Abstract
Reliable and accurate ultra-short-term prediction of wind power is vital for the operation and optimization of power systems. However, the volatility and intermittence of wind power pose uncertainties to traditional point prediction, resulting in an increased risk of power system operation. To represent the uncertainty of wind power, this paper proposes a new method for ultra-short-term interval prediction of wind power based on a graph neural network (GNN) and an improved Bootstrap technique. Specifically, adjacent wind farms and local meteorological factors are modeled as the new form of a graph from the graph-theoretic perspective. Then, the graph convolutional network (GCN) and bi-directional long short-term memory (Bi-LSTM) are proposed to capture spatiotemporal features between nodes in the graph. To obtain high-quality prediction intervals (PIs), an improved Bootstrap technique is designed to increase coverage percentage and narrow PIs effectively. Numerical simulations demonstrate that the proposed method can capture the spatiotemporal correlations from the graph, and the prediction results outperform popular baselines on two real-world datasets, which implies a high potential for practical applications in power systems.
NORMALLY, the ultra-short-term prediction of wind power refers to the estimation of wind power with the time horizon, which ranges from a few minutes to several hours [
Generally, ultra-short-term prediction of wind power consists of two components: deterministic point prediction and error estimation. The works of deterministic point prediction fall under three headings: physical methods, statistical methods, and artificial intelligence (AI) based methods.
1) Physical methods rely on the information of surrounding wind field (e.g., obstacle, surface roughness, and terrain) and numerical weather prediction (NWP) data (e.g., humidity, pressure, wind speed, and temperature) to model the relationship between wind power and wind speeds [
2) Statistical methods mainly include auto-regressive (AR), auto-regressive integrated moving average (ARIMA), auto-regressive moving average (ARMA), and gray methods [
3) Support vector machine (SVM), light gradient boosting machine (LightGBM), and multi-layer perceptron (MLP) are widely-used AI-based methods for ultra-short-term prediction of wind power in the last 20 years [
The traditional point prediction aims to generate deterministic prediction values, which cannot represent the prediction error caused by various reason such as volatility and intermittence of wind power. Further, interval prediction is one of the mainstream ways to estimate the error by adding lower and upper boundaries to each deterministic prediction value. The popular methods to construct prediction intervals (PIs) mainly include the Delta [
In a broad sense, the inputs of wind power prediction should be considered as a graph [
There has been increasing interests in generalizing traditional DNNs into graph neural networks (GNNs) in recent years. In particular, graph convolutional networks (GCNs) have been widely used in different fields (e.g., link prediction, drug synthesis, and traffic flow prediction) due to their superiority in modeling the spatial correlation between nodes [
Based on the above discussion, this paper proposes a novel GNN-based point prediction model and an improved Bootstrap technique for ultra-short-term interval prediction of wind power. Specifically, a GCN is employed to model the spatial correlation between nodes, and a more recent advanced model named bi-directional long short-term memory (Bi-LSTM) is utilized to capture the temporal correlation of time-series curves. Then, an improved Bootstrap technique is designed to balance the coverage percentage and width of PIs. Finally, the effectiveness of the proposed method is verified through real datasets. The main difference between this paper and previous publications involving GNNs lies in:
1) The nodes are generalized from adjacent wind farms into both wind farms and meteorological factors.
2) Different from previous publications [
3) The performance of point prediction model is improved by applying bidirectional learning techniques into the traditional LSTM, i.e., Bi-LSTM replaces the traditional LSTM to capture temporal correlations.
The key contributions of this paper are summarized as follows.
1) Without simplifying the inputs of ultra-short-term prediction of wind power into Euclidean data, this paper innovatively attempts to model the inputs as the new form of a graph from a graph-theoretic perspective. The spatial correlation between nodes is represented by an adjacency matrix. The historical data are viewed as the features of nodes to describe the temporal correlation of the wind power generation curves and meteorological factors.
2) To improve the accuracy of the point prediction, a novel GNN combining the GCN and Bi-LSTM is proposed to capture spatiotemporal correlations without artificial feature engineering.
3) As a flexible and efficient way, the improved Bootstrap technique is proposed to balance the coverage percentage and width of PIs. Besides, it is free of any distribution assumptions of prediction errors.
4) Extensive numerical simulations on two real-world datasets are performed to validate the effectiveness of the proposed method for ultra-short-term interval prediction of wind power.
The rest of this paper is organized as follows. Section II proposes a novel GNN for wind power prediction. Section III presents the improved Bootstrap technique and introduces the commonly-used evaluation indices of PIs. Section IV tests the proposed method and popular baselines on real datasets. Section V discusses the proposed method. Finally, the conclusion is given in Section VI.
Normally, interval prediction includes two steps: deterministic point prediction and error estimation. In this section, the predictive information (i.e., wind power of multiple wind farms and nearby meteorological factors) is represented as an undirected graph. Then, a GCN and a Bi-LSTM are integrated to model spatiotemporal correlations for point predictions, whose prediction errors are represented in the next section through the improved Bootstrap technique.
Normally, ultra-short-term prediction of wind power is performed using wind power of multiple wind farms and surrounding meteorological factors as inputs to a point prediction model. In other words, each wind farm is represented by its wind power, rather than the physical model.
As one of the innovations, this subsection employs a simple undirected graph to represent multiple wind farms and surrounding meteorological factors [

Fig. 1 Simple undirected graph to represent multiple wind farms and surrounding meteorological factors.
In practice, the predictive information is not always available. For example, some datasets without meteorological factors only include real nodes, and some datasets with one wind farm and surrounding meteorological factors only include a real node and multiple virtual nodes.
With respect to social networks, the correlation between nodes is generally described by an adjacency matrix A consisting of 0 and 1, where 0 means there is not an edge; and 1 means there is an edge. Similarly, the adjacency matrix of graph for wind farms and surrounding meteorological factors can be emulated with a correlation matrix to model the spatial dependence between nodes. There may exist different ways to construct graphs, which may be explored in future works due to page limits. For example, multiple wind farms can be constructed as a directed graph if the dataset includes only wind power without meteorological factors. However, the inputs of wind power prediction normally include wind power and meteorological factors. The wind power of the wind farm is viewed as a real node, and each meteorological factor is considered as a virtual node. It is difficult to describe the direction between real nodes and virtual nodes. Therefore, the undirected graph is constructed to describe the correlation (i.e., edge) between nodes.
