Comprehensive evaluation of ARMA–GARCH(-M) approaches for modeling the mean and volatility of wind speed Heping Liu, Ergin Erdem, Jing Shi ⇑ Department of Industrial and Manufacturing Engineering, North Dakota State University, Dept. 2485, PO Box 6050, Fargo, ND 58108, USA article info Article history: Received 8 August 2010 Received in revised form 27 September 2010 Accepted 27 September 2010 Available online 25 October 2010 Keywords: Wind speed Forecasting ARMA GARCH GARCH-M abstract Accurately modeling the mean and volatility of wind speed can be beneficial to effective wind energy uti- lization. For this purpose, this paper evaluates the effectiveness of autoregressive moving average– generalized autoregressive conditional heteroscedasticity (ARMA–GARCH) approaches for modeling the mean and volatility of wind speed. Five different GARCH approaches are included, and each consists of an original form and a modified form, GARCH-in-mean (GARCH-M). As a result, 10 different model struc- tures are evaluated, based on the 7-year hourly wind speed data collected at four different heights from an observation site in Colorado, USA. Multiple evaluation methods of modeling sufficiency are used. The results show that the ARMA–GARCH(-M) approaches can effectively catch the trend change of the mean and volatility of wind speed. Also, the volatility of wind speed has the nonlinear and asymmetric time- varying feature, and the ARMA–GARCH-M structures can consistently improve the modeling sufficiency of mean wind speed. As the height increases, the explanatory power of all ARMA–GARCH(-M) models slightly deteriorates. On the other hand, no single model structure outperforms the others at all heights, and this confirms that for any wind speed dataset, the potential models should be evaluated to find the most appropriate one for the highest modeling sufficiency. Ó 2010 Elsevier Ltd. All rights reserved. 1. Introduction Wind energy is one of the fastest growing energy sources. The World Wind Energy Report [1] indicates that the total installed wind power capacity reached to 160,000 MW in 2009, and it is ex- pected to double every 3 years. A US Department of Energy report [2] estimates that wind energy could contribute to 20% US electric- ity supply by 2030 even with a modest growth trajectory of wind power installation. However, in the utilization of wind energy, one critical difficulty is that the electricity generated by wind power systems is not as stable as those generated by other sources, and thus it is challenging to integrate wind energy into traditional electricity systems. This problem can be effectively mitigated if the operation of wind farms can be manipulated according to the accu- rate information of the mean and turbulence of wind speed. As such, modeling the short-term mean wind speed and its turbu- lence has been a crucial issue in wind power industry. For the short-term mean wind speed forecasting, there are a number of methods which mainly include physical models [3], conventional statistical models such as autoregressive (AR) mod- els, autoregressive moving average models (ARMA) and autore- gressive integrated moving average (ARIMA) models [4,5], Bayesian methods [6], neural networks [7–10], Kalman filter tech- niques [11], vector autoregression (VAR) and generalized impulse response analysis method [12], other non-conventional time-series models [13,14], and some hybrid approaches [15]. These methods provide accurate prediction for the mean wind speed. However, an evident deficiency is that they do not consider the turbulence (vol- atility or heteroskedasticity) of wind speed. In other words, they assume that the turbulence is homoscedastic. Heteroskedasticity is a critical aspect of data nonstationarity in time series forecasting. It implies that different observations in time series have different variances. Heteroskedasticity can pose some problems. For exam- ple, in the ordinary least squares (OLS) estimate, the presence of heteroskedasticity gives a false sense of precision, and the standard errors and confidence intervals estimated by OLS will be too nar- row although the regression coefficients of OLS are still unbiased [16]. In the prediction of mean wind speed, it is necessary to inves- tigate and model the volatility of wind speed. Actually, the mean- ing of modeling the volatility of wind speed is far beyond the demand of improving the modeling approach of mean wind speed, and it bears practical considerations of optimizing the operations of wind plants. Not only can the volatility modeling assist engi- neers in the operation of wind plants in a number of areas from the scheduling of peak load to the design of optimal cut-in and 0306-2619/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2010.09.028 ⇑ Corresponding author. Tel.: +1 701 231 7119; fax: +1 701 231 7195. E-mail address: jing.shi@ndsu.edu (J. Shi). Applied Energy 88 (2011) 724–732 Contents lists available at ScienceDirect Applied Energy journal homepage: www.elsevier.com/locate/apenergy