1.1. Time Series Forecasting and Wind Energy
Time series forecasting plays an essential role in many fields, especially in meteorology, economics and energy. Time series models produce forecasts by discovering the inner relationships within historical records. This paper focuses on wind speed forecasting, which is crucial in the whole life-cycle of wind farm construction and operation and is also the basic technique to guarantee the grid security of a wind-connected system. Wind power is economic and ecologically friendly, which makes it one of the most popular and promising alternative energy sources. Wind power accounts for approximately 10% of the national power use in many European countries, and for more than 15% in Spain, Germany and the US [
1]. However, the main obstacle for wind industry development is the variability of output power, which seriously prevents wind power penetration and threatens grid security. To guarantee the security of the grid system, the dispatching department have to balance the grid’s consumption and production within very small time intervals [
2]. Moreover, because the lack of accurate information on wind occurrence, the efficiency of wind turbine may also be limited [
3]. In actual power generation, wind predictions—especially the short-term forecasts—are important for scheduling, controlling and dispatching the energy conversion systems [
4]. However, as the most important characteristic of wind, speed can be easily influenced by other meteorological factors, such as air pressure, air temperature and terrain [
5]. Thus, wind speed prediction is not easy to address. Moreover, wind speed modelling has become one of the most difficult problems [
6,
7].
1.2. Wind Speed Forecasting: Existing Works
Many methods have been attempted to forecast wind speed. In general, they can be classified into two categories: physical and statistical methods. Physical methods are always referred to as meteorological predictions of wind speed, including the numerical approximation of models that describe the state of the atmosphere [
8], such as the Weather Research and Forecasting (WRF) model [
9]. These models always choose physical data such as topography information, pressure and temperature to forecast wind speed in the future [
10,
11]. As one of the current-generation physical models, WRF [
9] is widely used in both research [
12,
13,
14] and operational forecasts. Reference [
13] used the WRF model to manage ocean surface wind simulations forced by different initial and boundary conditions. Reference [
14] compared WRF with the Wind Atlas Analysis and Application Program (WAsP) model, to test the performance in terms of flow characteristics and energy yields estimates. Considering numerical weather prediction (NWP) models, one important issue is downscaling. Generally, two categories are focused on: dynamic and statistical downscaling methods. Dynamical downscaling methods have clear physical meanings and are unaffected by the observation data. However, they require large computational costs. Being different, statistical downscaling—including transfer function method (TFM), weather pattern method (WPM) and stochastic weather generator (SWG)—is simple to establish and needs a small amount of calculation, but it may be influenced strongly by observations [
15]. Recently, many new statistical downscaling techniques have been developed, such as the similarity method, hidden Markov model (HMM), generalized linear model (GLM) and others [
15].
Unlike physical models, statistical methods make forecasts by discovering the relationships in historical wind speed data and sometimes other variables (e.g., wind direction or temperature). The data used is recorded at the observation site or other nearby locations where data are available. Moreover, many statistical methods have been applied, such as the auto-regressive integrated moving average (ARIMA) model, Kalman filters, and the generalized auto-regressive conditional heteroscedasticity (GARCH) model,
etc. The statistical models can be used at any stage during modelling, and they often merge various methods into one. Physical and statistical models each have their own advantages for wind speed prediction, but few forecasts use only one of them. The physical prediction results are just the first step of wind forecast; then, the physically predicted wind speed can be regarded as an auxiliary input to other statistical models [
16,
17,
18]. Currently, grey models (GM) [
19,
20] and models based on artificial intelligence (AI) techniques [
21,
22] have been developed for this area, containing the artificial neural networks (ANNs) of multi-layer perceptrons (MLP) [
23], radial basis function (RBF) [
24], recurrent neural networks [
25,
26], and fuzzy logic [
27,
28].
