Gradient boosting dendritic network for ultra-short-term PV power prediction

Abstract

To achieve effective intraday dispatch of photovoltaic (PV) power generation systems, a reliable ultra-short-term power generation forecasting model is required. Based on a gradient boosting strategy and a dendritic network, this paper proposes a novel ensemble prediction model, named gradient boosting dendritic network (GBDD) model which can reduce the forecast error by learning the relationship between forecast residuals and meteorological factors during the training of sub-models by means of a greedy function approximation. Unlike other machine learning models, the GBDD proposed is able to make fuller use of all meteorological factor data and has a good model interpretation. In addition, based on the structure of GBDD, this paper proposes a strategy that can improve the prediction performance of other types of prediction models. The GBDD is trained by analyzing the relationship between prediction errors and meteorological factors for compensating the prediction results of other prediction models. The experimental results show that the GBDD proposed has the benefit of achieving a higher PV power prediction accuracy for PV power generation and can be used to improve the prediction performance of other prediction models.

Graphical abstract

Keywords

photovoltaic (PV) power prediction / dendrite network / gradient boosting strategy

1 Introduction

The vigorous development of renewable energy technologies is an effective way to address climate and energy problems. Photovoltaic (PV) power generation is a competitive alternative to fossil fuels because it generates electricity without any polluting emissions and noise, and is safe and reliable [1]. The installed PV capacity is growing at the fastest rate in the last two decades [2]. However, the fluctuating nature of solar energy results in a negative effect on PV power generation, which may further degrade the stability of the grid [3]. Therefore, there is an urgent need for accurate PV power prediction to enhance the utilization of solar resources and improve the optimal dispatch of power systems [4–8].

Various methods for building PV power prediction models have been proposed nowadays. In terms of technology [9,10], prediction methods can be classified into persistent methods [11], physical methods [12], statistical methods [13,14], and soft-computing methods. Soft-computing methods [15] are a class of methods that use artificial intelligence (AI) algorithms for PV power prediction. Such methods can deal with multivariate inputs and construct nonlinear mapping relationships between inputs and outputs by extracting high-dimensional features and have gradually become mainstream methods for PV power prediction [16,17].

The main AI algorithms used in PV power prediction models include back propagation neural network (BPNN) [18], support vector machine (SVM) [19], Elman neural network (Elman NN) [20,21], radial basis function neural network (RBFNN) [22], extreme learning machine (ELM) [23,24], long short-term memory (LSTM) neural network [25], and convolutional neural network (CNN) [26], etc. However, the strong stochastic nature of PV power and the complex and empirical hyperparameter settings of the model limit the prediction performance.

To address the above issues, more researchers have focused on hybrid prediction models that combine multiple models. There are many types of hybrid prediction models, such as hybrid models using metaheuristic algorithms, which solve the problem of finding the optimal hyperparameters of the model through the mechanism of computational intelligence [27–31]. These hybrid models can improve the performance of the PV power prediction model. However, they make the computational cost significantly higher, especially for multi-objective optimization. Another class of hybrid model decomposes PV power into multiple frequency components and models them separately for prediction to obtain more patterns of PV curve variation, as in Refs. [32–37]. However, modeling each PV power component data independently leads to increased complexity and is time-consuming during the modeling process. In addition, data decomposition requires the use of signal processing knowledge to set reasonable parameters.

Compared with the first two types of hybrid models, the ensemble model is a more practical solution. Ensemble models form powerful predictors by aggregating multiple weak predictors together. Wu et al. [38] builds an ensemble model based on the bagging strategy. The model utilizes the autoregressive integrated moving average (ARIMA) method, SVM, artificial neural network (ANN), and adaptive-network-based fuzzy inference system (ANFIS) for training and prediction, respectively, followed by assigning weights to the four methods by genetic algorithm (GA) to obtain the final prediction results. de Jesus et al. [39] builds an ensemble model based on a stacking strategy. The model utilizes the autoregressive moving average with exogenous inputs ARMAX and ANFIS for training and prediction, respectively, and inputs the two predictions into a hybrid deep network combining full convolutional network (FCN) and LSTM network to derive the final prediction.