As a simple example, the widely-used Pearson correlation coefficient is employed to represent the distance (i.e., edge) between the
(1) |
where and are the historical features of the
So far, the inputs of the point prediction model have been modeled as an undirected graph to capture the correlation between wind farms and surrounding meteorological factors.
Ultra-short-term point prediction aims to predict the wind power of the
(2) |
(3) |
where k is the time horizon; is the predicted wind power of the
In the next subsections, a novel GNN is proposed to model the spatiotemporal correlations of wind farms and meteorological factors, as shown in

Fig. 2 Framework of proposed GNN.
It is a vital issue to model the complex spatial dependencies between nodes for ultra-short-term prediction of wind power. Traditional CNN can only extract local spatial features of data (e.g., images) defined in the Euclidean domain, while the input data of ultra-short-term prediction of wind power are a graph rather than 2-dimensional matrices, which means that traditional CNN cannot capture complex topological information and spatial dependencies between nodes of the graph. Fortunately, the traditional CNN has been extended into the GCN defined in the graph domain to handle graph-structured data, and has received more and more attention because of its powerful performances.
There are many variants of GCN, which is mainly classified into two broad categories [
Given a correlation matrix and a feature matrix , the graph convolutional layer captures the spatial features between nodes through its first-order polynomial in the Laplacian after constructing a filter in the Fourier domain. As shown in
(4) |
(5) |

Fig. 3 Framework of spectral-based GCN consisting of multiple graph convolutional layers.
where is the identity matrix; is a new form of correlation matrix with self-loop (the correlation matrix of each graph convolutional layer is the same); is the degree matrix of the correlation matrix; ng is the number of graph convolutional layers; is the activation function of graph convolutional layers; represents the parameters to be optimized through supervised training of the
Another key issue to ultra-short-term prediction of wind power is modeling temporal dependence. Traditional DNNs (e.g., MLP) are incompatible for modeling time-series data, while RNN is a very promising algorithm, which is proficient in processing time-series data such as audio signals. Considering the traditional RNN involves vanishing gradient problems, some excellent variants have been proposed and show outstanding performance in different fields [

Fig. 4 Structure of LSTM unit.
(6) |
where , , , and are the activation vectors of the output gate, input gate, forget gate, and cell input activation, respectively; and are the sigmoid function and hyperbolic tangent function, respectively; is the weight of the output gate; is the weight of the forget gate; is the weight of the cell state; is the weight of the update gate; , , , and are the bias vectors of the forget gate, input gate, output gate, and cell state, respectively; is the latent state vector at time t; is the feature information at time t; is the cell state vector at time t; and is the Hadamard product.
Bidirectional learning is a widely-used technique to improve the prediction accuracy of traditional LSTM for sequence learning tasks, since the output of time-series prediction is not the only product of the previous input data, but a continuously correlated component. Bidirectional learning can help LSTM capture the temporal features in bidirectional aspects (i.e., the forward and reverse paths), while traditional LSTM is trained to model temporal features in one-way data flow (i.e., the forward path) only. The LSTM with bidirectional learning technique shows higher performance than traditional LSTM in various sequence learning tasks such as audio signal processing. Therefore, the Bi-LSTM is presented to capture the temporal features of time series from the last GCN layer.
As shown in
(7) |

Fig. 5 Structure of Bi-LSTM unit.
where is the latent state vector at time t of the forward LSTM; is the latent state vector at time t of the backward LSTM; is the latent state vector at time t of the Bi-LSTM; and is a mathematical operator (e.g., summation, multiplication, and concatenation) that is used to combine and .
Finally, the temporal features obtained from multiple Bi-LSTM layers are used as the inputs of a dense layer at the end of the GNN, which outputs the predicted wind power of the wind farm at time :
(8) |
where is the vector of inputs of the dense layer; and are the vectors of weights and biases of the dense layer, respectively; and is the activation function of the dense layer.
A deterministic point prediction model is proposed in the previous section. In this section, the traditional Bootstrap technique is improved to represent the prediction errors using narrow PIs. Then, several evaluation indices of PIs are presented.
A traditional deterministic point prediction only provides a single point that hides the error of wind power from noises of the dataset and the model itself, while interval prediction is an effective way to quantify the uncertainty through a lower and upper boundary. The PIs surround the prediction value from the deterministic point prediction model and cover the real value with high probability.
Bootstrap is a robust technique for error estimation, which can be used to generate PIs without making any assumption about the functional form of the probability distribution of prediction errors. Specifically, the construction of PIs for ultra-short-term prediction of wind power using the traditional Bootstrap technique mainly includes two steps [
A pre-trained point prediction model and real wind power are used to obtain the prediction errors of the training set. Then, the prediction errors of the training set are employed to construct the PIs of point predictions for the test set. In particular, the prediction errors of the training set are randomly sampled nt times into the group 1, where nt is the number of Bootstrap repeats, which should be large enough to ensure meaningful statistics. Ideally, the Bootstrap repeats are often hundreds or thousands given the time resources. In this paper, nt is 5000. A prediction error is obtained for each sampling process. Note that the Bootstrap technique allows a prediction error of the training set to be sampled more than once (i.e., sampling with replacement).
The prediction errors in group 1 are sorted in descending order, and the values at the given percentile are considered as the PI nominal confidence (PINC). For instance, when is equal to 0.9, a confidence interval of 90% PINC can be obtained by selecting the errors at the 95% percentile as the upper boundary and the 5% percentile as the lower boundary.
Although Bootstrap is a widely powerful and applicable statistical technique for quantifying uncertainty, its PIs are too conservative.

Fig. 6 A simple example of wind power interval prediction.
For each wind power, the traditional Bootstrap technique constructs PIs with a fixed interval width. The wide PIs are suitable for the periods when the wind power generation curve is highly volatile, as shown in the area surrounded by a rectangular. However, these fixed PIs are obviously too wide for wind power with small volatility (e.g., the area enclosed by ellipses), which will lead to the lack of concentration of PIs. Too wide PIs are also called conservative PIs, that is when wide PIs are used for risk-based decision-making (e.g., interval optimization) of power systems, their solutions require more reserve capacity of generation sides, giving a negative impact on economics. In short, the fixed PIs of the traditional Bootstrap technique is a great limitation, which remains to be solved.
