Power forecasting method of ultra-short-term wind power cluster based on the convergence cross mapping algorithm☆

0 Introduction

Under the overarching framework of‘‘carbon peak,carbon neutrality,” wind power generation has emerged as a critical development focus due to its advantages,including the elimination of long-distance transmission requirements,reduced land costs,and high social acceptance.

Wind power forecasting(WPF)is an important aspect of wind farm grid connections, and its accuracy is critical for enhancing the safety and economic operation of power grids[1].However,as the spatial coverage of wind power clusters expands, the input data required for site-level forecasting not only changes dynamically with wind speed fluctuations but also exhibits increased variability in its effectiveness for cluster total power, thereby complicating ultra-shortterm power forecasting of cluster wind power[2,3].

The dynamic changes and increasing disparity in the effectiveness of site-level forecast input data, coupled with varying wind speed fluctuation processes, present significant limitations to further improvements in forecasting accuracy.Consequently, there is an urgent need for highprecision power forecasting to provide reliable information regarding output variations [4].Forecasting methods grounded in statistical model frameworks, such as those leveraging deep learning,have gained prominence in recent years, driven by the continuous expansion of wind power operation data storage and the rapid advancements in AI technologies.[5-8].

Current research on deep learning-based prediction methods can be broadly categorized into two approaches.The first focuses on enhancing the mechanisms of deep learning networks to improve their ability to extract time-series features from data.The second approach aims to identify the characteristics of fluctuation processes and output scenarios, classify them accordingly, and then establish forecasting models for each category of weather processes or individually correct forecasting errors [9,10].

The first method[11]integrates the mutual information method with the false nearest neighbor method to reconstruct the phase space of wind power sequences,proposing an innovative chaotic time-series forecasting framework using a kernel function switching mechanism.In[12],convolutional neural networks (CNNs) and long short-term memory models (LSTMs) were combined, where CNNs extracted the spatial characteristics of long-term wind speeds, and LSTMs focused on short-term time-series features, allowing for wind speed forecasts across multiple locations.In [13], wind speed forecasting accuracy was improved by refining the LSTM-TCN model by considering turbine wake effects, unit status, and spatial distribution in wind farms.The second type of method [14] uses the wind speed vector and the daily change in pressure in numerical weather prediction (NWP)as the basis for cluster analysis to classify the types of weather and establish forecasting models for different weather types to improve the accuracy of forecasting.In [15], a short-term wind power forecasting method was proposed, combining variable-scale time windows and fluctuation feature extraction.It uses a variable-scale sliding window algorithm to dynamically extract features based on current wind speed characteristics, classify historical wind power data, and select specific parameters to build forecasting models.Reference [16] introduced a segmented shortterm prediction method for wind power clusters by adaptively dividing transition weather periods.Different forecasting methods have been applied to transition and non-transition periods to enhance the adaptability of wind power forecasting under extreme weather conditions.

The literature explores various approaches to improve ultra-short-term power forecasting accuracy.However,analyses of the relationships between meteorological factors and cluster power changes predominantly rely on linear or nonlinear correlation analyses, which yield symmetrical but non-directional results.Consequently,ultrashort-term forecasting models based on strongly correlated meteorological factors often exhibit low interpretability and limited generalization performance[17-20].

This study proposes applying a convergent crossmapping (CCM) algorithm-based causal connection analysis to quantitatively assess the causal relationship between cluster power and multiple meteorological factors.Compared with correlation analysis, causal analysis delves deeper into the statistical and physical relationships between power changes and meteorological factors,emphasizing the directionality of time-series data feature extraction.This approach significantly reduces the influence of redundant factors during training and modeling,thereby markedly enhancing power forecasting accuracy.Furthermore, given the varying causal relationships between multiple meteorological factors and cluster total power fluctuations under different fluctuation processes,a clustering model is developed to perform cluster analysis on multiple fluctuation processes.This model accurately identifies forecasting factors under distinct fluctuation processes.Based on the CCM algorithm,the study introduces a refined modeling technique for ultrashort-term wind power cluster power forecasting.

The main contributions of this method are as follows:

1) A cluster-level wind process description index system and classification method, considering wind speed and wind direction characteristics at multiple points within the wind power cluster,are proposed.This system characterizes cluster output variation under different wind processes, establishing a foundation for refined wind farm cluster power forecasting modeling.

2) A ‘‘multivariate meteorological forecasting factorcluster total power fluctuation” causal relationship analysis method, integrating the CCM algorithm, is suggested.This method quantitatively evaluates the causal mapping relationship between different meteorological forecasting factors and cluster power,emphasizing the directionality of time-series data feature extraction.It transforms traditional correlation-based modeling into deep causal relationship modeling, thereby improving model robustness.

3) A refined modeling approach for wind power cluster power forecasting,aligned with fluctuation processes,is proposed.This approach utilizes strong causal meteorological factors identified under different typical fluctuation processes as inputs to develop targeted forecasting models for these processes, thereby enhancing the refinement level of forecasting models.