The ensemble models of bagging and stacking strategies obtain more accurate prediction results by the complementary nature of different algorithms. However, the prediction results of the sub-prediction models of these different algorithms cannot achieve complete complementarity. Therefore, there will be some prediction results that are inferior to those of the sub-models. In addition, the sub-models with poor prediction performances may not participate in the final prediction results when the ensemble model is used, which actually causes a waste of computational resources.

In summary, traditional neural network algorithms, due to their complex structures, pose challenges to setting hyperparameters, thus limiting the predictive performance. While hybrid models can effectively tune the hyperparameters of individual models, doing so increases complexity and demands substantial modeling time. Ensemble models achieve accurate results by integrating various algorithms in a complementary manner. However, sub-models in traditional ensemble models may not always complement each other due to the use of different algorithms. Additionally, sub-models that do not contribute to result complementarity can lead to computation resource wastage. In contrast, the ensemble model of boosting strategy combines sub-models by means of the function greedy approximation, which has a higher utilization of computational resources. A comparison of the advantages and disadvantages of existing methods is shown in Tab.1.

Persson et al. [40] and Wang et al. [41] used gradient boosting decision trees (GBDTs) for PV power prediction. The prediction model based on GBDT uses historical weather data and PV power output data to iteratively train the model, which is used to predict the future PV power output based on numerical weather forecasting (NWF) data. Compared to other prediction models, GBDT does not need to study the temporal correlation of the feature variables on PV power before modeling, and the program performs feature selection itself internally after inputting the variables to GBDT. In Chen et al. [42], an extreme gradient boosting (XGBoost) is used to predict PV power generation. XGBoost is an improved model of GBDT with a better prediction performance.

Both GBDT and XGBoost are known as gradient boosting machines (GBMs) with classification and regression trees (CARTs) as base learners. Compared with other prediction models, GBM has the advantages of a strong model explanatory power, high accuracy, and stable error performance. However, most GBMs in current research use tree models as the base learners, which makes the prediction models require very many iterations of training. If an AI algorithm with a strong nonlinear mapping capability is used as the base learner of GBM, GBM may obtain high prediction results with a very small number of iterations.

An ultra-short-term prediction model for PV power based on a dendritic network (DD) was proposed [43]. It was experimentally verified that the DD had a high prediction accuracy and good generalization. Moreover, due to the replacement of the traditional activation function with Hadamard product in neurons, this makes DD a white-box model with a faster computation speed. The prediction model built based on DD has a strong explanatory power. However, over time, challenges such as an increase in data volume and equipment aging can cause a gradual decline in the accuracy of prediction models. Traditional solutions involve retraining prediction models using more historical data. Yet, this retraining process involves modifying model parameters and demanding substantial computational resources and time. Furthermore, the improvement in prediction performance is not assured. To achieve a more accurate PV power prediction, a GBDD prediction model with a new structure is proposed based on the computational logic of gradient boosting strategy and the characteristics of DD.

In GBDD, DD is used as the base learner of GBM to correct the prediction results by learning the relationship between prediction error and multiple meteorological factors during the iterative process to achieve a more accurate PV power prediction. In addition, with some modifications in the structure of GBDD, another use of the model was found and a strategy was proposed to improve the prediction performance of other types of prediction methods. A GBDD model is trained to compensate for the prediction results of other prediction models by analyzing the relationship between prediction errors and meteorological factors. This strategy is straightforward and effective compared to strategies that improve prediction performance by adjusting parameters.

To address issues such as data gaps and the high complexity of meteorological factors in ultra-short-term PV forecasting, this paper proposes a GBDD model for ultra-short-term PV power prediction. This will provide a data reference for the intraday dispatch of power systems containing PV generation. This paper is contributive because it proposed a GBDD model for ultra-short-term PV generation prediction, which could make a fuller use of the data from all multiple meteorological factors to achieve a more accurate PV power prediction. In addition, it proposed a new improvement strategy based on GBDD to enhance the forecasting performance of other forecasting models. The strategy is straightforward and efficient and is able to compensate for the prediction results by analyzing the relationship between the prediction residuals and meteorological factors. Moreover. it verified the effectiveness of the model and strategy proposed by experimental results, and explored the conditions required to combine the gradient boosting strategy with AI algorithms based on the results of benchmark models.

