1 Introduction

As nations around the world push for cleaner energy, the demand for renewable energy is obviously increasing, which highlights the importance of technological and economic issues [1]. Wind energy system is one of renewable energy and widely employed in the world. Wind turbines are mechanical devices that convert wind energy into electrical energy. The wind turbines is to convert the kinetic energy of the wind into mechanical energy, which is the motion of the shaft. This mechanical energy is converted into electrical energy in a turbine generator, and the resulting electrical energy can be used directly or stored in batteries. Recently, wind energy systems have been investigated, such as reverse supply network design [2] and optimal dynamic imperfect preventive maintenance [3]. The continuous innovation of wind energy technology has made this abundant renewable energy show its splendor. Therefore, the power output of wind energy prediction has become an essential issue for cleaner production [4, 5].

However, the power of wind energy is greatly affected by seasonal and regional factors, and the seasonal time series in the real world is a complex and nonlinear dynamical system. Seasonal time series prediction is an important field of prediction, especially for observing variables before collection and analysis to develop models that can describe potential relationships [6]. Some important factors must be considered in the process of wind power generation, such as wind speed, wind direction, air pressure, and temperature, as these factors can cause signal instability. Therefore, the data must be pre-processed before prediction. The fluctuation of the data signal can be improved using decomposition techniques to obtain a better characterization of the original data.

Moreover, it can be found from the literature that data usually contain many factors that cause signal instability. Previous scholars have solved this problem using decomposition techniques. Among them, wavelet transform technology is considered to be one of the best decomposition technologies. In addition, if the data will have different effects due to the season, seasonal factors should be identified and predicted by deseasonalized data. In the field of deep learning, LSTM has become one of the most frequently used methods for prediction because of its characteristics. Most current studies focus on LSTM combined and wavelet transform or LSTM combined with seasonal factors; however, Nourano and Behfat [7] and Ma et al. [8] explored the combination of LSTM with wavelet transform. Scholars regard the results of wavelet transform as seasonal results, not to find out its seasonal factors. Therefore, this study developed an LSTM with seasonal wavelet transform forecasting method for time series problems. Furthermore, the proposed method is examined for long-term wind farm electrical system power output forecasting. The long-term forecasting involves analyzing intricate systems with numerous interdependencies, such as economic, environmental, social, and technological systems. It is difficult to accurately capture all relevant variables and their interactions. This study attempts to propose an accurate forecasting model for long-term wind farm electrical system power output forecasting problem.

The main contributions are listed below:

  1. (1)

    The LSTM with seasonal wavelet transform forecasting adopted seasonal decomposition, wavelet decomposition, and LSTM to improve the time series prediction problem.

  2. (2)

    Considering the complementary characteristics of the regional resources, wind power can be delivered to a load canter through a large-scale and long-term power trade. The monthly wind power output forecasting can provide the evaluation of long-term power trade for decision-makers. Moreover, the performance of monthly wind power output forecasting could be effectively improved using LSTM with seasonal wavelet transform forecasting.

  3. (3)

    The LSTM can be extended to handle more complex time series problems using the seasonal wavelet transform method.

The remainder of this paper is organized as follows. Section 2 investigates several recent related studies on wind power forecasting, seasonal time series forecasts, pre-processed techniques, and LSTM. The assumptions and limitations of this study are presented in Sect. 3. LSTM with the seasonal wavelet transform forecasting method, including seasonal decomposition, wavelet decomposition, and LSTM technologies is introduced in Sect. 4. Section 5 presents the experimental setup for wind farm electrical system power output forecasting. Section 6 shows the experimental results of the LSTM with the seasonal wavelet transform forecasting method for wind farm electrical system power output dataset prediction. Section 7 discusses and analyzes our findings. Finally, this study draws conclusions, management insights, and proposes further work in Sect. 8.

2 Related Works

This section reviews recent studies on wind power forecasting, wind power forecasting with seasonal factors, pre-proceed techniques on wind power forecasting, and long short-term memory.