4) Using data from an offshore wind power cluster in China, the forecasting accuracies of various deep learning models applying the proposed method were compared and tested.Example analysis results demonstrate that the proposed method improves forecasting accuracy by 1.57%compared with traditional methods.

1 Methodology

The main contents of this study revolve around four research tasks:cluster analysis of historical meteorological data, causal analysis based on the CCM algorithm, establishment of an ultra-short-term power forecasting model,and error analysis of the forecasting results.The overall process of this study is illustrated in Fig.1.

The data used in this study were the historical data of a wind farm cluster consisting of five wind farms over the past three months, including the wind speed, wind direction, power of each wind farm, and cluster total power.

1.1 A classification method for typical fluctuating processes

This section explores a classification method for typical fluctuation processes with multi-temporal and spatial scale characteristics.First, a fluctuation process description index system for power fluctuations is proposed.Five indicators were used to cluster the power fluctuations under different fluctuation processes.Finally,several typical fluctuation processes with similar power fluctuation characteristics are obtained, and several typical output change characteristics are summarized.

This section first presents a study on the establishment of typical fluctuation processes.Cluster analysis was performed based on the fluctuation feature dataset, and several typical fluctuation processes were extracted from a large number of fluctuation processes.Fig.2 shows a flowchart for constructing the weather fluctuation characteristics dataset.

Fig.1 The overall flowchart of this study.

Fig.2 Flow chart of constructing weather fluctuation characteristics dataset.

The standard for ultrashort-term power forecasting involves predicting the cluster’s power in 15-minute intervals over the next 4 h [21].The fluctuations in meteorological data and power within this 4-h period are considered to be a fluctuation process.Time series data were sliced into 4-h segments, with each segment representing a fluctuation process that included historical meteorological data.

After completing the division of the fluctuation processes, it is necessary to quantitatively analyze the characteristic quantities of each fluctuation process.This study aims to describe the fluctuating characteristics of wind power clusters in terms of time and space for each fluctuation process, based on the causal mapping relationship between multiple meteorological factors and the total power of the cluster.The five wind farms comprising the wind power cluster have correlated wind speeds and directions.The degree of fluctuation in space is characterized by the wind speed variance and wind direction variance, while the degree of fluctuation in time is characterized by the total power variance of the wind power cluster.The total power change trend and the average fluctuation amplitude are used to characterize the output trend and amplitude and summarize the temporal dynamic changes in the output characteristics.Five meteorological factor fluctuation quantities, namely ‘‘wind speed variance, wind direction variance, total power variance, total power change trend,and average fluctuation amplitude [22],” are set to quantitatively characterize the spatial and temporal dynamic changes of multiple meteorological factors and the total power of the cluster during the fluctuation process.These five fluctuation quantities were used as characteristic quantities for the cluster analysis of each fluctuation process.The formula is as follows:

Wind speed, wind direction variance, and cluster total power variance value in each fluctuation process can be calculated, and three variance values of , and ,respectively, can be obtained as follows:

where N is the total number of wind speed/direction/power data during a fluctuation;i=1,2,...,N is the time point during a fluctuation; xi is the measured wind speed/direction/power data at i time point; μ is the average of wind speed/direction/power data during a fluctuation.

The total power change trend Px of the total cluster power in each fluctuation process is calculated as follows:

where P is the total power of wind farm cluster during a fluctuation; Max is the Maximum function; Min is the Minimum function.

The average fluctuation amplitude ΔP （t+Δt） in each fluctuation process can be quantitatively expressed as follows:

where t is the measured data recording time point;Δt is the selected time window (a time window of 4 h is selected in this study); P （t ） is the total power of wind farm cluster during a fluctuation at t time point.

For each fluctuation process, five meteorological factor fluctuation quantities, namelyand ΔP （t+Δt）, corresponding to the n fluctuation processes,are calculated using the above formula.After removing the abnormal data, five meteorological factor fluctuation quantities were used as column labels, and a weather fluctuation characteristic dataset with five columns and n rows was obtained.

Clustering is used to determine the internal connections between the data[23].Using cluster analysis,different data can be divided into multiple clusters according to the characteristic values of each dataset.Although the comparable data between various clusters are significantly different,the data within a cluster are similar.

The K-means clustering algorithm is a typical partitioning algorithm used in cluster analysis[24,25].The principle of the algorithm is as follows: k center points are initially selected randomly from the feature dataset.The Euclidean distances between all feature points and these center points were then computed, after which each feature point was assigned to the cluster of the nearest center point, and a new center point was calculated again in each cluster.This process is repeated iteratively until the calculated center point after each partition no longer changes within the allowable range, and the cluster analysis can be regarded as complete [26].

The objective function E during data partitioning and the coordinate of new center point ci were calculated as follows:

where i is the number of the cluster; k is the total number of clusters; Si is a set containing collections of all feature points of the i cluster; sx represents the coordinate of one feature point.

