With the development of wind power technology, the security of the power system, power quality, and stable operation will meet new challenges. So, in this paper, we propose a recently developed machine learning technique, relevance vector machine (RVM), for dayahead wind speed forecasting. We combine Gaussian kernel function and polynomial kernel function to get mixed kernel for RVM. Then, RVM is compared with back propagation neural network (BP) and support vector machine (SVM) for wind speed forecasting in four seasons in precision and velocity; the forecast results demonstrate that the proposed method is reasonable and effective.
Over the past decade, people in many countries worldwide have paid significant attention to wind power generation because of it being pollutionfree, clean, and renewable. At the end of 2012, worldwide installed capacity of wind power reached 282.2 GW, increased almost 20% compared to the previous year 2011 which of 240 GW. By the end of 2020, total global installed capacity will reach 1150 GW, and wind power will be over 2800 TWh, accounting for about 12% of global electricity demand; by the end of 2030, installed capacity will exceed 2500 GW, and wind power generating capacity will reach 6600 TWh, accounting for about 23% of global electricity demand [
We can cluster the wind forecasting techniques into two main groups; the first group are physical methods, taking physical considerations into account, such as temperature and local terrain. In [
Another group are statistical methods. Conventional ones are identical to the direct random timeseries model, such as autoregressive model (AR), moving average model (MA), autoregressive moving average model (ARMA), and autoregressive integrated moving average model (ARIMA). Kamal and Jafri [
Apart from the mentioned forecasting techniques, machine learning algorithms such as artificial neural network (ANN), Bayesian network (BN), and support vector machine (SVM) are usually adopted for time seriesbased wind prediction. Bilgili et al. [
However, the SVM had a number of significant and practical limitations. For example, we could not get probabilistic predictions and the kernel function must satisfy Mercer’s condition. In order to overcome these, Tipping [
This paper establishes RVM model to forecast dayahead wind speed in JiangSu, compared with BP and SVM models. It can show that the proposed method is more effective and robust and gets rid of the overfitting problem of traditional nonparametric regression models. The rest of the paper is organized as follows. A brief review of the theory of RVM learning for classification is provided in Section
Relevance vector machine (RVM), based on the overall Bayesian framework, is a sparse probability model and now is one of the hot research fields [
Given a training dataset
Consider
Therefore, the probability formula for relevance vector machine model is
Because we assume the targets are independent, thus the likelihood of the complete dataset can be defined as
Because there are so many parameters in the model as training examples, maximum likelihood estimation of
However, we cannot easily gain the full analytical solution to (
After defining the prior distribution and the likelihood distribution, according to Bayes’ theorem, we can obtain the posterior distribution of all unknown parameters:
Then the mean and the covariance of
In order to calculate
In related Bayesian models, this quality is referred to as “the marginal likelihood’’ [
Because we cannot obtain values of
When
Meanwhile, for the noise variance
In practice, because many of
At convergence of the hyperparameter estimation procedure, we use the posterior distribution over the weights for predictions, conditioned on the maximizing values
Because both terms in the integrand obey Gaussian distribution, we can easily compute the predictive distribution:
Eventually, we get the regression model of RVM in function (
In summary, the forecast process can be summarized as in the following steps:
initialize variances
compute posteriori statistics of weights
compute
if it is not convergent, then go back step (2); otherwise, go to step (5);
if
get the predictive mean intuitively from
In this study, wind speed values throughout 2008 on a wind farm in Jiangsu are taken as the training samples, and all the data have an interval period of 15 min. To evaluate the performance of proposed model, we establish RVM model for 96 points wind speed prediction, compared with SVM and BP in terms of forecast accuracy, model running time, and model complexity. Based on historical data in each quarter, we establish forecasting models to predict dayahead wind speed on March 25, June 26, September 29, and December 28.
Because the input vectors contain different kinds of physical quantities, in order to ensure the variables are comparable, but also solve problems such as the increasing of training time, first of all, we take normalization process on all input data. The normalized respective variables will be in
For evaluating the forecasting performance, mean relative error (MRE) and root mean square error (RMSE) are used; they are defined in functions
Because the kernel function of RVM does not need to satisfy Mercer’s condition, the selection of the kernel function has a certain degree of freedom. The basic idea of the hybrid kernel function [
In this paper, we combine Gaussian kernel function and polynomial kernel function to get mixed kernel of global and local nature:
To evaluate the performance of proposed model, RVM, BP, and SVM are established for dayahead wind speed predictions on March 25, June 26, September 29, and December 28. Table
Comparisons of forecast accuracy for three models.
Month  BP  SVM  RVM  

RMSE  MRE  RMSE  MRE  RMSE  MRE  
March  0.1123  0.0180  0.0925  0.0115  0.0782  0.0108 
June  0.1856  0.0520  0.2033  0.0428  0.1607  0.0347 
September  0.3521  0.0641  0.2297  0.0230  0.2451  0.0185 
December  0.1106  0.0136  0.0930  0.0057  0.0790  0.0046 
Avg. 






Comparisons of test time and vector number for three models.
Month  BP  SVM  RVM  

Test time (s)  Number of vectors or neurons involved  Test time (s)  Number of vectors or neurons involved  Test time (s)  Number of vectors or neurons involved  
March  677.56  300  639.24  216  302.48  33 
June  1237.43  380  857.25  306  288.87  40 
September  1002.86  340  641.91  320  293.85  35 
December  835.03  320  506.29  202  288.66  31 
Avg. 






Forecasting results with different models on March 25 (a), June 26 (b), September 29 (c), and December 28 (d).
It is found from the comparison results that the forecast accuracy of the proposed method is higher than that used by the other models. The average RMSE of RVM model is only 0.1408 m/s, lower than those of BP and SVM, which are 0.1902 m/s and 0.1546 m/s. The average MRE of RVM is 1.72%, while those of BP and SVM are 3.69% and 2.08%. In different seasons, forecast accuracy is different. In spring and winter, wind speed changes are relatively small, so higher precision can be got; in summer and autumn, wind speed at the coast has large fluctuations, so the accuracy will decrease.
Table
In this paper, RVM is proposed for dayahead wind speed forecasting. Firstly, we combine Gaussian kernel function and polynomial kernel function to get mixed kernel for RVM model. Then, we compare RVM with BP and SVM for wind speed forecasting in four seasons for precision and velocity. Finally, the simulation results show that the proposed method is more effective and robust and has better performance in terms of forecast accuracy, model running time, and model complexity than that used by BP neural network and SVM model. So the theoretical feasibility of RVM for the wind speed prediction has the some meaning.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China under Grants 51277052, 51107032, and 61104045.