2.1 Wind Power Forecasting

Li et al. [9] applied support vector machines (SVM) with the cuckoo search algorithm and verified the optimization of their method using wind power farms in China. Fu et al. [10] optimized the parameters in SVM using the improved chicken swarm algorithm. The La haute bone data were used to verify and improve the accuracy of improved chicken swarm algorithm optimization SVM model. Li et al. [11] considered that the parameters in the proposed algorithm are important factors that affect the prediction results. Therefore, the improved dragonfly algorithm-support vector machines model (IDA-SVM) was developed for the output of wind power of wind power farms prediction in France. Compared with the, SVM optimized by dragonfly algorithm, genetic algorithm, and grid search method, back propagation neural network (BPNN), and Gaussian process regression (GPR) proposed by the previous literatures, the IDA-SVM has a relatively small prediction error in short-term wind power forecasting. Hossain et al. [12] proposed a combined prediction model of the gated recurrent unit (GRU) and convolutional neural network (CNN) and optimized two parameters of convolution layers by the grid search technique. They used the data of a wind power farm in Australia to verify the better accuracy of the method. Li et al. [13] developed an enhanced crow search algorithm optimization-extreme learning machine (ENCSA-ELM) model for improving the utilization efficiency of clean energy. Over the past several decades, scholars have worked to develop and improve prediction models for wind power generation [14, 15]. Although there are many approaches, they all aim to minimize prediction errors as well as calculation costs [16].

2.2 Wind Power Forecasting with Seasonal Factor

Malhan and Mittal [17] believed that the calculation of seasonal factors can improve the accuracy of prediction. Therefore, a novel prediction model combining the auto-regressive integrated moving average (ARIMA) and bidirectional-LSTM was developed. The performance of the method was validated with data from wind power farms in India. Godinho and Castro [18] concluded that wind power generation has seasonal factors that cannot be ignored and found that the best performance of wind power in Portugal could be predicted by the adaptive neural fuzzy inference system (ANFIS). Liao et al. [19] developed the fuzzy seasonal long short-term memory network (FSLSTM). Using the example of a wind power farm in Taiwan, they verified that FSLSTM can effectively solve data containing seasonal factors and obtain the best prediction results among the methods proposed by previous scholars.

2.3 Pre-proceed Techniques on Wind Power Forecasting

Li et al. [11] proposed wavelet decomposition-improved atomic search algorithm-SVM to predict wind power generation, using wavelet decomposition data to reduce the error caused by unstable wind power generation. The performance of this model was verified by data from a wind power farm in France. Dong et al. [20] used complete symplectic geometry, ensemble empirical, wavelet packet, and variational decomposition models to decompose wind power generation data. It was found that better prediction results can be obtained using the WPD. Gupta et al. [21] believed that the decomposition technology of wavelet transform can reconstruct without data loss and predicted wind power generation in India through wavelet transform and ARIMA. Duan et al. decomposed the data by VMD and calculated the number of neurons in deep belief networks (DBN) by particle swarm optimization (PSO). Moreover, LSTM also has been used to predict wind power generation in Shaanxi, decomposed wind speed, wind power, and wind direction data by wavelet transform [22]. The wind direction and wind speed were first predicted, and the wind power generation was then predicted from these two prediction results. Taking the data of a wind power farm in Spain as an example and comparing the results with the methods proposed by previous scholars, it was verified that their method had better performance.

2.4 Long Short-Term Memory

LSTM was developed based on the recurrent neural network (RNN) [23]. When RNN memorizes information for the long term, it accumulates and calculates the previous information, causing the explosion or disappearance of neural network and inaccurate predictions. The purpose of LSTM is to memorize more previous information, including short-term information and long-term information. LSTM determines whether the information is useful before adding or deleting information to improve the reliability of the neural network. Punia et al. [24] successfully applied LSTM and random forests to forecast demand. The LSTM could model complex relationship between consideration time and regression which obtain good robustness of their forecasting proposition [25]. Numerous scholars also have combined LSTM with wavelet transformations and seasonal factors, respectively, and it has been applied in a wide range of fields.