2 Methodologies proposed

In this section, the DD, the gradient boosting strategy, and the implementation details of the GBDD are presented separately. In addition, a new strategy for improving the accuracy of other prediction models is proposed based on GBDD.

2.1 Strategy of gradient boosting

Gradient boosting is a classical strategy for boosting models. The core of this strategy is greedy function approximation [44]. This type of model has been used with good results in many fields. The gradient boosting model can be described as given by Eq. (1),

(1)f_m\left(x\right)=f_{m-1}\left(x\right)+v\cdot {\rho }_{m}h\left(x\right),

where f_m\left(x\right) and f_{m-1}\left(x\right) represent the mth iteration and (m−1)th iteration of the model, respectively; h(x) represents the base learner, which in the current study is mainly a decision tree; and v and ρ_m are the learning rate and shrinkage factor, respectively, which serve to avoid overfitting of the prediction model.

The most important step of the gradient boosting strategy is to fit the negative gradient by means of a base learner. In regression problems such as PV generation forecasting, half of the mean variance is usually used as the loss function, as shown in Eq. (2),

(2)L(y,f(x\left)\right)=\frac{1}{2}(y-f(x){)}^2.

For the prediction model f_{m-1}\left(x\right), the overall loss function is defined as J, as shown in Eq. (3),

(3)J={\sum }_i^{n}L(y_i,f_{m-1}(x_i\left)\right).

The gradient it produces for the current predicted value of each sample is shown in Eq. (4),

(4)\begin{array}{rl}\frac{\mathrm{\partial }J}{\mathrm{\partial }{f}_{m-1}\left(x_i\right)}=& \frac{\sum L\left(y_i,f_{m-1}\left(x_i\right)\right)}{\mathrm{\partial }{f}_{m-1}\left(x_i\right)}\\ =& \frac{L(y_i,f_{m-1}(x_i\left)\right)}{\mathrm{\partial }{f}_{m-1}\left(x_i\right)}=f_{m-1}\left(x_i\right)-y_i.\end{array}

The corresponding negative gradient is shown in Eq. (5),

(5)-\frac{\mathrm{\partial }J}{\mathrm{\partial }{f}_{m-1}\left(x_i\right)}=y_i-f_{m-1}\left(x_i\right),

where y_i-f_{m-1}\left(x_i\right) is the residual of the current model on the target data. Therefore, for a gradient boosting model for a regression problem such as power prediction, the basis function h(x) added each time is only needed to fit the residuals left over from the last time.

2.2 Dendrite network

DD is a basic machine learning algorithm with a new structure [45]. Unlike traditional neural networks with multilayer perceptron (MLP) structures that simulate the functions of cell body parts of the biological nervous system, DD implements multiple logical operations through Hadamard products to simulate the information processing of dendrites in the biological nervous system. The structure of MLP and DD is shown in Fig.1, in which, MLP and DD share a similar fully connected structure. However, DD employs matrix operation functions as the activation function for its neurons, whereas traditional MLPs and other neural networks use nonlinear mapping functions. In computer computations, matrix operation functions are simpler, more cost-effective, and derivable compared to nonlinear mapping functions. This implies that the structure of DD is a derivable white-box model, whereas traditional MLP structures are black-box models with unknown structures.

2.3 GBDD structure

In this paper, based on the first two parts, a GBDD model is proposed for PV power ultra-short-term prediction using DD as the base learner and half of the mean variance as the loss function. GBDD is a prediction model based on multiple weather factor data, which is trained with historical PV generation and corresponding weather factor data. Then the PV generation is predicted by utilizing NWF data.

As with most predictive models for neural networks, GBDD requires that the input variables to the model be selected first. Previously, ultra-short-term power prediction for a PV plant with an installed capacity of 263 kW was performed. In Lu et al. [43], the effect of three methods of Pearson correlation coefficient (PCC), grey relation analysis (GRA), and maximum information coefficient (MIC) was verified for variable selection on the prediction results by conducting several experiments with different four data sets. The experimental results show that the use of MIC can achieve smaller prediction errors (using the mean absolute error (MAE) as the evaluation index and DD with the same parameters as the prediction model, the prediction model errors using MIC, PCC, and GRA are 4.6792, 5.8881, and 6.3889 kW, respectively). Based on the experimental results and the analysis of the MIC calculation characteristics, it can be considered that it is due to the ability of MIC to detect both linear and nonlinear data relations, which fits exactly into the DD calculation. Therefore, MIC is used to select variables that have a significant effect on the PV output power as inputs to the initial model.

