An alternative electric power source, such as wind power, has to be both reliable and autonomous. An accurate wind speed forecasting method plays the key role in achieving the aforementioned properties and also is a valuable tool in overcoming a variety of economic and technical problems connected to wind power production. The method proposed is based on the reformulation of the problem in the standard state space form and on implementing a bank of Kalman filters (KF), each fitting an ARMA model of different order. The proposed method is to be applied to a greenhouse unit which incorporates an automatized use of renewable energy sources including wind speed power.
Energy is considered amongst the most significant factors that are closely related to both economic and social developments. It is also a fact that nowadays the majority of the electrical energy production is based on the fossil fuels, which on one hand are, without any doubt, highly efficient but on the other are responsible for the emission of greenhouse gases and their reserves are limited.
Consequently renewable sources of energy, such as wind, biomass, solar power, and wave power, have been already adopted for electric power production. It is well known that the wind power generation raises issues of reliability due to the fact that the wind speed is significantly and directly affected by various factors such as the type of the terrain, the height, season of the year, atmospheric conditions, obstacles present, and many more. This leads to the conclusion that unless the reliability of the wind power generation is at an acceptable level, wind power is not eligible for constant electrical energy supply to the power system [
Recent studies have shown that combined forecasting methods can offer robust solutions and can be efficiently implemented to various real-life problems in diverging fields such as chemical processes, economics, load forecasting, tourism demand, environmental issues, medicine, and many more [
In this study a hybrid model is presented that reveals the advantages of an ARMA and SVM model in wind speed modelling and prediction problem. Initially successful model identification and parameter estimation have to be performed in order to choose the most appropriate ARMA models. For tackling this task the well-established MMPA was used. This approach was introduced by Lainiotis [
In this research real data were used, provided by Vestas Hellas; the simulation results appear to be very promising.
The problem of fitting an ARMA model in a given time series is present for more than half a century and is still appearing in many different fields such as in remote monitoring of civil infrastructure [
Considering the general case an
It is obvious that the problem requires both the predictor’s order
The major disadvantage of the ARMA models is that their performance can be limited by any significant data nonlinearities.
Due to the fact that the wind speed does not have a constant or periodic behaviour, it was noted, by trial and error, that not a single ARMA model that was able to describe the whole data set satisfactory. It is actually the combination of various ARMA models, each one used for different time intervals and applied for different time durations that describes in the best manner the existing data. So instead of having various ARMA models of different order
If we assume that the model order fitting the data is known and is equal to
Now assign a new variable
If the general form of the matrices
Assuming that the system model and its statistics were completely known, the Kalman filter (KF) in its various forms would be the optimal estimator in the minimum variance sense.
However, if the system model is not completely known the MMPA, introduced by Lainiotis [
In the case under consideration assume that the model uncertainty is the lack of knowledge of the model order
A bank of Kalman filters is then applied, one for each model, which can be run in parallel, thus saving enormous computational time. At each iteration, the MMPA selects the model that corresponds to the maximum a posteriori probability as the correct one. This probability tends (asymptotically) to one, while the remaining probabilities tend to zero. The overall optimal estimate can be taken either to be the individual estimate of the elemental filter exhibiting the maximum posterior probability, for example, a value of 0.9 or higher [
The probabilities are calculated on-line in a recursive manner as it is shown by
For equations (
In support vector machines, as they were proposed in [
A mathematical representation of the SVM function is
By considering the above slack variables and in order to include any extra cost of the training errors, (
Finally by introducing positive Lagrangian multipliers and maximizing (
The Lagrangian multipliers
In this study the Gaussian kernel function (
The wind speed behaviour is unpredictable and it is difficult to be represented. This is the reason for combining two different techniques for modelling the linear and the nonlinear parts of the series. The hybrid model proposed is based on a linear pattern,
If
Consequently the forecast of the hybrid model is
Proposed method.