LSTM networks have been applied successfully for the prediction of various natural quantities, after applying suitable decomposition procedures on the raw data, prior to the training procedure. For example, Liu et al. [26] used wavelet transform to pre-process the data, used LSTM to predict the power output of three wind power farms in China, The Netherlands, and Mongolia, and verified the accuracy of the prediction. Li et al. [27] predicted the displacement behavior of dams in China by LSTM, in which the influencing factors included the water level and temperature, both of which vary significantly depending on the season. Therefore, the scholars used the seasonal-trend decomposition procedure based on loess (STL) to decompose the data and predicted the seasonal factors by the extremely randomized trees method (ExtRa-Trees). They verified the stability and efficacy of the method after comparing it with seven commonly used methods. Sun et al. [28] decomposed the sea-level data of China into the trend, seasonal, and random by wavelet transformation, and then predicted the trend and season by seasonal ARIMA and the random by LSTM. The results verified that the accuracy of seasonal ARIMA + LSTM is better than ARIMA and LSTM. Yin et al. [29] proposed LSTM based on STL for the price prediction of vegetables, and the results confirmed that seasonal LSTM improves the prediction accuracy of LSTM. Al-Janabi et al. [30] utilized PSO to determine the optimal structure of the LSTM to predict the air pollution concentration. The results show that this model increases the complexity of the LSTM but reduces the execution time. Dudek et al. [31] decomposed the data using error, trend, and seasonality exponential smoothing (ETS) to identify seasonal factors, and used LSTM to forecast the electricity demand for 35 countries in Europe. The results were compared with 15 methods of machine learning to validate the effectiveness of the method. Lv et al. [32] used VMD to decompose the power grid security data of the US and Singapore into seasonal factors and used LSTM to make predictions for every month. The proposed method found that the predictions could meet the actual demand. Nourani and Behfar [7] decomposed the data by wavelet transform. The WLST method was verified to have the superior accuracy using three runoff-sediment in rivers in the US Okedi and Fisher [33] used STL to decompose data on the density of bio-photovoltaics and used LSTM to make predictions and verify the accuracy of the model. Shahid et al. [34] proposed a genetic algorithm-LSTM (GALSTM) to optimize the parameters in LSTM and validated the results of the proposed method using wind power farms in Europe. Lu et al. [35] used VMD to decompose the raw data into high, medium, and low complexity data. The features in the data were identified by CNN as the inputs in the LSTM prediction model. The performance of the proposed model was validated through real cases of four wind power farms in China and was compared with eight methods proposed by previous scholars. Table 1 shows the related literature for LSTM and power output forecasting.

Table 1 Related literatures for LSTM forecasting
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3 Hypothesis and Limitations

Our research is based on three hypotheses and limitations. First, this study exclusively focuses on cases from Taiwan, ensuring geographic consistency and relevance of our data. Second, this study operates under the presumption that wind power forecasting in Taiwan exhibits seasonality, suggesting that different seasons could influence wind power outputs in distinct ways. Finally, given the characteristics of power forecasting, this study hypothesizes that it inherently encompasses both high-frequency and low-frequency components. Consequently, this study employs wavelet transform to decompose these frequency components, aiming to capture and predict variations in wind power outputs more effectively. These assumptions provide a clear direction and emphasis for this study, ensuring the proposed method and findings are pragmatically valuable.

4 LSTM with Seasonal Wavelet Transform Forecasting Method

The time series problem has been investigated in a lot of research. Recently, deep learning LSTM has been employed to handle the time series problem. However, improved deep learning methods for time series problems should be developed. This study developed an LSTM with seasonal wavelet transform forecasting model that could effectively handle the time series problem by novel multi-decompositions. The novel multi-decomposition deep learning models could obtain the advantages of wave transform, seasonal decomposition, and LSTM in actual time series problems. The flowchart of the proposed model is in Fig. 1. This study presents four stages for building LSTM with seasonal wavelet transform model which explained in algorithm as follows. The procedure of the LSTM with seasonal wavelet transform is as follows.

Fig. 1
figure 1

Flowchart of the LSTM with seasonal wavelet transform forecasting method for time series problems

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Step 1. Collect the dataset. The raw dataset is collected and arranged into a database and divided it into the training, validation, and testing datasets, respectively.

Step 2. The first-level seasonal. This study adopted the first-level decomposition is seasonal decomposition method. The raw dataset is divided into the trend and seasonality of the time series, respectively.

Step 3. Perform wavelet analysis. The trend dataset is divided into low- and high-frequency series datasets using wavelet analysis. The wavelet analysis identifies where a certain frequency exists in the temporal or spatial domain. Wavelet analysis can achieve that the following forecasting method easily approach series datasets.

