1. Introduction

Wind energy is an inexhaustible energy source and it has been paid more attention to worldwide in recent years [1-4]. Wind power prediction is an indispensable requirement for grid-connection operation because of the randomness of wind power.

The prediction of wind power has become an important issue on power system planning and many researchers have established mathematical models with which to predict in recent years [5-8]. But the most models to study wind power are based on linear theory. Because of its complicated and nonlinear characteristics, the results will be inaccurate with wind power predicted by linear theory.

Chaos theory has become a powerful theory for nonlinear science research [9, 10], by which the order and regularity hidden behind the disordered and complex phenomena can be revealed. In predicting wind power, a reconstruction of the phase space of one dimensional time series is made and the laws hidden behind it are revealed with this theory.

BP-ANNs, often referred to as ‘black boxes’ in which the parameters are difficult to obtain, provide a powerful method of identifying highly complex traits in data sets. It can be widely used in time series forecasting due to its characteristics of extreme computational power, massive parallelism, and fault tolerance [1, 11-13]. And it is more efficient through learning without enormous programming. BP-ANNs is not only can learn the smooth prediction function but also can be trained to enumerate unexpected short term regularities in time series.

CBPANNs approach is employed for wind power forecasting in the paper. A calculation of the largest Lyapunov exponent of the time series of wind power and a judgment of whether wind power has chaotic behavior are made at first. Secondly, the best delay time and best embedding dimension are calculated to reconstruct the phase space and determine the structure of CBPANNs. In CBPANNs, the weights and thresholds are optimized by GA. Finally, the output of a wind farm, as an example, is simulated by CBPANNs and BP-ANNs approaches. The result shows that the CBPANNs algorithm has great advantages over that of BP-ANNs in accuracy.

2. Proposed Methodology

The CBPANNs modeling is based on chaos theory, BPANNs theory and genetic optimization.

2.1 Chaos determination

The Lyapunov exponent is the average rate of exponential separation in adjacent orbit of phase space and it is the most important characteristic of chaotic dynamical system. The system with the time series having at least one positive largest Lyapunov exponent is a chaotic system. Wolf method [14] is used to calculate the largest Lyapunov exponent.

2.2 Phase space reconstitution

The output of wind power, sampled at the given intervals, is a discrete time series. The phase space of this time series can be reconstructed according to Pakard and Takens theory [15]. On the basis of the theory, all the dynamic information determining the system state, are contained in time series of the system variables. The system state orbit will still retain the main features of the original system state when the time series of single system variable is embedded to a new coordinate system. During the process of the phase space reconstruction, the evolution information of system variables can be extracted from one-dimensional time series of the system variables. That is to say, the time series x0, x1, x2, ..., xn can be transformed into xn(m, τ) = (xn, xn+τ,..., xn+(m-1)τ) . Where m is the embedded dimension, τ is the time delay, which is an integral multiple of the sampling time interval.

2.3 Calculation of best embedding dimension and best delay time

The best embedding dimension and best delay time can be calculated by C-C method [16], which can be described as follows:

The time series {x(i), i = 1,2,..., n} is decomposed into n non-overlapping subsequences at first.

where n is the length of the time series and it is an integral multiple of τ .

Block averaging strategy is used to calculate test statistics as follows.

where r is the search radius. If the time series {x(t)} distributes independently and identically, S2 (m, r, τ) will equal zero for all the r when both τ and m are fixed and n tends to infinity. The actual time series is finite and interrelated, however, S2 (m, r, τ) is not always equal to zero. τ ~ S2 (m, r, τ) expresses the autocorrelation of time series. The first zero point in τ ~ S2 (m, r, τ) or the time of the minimum difference for all the radiuses can be selected as the optimal time delay τd .

The parameters for calculating τd and m are taken as: n = 120 , m = 2,3,4,5 , ri = i × 0.5σ , i = 1,2,3 where σ = std(x) is the standard deviation of time series.

The first zero point or the first local minimum in is taken as τd .

Define

And the global minimum in S2cor(t) is taken as embedded window τw . Then m can be obtained by τw = (m−1)τ .

2.4 BP-ANNs

The BP-ANNs model contains three layers: one input, one hidden and one output layer. If the model has i input nodes, j hidden nodes, and k output nodes, there will be weights of N = i×j between input layer and hidden layer, thresholds of j in hidden layer, weights of M = j×k between hidden layer and output layer and thresholds of k in output layer.

2.5 BP-ANNs

The initial weights and thresholds of BP-ANNs are optimized by GA, which can improve network performance to make forecasting more accurate.

2.5.1 Population Initialization

The four variables of BP-ANNs, including the weights between the input layer and hidden layer, the thresholds of the hidden layer, the weights between the hidden layer and output layer, and the thresholds of the output layer, are encoded as binary string. And all of them are sequentially combined as an individual code.