When dealing with large data, the K-means approach offers advantages such as simplicity and rapid calculation velocity and is generally considered the preferred clustering algorithm [27].In this study, fluctuations were clustered using the k-means technique.

1.2 Causality analysis based on CCM algorithm

Considering the causal relationship between multiple meteorological factors and the total power fluctuation of a cluster, a causal relationship analysis method based on the CCM algorithm is proposed in this section.Aiming at the above typical fluctuation processes, the causal relationship intensity between the meteorological factors and the total power of the cluster in each typical fluctuation process was deeply excavated,and a strong causal relationship between the meteorological factors was screened using the CCM algorithm.

In contrast to the correlation analysis method [28], the causality analysis method aims to determine the cause of a result and then build a directed causal network structure among multiple system variables.Fig.3 shows the causal diagram among multivariate time-series variables; that is,through causality analysis among multiple variables, the correlation causal diagram can be obtained.Through causality analysis, redundant and outcome variables can be effectively eliminated; that is, meteorological factors unrelated to the total power of the cluster can be effectively eliminated to screen out the cause variables in the process of wind power fluctuation and conduct better data screening for later power forecasting [29].At present, causality analysis has been applied in many fields and has yielded good results.

Causality analysis commonly used algorithms have causality analysis methods based on information entropy,the method of Granger causality,the CCM algorithm,etc.Based on the large time-series data with multiple meteorological factors and the total power of the cluster used in this study,the CCM algorithm was adopted.The Granger causality and information-entropy-based causality analysis methods require a large amount of high-quality data for effective causality analysis.Moreover, the Granger causality method can only yield qualitative results.However, the CCM algorithm is more adaptable to noisy data[30]and has a strong causal association mining ability[31];therefore, it is more suitable for the causality analysis of relevant data in this study.

Fig.3 Causal diagram of multivariate time series variables.

Fig.4 CCM algorithm schematic.

The CCM algorithm is founded on the entire embedding theorem and the shadow manifold.As illustrated in Fig.4, two variables, C and R, are used as an example.According to Takens’ embedding theorem, a vector with a lag value can be employed to generate a shadow manifold of a true manifold.The points in the shadow manifold exhibit a one-to-one correspondence with the points in the true manifold[32].Consequently,R may be utilized to construct the shadow manifold MR,while C may construct the shadow manifold MC.If C and R share a causal relationship, then the neighboring points surrounding a point in MR will more accurately identify the neighboring points of the corresponding point in MC.As the time series becomes longer, the shadow manifolds MC and MR become ‘‘denser,” resulting in more ‘‘compact” neighboring points when a point is identified.Consequently, the error in identification is reduced,and the correlation coefficient between the true value of the C variable and the estimated value obtained through cross-mapping is higher.

In this method,the total power of the cluster and different meteorological factors are two time series x（t ）and y （t ）of length L, which can be regarded as two shadow manifolds obtained by projecting the system M in onedimensional space.Assuming the dimension D of shadow manifold is 2 and the sampling interval τ is 1,then the two time series can be reconstructed into two state spaces, and the state vectors X （t ） and Y （t ） at time point t in the two state spaces can be expressed as follows:

where D is the embedding dimension; τ is the sampling interval; x（t ） is the value of the meteorological time series x（t ）at time t;y （t ）is the value of the total power time series y （t ） at time t.

Then the two state vectors X （t ） and Y （t ） at all times constitute two state spaces, which can be regarded as two shadow manifolds Mx and My of the projection of the system M in one-dimensional space.

According to Takens’ embedding theorem, for any point on the shadow manifold, a number D of adjacent points can be found in each dimension.The set Gx,i,D of all adjacent points for point X （ti ）on the shadow manifold Mx at ti time point can be expressed as follows:

where ti is the i time point of the time series; X （ti ）D is the adjacent point of X （ti ） on the dimension numbered D.

From the adjacent points in Gx,i,D above, we can map the adjacent points to the shadow manifold My and get the point set Gy,i,D as follows:

where Y （ti ）1,Y （ti ）2,...,Y （ti ）D is the points obtained by mapping X （ti ）1,X （ti ）2,...,X （ti ）D onto My.

For the point Y （ti ） on the shadow manifold Mx at ti time point corresponding to X （ti ）, we can combine the weight coefficient wi and the point set Gy,i,D to calculate the estimated value ^Y （ti ）|My as follows:

where j is the number of the adjacent point, and j=1,2,...,D; L is the time series length; i is the number of the time point,and i=1, 2,..., L;ti is the i time point of the time series;wi is the weight coefficient;mi is the intermediate variable;X （ti ）j is the adjacent point of X （ti ）on the dimension numbered j.

According to the above calculation method, we can obtain the estimated value ^Y （ti ）|My at any time ti ∈t.Define the CCM value rx→y from x（t ） to y （t ） as follow:

where L is the time series length; i is the number of the time point, and i=1, 2,..., L; is abbreviated to is the average value of Y （t i）; is the average value of .

If there is a causal relationship from x（t ）to y （t）, a convergent rx→y will be obtained.This value range is (0, 1),and the larger the rx→y value, the stronger the causal relationship.