Step 4. Perform LSTM. LSTM is an effective prediction model in the deep learning field. This study adopted LSTM to forecast the low- and high-frequency series datasets. Furthermore, the training and validation dataset adopted Adam optimization [36] to find the optimal parameters of LSTM, which is a method for stochastic optimization.

Step 5. Perform wavelet synthesis. The prediction values of the low- and high-frequency series datasets are synthesized. The seasonal index also is synthesized the prediction values.

Step 6. Present the final forecasting result. This study used the mean absolute percentage error [MAPE (%)] to measure the result of forecasting and provide a report for the decision-makers.

Algorithm
figure a

LSTM with seasonal wavelet transform

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4.1 LSTM with Seasonal Wavelet Transform Forecasting Method

  1. (1)

    1-Level seasonal decomposition: in seasonal decomposition, a time series is described using the additive and multiplicative decomposition models. In the additive decomposition model, the linear trend is a straight line and the linear seasonality has the same frequency and amplitude. In the multiplicative decomposition model, it is common with economic time series. The trend is a curved line and the nonlinear seasonality has an increasing or decreasing frequency or amplitude over time. The decomposition model can be referred to [37]. The additive and multiplicative decomposition models could be defined as follows:

    $$ {\text{Multiplicative}}\,{\text{decomposition}}\,{\text{model}}:\,\hat{Y}_{t} = f(T_{t} ) \times S_{t} \times \varepsilon , $$
    (1)

    where\(\hat{Y}_{t}\) is the value of forecast of the time series at time t; f(Tt) is the estimated value of trend Tt at time t; St is the seasonal influence at time t; and ε is the model noise.

  1. (2)

    2-level wave transformer: The new deep learning forecasting method was based on wavelet analysis and seasonal decomposition. Wavelet transform is the result of wavelet analysis. Although the theoretical basis of wavelet is difficult to understand, in the application, wavelet transformation has great similarities with subband coding, which is not difficult to implement. Numerous researchers have proposed various compression and decompression methods. Famous methods include the set partitioning in hierarchical trees, the embedded zerotree wavelets, and embedded block coding with optimized truncation compression. The two compression methods of zero tree compression and architecture tree set segmentation compression are based on the concept of spatial orientation trees to describe the wavelet coefficients of two-dimensional images and use the characteristics of spatial orientation trees to obtain better compression performance. The difference between the two methods is the searching method of the spatial direction tree and coded symbols.

Analytic wavelet is a good method when using continuous wavelet transform (CWT) for time–frequency analysis, because the coefficients of wavelet are complex, and these coefficients’ information about the phase and amplitude of the signal being analyzed.

Analytic wavelets are well suited for studying how the frequency content of nonstationary signals evolves over time in the real world. Figure 2 shows the flowchart of the wavelet transform.

Fig. 2
figure 2

Wavelet transform flowchart with seasonal decomposition

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In Fig. 2, the low- (XWL) and high-frequency (XWH) series datasets can be formulated as follows (Percival and Walden [38]):

$$ X_{{{\text{WL}}}} = \sum\limits_{k = - \infty }^{\infty } {f(T_{t} )[k] \cdot h[t - k]} $$
(2)
$$ X_{{{\text{WH}}}} = \sum\limits_{k = - \infty }^{\infty } {f(T_{t} )(k) \cdot g_{W} [t - k]} , $$
(3)

where h[t-k] is the low-frequency analysis; and gW[t-k] is the high-frequency analysis.

  1. (3)

    This study adopted low- and high-frequency analysis, and employed LSTM to predict the value of trend with low and high frequency. \(\tilde{x}_{{{\text{WT}}}}\) has a rapidly decreasing function in frequency and is small outside of some intervals. Orthogonal and biorthogonal wavelets are usually designed to have compact support in a time series. The energy concentration of wavelets with compact support in time is lower than that of wavelet with rapid reduction in time. Most orthogonal and biorthogonal wavelets are asymmetric in the Fourier domain. Furthermore, the construction of the LSTM method for inputting low-frequency series dataset {(t, XWL), t = 1, 2, …, N} and the high-frequency series dataset {(t, XWH), t = 1, 2, …, N} can be represented as follows:

    $$ \left\{ {\begin{array}{*{20}c} {X_{{{\text{WL}}}} = Y_{{{\text{tL}}}} = o_{{{\text{tL}}}} \otimes \tan h(c_{{{\text{tL}}}} )} \\ {X_{{{\text{WH}}}} = Y_{{{\text{tH}}}} = o_{{{\text{tH}}}} \otimes \tan h(c_{{{\text{tH}}}} )} \\ \end{array} } \right., $$
    (4)

    where long-term states ctL and ctH and output gates otL and otH can be estimated, respectively. An LSTM unit contains four layers. The operation of an LSTM using a low-frequency series dataset can be written as the follows:

Input gate:

$$ I_{{{\text{tL}}}} = \sigma \left( {W_{{t_{{{\text{MI}}}} }}^{T} t + W_{{h_{I} }}^{T} h(t - 1) + b_{t} } \right). $$
(5)

Forget gate:

$$ f_{{{\text{tL}}}} = \sigma \left( {W_{{{\text{tf}}}}^{T} t + W_{{{\text{hf}}}}^{T} h(t - 1) + b_{f} } \right). $$
(6)

Output gate:

$$ o_{{{\text{tL}}}} = \sigma \left( {W_{{{\text{to}}}}^{T} t + W_{{{\text{ho}}}}^{T} h(t - 1) + b_{{{\text{to}}}} } \right). $$
(7)

Neuron input cell input:

$$ g_{{{\text{tL}}}} = \sigma \left( {W_{{{\text{tg}}}}^{T} x_{{{\text{Li}}}} + W_{{{\text{tg}}}}^{T} h(t - 1) + b_{{{\text{tg}}}} } \right), $$
(8)

where \({b}_{i},{b}_{f},{b}_{o},{b}_{g}\) are the deviation terms of each of the four layers; σ represents the sigmoid function 1/(1 + exp(−t))\(;\) and tanh is the hyperbolic tangent function (exp(t)−exp(−t))/(exp(t) + exp(−t)).

The long-term and short-term states are calculated as follows:

$$ {\text{Long}} - {\text{term state}}:c_{{{\text{tL}}}} = f_{{{\text{tL}}}} \times c_{{\text{tL - 1}}} + I_{{{\text{tL}}}} \times g_{{{\text{tL}}}} . $$
(9)

The LSTM with a trend low-frequency series can be represented as

$$ Y_{{{\text{tL}}}} = o_{{{\text{tL}}}} \times \tanh (c_{{{\text{tL}}}} ). $$
(10)

The LSTM with a high-frequency series can be approach based on the same procedures, as follows:

$$ Y_{{{\text{tH}}}} = o_{{{\text{tH}}}} \times \tanh (c_{{{\text{tH}}}} ). $$
(11)

The seasonal prediction values (\({\hat{\text{Y}}}_{{{\text{SWT}}}}\)) based on the low and high frequency found using LSTM can be synthesized by the following equation [23]:

$$ {\hat{\text{Y}}}_{{{\text{SWT}}}} = S_{t} \times (\tilde{y}_{L} + \tilde{y}_{H} {)}\,{ = }\,S_{t} \times ({\text{(Y}}_{{{\text{tL}}}} \tilde{ * }h) + ({\text{Y}}_{{{\text{tH}}}} \tilde{ * }g_{W} )), $$
(12)

where \(\tilde{ * }\) is solids of revolution using integration.

  1. (4)

    Furthermore, this study uses MAPE (%) to measure the prediction accuracy of the proposed model. The formula of MAPE (%) is shown in Eq. (13).

    $$ {\text{MAPE}}(\% ) = \frac{100}{M}\sum\limits_{t = 1}^{M} {\left| {\frac{{Y_{{{\text{Ct}}}} - Y_{C} (x_{t} )}}{{Y_{{{\text{Ct}}}} }}} \right|} , $$
    (13)

    where M is the number of forecasting periods, \(Y_{Ct}\) is the actual value, and \(Y_{C} (x_{t} )\) is the forecasting value at period t.

5 The Experimental Setup

5.1 The Data

The transition to an energy economy based on renewable resources involves difficult issues related to the capacity, reliability, and variability of non-dispatchable energy resources, such as tidal energy, solar, or wind. Renewable energy is planned to generate 20% of Taiwan’s electricity needs by the end of 2025. Therefore, the prediction of wind farm electrical system power output is an essential issue in Taiwan.