2.5.2 Fitness Selection

The error matrix norm between the prediction and actual data in samples is taken as the objective function, the sort of which is made by calling the ranking function FitnV=ranking (obj) (obj is objective function), and the results act as the fitness function.

2.5.3 Genetic manipulation

In genetic operation, the selection, crossing and mutation of individuals are respectively made by stochastic universal sampling, single point crossover and probability choosing.

2.6 CBPANNs algorithm

On the basis of the analysis above, the method can be descried as follows.

The data of wind power are sampled at intervals of 15 minutes and the sampled data are taken as the original time series at first. Then the largest Lyapunov exponent of the time series is calculated by Wolf method to determine whether the time series have chaotic behavior. If so, the best embedding dimension and best delay time are calculated based on C-C method, and then the phase space is reconstructed. Finally, a three-layer CBPANNs is constructed according to the phase space, at the same time, the weights and thresholds of CBPANNs are optimized by GA. The algorithm flow chart is shown in Fig. 1.

Fig. 1.Flow chart

3. Simulation

In order to verify the validity of CBPANNs method, a sampled data of wind power is simulated and analyzed.

The wind power data are sampled at intervals of 15 minutes from 0:00 on 5/10/2012 to 23:00 on 6/6/2012 in a wind farm, which are taken as the original time series, and shown in Fig. 2.

Fig. 2.Time series of wind power

The largest Lyapunov exponent λmax = 0.1943 can be obtained. Therefore, the time series of wind power have chaotic behavior. The variation of , and S2cor(t) with time can be drawn in Fig. 3. Then τd = 12 and τW = 72 can be obtained, thus having m = 7 .

Fig. 3.Variation of , and S2cor(t) with time

τd and m are used to reconstruct the phase space of X = {x(1), x(2),...,x(2688)} . A reconstructed phase space which is a matrix of (n-τw)×m, is established by taking the data in X. The elements of the row are composed of the data taken at the interval of τd and those of the column of the data taken from x(i) to x(i+τw) (i=1,2,…,(n-τw)) successively. Therefore, the reconstructed phase space can be expressed by:

The hidden node n2 and the input node n1 in three-layer BP-ANNs are related by n2=2n1+1. Therefore, a three-layer CBPANNs model with 7 (n1=m) input nodes, 15 hidden nodes and 1 output node (7-15-1) is obtained, which is shown in Fig. 4. The following fitting function can be achieved by the model.

Fig. 4.CBPANNs model

The original time series that come from the 73th to 2208th data are taken as training samples and the 2209th to 2688th data are taken as testing samples. Let the error between training output and expected output (actual output) be 0.001, learning rate be 0.9, momentum factor be 0.95, the training time be 1000, and the parameters of GA be as shown in Table 1.

Table 1.Parameters of GA

In order to compare the results forecasted by BP-ANNs and CBPANNs methods, a simulation is performed under the same conditions, and the results are shown in Fig. 5.

Fig. 5.Comparison output of BP-ANNs and CBPANNs

The absolute error (AE), relative error (RE), correlation coefficient(R), mean square error (MSE), root mean square error (RMSE), mean average error (MAE), mean average percentage error (MAPE) and sum of squared error (SSE) are used to evaluate the forecasting effect of CBPANNs. They are defined as shown in Eqs. (8)-(15).

where ŷi are the forecasting values and yi are the actual values.

AE and RE are respectively plotted in Figs. 6 and 7. From the figures, it can be seen that the forecasted results by CBPANNs method have higher accuracy and smaller AE and RE than those by BP-ANNs approach. When the wind power fluctuates drastically, both the AE and RE calculated by the two methods will increase. But they are much smaller by CBPANNs than by BP-ANNs.

Fig. 6.Comparison of absolute errors

Fig. 7.Comparison of relative errors

In Table 2 are listed the main evaluation indexes calculated by these two methods. It can be seen that CBPANNs method has advantages over BP-ANNs in all the evaluation indexes, thus proving the effectiveness of the prediction by CBPANNs.

Table 2.Comparison of evaluation indexes

The evaluation indexes will become poor with the increasing length of testing samples. In order to analyze the variation of the evaluation indexes with training samples and testing samples, four groups of data are taken which are shown in Table 3, and a simulation of them is performed. The RE of prediction of the final original time series that come from the 2209th to 2688th data（from 0:00 on 6/1/2012 to 23:00 on 6/6/2012）are compared, which are plotted in Fig. 8. It can be seen that the shorter the length of the training samples and the longer the length of the testing samples, the greater the RE will be.

Table 3.Four groups of data taken for simulation

Table 4.Comparison of evaluation indexes under the condition of the different lengths of the training samples and testing samples

Fig. 8.Variation of relative errors with testing data

4. Conclusion

The research proposes CBPANNs, a new method for forecasting wind power, of which the initial weights and thresholds are optimized by GA. The results obtained show that this method is characterized by high precision.