Ideally, appropriate PIs should be narrow when the wind power is weakly volatile, as weak volatility tends to imply small prediction errors. Relatively, PIs should be wide, when wind power is highly volatile, because highly volatile often means large prediction errors. To evaluate the volatility of predicted wind power at time , the widely-used standard deviation is employed as:
(9) |
where is the standard deviation of predicted wind power of the wind farm from time to time Note that the size of q is determined to be 7 in this paper by analyzing the Pearson correlation coefficient between standard deviation and prediction errors. For other datasets, q may vary, but it can also be determined by analyzing the Pearson correlation coefficient.
In the previous paragraph, the standard deviations of point predictions are employed to reflect the volatility and prediction errors of wind power. In other words, if the standard deviation is large, the prediction error is also large. If the standard deviation is small, the prediction error is also small.
To valid the relationship between the prediction errors and the standard deviation of point predictions, the proposed GNN is used to obtain point predictions for the test set. Then, the standard deviation of each point prediction is calculated by (9), and the prediction error of each point prediction can be obtained by calculating the absolute error. Finally, the standard deviation of each point prediction is considered as the x-axis, and the prediction error of each point prediction is regarded as the y-axis, as shown in

Fig. 7 Prediction error and standard deviation.
From
1) When the standard deviation is small (i.e., the volatility is weak), the point prediction errors are also small. In other words, a small standard deviation implies a small prediction error. For example, for the region surrounded by ellipses in
2) As the standard deviation becomes larger, some large prediction errors start to appear, and the large prediction errors are distributed at the locations where the standard deviation is greater than 0.03 p.u.. At these points, PIs should be wide to cover real values.
Generally, point prediction errors are strongly correlated with standard deviations, which should have the potential to guide the design of PIs. In brief, when the standard deviations of point predictions are small, the PIs should be narrow, because the prediction errors are small. When the standard deviations of point predictions are large, the PIs should be wide, since the prediction error may be large.
Based on the above analysis, this paper improves the traditional Bootstrap technique using standard deviation to obtain appropriate PIs. Firstly, the prediction errors are grouped based on the standard deviation of point predictions. For point predictions in the test set, standard deviations are calculated to determine which group they belong to. Then, this paper resamples the prediction errors from the same grouped validation set for point predictions of the test set.
A vivid explanation is that the improved Bootstrap technique narrows the PIs of the ellipse-enclosed regions in
Normally, compared with the training set, prediction errors of the validation set are closer to those of the test set. Therefore, the prediction errors of validation set are used to estimate the prediction errors of test set in the improved Bootstrap technique. For each point prediction of the validation set, its prediction error and standard deviation are calculated. This prediction error is put into group 1. Meanwhile, if the standard deviation is lower than s1, this prediction error is also placed in group 2. Obviously, group 2 is the proper subset of group 1.
For each point prediction of the test set, its standard deviation is calculated using (9). If this standard deviation is less than s2, assign all prediction errors in group 2 to group 3; otherwise, assign all prediction errors in group 1 to group 3. Then, the prediction errors in group 3 are randomly sampled nt times, and the sampled prediction errors are put into group 4. Note that the prediction error in group 3 can be sampled more than once (i.e., sampling with replacement).
Theoretically, s1 should be greater than s2. If s1 is smaller than s2, prediction errors of group 1 are smaller than the real point prediction errors for the test set, so the constructed PIs will be too narrow to cover real values.
For example, if s1 is 0.01, all prediction errors in group 1 are less than 0.15, which can be observed from
The prediction errors in group 4 are sorted in descending order, and the values at the given percentile are considered as the confidence interval. Specifically, the error at the ()
For ease of understanding and reproducing the improved Bootstrap technique, the Algorithm 1 in Appendix A shows the codes and comments directly with MATLAB.
The evaluation of PIs is often considered in terms of reliability and sharpness [
(10) |
(11) |
where Ui is the upper boundary of the
Generally, the larger the PICP is, the more reliable the PIs are. With the same PICP, a smaller PINAW indicates narrower PIs and higher quality of PIs. In addition, PICP and PINAW are two conflicting metrics, so previous research often employs the coverage width criterion (CWC) to balance them [
(12) |
(13) |
where is a penalty coefficient. The larger is, the greater contributions of the PICP to the CWC are. Generally, is determined by the system operator. As an example, is equal to 5 in this paper according to [
To fully test the performance of the proposed GNN and improved Bootstrap technique, different features of two wind power datasets are used for simulation and discussion. Specifically, the first dataset includes the real wind power of 16 geographically adjacent wind farms without meteorological factors, so the first dataset can be considered as a graph with 16 real nodes whose features are the historical wind power. The second dataset consists of real wind power of a wind farm and local meteorological factors (e.g., wind direction, wind speed, temperature, pressure, and density), and the second dataset can also be viewed as a graph with 6 nodes (1 real node and 5 virtual nodes), whose features are represented by historical wind power or historical meteorological factors.
Wind power prediction is performed based on the following information: ① NWP information (i.e., forecasts of meteorological factors); ② historical meteorological factors and wind power. Their respective importance is highly dependent on the predicted time horizon. The shorter the time horizon is, the more impact historical measurements have, and vice versa for NWP information. Normally, the equilibrium is about 2 hours according to [
Both datasets are collected by the National Renewable Energy Laboratory [
To validate the performance of the proposed method, some advanced deterministic point prediction models (e.g., MLP in [
1) The middle layer of the MLP includes 3 dense layers, whose numbers of neurons are 30, 35, and 20, respectively.
2) For the LightGBM, the maximum tree depth is 5, and the number of the maximum tree leaves is 25; the number of boosted trees is 1000, and the learning rate is 0.001; the minimum number in a child is 80, and the subsample ratio is 0.8.
3) The middle layer of the GCN includes 3 GCN layers, whose numbers of output channels are 32, 16, and 16, respectively.