The wind farm electrical system power output is greatly affected by seasonal and regional factors, and the real-world seasonal time series is a nonlinear, dynamic, and complex system. Seasonal time series prediction is an important field of prediction in which previous observations of the same variable are collected and analyzed to develop a model to describe the potential relationship. Data on the monthly wind power output in Taiwan were collected from the website (https://www.taipower.com.tw/en/index.aspx), as shown in Fig. 3. This study used the monthly wind power output (time series dataset) of Changgong, Mailiao, Guangyuan, and Wanggon wind farms, which have the top four wind generating capabilities in Taiwan, to examine the proposed forecasting method.

Fig. 3
figure 3

The wind farm electrical system power output of different regions in Taiwan

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The wind farm electrical system power output can be divided into different regions and set the collection period in Taiwan. In this study, the training set was designed from January 2016 to October 2018. The validation set was designed from November 2018 to November 2019. The testing set was designed from December 2019 to March 2021.

5.2 The Parameters of LSTM with Seasonal Wavelet Transform Forecasting Method

In this study, the parameters of the validation and testing set for the LSTM with seasonal wavelet transform forecasting method are as shown in Table 2. The seasonal index (trend multiplied cycle) can be calculated by the three quarter moving averages. Therefore, the index 1 to 3 in Table 2 represents the seasonal index for 1 to 3 quarter, respectively. The learning rate, number of hidden neurons, and termination condition of the proposed method were 0.05, 200, and 250 epochs, respectively.

Table 2 The parameters of proposed LSTM with seasonal wavelet transform forecasting method
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6 Results for Various Wind Farm Electrical System Power Output Forecasting Examples

This study designed an LSTM with seasonal wavelet transform forecasting method for forecasting wind power output in Taiwan. In this study, the SLSTM [19], WLSTM [39], SARIMA [37], BPNN [40], GRNN [41], LSSVR [42], and SVR [43] methods were examined. The experimental results and average MAPE (%) obtained using various models in different wind power farms in Taiwan are shown in Table 3. The MAPE (%) was ranked in the order of the proposed method, followed by WLSTM, SLSTM, BPNN, LSSVR, GRNN, SARIMA, and SVR. It can be observed that the proposed LSTM with seasonal wavelet transform forecasting model is superior to the other models in forecasting accuracy, with an average MAPE of 6.237%. The WLSTM and SLSTM model performances are relatively similar and ranked second and third, respectively. The SVR model performed worst on all datasets, with an average MAPE of 92.816%. The superiority of the proposed model was mainly due to its ability to multi-decomposition deep learning models. The multi-decomposition deep learning models had the advantages of wave transform, seasonal decomposition, and LSTM in actual Taiwan wind power output forensicating. Table 3 also shows that the performance of the proposed method is robustly better than that of the traditional methods and could obtain better performance.

Table 3 Forecasting performances of various models
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The Mann–Whitney test is a nonparametric statistical test used to determine the differences between two independent samples. The W-value is the test statistic, and the p-value represents the significance of the statistic. When the p-value is smaller than a specific threshold (e.g., 0.05), the two samples are usually considered to be significantly different. The Mann–Whitney test was implemented to test the forecasting results, as shown in Table 4.

Table 4 Mann–Whitney test for the forecasting results
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The proposed method often has notably low p-values compared with the other methods, indicating significant differences in their performance. Various forecasting methods, such as GRNN and LSSVR, have multiple instances with a p-value of 0.000, implying extremely substantial differences compared with other techniques. The test results exhibit considerable differences among the proposed method, SLSTM, WLSTM, SARIMA, BPNN, GRNN, LSSVR, and SVR methods, indicating that the forecasting results of the performance of the proposed method, SLSTM, and WLSTM methods are superior to those of the traditional approaches based on MAPE (%), especially in the Guangyuan wind power farm example.

Figures 4, 5, 6 and 7 present point-to-point comparisons of the actual and forecasted values of the wind power outputs of different farms. The actual values, predicted values, and errors for Changgong wind power farm are shown in Fig. 4. As shown in Fig. 4a–h, the error values of the SLSTM, WLSTM, SARIMA, BPNN, GRNN, LSSVR, and SVR methods were higher from October 2020 to March 2021. SLSTM cannot efficiently capture the trend of the data of the Changgong wind power farm. WLSTM could not efficiently capture the seasonal data of the Changgong wind power farm. SARIMA, BPNN, GRNN, LSSVR, and SVR methods could not obtain outperformance from October 2020 to March 2021 than the LSTM with seasonal wavelet transform forecasting method in the Changgong wind power farm.

