WIND POWER FORECASTING BASED ON REFINED LSTAR-GARCH MODEL

Wind Power forecasting(WPF) plays a crucial role in the secure and economic operation of power systems. To improve precision of the wind power forecasting model, the latent information in volatility characteristics of the wind power time series needs to be further explored. In this paper, with a novel Smooth Transition Autoregressive (STAR) type model, the multiple regime switching effect on the volatility of wind power time series is investigated. First, the Threshold Auto-Regressive (TAR) and the traditional Logistic STAR is provided, and the separation of different regime in the volatility of time series is discussed preliminarily. Second, a Refined STAR (RSTAR) structure with the novel smooth transition function is presented to depict the double asymmetric effect. By combining the RLSTAR structure and the GARCH framework, a prospective Refined Logistic Smooth Transition Auto-Regressive GARCH (RLSTAR-GARCH) wind power forecasting model is proposed. Moreover, considering the fat-tail effect in the volatility of wind power time series, RLSTAR-GARCH models with fat-tail distribution are proposed for generalization. Case studies on a practical sample for wind power forecasting validate the feasibility and effectiveness of this method. With the smooth transition structure, the behaviours of RLSTARGARCH type model near the threshold are highlighted. Comparison of the forecasting performance between the proposed models and the classical model shows that the RLSTAR model is promising for wind power forecasting.


INTRODUCTION
In recent years, power systems witness the integration of large production renewable generation into grid.Wind power plays a crucial role in the secure and economic operation of power systems.It is reported that by the end of 2014, the global total installed wind power capacity reached to 369.6 GW, and 24 countries had the installed wind power capacity of more than 1 GW [1].As an extensively concerned issue from engineers and researchers, many effective wind power forecasting methods have been proposed.Generally, there are physical models [2], Auto-regressive Moving Average (ARMA) models [3], Generalized Auto-regressive Conditional Heteroskedastic (GARCH) models [4], Artificial Neural Network (ANN) methods [5,6], spatial models [7], Kalman filter techniques [8], Grey model [9], hybrid approaches [10,11], and volatility model represented by GARCH (Generalized Autoregressive Conditional Heteroskedasticity) model [11].However, due to the intrinsically nonlinear characteristics of wind power time series, the latent information in volatility characteristics of the wind power time series needs to be further explored.Based on the analysis of large amounts of real-world historical wind power data, it has been noticed that different shocks have different influence on the volatility of wind power time series and the mechanism of the switching-regime is complicated.Considering that Smooth Transition Auto-Regressive (STAR) model is good at depicting nonlinear characteristics of time series, a kind of improved STAR -RLSTAR model is proposed on the second order moment level, and the mechanism of multiple regime switching effect is discussed.Furthermore, extended RLSTAR models following non-Gaussian distributions are proposed and applied in the case study.The remainder of the paper is organized as follows.First the traditional TAR and LSTAR models are described, and the limitations of TAR and LSTAR are discussed.Then the RLSTAR model is proposed to depict the multiple regime switching effect.Fat tail distribution methodology that are employed to for model generalization is presented.A case study is demonstrated to validate the proposed model.

TAR Model
Threshold Autoregressive (TAR) model, a classical nonlinear model, can realize the regime switching by means of threshold space [12].As a commonly used model, two regimes Self-Exciting TAR (SETAR) model can be expressed as follows where y t follows an threshold autoregressive of order p process.Note that () td I y c   is a dummy variable with threshold c which is defined as

LSTAR Model
TAR model has non-continuous point at the threshold, and achieves switch between different regimes by means of step function.However, for some specific systems, it is more reasonable if gradual switching is adopted.Terä svirta proposed the Smooth Transition Autoregressive (STAR) Model, which was a nonlinear model and had smooth transition between different regimes [13,14].If the time series { y t }satisfies (3) then{ y t } follows the STAR(p) model of two regimes.where d is the delay parameter, , s  are parameters representing the location and scale of model transition, respectively.The smooth transition function ， () In practice, () F  is assumed to be in some classical forms.One of the popular used functions is a logistic function which is written as 1 ( , ) 1 exp( ( where

RLSTAR Models
The classical regime switching model deals with the two regimes problem.In practice, the regime switching mechanism of time series might be more complicated.Moreover, some regimes are nested in the regimes.Based on the analysis of large amounts of real-world historical wind power data, there is multiple regime switching effect in the volatility of some time series [15].To depict multiple regime switching effect, a kind of Refined LSTAR(RLSTAR) model is proposed in this study.The specification of RLSTAR is as follows Where, 1 ( ; , ) 1 exp( ( )) When n=2, the model is reduced to the double version of RLSTAR -Double LSTAR (DLSTAR) model.The specification of DLSTAR is shown as: ( ; , ) ( ; , )

RLSTAR and Its Combination with Volatility Model
GARCH type model is a popular model to describe the volatility of time series [16,17].As the classical form of GARCH model, GARCH(p, q) model is expressed using conditional mean equation 1 () and conditional variance equation where Ey  is the conditional mean taking into account information at t-1.
RLSTAR and GARCH can be prospectively combined to obtain the RLSTAR-GARCH model with () ; Where the asymmetric parameter 1  depicts the classical asymmetric effect, and the incremental asymmetric parameter 2  depicts the asymmetric effect in the incremental level.From the analysis above, the RLSTAR-GARCH model can realize the smooth transition between different regimes, depict the multiple regime switching effect, and ensure the continuity near threshold point.