According to this principle, based on the quantitative value, meteorological factors with strong causal relationships in various types of fluctuation processes can be selected as input data for subsequent ultrashort-term power forecasting modeling.

1.3 Ultra-short-term wind power forecasting method for multivariate weather fluctuation processes

Taking into account the differences in meteorologicalpower mapping relationships under different fluctuation processes[33],this method first classifies the data into fluctuation processes with multi-temporal and spatial scale characteristics and then performs a causal relationship analysis between meteorological factors at multiple points and the total cluster power based on the CCM.Training and test sets were established based on meteorological factor data with strong causal relationships, and deep learning models were used to model different fluctuation processes.Finally, the root mean square error (RMSE)and mean absolute error (MAE) were used to assess the prediction outcomes.Fig.5 and Table 1 show the entire ultra-short-term power forecasting refined modeling process.

As the data used in this study includes 6 types of typical fluctuation processes, the corresponding strong causal meteorological factor data is the first input for different typical fluctuation processes,and the input data is divided into two sets; subsequently, the deep learning model is used to train and test the model for the 6 typical fluctuation processes,which includes setting the number of training steps, loss function, and other steps; after completing the modeling process of the 6 types of models, the ultrashort-term power forecasting refined model is finally obtained.

In this study, the input is set as strong causal wind speed and direction and historical total power, and the output is predicted power, so the number of input layers is set to 3 and the number of output layers is set to 1.During model training, we can set the loss function J,and the process of backpropagation through J is as follows:

Fig.5 Ultra-short-term power forecasting refined modeling process.

Table 1 Ultra-short-term power forecasting refined modeling methodology.

Input:X: All historical data of wind Farm clusters,X= x（t）1,x（t）2,...,x（t ）5,x（t ）all}.Where x（t）1,x（t）2,...,x（t）5 is the time series of all historical wind speed and wind direction of the wind farm numbered 1,2,...,5 in the wind farm cluster;y（t）all is the time series of all total power of the wind farm cluster;k: The number of the fluctuation process;Xrd: Randomly selected historical meteorological factors,Xrd = x（tr ）1,x（ tr ）2,...,x（tr ）5,y（t）rd{}.Where x（tr ）1,x（tr ）2,...,x（tr ）5 is the time series of historical wind speed and wind direction of the wind farm numbered 1,2,...,5 in the wind farm cluster every 15 minutes within 4 hours before the forecast period; y（t）rd is the time series of total power of the wind farm cluster.Output:modelccm: Power forecasting refinement model based on CCM,modelccm = model1,model2,...,modelk{{};ppred: Power forecast results.0Strat 1Input: X, k, Xrd 2Sindex = S1,S2,S3,...,Sk } is the fluctuation process characteristic data set based on five indicators 3X → X1,X2,X3,...,Xk{} is divided into k categories by clustering based on Sindex 4For i: 1 to k 5 x（t ）i,CCM =CCM x（t ）=x（t ）1,x（t ）2,...,x（t）5,y（t）=y（t ）all{（）is the strong causal meteorological factors 6 Divide x（ t ）i,CCM into training set: x（t）i,train and testing set:x（t ）i,test 7modeli =train model x（t）i,train,x（t ）i,test（）■8s= σ2s,σ2d,σ2p,Px,ΔP （t+Δt）■is the fluctuation process characteristic from Xrd 9For i: 1 to k 10If s ∈Si 11ppred =modeli x（t）i,CCM（）12 Output: modelccm, ppred 13 End

where p is the measured total power;ppred is the power prediction result calculated by the model; f is the activation function;m is the number of hidden layer;n is the number of input layer; ωi is the weight matrix from the i hidden layer to the output layer; υji is the weight matrix from the j input layer to the i hidden layer;xj is the input meteorological factor time series of the j input layer; Δωi and Δυij is the modified value of the ωi and υji;η is the learning rate;∂J/∂ωi and ∂J/∂υij is the derivative of loss function.

Through the above loss function, weight matrix is iteratively updated, and finally obtain the optimal prediction result and complete the model training.

In this study, six deep learning models-LSTM, Bi-LSTM, CNN, N-Beats, DLiner, and Transformer models-were used to model the ultra-short-term wind power forecasting model for six typical fluctuation processes.Each model has the following distinct guiding concepts:

1) Short-term and long-term memory network(LSTM)

LSTM is an important method for DL LSTM,which is a special type of RNN;LSTM leverages the time sequence of the input for calculations and analyses [3,36-38].Similar to an RNN, the LSTM also has a chain structure, as shown in Fig.6, as an LSTM neural network unit.

Fig.6 LSTM neural network unit.

The components of the LSTM unit are the forget gate,input gate, and output gate.In addition to the four gate structures, LSTM also adds the unit state of the neuron,which may be expressed using a vector to depict the‘‘memory” of the input information after t time.This vector includes the ‘‘summary” of all previous input information of the neural network before that moment.This can prevent the long-term dependence caused by long time series,resulting in large errors in the forecasting results.

