is 0.8; the correlation of IMF4, IMF5, and IMF6 subseries of EEMD are 0.61, 0.92, and 0.47, respectively; the correlation of IMF5 and IMF6 subseries of CEEMDAN are 0.68 and 0.57, respectively. Therefore, these subseries are selected to form the new feature matrix and the rest of the subseries are excluded from the model.

3.5. Model parameters setting

After several experiments, the prediction model was created by dividing the initial 66870 sets of data into a training set and a test set by a ratio of 7:3, then feeding the results into a NARX neural network model with 12 hidden layer neurons and the order of the time delay being 8. The output 20059 PV power test set and residual vector are obtained and the latter 20059 sets of the original data are jointly input to the LSTM neural network and LightGBM algorithm, and the correlation analysis shows that the test set power and residual vector have a high correlation of 0.87 and 0.51. The LSTM neural network model uses the ReLU function as the activation function, the optimizer is Adam, batch_size is 32, the maximum number of iterations is 32, and the learning rate is 0.1; The optimization of the LightGBM algorithm hyperparameters is performed using the grid search algorithm to obtain a learning rate of 0.01, the number of base learners n_estimators of 15000, and the number of leaf nodes num_leaves of 31 by default.

Using the data from February 6, 2014, to December 17, 2014, as training samples, the PV power is shown in Figure 8, to avoid chance error and increase the generalization of the model selected December 23 (sunny day 1) and December 25 (sunny day 2); December 18 (cloudy day 1) and December 20 (cloudy day 2); December 17 (rainy day 1) and December 31 (rainy day 2) a weather type two days in total six days were selected as the sample test set.

3.6. Validation of combined modal decomposition

This paper adopts the method of Combined Decomposition(CD) to deeply explore the intrinsic connection between PV power and historical time series to decompose PV power. To verify the effectiveness of combined modal decomposition, four prediction models, LSTM, CD -LSTM, NARX-LSTM-LightGBM, and CD -NARX-LSTM-LightGBM, are selected for comparison in this section. Predictions were made for the above test days, and by observing Figure 9, it can be seen that the four models match the actual PV power prediction curves under sunny weather. The prediction models with combined modal decomposition in cloudy and rainy weather perform better, the prediction curves fit better with the real values, and the overall curves are consistent.

Table 3 shows the results of the error-index comparison table, and Figure 10 shows the prediction error histogram pile-up for the four models predicting PV power (different colored areas indicate different prediction methods, and their smaller areas indicate smaller errors of the corresponding methods). Observing Table 2 and Figure 9, it is concluded that the model with PV power decomposed by the combined modal decomposition method performs better than the model with PV power decomposed by the no combined modal decomposition method on all three types of test days. The CD-LSTM model reduces 24.68%, 29.82%, and 29.82% over the LSTM model $e_{rmse}$ , $e_{mae}$ and $e_{mape}$ , respectively; The CD-NARX-LSTM-LightGBM model reduces 56.30%, 58.45%, and 63.04% over the NARX-LSTM-LightGBM, respectively. The experimental results demonstrate that the prediction accuracy can be improved by the combined modal decomposition method.

3.7. Validation of NARX-LSTM-LightGBM model

To further verify how well the suggested CD-NARX-LSTM-LightGBM method predicts outcomes For experimental comparison of PV power prediction, six models of NARX, LSTM, LightGBM, RNN, Gated Recurrent Unit (GRU), and NARX-LSTM-LightGBM with combined modal decomposition are created for three weather types: sunny, cloudy, and rainy days. A comparison of the model prediction PV generation curves is shown in Fig. 11. From Figure 11, it is known that all six prediction methods perform better on sunny days, but the CD-NARX-LSTM-LightGBM model performs better on cloudy and rainy days where the environmental factors fluctuate a lot. PV power has a large amplitude variation at times because of cloudy and rainy days. Environmental variables have characteristics such as high volatility and strong non-linearity, which have a direct impact on PV power production. Overall, the NARX-LSTM-LightGBM model after combined modal decomposition performs the best. Among the predictions of the three weather types, the predicted values of the method in this paper are closer to the actual power, and the prediction performance of the PV power is superior. Especially in cloudy and rainy weather types, which are more volatile, the predicted power curve of this paper's method fits better with the real power curve.

Figure 12 shows the error histogram of the six prediction methods under each weather type, and Table 4, Table 5 and Table 6 show the comparison table of prediction errors, from the figure, it can be seen that the error area of PV power prediction by the methods in this paper are the best performance of the minimum, from the table, it can be seen that, taking Sunny1, Cloudy1 and Rainy1 as examples, the $e_{rmse}$ of the combined modal decomposition model under Sunny1 weather are 2.011, 0.913, 4.095 , 0.735, 0.897, 0.465kw, $e_{mae}$ are 1.298, 0.636, 3.233, 0.491, 0.616, 0.213kw, $e_{mape}$ are 0.149, 0.074, 0.371, 0.056, 0.07, 0.025kw, respectively; $e_{rmse}$ under cloudy1 weather are 4.655, 3.443, 2.773, 3.922, 3.69, 1.645kw, $e_{mae}$ of 3.032, 2.092, 2.363, 2.366, 2.155, 0.892kw, $e_{mape}$ of 0.516, 0.36, 0.46, 0.4, 0.365, 0.153kw, respectively; $e_{rmse}$ under rainy1 weather are 4.655, 3.443, 2.773, 3.922, 3.69, 1.645kw, $e_{mae}$ are 3.032, 2.092, 2.363, 2.366, 2.155, 0.892kw, and $e_{mape}$ are 0.516, 0.36, 0.406, 0.4, 0.365, 0.153kw, respectively, compared to the combined decomposed single-model prediction method, the combined modal decomposition followed by combined prediction model has the smallest error index and the best prediction performance under different weather types.

4. Conclusions

This paper proposes a combined prediction model based on CD-NARX-LSTM-LightGBM for PV power prediction, and the following conclusions are obtained:

• The combination of EMD, EEMD, and CEEMDAN decomposes the original PV power, which can effectively reduce the original curve's nonlinearity and complexity, increase the positive correlation features, and improve the accuracy of PV power prediction.
• NARX, LSTM, and LightGBM models are based on different principles and mathematical models, each with excellent performance in time series data forecasting problems, and the combined NARX-LSTM-LightGBM forecasting model is better able to fully exploit the intrinsic information linkage of historical time series data.
• Compared with other single models, the combined prediction model based on CD- N.M. NARX-LSTM-LightGBM proposed in this paper has obvious advantages for the prediction of PV power, which has better and excellent prediction accuracy in both steady and non-steady weather, and it has the prospect and significance for application in other fields.