4) The middle layer of the Bi-LSTM includes 3 Bi-LSTM layers, whose dimensions of the output space are 25, 25, and 20, respectively.
5) As shown in

Fig. 8 Parameters of proposed GNN.
6) For the GNN in [
Besides, the following parameters are common to these models: the optimizer is the Adam algorithm, and the loss function is the mean absolute error. The training epoch is 200, and the batch size is 32. The activation function of each middle layer is rectified linear unit (ReLU) function. The output layer of each model is a dense layer with 1 neuron, and its activation function is the sigmoid function.
All the above-mentioned models are tested in the Spyder 4.1.5 with the Spektral 1.0 and Tensorflow 2.0, which are popular libraries of deep learning. The key parameters of computer are as follows: CPU 1.60 GHz, 8 GB RAM, Intel Cor
To discuss the key parameters of the improved Bootstrap technique, two cases from the spring are used as simple examples to show how to select s1 and s2. This selection process provides general guidance. Repetition can reduce anomalous results. Considering limited time resources, the proposed GNN is independently trained 30 times. In addition, the difference of metrics between the traditional and improved Bootstrap techniques is presented to analyze whether key parameters have a positive or negative impact on performance.
Specifically, the traditional Bootstrap technique is used to obtain the average metrics of the validation set, recorded as PICP1 and PINAW1. Then, the improved Bootstrap technique with different parameters (s1 and s2 varying from 0.004 to 0.1) is employed to obtain the average metrics of the validation set, recorded as PICP2 and PINAW2. Finally, the differences between metrics and are visualized, as shown in

Fig. 9 Average metrics of validation set. (a) of the first dataset. (b) of the second dataset. (c) of the first dataset. (d) of the second dataset. (e) CWC of the first dataset. (f) CWC of the second dataset.
If , PICP2 is also smaller than PICP1, even though PINAW2 is smaller than PINAW1. In other words, narrow PIs come at the expense of PI accuracy. Relatively, when , PICP2 is very close to PICP1 and most of the PINAW2 is smaller than PINAW1 as we expect, since the improved Bootstrap technique outperforms the traditional one.
Further, the partial CWCs of the validation set are shown in
The main goal of wind power interval prediction is to derive reliable and narrow PIs. Compared with low-confidence-level PIs, high-confidence-level PIs are more practically meaningful for economic and safe operation of power systems, so different confidence levels varying from 90% to 99% will be considered in following simulations.
To test the performance of the improved Bootstrap technique, the traditional Bootstrap technique [
Method | PINC (%) | The first dataset | The second dataset | ||||
---|---|---|---|---|---|---|---|
PICP (%) | PINAW (p.u.) | CWC (p.u.) | PICP (%) | PINAW (p.u.) | CWC (p.u.) | ||
Gaussian method | 90 | 42.1 | 0.090 | 1.074 | 58.8 | 0.208 | 1.196 |
95 | 49.2 | 0.107 | 1.164 | 68.7 | 0.248 | 1.176 | |
99 | 59.3 | 0.141 | 1.167 | 78.9 | 0.327 | 1.220 | |
Traditional bootstrap | 90 | 84.3 | 0.300 | 0.700 | 80.4 | 0.309 | 0.808 |
95 | 91.1 | 0.424 | 0.939 | 87.1 | 0.404 | 1.004 | |
99 | 99.1 | 0.847 | 0.847 | 93.8 | 0.772 | 1.774 | |
Improved bootstrap | 90 | 84.4 | 0.274 | 0.637 | 81.1 | 0.283 | 0.725 |
95 | 91.1 | 0.371 | 0.822 | 87.0 | 0.372 | 0.926 | |
99 | 98.7 | 0.627 | 1.262 | 96.1 | 0.633 | 1.364 |
Method | PINC (%) | The first dataset | The second dataset | ||||
---|---|---|---|---|---|---|---|
PICP (%) | PINAW (p.u.) | CWC (p.u.) | PICP (%) | PINAW (p.u.) | CWC (p.u.) | ||
Gaussian method | 90 | 60.3 | 0.024 | 0.129 | 60.9 | 0.163 | 0.862 |
95 | 64.2 | 0.028 | 0.161 | 68.6 | 0.195 | 0.929 | |
99 | 69.7 | 0.037 | 0.199 | 78.9 | 0.257 | 0.959 | |
Traditional bootstrap | 90 | 92.7 | 0.168 | 0.168 | 92.1 | 0.347 | 0.347 |
95 | 97.0 | 0.248 | 0.248 | 96.8 | 0.466 | 0.466 | |
99 | 100.0 | 0.553 | 0.553 | 99.7 | 0.858 | 0.858 | |
Improved bootstrap | 90 | 92.3 | 0.109 | 0.109 | 91.9 | 0.323 | 0.323 |
95 | 97.6 | 0.160 | 0.160 | 97.2 | 0.445 | 0.445 | |
99 | 99.7 | 0.314 | 0.314 | 99.2 | 0.716 | 0.716 |
Method | PINC (%) | The first dataset | The second dataset | ||||
---|---|---|---|---|---|---|---|
PICP (%) | PINAW (p.u.) | CWC (p.u.) | PICP (%) | PINAW (p.u.) | CWC (p.u.) | ||
Gaussian method | 90 | 57.8 | 0.127 | 0.761 | 51.7 | 0.173 | 1.348 |
95 | 64.4 | 0.152 | 0.853 | 60.8 | 0.207 | 1.352 | |
99 | 73.7 | 0.200 | 0.907 | 76.6 | 0.272 | 1.104 | |
Traditional bootstrap | 90 | 85.7 | 0.215 | 0.482 | 82.5 | 0.314 | 0.771 |
95 | 91.9 | 0.345 | 0.748 | 92.6 | 0.444 | 0.945 | |
99 | 98.3 | 0.637 | 1.296 | 98.1 | 0.741 | 1.518 | |
Improved bootstrap | 90 | 84.2 | 0.164 | 0.383 | 82.9 | 0.262 | 0.635 |
95 | 92.6 | 0.268 | 0.571 | 93.0 | 0.391 | 0.823 | |
99 | 98.3 | 0.457 | 0.930 | 98.1 | 0.585 | 1.199 |
Although the PINAW of the Gaussian method is smaller than those of Bootstrap techniques, its PICP is much smaller than the PINC. The reason for this phenomenon is that the prediction errors of deterministic point prediction models do not follow the Gaussian distribution. From the above four tables, it is found that both traditional and improved Bootstrap techniques provide more reliable PIs of the measured wind power than the Gaussian method.