2) Bi-directional long short-term memory (Bi-LSTM)

A single-layer Bi-LSTM is composed of forward and backward LSTM.Fig.7 shows the operation of the Bi-LSTM model.Compared with the LSTM model, Bi-LSTM can receive the previous and next information simultaneously and integrate the information after all time steps, making the results more comprehensive.

Fig.7 Bi-LSTM operation flow.

3) Convolutional neural networks (CNNs)

Compared to traditional fully connected neural networks,CNNs have stronger feature extraction capabilities and spatial invariance.Convolutional layers, activation functions, pooling layers, fully linked layers, and normalizing layers are among the most common components.The basic component of a CNN is the convolutional layer.As shown in Fig.8, the convolutional layer generates feature maps by sliding on the input data and performing dotproduct operations.These feature maps retained the spatial relationships and important features of the input data.

Fig.8 One-dimensional convolution.

4) Neural basis expansion analysis (N-Beats)

The N-Beats model is composed of multiple stacked feedforward networks, each of which is called a ‘‘block.”Each block contains four full connected layers.The model input xℓ passes through four full connected layers and a linear layer LINEAR to obtain two outputs and .Among them, is the forward forecasting value, which is used to generate the final forecasting result, and is the backward forecasting value,which is used to represent part of the time series that is not explained by the model. will be passed as input to the next block, such that the model can be gradually corrected and the forecasting accuracy can be improved.The formula used to calculate each full connected layer is as follows:

where FCℓ,i is the activation function; ℓ is the block number; i is the number of fully connected layers contained in each block, and i=1,2,3,4; wℓ,i is the weight matrix vector; hℓ,i is the calculation result of the i block; bℓ,i is the deviation vector.

After four layers of fully connected layers,the final output is hℓ,4.Input hℓ,4 into the linear layer and calculate as following:

where andis the weight matrix vector; andis the expansion coefficient.

and are input into two base layer activation functions and for calculation.The formula is as follows:

where is the output of backward forecasting; is the output of forward forecasting value.

After multiple blocks are processed, the final output of N-Beats is the sum of each .The formula is as follows:

where is the final output of N-Beats; ℓ is the number of blocks.

5) DLiner

The DLiner framework is founded on linear regression and comprises multiple stacked linear layers.Each layer performs a linear transformation of the input data, allowing the model to capture intricate linear relationships within the time series while retaining the interpretability of linear models.

Compared to general deep neural networks,the DLiner framework offers greater transparency, as the weights of each linear layer can be analyzed to understand the model’s output.

6) Transformer

The transformer architecture is built entirely on a selfattention mechanism,eliminating the reliance on sequence order.This design enables parallel processing of sequence data, thereby significantly improving computational effi-ciency.The transformer consists of two primary components: the encoder and the decoder.The encoder transforms the input sequence into an alternative representation,which the decoder subsequently uses to generate the target sequence.Both components consist of multiple identical layers, each comprising a multi-head self-attention sublayer and a feedforward neural network sublayer.The multi-head attention mechanism allows the model to compute attention across different subspaces simultaneously,thereby extracting a rich set of features from the data.

2 Case analysis

2.1 Dataset

The data used in this study are historical data of a wind farm cluster in China that includes five wind farms.Existing time series of the historical wind speed,wind direction,power,and total power of the cluster for each wind farm in the past three months.For problematic data, this study uses conventional data cleaning methods and the quartile method to identify outliers, restores them through linear regression methods, and finally obtains a dataset every 15 min at five points.

Fig.9 Model input and output.

The dataset was divided into multiple fluctuation processes in units of 4 h(16 points),and the input and output of the prediction model were in units of the fluctuation processes, as shown in Fig.9.The meteorological values contained in multiple historical fluctuation processes were the input,and the output was the total power value of the cluster contained in a future fluctuation process.

During the training and testing processes of the model,no NWP data were input.Only the meteorological data of the n fluctuation processes before the intended prediction period were input, and the meteorological data were obtained from the wind tower data of each wind farm.

For the input data,this study screened according to the strength of the causal relationship between the meteorological factors of different wind farms and the total power of the cluster.Considering that different typical fluctuation processes have different inputs and outputs, a fluctuation characteristic dataset was first established according to the fluctuation characteristics of the total power of the cluster for cluster analysis to further screen meteorological factors with strong causal relationships as input data.

For each fluctuation process, this study calculated the fluctuation values of five meteorological factor fluctuation quantities to n fluctuation processes and used the fluctuation values of the five meteorological factors as column labels.Finally,a weather fluctuation characteristic dataset with n rows and five columns was obtained.Table 2 shows part of the weather fluctuation characteristic dataset.

After sorting the fluctuation characteristics dataset, all fluctuation characteristics data were normalized using max-min normalization to prevent errors caused by different dimensions in the cluster analysis.