Compared with the traditional GBM, GBDD has a novel structural design. Compared with other studies that combine GBM with neural networks, GBDD is a new algorithm designed based on the gradient boosting theoretical framework, rather than a simple combination of algorithms. Specifically, there are several differences as follows:

First, the traditional GBM initial model is set to the average of the output data, while the initial model of GBDD is set to f₀(x) = DD₀(x), which allows the initial model itself to make power predictions, which will reduce the number of iterations required for the model.

Secondly, since DD itself is a strong learner, both the learning rate and the shrinkage factor are set in the range of 0 to 1 in order to avoid overfitting the prediction model. The shrinkage factor is set according to Eqs. (6) and (7),

(6){\rho }_m=\text{arg}\underset{\rho }{\mathrm{m}\mathrm{i}\mathrm{n}}{\sum }_{i=1}^{n}L(y_i,f_{m-1}(x_i)+\rho h(x_i\left)\right),

(7)

where ρ represents the step size of the function f_m−1(x), the base learner h(x) denotes the direction of advancement taken by this step size, and n and m mean the number of samples and iterations. As the dependent variable of the ith sample, y_i is used as the input of the loss function L(y, f) along with the model output f_m after the mth iteration training. Moreover, f_m can be obtained from f_m−1 through ρh(x) stepping. The step size ρ will determine the value of the sum of the final loss functions. To achieve the minimum output error of the model, it is natural to expect the sum of the final loss functions to be the smallest. Therefore, the step size ρ obtained when the sum of the final loss functions is the smallest is called ρ_m, which means that the step size ρ_m that occurs during the mth iteration is the optimal step size. To prevent model overfitting and maintain accuracy, ρ_m is usually bounded between 0 and 1. Therefore, Eq. (7) is employed to limit the value of ρ_m.

In the GBM algorithm, the learning rate (v) requires manual adjustment, involving multiple experiments to propose an empirical setting for v. The learning rate for the sub-model in the first iteration is set to 1, and subsequently halved in each subsequent iteration. The learning rate obtained through this method are presented in Tab.2 in Section 4.

In addition, in order to avoid overfitting, the model to predict the residuals generated in the training phase of the model, especially the larger residual samples, this paper introduces a Hampel filter to perform a smoothing filter on the residuals, which has the advantage that the base learner DD(x) can focus more on the trend of the residuals as a whole rather than on the individual residual data with larger values.

Finally, according to the different settings of the input parameters of the base learner, two different strategies are presented in this paper for the training of the subsequent iterative models, i.e., the same input variable gradient boosting strategy (Strategy No. 1) and the different input variables gradient boosting strategy (Strategy No. 2). Section 3 will provide a detailed description of these two strategies. The structure of the GBDD with the two strategies is shown in Fig.2, in which, DD_i (i = 1, 2, …, n) refers to the network model trained in the ith iteration.

The red line represents the gradient boosting strategy with the same input variables, while the blue line represents the gradient boosting strategy with different input variables. When one of the strategies is executed, the line representing the other strategy is considered not to exist.

2.3.1 Gradient boosting strategy with same input variables

Training and prediction of subsequent iterations of the base learner using the same input data as the initial base learner is called the same input variables gradient boosting strategy which considers that the errors generated by the base learner prediction are influenced by the initial weights and parameters of the DD. Therefore, the prediction effect can be improved by adjusting the parameters of the subsequent DDs without adjusting the input data.

2.3.2 Gradient boosting strategy with different input variables

Based on Strategy No. 1, the subsequent iterations of the base learners are trained and predicted using input data that are not identical to the first layer of base learners. This approach is called the heterogeneous input gradient boosting strategy. This strategy considers PV generation as a complex process that is affected by many factors. The set of variables that have the most significant impact on PV generation is selected as the input in the initial base learner, while the error generated by the model is more likely to be caused by other variables that are not used as input. By using correlation analysis to find the most significant set of variables for the error as the input conditions of the subsequent base learner, it is possible to use all meteorological factor data more effectively and improve the prediction performance.