Note that the partial CWCs of the Gaussian method are smaller than those of Bootstrap techniques in some scenarios (e.g., the first dataset with 99% PINC in summer), but it does not mean that the Gaussian method is better than Bootstrap techniques. The reason is that the PICP of the Gaussian method is much smaller than the PINC. For example, in the first dataset with 99% PINC in summer, the PICP of the Gaussian method is 69.7%, whereas the expected probability to cover real values is 99%. Relatively, the PICPs of Bootstrap techniques are greater than the PICNs in most scenarios, which indicates that Bootstrap techniques can ensure the security of the power system with the expected probability.
Method | PINC (%) | The first dataset | The second dataset | ||||
---|---|---|---|---|---|---|---|
PICP (%) | PINAW (p.u.) | CWC (p.u.) | PICP (%) | PINAW (p.u.) | CWC (p.u.) | ||
Gaussian method | 90 | 47.8 | 0.100 | 0.921 | 59.5 | 0.115 | 0.646 |
95 | 54.3 | 0.119 | 1.031 | 67.8 | 0.138 | 0.675 | |
99 | 63.8 | 0.157 | 1.069 | 77.9 | 0.184 | 0.712 | |
Traditional Bootstrap | 90 | 81.8 | 0.267 | 0.669 | 90.0 | 0.280 | 0.560 |
95 | 91.5 | 0.368 | 0.806 | 94.9 | 0.425 | 0.852 | |
99 | 99.2 | 0.822 | 0.822 | 98.8 | 0.805 | 1.617 | |
Improved Bootstrap | 90 | 82.4 | 0.236 | 0.579 | 89.5 | 0.234 | 0.474 |
95 | 91.5 | 0.316 | 0.692 | 94.2 | 0.331 | 0.676 | |
99 | 99.2 | 0.612 | 0.612 | 99.0 | 0.562 | 0.562 |
Further, the comparison of the PICP and PINAW between the traditional Bootstrap technique and the improved Bootstrap technique shows that their PICPs are very similar, but the PINAW of the latter is smaller, suggesting that the improved Bootstrap technique can effectively narrow the width of PIs with negligible reduction of PICP. For example, in the first dataset in winter, the PICP is 0.992 for both the traditional and improved Bootstrap techniques when the PINC is 99%, and the PINAW for the improved Bootstrap technique is reduced by 25.54% compared with the traditional Bootstrap technique.
In the first dataset with in spring, the CWC of the improved Bootstrap technique is larger than that of the traditional Bootstrap technique. This is because the definition of CWC derived from [

Fig. 10 Comparison of traditional and improved Bootstrap techniques using two samples from the first and second datasets. (a) Interval prediction using traditional Bootstrap technique in the first dataset. (b) Interval prediction using improved Bootstrap technique in the first dataset. (c) Interval prediction using traditional Bootstrap technique in the second dataset. (d) Interval prediction using improved Bootstrap technique in the second dataset.
It is clear that the traditional and improved Bootstrap techniques have similar PIs for the strong volatile regions (e.g., steep ramps, prominent peaks and valleys). This is because prediction errors in strong volatile regions tend to be large, and narrowing the PIs in these regions may cause the PICP to drop. The improved Bootstrap technique aims to keep the PIs in strong volatile regions and reduce the width of PIs in the weak volatile regions. For example, elliptical enclosed regions in
In order to validate the superiority of the proposed method, the popular deterministic point prediction methods (e.g., MLP in [
Dataset | Season | PINC (%) | MLP | LightGBM | GCN | Bi-LSTM | GNN | Proposed method | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
PICP (%) | PINAW (p.u.) | CWC (p.u.) | PICP (%) | PINAW (p.u.) | CWC (p.u.) | PICP (%) | PINAW (p.u.) | CWC (p.u.) | PICP (%) | PINAW (p.u.) | CWC (p.u.) | PICP (%) | PINAW (p.u.) | CWC (p.u.) | PICP (%) | PINAW (p.u.) | CWC (p.u.) | |||
First | Spring | 90 | 80 | 0.282 | 0.751 | 80 | 0.286 | 0.753 | 84 | 0.316 | 0.738 | 86 | 0.290 | 0.648 | 85 | 0.285 | 0.649 | 84 | 0.274 | 0.637 |
95 | 92 | 0.456 | 0.980 | 94 | 0.449 | 0.913 | 91 | 0.407 | 0.900 | 92 | 0.404 | 0.865 | 92 | 0.394 | 0.861 | 91 | 0.371 | 0.822 | ||
99 | 99 | 0.780 | 1.574 | 98 | 0.769 | 1.561 | 99 | 0.726 | 1.456 | 98 | 0.668 | 1.386 | 98 | 0.645 | 1.313 | 99 | 0.627 | 1.262 | ||
Summer | 90 | 91 | 0.144 | 0.144 | 92 | 0.135 | 0.135 | 94 | 0.116 | 0.116 | 93 | 0.124 | 0.124 | 92 | 0.121 | 0.121 | 92 | 0.109 | 0.109 | |