2.2 Characteristic analysis of typical fluctuation process

In this study,the k-means technique was utilized to create a clustering model, and multiple fluctuation processes of the wind power cluster were clustered and analyzed based on a weather fluctuation characteristic dataset.The n fluctuation processes were divided into several typical fluctuation processes, and as shown in Table 3, the number of fluctuation processes contained in each type of fluctuation process was used as the output.Simultane-ously, a frequency histogram was used to visualize the characteristics of each type of fluctuation process, and the power fluctuation characteristics of each type of typical wind power fluctuation process were observed.

Table 2 Part of the weather fluctuation characteristic dataset.

Fluctuation Process Mean Fluctuation Amplitude 1 0.22913,503.5136,852.1119,599.7-0.993 2 0.10112,333.923,287.065,332.7-0.781..................2,1270.12019,266.61,907.919,015.40.760 2,1280.25363,076.27,434.731,300.6-0.999 Wind Speed Variance Wind Direction Variance Wind Power Variance Total Power Variation Trend

Table 3 Indicates the number of fluctuation processes included in typical fluctuation processes.

Fluctuation Process CategoryType 1Type 2Type 3Type 4Type 5Type 6 The number of Fluctuation Processes470363412161295427

The clustering results were visualized,and an eigenvalue frequency distribution histogram of each fluctuation process in each class was drawn to analyze the power characteristics of each fluctuation process.The characteristics of wind speed and wind direction fluctuations in six types of fluctuation processes were analyzed.

Fig.10 Histogram of frequency of fluctuation characteristics of typical fluctuation processes.

As shown in Fig.10,there is minimal difference between the typical fluctuation processes in terms of the wind speed variance and power variance.From the perspective of wind direction variance, the wind direction variance of the fluctuation processes of Types 1,3,4,and 6 are mostly distributed within the interval (0, 0.2), indicating that the wind direction fluctuation of each type of fluctuation process was relatively small.For the type 2 and 5 fluctuation processes, the wind direction variance was mostly distributed in the interval(0.5,1.0),and the left interval value of the maximum frequency interval was more than 0.5,indicating that the wind direction fluctuation of each type of fluctuation process was relatively large.From the perspective of the total power variation trend,the power variation trend value of the Type 6 fluctuation process is the smallest and is mostly distributed in the interval of (0,0.25),whereas the power variation trend value of the Type 4 fluctuation process is mostly distributed in the interval of(0.5, 1.0), which is the largest variation trend of all fluctuation processes.

The average fluctuation amplitudes of each fluctuation process exhibited significant differences.As shown in Fig.11, the first and second types of fluctuation processes are (0, 0.05) in terms of the maximum frequency distribution.The fourth, fifth, and sixth types of fluctuation processes are (0.95, 1.00).The third type of fluctuation process is(0.45,0.50).From the perspective of distribution shape, the first, second, fourth, and sixth types of fluctuation processes all show an ‘‘L”-type distribution, and the frequency of the interval with the largest frequency is much higher than that of other intervals and is at the most edge of the whole distribution interval.However,after the interval with the highest frequency was excluded, the frequencies of the other intervals did not differ significantly.The third type of fluctuation process is a roughly inverted‘‘V”-shaped distribution.The fifth type of fluctuation process was evenly distributed throughout the distribution interval, and the frequency of the largest frequency interval was only slightly higher than that of the others.

Fig.11 Histogram of average fluctuation amplitude and frequency of typical fluctuation processes.

Overall,considering Type 1 and Type 4 typical fluctuation processes as examples,the wind speed and wind direction variances of the two types of typical fluctuation processes are small, that is, the fluctuation degree of the two types of typical fluctuation processes is small in time and space; however, the concentration degree of the wind speed and wind direction variances of Type 1 typical fluctuation processes is significantly greater than that of the typical fluctuation processes of Type 4 in the low range.Therefore, the difference between the two typical fluctuation processes can be seen from the output fluctuation:The Type 1 typical fluctuation process’s total power variation trend of wind power clusters is mostly focused in the period (0, 0.5), where the interval (0.5, 1) is mostly where type 4 typical fluctuation processes are focused,and its average fluctuation range is more evident.The region between 0 and 0.25 is mostly where Type 1 typical fluctuation processes occur.The interval (0.5, 1) contains the most typical Type 4 fluctuation processes.This classification method can effectively distinguish between the typical fluctuation processes.

2.3 Causal relationship analysis based on CCM

In causality analysis, the parameters must first be set.As shown in Table 4,the basic parameters were set before the causality analysis based on the CCM algorithm.

The CCM algorithm was used to establish the causality analysis model.Finally, for each type of fluctuation, the wind speed and wind direction of each wind farm were taken as input variable X, and the total cluster power was taken as input variable Y.Causality analysis was performed using a causality analysis model.The CCM algorithm can quantitatively determine the strength of the causal relationship.During the process of various fluctuations, the stronger the causal relationship between the meteorological data and total cluster power, the larger the causal relationship strength value; thus, the meteorological factors of a strong causal relationship can be screened accordingly.After the causality analysis based on the CCM algorithm, the results were output and collated.Fig.12 presents a line diagram of the causal relationship intensity between the meteorological data and the total cluster power in each fluctuation process.