2.4 Strategy for improving other prediction methods based on GBDD

Improving the prediction performance of other prediction methods by GBDD is another way of using the GBDD proposed in this paper. In this use, GBDD is no longer used as a separate PV power prediction model but as a supplementary model to the prediction model. According to the operational logic of the gradient boosting theory, if the initial model f₀ of GBDD is replaced with the prediction model of other methods, the subsequent GBDD sub-models can improve the prediction performance of the prediction model by analyzing the relationship between the prediction error and multiple meteorological factor data to correct the prediction results during the training process.

The novel strategy proposed is simpler, more efficient, and requires less computational cost than strategies that use metaheuristic algorithms to optimize the hyperparameters of the prediction model. No adjustments to the parameters or structure of the original prediction model are required in the process of executing the strategy. Moreover, the time at which the complementary model is added to the prediction model can be controlled by the user, increasing the manipulability. Specifically, the novel strategy proposed consists of the following steps:

First, the training of this supplementary model requires collecting the prediction results and actual power generation of the prediction model for the recent period and the corresponding multivariate meteorological factor data.

Then, the residual data of the prediction model are calculated from the prediction results and the actual PV power generation, and the multivariate meteorological factors that are significantly correlated with the residual data are selected as the input variables of GBDD by MIC analysis.

Finally, the residual data and input variable data are used for network training. While the prediction model makes subsequent predictions, the trained GBDD predicts the residual data. The prediction results of the GBDD are added to the PV generation predicted by the prediction model to obtain the final corrected prediction results.

The overall structure of the strategy is shown in Fig.3.

3 Experiment data

In this section, the data set used in this paper is described in detail. The raw data set is pre-processed and the data set is partitioned. In addition, the evaluation metrics used for the experiments are described.

3.1 Data set analysis

The experimental PV power output data set from the Desert Knowledge Australia Solar Centre , which records the power generation of a 6.05 kW PV plant located in Alice Springs, was randomly selected from April to May 2022 for the experiments in this paper. Installation parameters of the adopted PV plants are introduced in Tab.3. The historical multiple meteorological factor data and the NWF data of the PV plant were obtained from the short-term forecasts based on historical data in Alice Springs Weather Station. The PV plant and weather station are shown in Fig.4.

Within the data set, there are nine meteorological factors relevant to PV power: diffuse horizontal radiation (DHR), global horizontal radiation (GHR), radiation diffuse tilted (RDT), radiation global tilted (RGT), daily rainfall (DR), relative humidity (RH), temperature Celsius (TC), wind direction (WD), and wind speed (WS). Establishing a predictive model based on DD requires the selection of input variables, and different choices of input variables can impact the predictive performance of the model. This paper employs three methods to analyze the aforementioned meteorological factors, MIC, PCC, and GRA. MIC can detect both linear and nonlinear data relationships, PCC can detect linear data relationships and indicate positive or negative correlations by the sign, while GRA can detect linear data relationships even with a limited data volume. By utilizing these three methods to analyze the correlation between PV power and meteorological factors, the obtained results are presented in Tab.4 and Fig.5, from which it can be observed that a meteorological factor with a higher correlation coefficient has a greater impact on the outcome.

However, the patterns of the same meteorological factor vary in these three correlation calculation methods. Therefore, it is necessary to use ablation experiments to determine the optimal combination of input variables. Based on the results of the three correlation analysis methods, the meteorological factors are ranked from the highest to the lowest correlation, and they are successively combined as inputs for the predictive model. The MAE is employed as the evaluation metric for predictive performance. The experimental results are depicted in Fig.6.

The experimental results indicate that when the first five meteorological factors arranged by MIC (DHR, GHR, RDT, RGT, and WS) are selected as input variables, the predictive model yields the lowest prediction error (4.6792 kW). Adding additional input variables with a low correlation might further decrease the prediction accuracy. Consequently, these five variables are the most suitable for serving as initial input variables for the predictive model. In GBDD, this paper introduces two distinct input strategies for training subsequent iterative models, denoted as Strategy Nos. 1 and 2 in Section 2. Strategy No. 1 involves employing the same input data (i.e., the aforementioned 5 meteorological factors) as the initial base learner to train and predict the subsequent base learners. This strategy facilitates the enhancement of predictive performance by adjusting the hyperparameters of DD without modifying the input data. In contrast, Strategy No. 2 builds upon Strategy No. 1 by training and predicting subsequent base learners with the input data that is not entirely identical. The underlying concept of Strategy No. 2 is that the residuals in the predictions of the initial base learner might stem from the influences of other meteorological factors that were not included as input. Thus, by utilizing the MIC to identify the group of meteorological factors that have the most significant impact on the residuals, Strategy No. 2 aims to fully leverage all meteorological data and subsequently improve predictive performance.