95 | 93 | 0.434 | 0.903 | 99 | 0.270 | 0.270 | 98 | 0.205 | 0.205 | 97 | 0.191 | 0.191 | 97 | 0.175 | 0.175 | 98 | 0.160 | 0.160 | ||
99 | 100 | 0.496 | 0.496 | 100 | 0.464 | 0.464 | 100 | 0.387 | 0.387 | 100 | 0.406 | 0.406 | 100 | 0.383 | 0.383 | 100 | 0.314 | 0.314 | ||
Autumn | 90 | 83 | 0.217 | 0.518 | 81 | 0.205 | 0.522 | 87 | 0.211 | 0.462 | 82 | 0.158 | 0.391 | 83 | 0.16 | 0.386 | 84 | 0.164 | 0.383 | |
95 | 87 | 0.368 | 0.921 | 89 | 0.340 | 0.827 | 91 | 0.330 | 0.740 | 92 | 0.376 | 0.803 | 92 | 0.333 | 0.716 | 93 | 0.268 | 0.571 | ||
99 | 91 | 0.562 | 1.397 | 95 | 0.550 | 1.215 | 98 | 0.564 | 1.166 | 98 | 0.469 | 0.964 | 98 | 0.464 | 0.954 | 98 | 0.457 | 0.930 | ||
Winter | 90 | 85 | 0.312 | 0.712 | 81 | 0.262 | 0.681 | 83 | 0.284 | 0.695 | 84 | 0.285 | 0.672 | 83 | 0.271 | 0.656 | 82 | 0.236 | 0.579 | |
95 | 95 | 0.499 | 1.010 | 91 | 0.412 | 0.921 | 92 | 0.419 | 0.911 | 88 | 0.348 | 0.833 | 90 | 0.329 | 0.760 | 92 | 0.316 | 0.692 | ||
99 | 98 | 0.688 | 1.405 | 99 | 0.710 | 0.710 | 99 | 0.695 | 0.695 | 99 | 0.697 | 0.697 | 99 | 0.672 | 0.672 | 99 | 0.612 | 0.612 | ||
Second | Spring | 90 | 79 | 0.302 | 0.820 | 81 | 0.320 | 0.815 | 81 | 0.306 | 0.779 | 79 | 0.275 | 0.762 | 79 | 0.281 | 0.758 | 81 | 0.283 | 0.725 |
95 | 88 | 0.448 | 1.097 | 86 | 0.415 | 1.083 | 89 | 0.447 | 1.048 | 88 | 0.413 | 1.005 | 87 | 0.394 | 0.973 | 87 | 0.372 | 0.926 | ||
99 | 96 | 0.717 | 1.569 | 96 | 0.633 | 1.375 | 98 | 0.652 | 1.352 | 98 | 0.643 | 1.324 | 98 | 0.64 | 1.323 | 97 | 0.631 | 1.318 | ||
Summer | 90 | 87 | 0.329 | 0.714 | 95 | 0.396 | 0.396 | 93 | 0.372 | 0.372 | 92 | 0.363 | 0.363 | 92 | 0.354 | 0.354 | 92 | 0.323 | 0.323 | |
95 | 94 | 0.427 | 0.878 | 98 | 0.509 | 0.509 | 98 | 0.518 | 0.518 | 97 | 0.485 | 0.485 | 97 | 0.477 | 0.477 | 97 | 0.445 | 0.445 | ||
99 | 100 | 0.784 | 0.784 | 100 | 0.764 | 0.764 | 99 | 0.741 | 0.741 | 100 | 0.748 | 0.748 | 100 | 0.742 | 0.742 | 99 | 0.716 | 0.716 | ||
Autumn | 90 | 83 | 0.329 | 0.805 | 83 | 0.287 | 0.692 | 84 | 0.295 | 0.692 | 80 | 0.248 | 0.666 | 80 | 0.248 | 0.653 | 83 | 0.262 | 0.635 | |
95 | 92 | 0.414 | 0.887 | 90 | 0.388 | 0.893 | 94 | 0.428 | 0.877 | 92 | 0.388 | 0.850 | 92 | 0.39 | 0.841 | 93 | 0.391 | 0.823 | ||
99 | 99 | 0.714 | 1.440 | 98 | 0.644 | 1.328 | 98 | 0.645 | 1.312 | 97 | 0.605 | 1.290 | 97 | 0.612 | 1.285 | 98 | 0.585 | 1.199 | ||
Winter | 90 | 87 | 0.255 | 0.560 | 87 | 0.251 | 0.550 | 88 | 0.236 | 0.502 | 83 | 0.200 | 0.489 | 84 | 0.223 | 0.483 | 90 | 0.234 | 0.474 | |
95 | 95 | 0.408 | 0.825 | 91 | 0.339 | 0.761 | 94 | 0.370 | 0.751 | 92 | 0.333 | 0.711 | 93 | 0.332 | 0.699 | 94 | 0.331 | 0.676 | ||
99 | 99 | 0.637 | 1.277 | 99 | 0.596 | 1.207 | 98 | 0.461 | 0.951 | 99 | 0.586 | 0.586 | 99 | 0.579 | 0.579 | 99 | 0.562 | 0.562 |
1) Strong prediction performance: it can be observed that the proposed method obtains the smallest CWCs under all PINCs and seasons for both two real datasets, proving the effectiveness for wind power interval prediction. For example, for the first dataset with 90% PINC in spring, the CWCs of the proposed method are 15.18%, 15.40%, 13.69%, 1.70%, and 1.85% lower than those of the MLP, LightGBM, GCN, Bi-LSTM, and GNN, respectively. Note that some methods may have higher PICPs compared with the proposed method, but they are at the cost of interval width, that is, their PINAWs are much larger than that of the proposed method. For example, for the firs
2) Spatiotemporal prediction capability: compared with the traditional MLP and LightGBM, neural network-based models (e.g., GCN, Bi-LSTM, GNN, and proposed GNN), which aim to model temporal features or spatial features, usually have better precision for ultra-short-term interval prediction of wind power. For instance, for the firs
3) An ablation study: to verify if the proposed method has the ability to capture spatiotemporal features from wind power data, an ablation study is conducted to analyze how each part (i.e., GCN and Bi-LSTM) of the proposed method works. The average CWCs of two datasets with 90% and 95% PINCs are visualized in

Fig. 11 Average CWCs of two datasets with different PINCs. (a) CWC of the first dataset with 90% PINC. (b) CWC of the first dataset with 95% PINC. (c) CWC of the second dataset with 90% PINC. (d) CWC of the second dataset with 95% PINC.