As illustrated in Fig.12,with the exception of the Type 4 fluctuation process, the causal influence of wind speed on the total power of the cluster is generally stronger than that of wind direction across the other fluctuation process types.Additionally, the wind direction at wind farm No.1 exhi-bits a significant causal relationship with the total power of the cluster under various fluctuation process scenarios.

Table 4 Causality analysis model parameter setting.

Model Parameter Name Parameter Setting Variable x（t）Wind speed and direction of each wind farm Variable y（t）Total cluster power The length of the time series L The total time series length of a typical fluctuation process

Fig.12 Line chart of causality strength.

2.4 Power forecasting results analysis

This study analyzed the power forecasting results from two perspectives.First,the effectiveness of causal screening of predictors was evaluated.The results demonstrated that this approach improved power forecasting accuracy compared to the traditional correlation-based screening method and the method without any screening.To validate this, two control groups were established: one using correlation-based screening and another without screening.To eliminate the influence of specific deep learning models, six deep learning models were employed for synchronous validation.

Second, the effectiveness of the refined modeling approach was examined.In this study, power forecasting models were tailored for six typical fluctuation processes.To demonstrate the superiority of the refined modeling approach over traditional direct modeling, control experiments were conducted with identical datasets for both approaches.The forecasting results were assessed using the RMSE and MAE indicators [10,34,35].

2.4.1 Evaluation indicators

In this study, strong causal meteorological factors were identified and incorporated into the deep learning model.These factors were divided into training and test sets specific to each type of fluctuation process.During model training, different learning rates were assigned to the strong causal meteorological factors for varying wind farms to optimize the forecasting model and improve forecasting accuracy.After generating the forecasting results, RMSE and MAE indicators were used to quantify forecasting errors.The RMSE and MAE are calculated as follows:

where Pture,i is the measured time series total power;Ppred,i is the predicted time series total power value; L is the length of Pture,i; i is the number of the time point, and i=1,2,...,L.

For the cluster total power prediction results, every 15 min in the next 4 h,an error evaluation was performed,and the average value of six groups of RMSE or MAE values obtained from six types of typical fluctuation processes at each time point was calculated to obtain the comprehensive RMSE or MAE value of the entire time series;that is,the error evaluation of the cluster total power prediction results for multiple typical fluctuation processes was performed from the overall time.

2.4.2 Validity analysis of causal screening of predictors

To evaluate the effectiveness of causal screening of predictors(Strong Causality),two control groups were established for comparison.In the first group, a traditional correlation analysis method was applied, where strongly correlated meteorological factors (Strong Correlation)were used as input data while keeping other parameters unchanged.In the second group, all meteorological data were used as input without any screening (All Factors).Fig.13 illustrates the RMSE values of the prediction error under the CCM method and the two control groups across six different deep-learning models.

Fig.13 Comprehensive RMSE value (including two control groups).

As depicted in Fig.13,the RMSE forecasting error of the ultra-short-term wind power cluster forecasting model based on the CCM method was consistently smaller than that of the two control groups across all deep learning models.Taking the LSTM model as an example, the RMSE value of the method without screening forecasting factors(All Factors)exhibited a significant increase over time,with overall fluctuations exceeding 0.15.In contrast,the forecasting errors for the causal analysis method(Strong Causality)and the correlation analysis method (Strong Correlation)increased more gradually,with overall fluctuations remaining below 0.07.From the perspective of forecasting accuracy, the causal analysis method (Strong Causality)outperformed the control methods.Compared to the method without screening forecasting factors(All Factors),it achieved a 7.32%improvement in accuracy.Furthermore,compared to the correlation analysis method(Strong Correlation),it achieved a 1.57%improvement.These results confirm that the proposed CCM-based causal screening method significantly enhances the precision of ultra-short-term wind power cluster forecasting.

The forecasting accuracy varied depending on the deep learning model used with the causal screening method.Table 5 presents the average MAE values obtained using the refined modeling approach with different deeplearning models.

It can be observed from the table that the forecasting accuracy of this method was higher when using the LSTM,Bi-LSTM,and DLinear models,and the training time was approximately 40 s.Taking the forecasting error modeled using the LSTM model as an example, Table 6 shows the RMSE values every 15 min for the next 4 h predicted by the causal screening method and the two control groups.

From the comprehensive RMSE values, the maximum RMSE value of the forecasting result based on the causal analysis method (Strong Causality) was 0.160, and the minimum was 0.099, indicating a good forecasting effect.

2.4.3 Analysis of effectiveness of modeling methods

To demonstrate that the refined modeling approach proposed in this study for ultra-short-term power forecasting—categorized based on six distinct typical fluctuation processes—improves forecasting accuracy,a control group utilizing direct modeling (uncategorized) was established for comparison.The categorized power forecasting results were evaluated against those of the control group.Fig.14 presents a comparison of the actual cluster total power and the power forecasting values for the two approaches.