3.2 Data set processing

3.2.1 Abnormal data processing and data normalization

The abnormal data processing is done to eliminate the negative impact of the anomalous data on the training phase of the prediction model. The improved Hampel filter based on coupled information analysis [43] is used to replace outliers and fill in missing values for the data in the data set used for the experiments.

Data normalization aims to treat data entries in different ranges as consistent ranges, thus minimizing regression errors while maintaining the correlation between data sets. In this paper, min-max normalization is used to restrict the data between −1 and 1. All feature data are normalized before being input into the prediction model.

3.2.2 Division of data set

To construct and validate the GBDD model proposed, the records from May 22 to May 31, 2022, are used for testing, while the other records are used as training samples. Since PV cells do not generate electricity at night, only data from the daytime portion (7:10 am–18:40 pm) are considered. Based on the method proposed and the benchmark method, the performance is evaluated for the intraday (within 15 min) PV power prediction scenario.

3.2.3 Evaluation metrics

To evaluate the performance of the model proposed, MAE, root-mean-square error (RMSE), and mean absolute percentage error (MAPE) are used as evaluation metrics, whose equations are

(8)\mathrm{M}\mathrm{A}\mathrm{E}=\frac{1}{n}{\sum }_{i=1}^n\left|P_{\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{u}\mathrm{a}\mathrm{l}}\right(i)-P_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{t}}(i\left)\right|,

(9)\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{\frac{1}{n}{\sum }_{i=1}^n\left(P_{\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{u}\mathrm{a}\mathrm{l}}\right(i)-P_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{t}}(i){)}^2},

(10)\mathrm{M}\mathrm{A}\mathrm{P}\mathrm{E}=\frac{1}{n}{\sum }_{i=1}^n\left|\frac{P_{\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{u}\mathrm{a}\mathrm{l}}\left(i\right)-P_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{t}}\left(i\right)}{P_{\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{u}\mathrm{a}\mathrm{l}}\left(i\right)}\right|\times 100\mathrm{\%},

where P_actual and P_predict represent the actual and predicted PV power, respectively; and n represents the sampling points of the PV power generation period.

4 Experiment and discussion

In this section, the benchmark models for the experimental comparison are first presented. Then a case study is conducted to compare the results of PV power prediction with different algorithms. Finally, the feasibility of GBDD to improve the accuracy of other prediction methods is experimentally verified. All experiments are conducted utilizing the same processor Intel^® Core™ i7-7700HQ CPU @ 2.80GHz 2.81GHz and simulation environment MATLAB R2021a.

4.1 Benchmark models

The GBDD mode proposed is compared with the following eight benchmark models, i.e., single model: support vector regression (SVR), LSTM, and BP; ensemble model: XGBoost, particle swarm optimization (PSO)-LSTM-SVR-BP, gradient boosting support vector regression (GBSVR), gradient boosting long-short term memory (GBLSTM), and gradient boosting back propagation (GBBP).

PSO-LSTM-SVR-BPNN adopts an ensemble approach similar to that in Wu et al. [38]. The model is first trained with three seed methods for prediction, and then the final prediction results are obtained by assigning weights to the three methods through PSO.

The three benchmark models, GBSVR, GBLSTM, and GBBP adopt a model structure similar to that of GBDD. These three benchmark methods are used to evaluate the performance of GBM using other AI algorithms. The hyperparameters of the benchmark models and the GBDD model proposed were determined based on the grid search method, as shown in Tab.2.

4.2 Case study results and discussion

To illustrate the validity of the model proposed, ablation experiments are performed. The prediction results of GBDD are shown in Fig.7. The total accuracy of the model of both strategies is shown in Fig.8, where GBDD1 indicates that the model performed one iteration and GBDD2 indicates that the model performed two iterations.