It is clear that the proposed method based on the spatiotemporal features has a smaller CWC than others based on a single feature, implying that the proposed method is able to model spatiotemporal features from wind power data and meteorological factors accurately. For instance, for the first dataset with 90% PINC in summer, the CWC of the proposed method is reduced by approximately 12.10% compared with the Bi-LSTM that considers only temporal features. The CWC of the proposed method is reduced by 16.23% for the first dataset with 95% PINC in summer, indicating that the proposed method can portray temporal dependence of wind power data. Compared with GCN, which considers spatial features and ignores temporal features, for the first dataset with 90% PINC and 95% PINC in summer, the CWCs of the proposed method are decreased by approximately 6.03% and 21.95%, respectively, implying that the proposed method can portray spatial dependence well.
Further, the time complexity is tested by performing each method 30 times. Specifically, the samples from the first dataset in spring are used as a simple example. The average training time (i.e., the time to train a model) and inference time (i.e., the time to obtain PIs of a sample using the trained model) of different methods are listed in
Method | Training time (s) | Inference time (s) |
---|---|---|
MLP | 14.836 | 0.002 |
LightGBM | 2.783 | 0.001 |
GCN | 29.514 | 0.003 |
Bi-LSTM | 196.731 | 0.004 |
Proposed method | 393.638 | 0.008 |
GNN | 262.438 | 0.006 |
The training time of the proposed method is relatively long, which is the main disadvantage of the proposed method. It has to be mentioned that a few minutes of training time is acceptable in practical engineering. In addition, the inference time of each method is far less than 1 s, which can meet the real-time requirement of ultra-short-term prediction of wind power.
In fact, the multi-step wind power interval prediction with one-hour-ahead and two-hour-ahead has been implemented to obtain satisfactory PIs based on the proposed method. Besides wind power interval prediction with the hourly horizon, the wind farm controller and transmission system operator are also highly interested in intra-hour PIs. For example, the 30-min measures are indispensable to reserve dispatch, continuous generation, wind farm control, and so on. In addition to the 1-hour time horizon in previous sections, this subsection further tests the performance of the proposed method for different look-ahead horizons, e.g., 0.5, 1.5, and 2 hours, and average prediction results of the test set with 95% PINC are given in
Dataset | Season | Horizon (hour) | MLP | LightGBM | GCN | Bi-LSTM | GNN | Proposed method | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
PICP (%) | PINAW (p.u.) | CWC (p.u.) | PICP (%) | PINAW (p.u.) | CWC (p.u.) | PICP (%) | PINAW (p.u.) | CWC (p.u.) | PICP (%) | PINAW (p.u.) | CWC (p.u.) | PICP (%) | PINAW (p.u.) | CWC (p.u.) | PICP (%) | PINAW (p.u.) | CWC (p.u.) | |||
First | Spring | 0.5 | 90 | 0.345 | 0.791 | 94 | 0.336 | 0.690 | 90 | 0.296 | 0.670 | 92 | 0.285 | 0.614 | 92 | 0.274 | 0.597 | 93 | 0.268 | 0.569 |
1.5 | 95 | 0.560 | 1.135 | 94 | 0.542 | 1.112 | 90 | 0.498 | 1.130 | 92 | 0.489 | 1.055 | 91 | 0.476 | 1.049 | 92 | 0.450 | 0.977 | ||
2.0 | 95 | 0.652 | 1.320 | 95 | 0.630 | 1.272 | 88 | 0.572 | 1.368 | 93 | 0.565 | 1.202 | 90 | 0.527 | 1.200 | 91 | 0.511 | 1.149 | ||
Summer | 0.5 | 95 | 0.204 | 0.204 | 98 | 0.198 | 0.198 | 98 | 0.148 | 0.148 | 98 | 0.151 | 0.151 | 98 | 0.148 | 0.148 | 97 | 0.125 | 0.125 | |
1.5 | 94 | 0.435 | 0.895 | 97 | 0.355 | 0.355 | 98 | 0.252 | 0.252 | 98 | 0.237 | 0.237 | 97 | 0.229 | 0.229 | 97 | 0.196 | 0.196 | ||
2.0 | 93 | 0.434 | 0.905 | 95 | 0.384 | 0.384 | 98 | 0.293 | 0.293 | 97 | 0.276 | 0.276 | 96 | 0.271 | 0.271 | 96 | 0.226 | 0.226 | ||
Autumn | 0.5 | 85 | 0.299 | 0.793 | 87 | 0.314 | 0.786 | 91 | 0.205 | 0.462 | 93 | 0.196 | 0.412 | 92 | 0.189 | 0.406 | 92 | 0.180 | 0.388 | |
1.5 | 94 | 0.817 | 1.685 | 86 | 0.428 | 1.104 | 89 | 0.456 | 1.061 | 89 | 0.425 | 0.997 | 88 | 0.387 | 0.936 | 87 | 0.349 | 0.870 | ||
2.0 | 94 | 0.818 | 1.692 | 93 | 0.580 | 1.222 | 89 | 0.513 | 1.202 | 88 | 0.491 | 1.182 | 88 | 0.468 | 1.149 | 88 | 0.414 | 0.996 | ||
Winter | 0.5 | 95 | 0.367 | 0.738 | 92 | 0.285 | 0.610 | 92 | 0.302 | 0.645 | 91 | 0.228 | 0.503 | 92 | 0.221 | 0.479 | 92 | 0.212 | 0.454 | |
1.5 | 94 | 0.706 | 1.468 | 91 | 0.524 | 1.162 | 88 | 0.458 | 1.114 | 93 | 0.518 | 1.082 | 91 | 0.428 | 0.951 | 91 | 0.406 | 0.891 | ||
2.0 | 94 | 0.732 | 1.499 | 93 | 0.628 | 1.335 | 88 | 0.535 | 1.311 | 94 | 0.624 | 1.290 | 89 | 0.543 | 1.269 | 91 | 0.550 | 1.227 | ||
Second | Spring | 0.5 | 88 | 0.328 | 0.791 | 89 | 0.325 | 0.772 | 86 | 0.287 | 0.747 | 89 | 0.312 | 0.726 | 88 | 0.301 | 0.726 | 87 | 0.284 | 0.700 |
1.5 | 89 | 0.575 | 1.339 | 89 | 0.573 | 1.359 | 85 | 0.496 | 1.336 | 86 | 0.523 | 1.338 | 86 | 0.497 | 1.280 | 86 | 0.470 | 1.220 | ||
2.0 | 88 | 0.676 | 1.616 | 87 | 0.647 | 1.592 | 85 | 0.600 | 1.570 | 86 | 0.576 | 1.484 | 85 | 0.563 | 1.477 | 86 | 0.548 | 1.390 | ||
Summer | 0.5 | 95 | 0.294 | 0.596 | 99 | 0.394 | 0.394 | 99 | 0.375 | 0.375 | 98 | 0.373 | 0.373 | 98 | 0.355 | 0.355 | 98 | 0.313 | 0.313 | |