In this method the weighted average of the estimates produced by the elemental ARMA filters was used as a data preprocessor in order to detect the data’s linearities. This was succeeded using a bank of 10 Kalman filters of order
This research was conducted based on the hour average of daily wind speed recorded by the Vestas Hellas from November 2010 up to February 2011. The obtained time series did not follow any periodic pattern and it was also presenting irregular amplitudes, making it hard to both model and predict (Figures
November 2010: predicted and observed (raw) time series.
December 2010: predicted and observed (raw) time series.
January 2011: predicted and observed (raw) time series.
February 2011: forecasted predicted and observed (raw) time series.
The aim of this work is to generate a single-step prediction based on past observations. The data were normalized to take values from zero to one, before using them as input data to the hybrid model.
From the 2725 available data points, 720 were for November, 744 for December, 744 for January, and 517 for February. For each month 20% of the available data was used for training, 20% for validation, and 60% for testing.
The performance of the hybrid method is judged by (a) comparing the predicted and the observed (raw) wind time series (Figures
Performance of the hybrid model.
Month | MAPE % |
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November | 2.78 | 0.8815 |
December | 3.27 | 0.8533 |
January | 3.19 | 0.8635 |
February | 3.02 | 0.8758 |
Average |
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November 2010: scatter of predicted and observed wind speed time series.
December 2010: scatter of predicted and observed wind speed time series.
January 2011: scatter of predicted and observed wind speed time series.
February 2011: scatter of predicted and observed wind speed time series.
Figures
For the sake of completeness of this research the results of forecasting using only the adaptive combination of MMPA with the ARMA models as well as using only the ANN architecture with the SVM are also presented. The first method was applied successfully in [
Using the data for November, Figures
Summary of results.
Month | MAPE % |
| ||||
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Hybrid | MMPA-ARMA | SVM | Hybrid | MMPA-ARMA | SVM | |
November | 2.78 | 4.93 | 4.73 | 0.8815 | 0.7462 | 0.7861 |
December | 3.27 | 4.75 | 4.32 | 0.8533 | 0.7225 | 0.7689 |
January | 3.19 | 5.23 | 4.12 | 0.8635 | 0.7349 | 0.7921 |
February | 3.02 | 5.64 | 4.43 | 0.8758 | 0.7182 | 0.7783 |
Average |
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November 2010: predicted and observed (raw) time series, MMPA-ARMA.
November 2010: predicted and observed (raw) time series, SVM.
November 2010: scatter of predicted and observed wind speed time series, MMPA-ARMA.
November 2010: scatter of predicted and observed wind speed time series, SVM.
At this point it should be mentioned that when dealing with the wind speed forecasting problem it is quite difficult to attain very high prediction accuracy. The hybrid method proposed gave some very satisfactory results and the accuracy level reached can be considered sufficient for decision making as far as electric power production is concerned.
Last but not least it should be also mentioned that a designer’s crucial task when using MMPA is to assign a proper value for the cardinality the quality of the overall MMPA estimate (in terms of MAPE) increases with the number of the elemental filters applied, so a small value of the computational load of the MMPA is proportional to the number of filters implemented; so a large value
A way of assigning the proper value of
December 2010: MMPA probability sequence (ARMA model order selection) for the whole data set provided.
It can be seen that the highest order
The area of forecasting is very demanding and it is over 50 years that ARMA models were exclusively used for tackling real-life problems. Recently ANN were applied in difficult prediction problems showing very satisfactory results especially due to their ability of manipulating the nonlinearities of the dataset. The aim of this work was not to just add yet another technique of wind speed prediction but to actually validate the fact that different forecasting methods fulfil each other and lead to accurate results. As it was shown the two individual forecasting methods, adaptive MMPA-ARMA and SVM, cannot match the performance of the hybrid method proposed. The results of the proposed method are to be applied to a greenhouse unit which incorporates an automatized use of renewable energy sources including wind speed power. Future work can include adjustments for wind speed prediction for time intervals smaller than 1 hour, say ten-minute intervals, and also for on-line wind speed prediction.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported in part by the European Union (European Social Fund-ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF), Research Funding Program: ARCHIMEDES III. Investing in knowledge society through the European Social Fund.