Time Series and Renewable Energy Forecasting Time Series and Renewable Energy Forecasting

Renewable energy generation has been constantly increasing during recent years. Wind and solar have had the most significant growths among all renewable resources. Wind and solar resources are highly intermittent and dependent on meteorological parameters and climatic conditions. The power output of wind turbines is subject to various meteorological parameters, such as wind speed, wind direction, air tempera- ture, relative humidity, etc., among which the wind speed is the most direct and influential factor in wind power generation. Solar photovoltaic (PV) power is a func- tion of solar radiation. Wind speed and solar radiation time series data exhibit unique features which complicate their prediction. This makes wind and solar power fore- casting challenging. Accurate wind and solar forecasting enhances the value of renewable energy by improving the reliability and economic feasibility of these resources. It also supports integrating solar and wind power into electric grids by reducing the integration and operation costs associated with these intermittent generation sources. This chapter provides an overview of the time series methods that can be used for more accurate wind and solar forecasting. show the developed method based on support regression is more accurate than the persistence and


Introduction
Power generation forecasting is the fundamental basis in managing existing and newly constructed power systems. Without having accurate predictions for the generated power, serious implications such as inappropriate operational practices and inadequate energy transactions are inevitable. High penetrations of intermittent renewable energy sources such as wind and solar significantly increase uncertainties of power systems which in turn, complicate the system operation and planning. Accurate forecasting of these intermittent energy sources provides a valuable tool to ease the complication and enable independent system operators (ISOSs) to more efficiently and reliably run power systems.
There are three major methods for wind and solar forecasting; classical statistical techniques, computational intelligent methods, and hybrid algorithms. Each category includes several methods.
Time series methods are one of the most commonly used statistical techniques for forecasting. Time series can be defined as "the evolution of a set of observations sampled at regular intervals along time. The specificity of time series models, compared to other statistic methods, is that it introduces 'time' as one of its explicative variables" [1]. Time series develop mathematical models that can forecast future observations on the basis of available data. Section below provides definitions and explanations for time series methods commonly in use for forecasting.

Time series methods
This section provides an overview of the most commonly used time series methods for solar and wind forecasting. A brief description is provided for each method along with its mathematical representation.

Autoregressive (AR)
The autoregressive (AR) model presents a process whose current value can be represented as a linear combination of the past values and a signal noise ω t . The AR model of order m, AR(m), is described by [2] where x t is the time series values, ω t is the noise, Φ = (Φ 1 , Φ 2 , …, Φ m ) is the vector of model coefficients and m is a positive integer.

Moving average (MA)
Unlike the AR model that uses a weighted sum of past values (x tÀi ) to provide a time-series representation, the moving average (MA) model combines n past noise values (ω t , ω t À 1 , ω t À 2 , ω t À n ) to develop a time-series process. The MA model of order n, MA(n), is describes as, is describes as [3]: where θ = (θ 1 , θ 2 , …, θ n ) is the vector of model coefficients and θ 0 = 1.

Autoregressive moving average (ARMA)
The autoregressive moving average (ARMA) model is developed by combining AR and MA terms to provide a parsimonious parametrization for a process. The ARMA model of orders m and n, ARMA(m, n) is given by [3]: where Φ i and θ j are the autoregressive and moving average coefficients of the ARMA model.

Autoregressive moving average model with exogenous variables (ARMAX)
The auto regressive moving average model with exogenous variables (ARMAX) provides a multivariate time-series representation to enhance the accuracy of the univariate ARMA model by including relevant information in addition to the time-series under consideration. For example, climate information such as cloud cover, humidity, wind speed and direction can be included as exogenous variables in an ARMA model to develop an ARMAX for more accurate forecasting of solar radiation time series. The ARMAX model of orders m, n and p, ARMAX (m, n, p), is defined as [3]: where Φ i , θ j and λ k are the autoregressive, moving average and exogenous coefficients of the ARMAX model, and e t is the exogenous input term.

Autoregressive integrated moving average (ARIMA)
The autoregressive integrated moving average (ARIMA) model is used for non-stationary time series. Despite representing differences in local trend or level, different sections of nonstationary processes exhibit certain levels of similarity. A stationary ARMA (m, n) process with the dth difference of the time-series develops an ARIMA (m, d, n) model. The ARIMA (m, d, n) model is represented by [4]: where S = 1 À q À1 and Φ m (q) is a stationary and invertible AR(m) operator; x t , ω t , Φ i and θ j are the observed time series values, error, AR and MA parameters, respectively; d is the number of non-seasonal differences; m is the number of autoregressive terms, and n is the number of lagged forecast errors.