It can be seen that the solution using Strategy No. 2 provides a more significant improvement in the prediction performance. As an ablation experiment, the MAE, MAPE, and RMSE of GBDD (Strategy No. 2) decreased by 38.85%, 32.63%, and 30.89%, respectively, compared to the those of DD.

This result indicates that the GBDD model proposed can effectively improve the prediction performance. Additionally, it can also be concluded that there is a limit to the improvement of prediction performance when only the most important relevant input variables are selected in the prediction model. To more effectively improve the prediction performance, other variables not considered in the prediction model need to be included. The GBDD mentioned in the subsequent experiments are trained based on Strategy No. 2.

Furthermore, the comparative experiments between GBDD1 and GBDD2 suggests that GBDD experiences a significant improvement in predictive performance after one round of gradient boosting iteration, while the improvement in results after the second iteration is limited compared to that of the first one. The reason for this is that GBDD is an excellent strong learner. After the first iteration, the residuals generated by the model is reduced to a large extent. This indicates that even with the addition of more iterations, the improvement in predictive performance becomes negligible, which is also the reason for the limited enhancement observed in GBDD2.

To further validate this feature, the prediction results of GBDD under one, two, three, and four iterations in Strategy No. 2 are compared, as shown in Fig.9 and Tab.5. From Fig.9, it can be seen that as the number of iterations accumulates, the prediction error does not significantly decrease. When the number of iterations increased from one to four, the increment of RMSE (kW) from GBDD1 to GBDD4 is only 0.68% ((0.13592212−0.1350)/0.1359 × 100% = 0.68%), while the computational cost quadruples, making it not cost-effective. This experiment demonstrates that GBDD, as an effective predictive model, only requires one iteration to achieve an excellent predictive performance. Increasing the number of iterations not only leads to limited improvements in predictive performance but also increases computational costs.

To analyze the impact of different weather conditions on the forecast model, the forecast data are classified according to the weather on the forecast day for a more detailed analysis. The test data set are classified according to weather conditions as sunny (May 22, May 25, and May 26), cloudy (May 23, May 24, May 25, and May 28), and overcast (May 29, May 30, and May 31). The results of the analysis are shown in Tab.6 while the values of residual compensation are shown in Fig.10 and Tab.7.

The results of MAE and MAPE show that the prediction performance of GBDD for overcast days is significantly compensated for, even better than the prediction improvement on sunny and cloudy days. The reason for this is that DD_i (i = 1, 2) in GBDD has a higher attention to individuals with larger residual values. Therefore, after several iterations, the model has a better predictive performance for the more fluctuating PV output cases. The comparison of the prediction results with benchmark models is summarized in Tab.8, from which, it can be seen that the GBDD model proposed has the smallest MAE, MAPE, and RMSE compared to benchmark models. The prediction performance of BP is the best of the single models.

XGBoost as a mature application of the GBM performs better in the PV power prediction problem, and is second only to the model proposed in terms of MAPE results, only behind the model proposed and the PSO-BP-SVR-LSTM model in terms of RMSE results.

From the results of PSO-BP-SVR-LSTM, the prediction of this ensemble model based on the bagging strategy is more accurate than that of a single model, but this advantage is very small compared with BP, the best single prediction model, and even larger than the MAPE value of BP in terms of MAPE index. By analyzing the weights assigned by PSO to the three single models, it can be found that the weights are set as α_BP = 0.43, α_SVR = 0.57, and α_LSTM = 0. This combination of weights means that LSTM is not added to the prediction model, which actually causes a waste of computational resources.

Comparing the results of BP and GBBP, it can be observed that the GBM with BP as the base learner makes the prediction performance worse. The reason for this result is that BP has a strong nonlinear fitting ability, which makes the residuals generated by BP in the training phase relatively small compared to the residuals generated in the test set. In addition, BP overfitting occurs during the iteration process, making the residuals predicted by BP unable to match the residuals actually generated, thus causing an increase in error during the model overlay.

Comparing the results of SVR and GBSVR, it can be seen that GBM with SVR as the base learner can further improve the prediction performance, but this improvement is very limited. Since the solution of SVR is sparse, its solution is limited to the support vector, which is a fraction of the full training sample. When training with residuals as the target, the training samples are highly volatile, which makes regression of SVR for more volatile samples not good enough. As a result, the improvement of prediction performance after gradient boosting iterations is limited.