Fig. 4
figure 4

Forecasting results of various models for the Changgong wind power farm. a Proposed method, b SLSTM, c WLSTM, d SARIMA (1, 0, 0) (1, 1, 0)12, e BPNN, f GRNN, g LSSVR, and h SVR

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Fig. 5
figure 5

Forecasting results of various models for the Mailiao wind power farm. a Proposed method, b SLSTM, c WLSTM, d SARIMA (1, 0, 0) (1, 1, 0)12, e BPNN, f GRNN, g LSSVR, and h SVR

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Fig. 6
figure 6

Forecasting results of various models for the Guangyuan wind power farm. a Proposed method, b SLSTM, c WLSTM, d SARIMA (1, 0, 0) (1, 1, 0)12, e BPNN, f GRNN, g LSSVR, and h SVR

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Fig. 7
figure 7

Forecasting results of various models for Wanggong wind power farm. a Proposed method, b SLSTM, c WLSTM, d SARIMA (1, 0, 0) (1, 1, 0)12, e BPNN, f GRNN, g LSSVR, and h SVR

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The actual values, predicted values, and errors for Mailiao wind power farm are shown in Fig. 5. As shown in Fig. 5a–h, the error values of the WLST and SARIMA methods were higher from October 2020 to March 2021. The proposed method and SLSTM could capture the trend and seasonal data efficiently of the Mailiao wind power farm. WLSTM could not capture the seasonal data efficiently of the Mailiao wind power farm. SARIMA, BPNN, GRNN, LSSVR, and SVR also could not capture the trend of the data of the Mailiao wind power farm. The proposed method could obtain outperformance than other methods for the Mailiao wind power farm.

The actual values, predicted values, and errors for Guangyuan wind power farm are shown in Fig. 6. As shown in Fig. 6a–h, the error values of the SLSTM, SARIMA, BPNN, GRNN, LSSVR, and SVR methods were higher from October 2020 to March 2021. The proposed method and WLSTM could efficiently capture the trend and seasonal of the data from the Guangyuan wind power farm. SLSTM could not efficiently capture the trend of the data of the Guangyuan wind power farm, and SARIMA, BPNN, GRNN, LSSVR, and SVR could not capture the trend of the data. The proposed method could successfully obtain outperformance than the other methods for the Guangyuan wind power farm.

The actual values, predicted values, and errors for Wanggong wind power farm are shown in Fig. 7. As shown in Fig. 7a–h, the error values for the SLSTM, WLSTM, SARIMA, BPNN, GRNN, LSSVR, and SVR methods were higher from October 2020 to March 2021. The proposed method could efficiently capture the trend and seasonal of the data of the Wanggong wind power farm. SLSTM could not capture the trend of the data efficiently of the Wanggong wind power farm. The WLSTM could not capture the seasonal data efficiently of the Wanggong wind power farm. SARIMA BPNN, GRNN, LSSVR, and SVR could not capture the trend and seasonal of the data of the Wanggong wind power farm. The proposed method could also successfully obtain outperformance than other methods for the Wanggong wind power farm.

7 Discussion

In this study, the LSTM with seasonal wavelet transform forecasting method is proposed and compared with seven other methods. The LSTM-based methods, including the proposed method, SLSTM, and WLSTM, can obtain better performance than traditional SARIMA, BPNN, GRNN, LSSVR, and SVR. The analysis of the four wind farms power output forecast models using the LSTM with seasonal wavelet transform forecasting method revealed a number of phenomena:

  1. (1)

    SARIMA can only obtain good performance in Changgong wind power farm. The reason may be the power output in Changgong wind power farm has a strong seasonal influence. Smaller wind farm power output cannot obtain superior performance.

  2. (2)

    In comparison of LSTM-based methods, the performance ranking is proposed method > WLST > SLSTM. The wavelet transform can be evinced in the four examples that effectively handle wind power farm power output forecasting problem.