1.5 | 92 | 0.549 | 1.192 | 98 | 0.601 | 0.601 | 97 | 0.591 | 0.591 | 96 | 0.576 | 0.576 | 96 | 0.548 | 0.548 | 96 | 0.452 | 0.452 | ||
2.0 | 93 | 0.639 | 1.359 | 97 | 0.668 | 0.668 | 96 | 0.659 | 0.659 | 95 | 0.624 | 0.624 | 95 | 0.614 | 0.614 | 95 | 0.596 | 0.596 | ||
Autumn | 0.5 | 93 | 0.312 | 0.659 | 92 | 0.301 | 0.645 | 92 | 0.300 | 0.646 | 88 | 0.259 | 0.619 | 91 | 0.269 | 0.591 | 93 | 0.274 | 0.579 | |
1.5 | 88 | 0.496 | 1.206 | 94 | 0.537 | 1.101 | 94 | 0.511 | 1.045 | 88 | 0.434 | 1.049 | 91 | 0.465 | 1.039 | 94 | 0.495 | 1.027 | ||
2.0 | 94 | 0.635 | 1.298 | 89 | 0.536 | 1.279 | 90 | 0.563 | 1.282 | 95 | 0.619 | 1.252 | 93 | 0.593 | 1.252 | 93 | 0.596 | 1.240 | ||
Winter | 0.5 | 91 | 0.226 | 0.497 | 94 | 0.225 | 0.468 | 95 | 0.267 | 0.535 | 96 | 0.266 | 0.266 | 95 | 0.256 | 0.256 | 95 | 0.230 | 0.230 | |
1.5 | 95 | 0.484 | 0.971 | 93 | 0.464 | 0.974 | 95 | 0.481 | 0.972 | 92 | 0.439 | 0.959 | 93 | 0.431 | 0.915 | 94 | 0.435 | 0.884 | ||
2.0 | 91 | 0.529 | 1.163 | 93 | 0.545 | 1.149 | 95 | 0.577 | 1.158 | 95 | 0.583 | 0.583 | 95 | 0.562 | 0.562 | 95 | 0.525 | 0.525 |
From
Practically, the wind farm controller and transmission system operator are likely to focus on system-level aggregated wind power. In this case, historical wind power of wind farms and surrounding meteorological factors can be taken as inputs to the proposed method with high flexibility to predict intervals of aggregated wind power based on the farm-level information. With the high precision and flexibility, the proposed method provides PIs of ultra-short-term wind power to facilitate various rise-based decision-making tasks (e.g., interval optimization and robust optimization of power systems) to determine the needed reserve [
In this paper, the goal is to propose a new GNN and the improved Bootstrap technique for ultra-short-term interval prediction of wind power. The key factors affecting the performance of the proposed method are the standard deviations to be initialized in the improved Bootstrap technique.
Both theoretical analysis and simulation suggest that the parameter s1 should be larger than the parameter s2 to obtain a wide enough PIs, which can cover real values with a specified probability (i.e., PI nominal confidence). When the proposed method is migrated into other datasets, these key parameters can be determined by similar simulation steps in Section IV-B.
To improve the precision of ultra-short-term prediction of wind power, this paper attempts to model the inputs as a graph from a new perspective. A GNN-based point prediction model is presented to model spatiotemporal features, and then an improved Bootstrap technique is proposed to obtain high-quality PIs. Through numerical simulation on two real-world datasets, the following conclusions are obtained.
1) The improved Bootstrap technique can effectively reduce the width of PIs with negligible reduction of PICP, especially for wind power generation curves with weak volatile regions.
2) Compared with other popular point prediction methods (e.g., MLP, LightGBM, GCN, Bi-LSTM, and GNN in [
3) No matter how the time horizon changes, the proposed method can obtain the superior performance to other baselines (e.g., MLP, LightGBM, GCN, Bi-LSTM, and GNN in [
Although the numerical simulation results show that the proposed method outperforms popular baselines, it still has some limitations to be addressed.
1) The traditional and improved Bootstrap techniques have similar PIs for the strong volatile regions, since the reduction of the interval width of these regions easily causes the PICP to drop, and the improved Bootstrap technique only aims to reduce the width of PIs for the weak volatile regions. In future works, the Bootstrap technique can be further improved to target regions with strong volatility.
2) In addition to hourly and intra-hourly wind power interval predictions, the proposed method may be extended to wind power prediction with a longer time horizon.
3) The widely-used PICP, PINAW, and CWC are used to test the performance of the proposed method. In the future, more metrics (e.g., pinball loss, Winkler score, and continuous ranked probability score) can also be used for further evaluation of models.
Appendix
Algorithm 1 : construction of PIs using improved Bootstrap technique |
---|
% Initialize parameters α, Bootstrap repeats Br, s1, s2: Alpha=0.95; % ; Br=5000; % Bootstrap repeats is 5000 s1=0.036; % s1=0.036; s2=0.024; % s2=0.024; % Calculate prediction errors and standard deviation of validation set Errors=Predictions-Real_values_of_validation_set STD_Error=STD(Point_predictions_of_ validation_set) % Assign errors to group 1 and group 2 k=1 for i=1:length(STD_Error) Group1(i)=Errors(i) if STD_Error<s1 Group2(k)=Errors(i) k=k+1 end end % Assign errors to group 3 if STD(Point_predictions_of_test_set)<s2 Group3=Group2 else Group3=Group1 end % Assign errors to group 4 for i=1:Br id=randperm(length(Group3)) Group4(i)=Group3(id(1)) end % Construct PIs ;
Upper=percentile(Group4, Alpha +(()/2)) % Output results sprintf(‘%.2f’, Lower) sprintf(‘%.2f’, Upper) |
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