Autoregressive fractionally integrated moving average (ARFIMA)
The autoregressive fractionally integrated moving average (ARFIMA) model is used for longmemory forecasting. ARFIMA generalizes ARIMA by allowing the differencing to take fractional values. An ARFIMA model is given by [5]: where powers of L indicate a corresponding number of shifts backward in the time series, and (1 À L) d is the fractional differencing operator.

Autoregressive integrated moving average with exogenous variables (ARIMAX)
The autoregressive integrated moving average with exogenous variables (ARIMAX) includes the previous values of an exogenous time-series in the ARIMA to enhance its performance and accuracy. It is more applicable to time-series with sudden changes in trends. An ARIMA (m, d, n) process including the past p values of an exogenous variable e t develops an ARIMAX process of order (m, d, n, p). The ARIMAX (m, d, n, p) model is represented by [3]: where ω t is the white noise. Φ i , θ j and λ k are the coefficients of the autoregressive, moving average and exogenous inputs, respectively.

Vector autoregressive (VAR)
The vector autoregressive (VAR) model characterizes linear dependences between two or more time-series. VAR model uses multiple variables to generalize the univariate autoregressive model (AR model). A k-dimensional VAR model of order L is given by [6].
where x t and v are k Â 1 vectors of variables and constants, respectively. L is the maximum lag in the VAR model, A i is a k Â k matrix of lag order parameters, and ω t = (ω 1t , …, ω kt ) is the vector of white noise [6,7].

Autoregressive conditional heteroscedasticity (ARCH)-generalized ARCH (GARCH)
The autoregressive conditional heteroscedasticity (ARCH) is used for time series with specific variances for the error terms [7].
Estimated values are calculated using the following equations [8]: where x t is the observed time series values; ε t is the error; σ t is the conditional standard deviation; and a 0 is the constant added to the model.
The generalized ARCH (GARCH) model estimates the values by: By setting p = 0, the GARCH model reduces to an ARCH process with parameter q.

Performance metrics
The performance of the forecast methods is measured by various metrics related to the forecast error. Higher values of errors correspond to less forecast accuracies. This section provides the definitions and equations for performance metrics which are commonly used to calculate the forecast error. Note that x represents the observed value,x is the predicted value (forecast) and n is the total number of samples.

MSE
Mean square error (MSE) is calculated by:

NMSE
Normalized mean square error (NMSE) is calculated by normalizing the MSE as:

RMSE
Root mean square error is given by calculating the square root of the MSE as:

NRMSE
Normalized root mean square error (NRMSE) is calculated by normalizing the RMSE as:

MAE
Mean absolute error is calculated by:

NMAE
Normalized mean absolute error (NMAE) is calculated by normalizing the MAE as:

MRE
Mean relative error (MRE) is calculated by:

MBE
Mean bias error (MBE) is calculated by:

MAPE
Mean absolute percentage error is calculated by:

MASE
Mean absolute scaled error is calculated by:

MSPE
Mean square percentage error is calculated by:

Time series methods for solar energy/wind power forecasting
Time series methods have been extensively used to forecast solar radiation/power and wind speed/power. Typically, solar and wind data exhibit features such as non-linearity and nonstationarity which cannot be captured by most of the time series methods. To address this limitation, these methods are used in combination with other computational intelligent or data processing methods to take advantage of their capabilities to better characterize wind and solar data for more accurate forecasting. These combinations are referred to as hybrid methods which are proven effective for renewables forecasting.

Time series methods for solar energy forecasting
This section provides a review of the articles that use time series methods individually or in hybrid algorithms for solar radiation/power forecasting. The literature review provides a summary of the solar-related variable that is predicted, the horizon for which the variable is predicted, the performance metrics in use to calculate the forecast error, the time series methods and data in use, and the research findings of each article. Table 1 provides the summary.

Time series methods for wind power forecasting
This section provides a review of the articles that use time series methods individually or in hybrid algorithms for wind speed/power forecasting. The literature review provides a summary of the wind variable that is predicted, the horizon for which the variable is predicted, the performance metrics in use to calculate the forecast error, the time series methods and data in use, and the research findings of each article.

Conclusion
This chapter provides a comprehensive literature review to demonstrate the application of time-series methods for renewable energy forecasting. In spite of recent developments in intelligent methods and their extensive applications for more accurate solar energy/wind power forecasting, our literature review concludes that time-series methods, individually or in combination with intelligent methods, are still viable options for short-term forecasting of intermittent renewable energy sources due to their less computational complexities.