The real power is represented by the black solid line,the forecasting power of the refined model for the six distinct typical fluctuation processes is represented by the red solid line,and the predicted power of the direct modeling is represented by the blue dotted line.The forecasting results of the refined model(categorized)proposed in this study were the closest to the actual power of the forecasting results obtained from the six types of fluctuation processes.In contrast to the control group (uncategorized), based on causal analysis, the forecasting model that performs ultrashort-term power forecasting refined for six different typical fluctuation processes (categorized) has a better forecasting effect.

The MAE indicator was used to evaluate the error in cluster total power forecasting values at 15-min intervals over the subsequent 4-h period, as predicted by the ultra-short-term power forecasting models using the two methods.Table 7 summarizes the comprehensive MAE values for the two modeling methods across 16 future time points.

The data in Table 7 reveal that the refined modeling method (categorized) tailored to the fluctuation process consistently produced smaller prediction errors for total cluster power over the next 4 h compared to the directmodeling approach (uncategorized).Furthermore, as time progressed, the rate of increase in the MAE values for the categorized method was significantly slower than that observed for the control group (uncategorized).

Table 5 Average MAE of the refined modeling method under different deep learning models.

ModelLSTMBi-LSTMCNNN-BeatsDlinearTransformer Average MAE0.0890.0880.0990.1290.0870.259

Table 6 The comprehensive RMSE value at each time point.

Time Point Strong Causality RMSE Strong Correlation RMSE All Factors RMSE Time Point Strong Causality RMSE Strong Correlation RMSE All Factors RMSE 1 0.1220.1490.13990.1210.1380.216 2 0.0990.1160.118100.1270.1430.220 3 0.0970.1120.108110.1360.1520.221 4 0.1000.1120.127120.1410.1570.230 5 0.1010.1150.153130.1490.1640.255 6 0.1050.1180.175140.1560.1710.260 7 0.1070.1220.204150.1570.1720.265 8 0.1140.1300.21160.1600.1730.260

Fig.14 Power forecasting result curves of two modeling methods.

Table 7 The comprehensive MAE value of two modeling methods at each time point.

Time Point Uncategorized MAE 10.0860.07190.0940.124 20.0710.062100.0990.133 30.0710.070110.1070.139 40.0710.079120.1120.145 50.0750.092130.1170.152 60.0790.104140.1250.157 70.0820.112150.1260.164 80.0870.116160.1270.170 Categorized MAE Uncategorized MAE Time Point Categorized MAE

3 Conclusion

This study incorporates the causal relationships between multivariate meteorological factors and cluster total power fluctuations to propose an ultra-short-term wind power cluster forecasting method based on the CCM.The following conclusions are drawn:

1) By utilizing the causal relationship analysis approach of the CCM algorithm,this method identifies meteorological forecast factors with strong causal relationships and quantitatively evaluates the causal mapping relationships between these factors and cluster total power.Furthermore, it explores the underlying physical relationships,significantly reducing the impact of redundant meteorological forecast factors on cluster total power forecasting.Compared with traditional methods,this approach enhances the directionality and interpretability of the forecasting model.

2) Recognizing the variations in the causal mapping relationships between multivariate meteorological factors and the cluster total power under different typical fluctuation processes, a clustering model for typical fluctuation processes is established.This model effectively condenses a large number of fluctuation processes into several distinct types,each characterized by unique output change patterns.

3) By employing different strong causal meteorological forecast factors corresponding to various typical fluctuation processes as inputs to the power forecasting model, refined modeling for typical fluctuation process prediction is achieved.Comparative verification demonstrates that this method offers a measurable improvement in prediction accuracy over traditional approaches.Specifically, the prediction accuracy improves by an average range of 1.57 %to 7.32 % compared to ultra-short-term power forecasting models based on correlation analysis.

In summary,the proposed ultra-short-term wind power cluster forecasting method based on CCM can effectively enhance forecasting accuracy by incorporating causal relationships between meteorological factors and power fluctuations considering different typical fluctuation processes.This study not only improves prediction accuracy but also provides valuable insights into the underlying physical processes that govern power fluctuations.The analysis results underscore its potential for broader applications in energy forecasting and highlights the importance of causal analysis in enhancing model interpretability and robustness.

CRediT authorship contribution statement

Yuzhe Yang: Writing - original draft, Visualization,Methodology, Formal analysis, Conceptualization.Weiye Song: Writing - review & editing.Shuang Han: Writing -review & editing.Jie Yan: Writing - review & editing.Han Wang: Writing - review & editing.Qiangsheng Dai:Writing-review&editing.Xuesong Huo:Writing-review& editing.Yongqian Liu: Writing - review & editing.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This research was funded by the State Grid Science and Technology Project ‘‘Research on Key Technologies for Prediction and Early Warning of Large-Scale Offshore Wind Power Ramp Events Based on Meteorological Data Enhancement” (4000-202318098A-1-1-ZN).