As one of the most widely used time series approach, ARIMA has been used as an effective and efficient forecasting technique in many fields, including traffic, energy, and the economy. Generally, ARIMA is a linear model that represents both stationary and non-stationary series [
29] and uses historical time series patterns to make forecasts for the future data trend. In terms of the wind speed prediction problem, which is studied in this paper, the ARIMA models are effective and suitable for short-term and very-short-term predictions. References [
30,
31,
32] applied the auto-regressive moving average (ARMA) model to wind speed predictions with different time horizons. Furthermore, because wind-related data always show obvious periodicity, a seasonal ARIMA model can be defined with the consociation of a seasonal difference process [
33]. Later, the fractional-ARIMA model was proposed by Kavasser and Seetharaman [
34], which assumes that the differencing parameter
of ARIMA (
) is a fractionally continuous value in the interval (−0.5, 0.5). Their model was used for wind speed prediction on the day-ahead and two-day-ahead time horizons in North Dakota. When there is little knowledge available or there is no suitable model relating the predicted variables to other explanatory factors, the ARIMA model is particularly useful [
35]. Some articles made a hybrid approach by combining the ARIMA model with other methods. Studies take ARIMA as the first step of a hybrid method, and then the residual series of ARIMA can be regarded as the nonlinear part of the original series. Reference [
36] developed a hybrid ARIMA-ANN model for hourly wind speed prediction. In their method, the ARIMA model was first used for wind speed forecasting, while the ANN was chosen to reduce the errors from the ARIMA models. Later, a hybrid method combined the seasonal ARIMA, and the least square support vector machine (LSSVM) was developed in Reference [
5] for monthly wind speed prediction in the Hexi Corridor of China. Here, both ANN and LSSVM are quite effective for addressing series within nonlinear signals.
Improvement made on ARIMA has enhanced the model performance substantially. However, by considering either the improved ARIMA model or the ARIMA-combined hybrid methods for wind forecast, most approaches employ only the historical observations but not the factors of atmospheric dynamics. Some studies claimed that an accurate wind prediction method must include a numerical weather prediction (NWP)-based process [
37].
1.3. Original Contribution: Developed Self-Adaptive Wind Speed Forecasting Strategy
The original contribution of this paper is the development of a self-adaptive (SA) auto-regressive integrated moving average with exogenous variables (ARIMAX) model optimized by the chaotic particle swarm optimization (CPSO) algorithm called the SA-ARIMAX-CPSO approach, which is applied to wind speed prediction. Specifically, the applied ARIMAX model takes the WRF simulation as an exogenous part, which makes the forecasting model a combination of both statistical and physical information. Moreover, the CPSO-driven SA strategy enables the proposed method to syncretize the previous model and the recently updated information. In this paper, the self-adaptation contains two parts. The first one is new model fitting, when the recent measurements or WRF data are available. This paper updates the fitting coefficients every time-step, while the WRF model runs once a day. The second one is adaptation process, where the optimal adaptive weights are determined only based on the training set.
On the issue of very-short-term wind speed prediction, models were established generally based on a statistical process, while the NWP simulations were typically used for short-term predictions. This is mainly due to the model accuracy and calculation costs. This paper develops a hybrid approach for very-short-term wind speed prediction combining both statistical and physical models, which has an acceptable amount of calculation and effective model performance. Specifically, the WRF model is now the current generation physics-based atmospheric model, which is widely applied; the ARIMA process is the typical time series model, which emphasizes modelling the relationship among historical observations. Thus, the proposed ARIMAX model in this paper considers not only the statistical information from historical wind speed observations but also the physical process of atmospheric motion.
Furthermore, this paper develops a SA strategy to apply for the ARIMAX method. Model parameters are always fixed values that are determined by the training data set; this may be unreasonable in a dynamic process. When new information is obtained, the prediction system should be updated. In this paper, the new information includes two parts—the newly updated measurement records and the WRF simulation result. From this opinion, this paper develops a SA-ARIMAX model, which has adaptive model parameters when the new information is available. During this process, the CPSO algorithm is applied to obtain the optimized parameters. Simulation results show that the developed SA-ARIMAX-CPSO method in this paper performs considerably better than the original auto-regressive moving average with exogenous variables (ARMAX), ARIMAX, and adaptive ARMAX models.