Comparing the results of LSTM and GBLSTM, it can be seen that GBM with LSTM as the base learner can improve the prediction performance more significantly. This can be attributed to the fact that the poor parameter setting of the LSTM degrades the prediction performance, making it more of a weak learner than a strong one, and therefore the model has a better generalization, which results in a significant improvement in prediction performance after two GB iterations.

Furthermore, the computation time for each model is provided in Tab.8. It is obtained in the same environment for all cases. From the results, it can be observed that the model proposed has the best prediction accuracy while BP and SVR models have the shortest computation time, but a lower accuracy. For PV application, it is acceptable to optimize the prediction accuracy at the sacrifice of computational time within a certain range (e.g., less than 60 s in this paper).

In summary, it can be seen that the AI algorithm that can be applied to GBM should have both good fitting characteristics and generalization characteristics. From the experimental results, it is observed that DD is the AI algorithm that is more in line with the requirements.

4.3 Experiment with improvement of other prediction methods based on GBDD

The improvement of the existing method requires the collection of prediction results of the prediction method and true PV output data. In this part of the experiment, the prediction data of BP, SVR, and LSTM for the first 5 days of the test set (May 22 to May 26) and the prediction data of the training set are used as references. A GBDD model is trained to assist in predicting the PV power for the second 5 days of the test set (May 27 to May 31) by analyzing the relationship between the error generated by the model prediction and meteorological factors. The prediction evaluation results are shown in Fig.11 and Tab.9.

From the experimental results, it can be seen that an improvement in the prediction performance can be achieved by GBDD. Especially for the LSTM network, after optimization with GBDD2, its MAE decreases from 0.1058 to 0.0855 kW. For the BP and SVR networks, optimization with the GBDD network reduces the error by at least 3%.

Overall, the strategy is straightforward and effective by analyzing the relationship between forecast residuals and meteorological factors to compensate for the forecast results. Specifically, the strategy has the following advantages:

1) The whole process is similar to adding a plug-in model without interfering with the structure and operation of the original forecast model itself.

2) The strategy directly targets the prediction errors generated by the prediction model and thus has a significant performance improvement for PV power prediction.

3) Users can dynamically adjust the impact of GBDD on the prediction results of the original prediction model by changing the learning rate according to the prediction performance.

5 Conclusions

In this paper, to address the challenges posed by parameter optimization complexities, computational intensity, and resource waste in traditional short- and long-term PV power prediction methods, a novel GBDD model is proposed. Built upon the DD model as its base learner and utilizing matrix operation functions as neuron activation functions, the GBDD model proposed achieves cost-effective computations. Simultaneously, employing a gradient boosting strategy, it refines prediction outcomes through iterative learning of the relationship between prediction errors and various meteorological factors. As a result, it achieves more accurate PV power generation forecasts.

The gradient boosting strategy with different input variables is experimentally identified as the best model training scheme, which can make full use of the multivariate weather factor data available to predict PV output power. The results obtained verify that the model is effective in improving the prediction results, and the model proposed in this paper reduces the MAE, MAPE, and RMSE by 38.85%, 32.63%, and 30.89% respectively, compared to the initial model DD. Compared with other benchmark models, the model proposed can also achieve a better prediction performance. Compared to other AI algorithms that are GBM-based learners, DD is a more suitable AI algorithm as the base learner because of its good fitting and generalization properties.

In addition, based on the GBDD model proposed, a strategy to improve the prediction performance of other prediction models was suggested. Unlike the scheme of adjusting the model parameters and retraining to improve the model prediction accuracy, the method suggested does not require any adjustment to the original prediction model, but trains a new GBDD model to compensate for the residuals generated when the original prediction model predicts. The feasibility of the method is verified by experiments.

In summary, the GBDD model proposed cannot only achieve a good performance for ultra-short-term PV power prediction but also be used as an extension method for other prediction methods to improve the prediction performance of other prediction methods. If there are other prediction problems outside PVs that involve substantial feature selection or require further enhancement in predictive accuracy, the method proposed in this paper can also be adopted. In the subsequent research, extending the model proposed to other related fields shall be considered. In addition, further improvements or algorithmic additions can be applied to the GBDD model proposed.

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