  3. (3)

    The LSTM with seasonal wavelet transform forecasting method could efficiently handle seasonal influences. The seasonal influence of wind power output can be observed in the Top 4 wind power farms in Taiwan.

  4. (4)

    In all examples, the performance of the LSTM with seasonal wavelet transform forecasting method is better demonstrated and accurately captures the trends and seasonal of the wind power output. This is SLSTM and WLSTM cannot be observed in the four examples.

  5. (5)

    The proposed method combines seasonal decomposition, wavelet transform, and LSTM to improve the time series prediction problem comprehensively, which can perform better than the other methods. It means that it can capture the variations in the data due to seasonality and further decompose the instability signals. The proposed method could therefore be used as a forecasting model for the wind power output in Taiwan.

8 Conclusions, Managerial Insights, and Further Research

8.1 Conclusions

Due to worldwide concern regarding sustainable development goals (SDGs), wind power output forecasting has become an essential issue for cleaner production. In this study, an LSTM with seasonal wavelet transform forecasting method is developed to predict the power output of wind farms in Taiwan by exploiting the unique strengths of seasonal decomposition, wave transformer, and the LSTM technique. The experimental results indicate that the proposed method offers a promising alternative for predicting wind power output in Taiwan. The commendable performance of the proposed method stems from its dual capability: the fusion of seasonal decomposition and wavelet transform adeptly captures data trends and seasonality, while the LSTM ensures reliable forecasts. This innovative combination signifies a breakthrough, marking the first extension of LSTM to synchronize with wavelet transform and seasonal decomposition. The multi-layered mechanism of the proposed approach is distinctly equipped to tackle the intricacies of power output forecasting, consistently delivering precise predictions. Such a pioneering endeavor not only amplifies our understanding of wind power forecasting but also offers invaluable insights into management sciences, deep learning, and predictive analytics.

8.2 Managerial Insights

Nowadays, wind power output has become one of the core electrical systems. With its exemplary performance, the proposed forecasting method has significant practical implications for wind farm electrical systems. By employing LSTM combined with the seasonal wavelet transform, managers can achieve higher accuracy in forecasting wind farm electrical output. The accurate long-term wind farm electrical system power output forecasting (LSTM with seasonal wavelet transform) plays a crucial role in the planning, operation, and optimization of wind power generation. Long-term power output forecasting helps energy planners and grid operators estimate future electricity production from wind farms. This information is essential for determining the optimal mix of energy sources and planning the deployment of other power generation technologies to meet future electricity demand. It gives decision-makers a solid foundation to allocate resources judiciously and make enlightened investment choices.

Long-term forecasting also provides valuable insights into the expected power generation levels over extended durations, such as months or years. This information assists grid operators in managing the integration of wind power into the electricity grid and maintaining system stability. This predictive capability facilitates the effective scheduling and deployment of alternate power sources, such as conventional thermal plants or energy storage, to compensate for fluctuations in wind power output. For wind farm developers and operators, precise long-term output predictions are invaluable. It enables them to optimize the design, layout, and capacity decisions for their wind farms. Armed with projections of future energy production, they can make informed decisions about the quantity and variety of wind turbines to install, as well as the overall configuration of the wind farm. Additionally, long-term forecasting aids in identifying maintenance needs and scheduling preventive maintenance activities to ensure the long-term reliability and performance of wind farm assets.

Precision in long-term power output forecasting empowers wind farm operators to carve a prominent niche in the energy market. With accurate projections, they can effectively strategize around power purchase agreements, hedging tactics, and avenues for revenue maximization. By clearly understanding future energy production, operators can negotiate contracts and develop financial models that bolster revenue while mitigating risks associated with price fluctuations. Long-term power output forecasting for wind farms is crucial for achieving renewable energy integration targets and decarbonization goals. By accurately predicting future wind power production, policymakers and stakeholders can assess the contribution of wind energy to the overall energy mix and develop strategies to increase renewable energy penetration. Long-term forecasting helps plan the necessary infrastructure, transmission capacity, and storage solutions required to support a higher share of wind power in the grid.

8.3 Future Research

Future work can investigate using LSTM with seasonal wavelet transform forecasting method for other types of time series data. Future research could also consider using hybrid deep learning models, such as convolutional neural network-LSTM hybrids, to improve the accuracy of forecasting of the LSTM with the seasonal wavelet transform forecasting method.