Fat Tail Effect and Application of Non-Gaussian Distributions
Fat tail effect exists in the wind power time series.As a result, to improve the capability of depicting the fat tail of wind power time series it would help to extend from following normal distribution to non-Gaussian distributions [17,18].Most commonly used non-Gaussian distributions including t distribution, Generalized Error Distribution (GED), and Laplace distribution are employed in this work.

1) t Distribution
The probability density function of t distribution is where   is Gamma function, and n is the degree of freedom.
2) GED The probability density function of GED is where , which is a function of distribution shape parameter v; when 2 v  , GED will have fatter tail than normal distribution; when 2 v  , GED is reduced to normal distribution.
3) Laplace distribution Laplace distribution is also called the double exponential distribution.The probability density function is exp () 2 , the Laplace distribution is a standardized Laplace distribution yielding mean=0, variance=1.The standardized Laplace distribution processes fatter tail than normal distribution.In this work, RLSTAR-GARCH model following the standardized Laplace distribution is called a RLSTAR-Laplace model.

Parameter Estimation
Conditional Maximum Likelihood Estimation (CMLE) is employed to estimate parameters of the standard RLSTAR models.At the same time, Marquardt algorithm, a famous modified version of Gauss-Newton algorithm, is used to control the iteration process [19,20].Moreover, Parameters of the fat-tailed RLSTAR models are estimated also by CMLE, assuming that follows t distribution, GED, or Laplace distribution.

Data
The historical wind power data provided from a coastal wind farm groups in Yancheng, China, is used to examine the presented forecasting models.Collected from Jiangsu, a province with rich coastal wind resources in East China, the data samples can be representative of a typical wind power pattern in China.The samples are the 5-minute wind power data, containing 2016 points of data from April 1 to April 7, 2013.

Stationary Test Results
Stationary test is first carried out by ADF (Augmented Dickey Fuller) and PP (Phillips-Perron) test [20] to examine the stationarity of the wind power time series.The results of the two tests consistently report that the wind power series Y t is not stationary at 5% significance level.Then, differencing is used to obtain the first differenced series I t .
At this time, ADF and PP test are both statistically significant at the 5% significance level, indicating that it is stationary, and the stationary precondition of modelling is met.The following study is based on the series of I t .

Modelling with RLSTAR Models
Considering serial dependence in wind power series, autoregressive moving average (ARMA) structure is employed in the mean equation of GARCH-M type model.The mean equation of RLSTAR model is shown below: ) Based on the routine from [11], the orders of ARMA are determined as ARMA(4,5)-RLSTAR(1,1)-GARCH(1,1) (RLSTAR, for short) with the specification of ( 18), ( 11), (12), and (13).Consequently, the parameter is obtained by CMLE.Estimated parameters of RLSTAR are shown in Table I.

Generalizing RLSTAR to the fattail version
Furthermore, considering the fat tail effect in wind power time series, the RLSTAR model can be extended to follow non-Gaussian distribution.RLSTAR-t, RLSTAR-GED, RLSTAR-L models are established, and the corresponding parameters are also estimated by CMLE.The parameters of these fat-tail LSTAR type models are summarized in Table I.From Table I, we conclude the following results: 1) The significance levels of the parameters in the mean equation and variance equation in each RLSTAR model are all reached.The inherent structures of the parameters in mean equation and variance equation among the studied models are close to each other.2) Both the asymmetric parameters and the incremental asymmetric parameters of Standard RLSTAR, RLSTAR-t, RLSTAR-GED and RLSTAR-L models are all significant, and the meaning implicated by the signs of each parameter are consistent.All models witness the double asymmetric effect in the wind power time series volatility.For the same strength of positive and negative shocks (original stochastic disturbance), positive shocks have greater impacts on conditional variance than negative shocks.On the contrary, positive increments have weaker impacts on conditional variance than negative increments.As a result, it is well-founded to incorporate multiple regime switching in wind power time series volatility.
3) The shape parameter v of GED in RLSTAR-GED model is less than 2 ( [1.4,1.5]v  ), which is the case that GED has fat tail characteristics.Similarly, the degree of freedom n of t distribution in RLSTAR-t model is relatively small and the parameter is significant.Study results show that the depiction of fat tail effect in the wind power time series, these different RLSTAR model are similar.

Evaluation of Forecasting Performance
With all parameter estimated, wind power forecasting can be conducted based on the proposed models.II).Overall, RLSTAR type models perform better than LSTAR type models, GARCH model, and the TP model.In all, taking into consideration of the multiple regime switching effect in wind power time series, it is reasonable to convert non-regime switching model to multiple STAR models.

CONCLUSIONS
In this paper RLSTAR model is proposed for wind power forecasting.Multiple regime switching of wind power time series is investigated.Furthermore, fat tail effect is applied to the proposed model for wind power forecasting.RLSTAR models effectively can depict the multiple regime switching effect and resolved the discontinuity point issue theoretically, such that the forecasting result can be much closer to actual conditions.A case study clearly illustrates that standard version of RLSTAR provide the most promising forecasting results.Asymmetric parameters and incremental asymmetric parameters in the RLSTAR models show distinctive practical meanings.From the analysis on these parameters, it is indicated that volatility in wind power time series have significant double asymmetric effect.
It is important to concentrate on comprehensive description of volatility in wind power time series.Discussion and analysis on the multiple regime switching of volatility is under the way to improve wind power forecasting performance.
 , and order of RLSTAR is set to 2, the conditional variance equation of RLSTAR-GARCH model is

Table I
According to the forecasting results, It can be found that the forecasting performances of RLSTAR models are satisfying.From the point of E RMSE ， E MAE ， E MAPE , RLSTAR model excels other models (Table