Open access peer-reviewed chapter

Time Series and Renewable Energy Forecasting

By Mahmoud Ghofrani and Musaad Alolayan

Submitted: April 7th 2017Reviewed: September 5th 2017Published: December 20th 2017

DOI: 10.5772/intechopen.70845

Downloaded: 1299


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 temperature, 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 function of solar radiation. Wind speed and solar radiation time series data exhibit unique features which complicate their prediction. This makes wind and solar power forecasting 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.


  • forecasting
  • renewable energy
  • solar
  • time series
  • wind

1. 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.

2. 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.

2.1. 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 ωt. The AR model of order m, AR(m), is described by [2]:


where xt is the time series values, ωt is the noise, Φ = (Φ1, Φ2, …, Φm) is the vector of model coefficients and m is a positive integer.

2.2. Moving average (MA)

Unlike the AR model that uses a weighted sum of past values (x~ti) to provide a time-series representation, the moving average (MA) model combines n past noise values (ωt, ωt − 1, ωt − 2, ωt − n) to develop a time-series process. The MA model of order n, MA(n), is describes as, is describes as [3]:


where θ = (θ1, θ2,  …, θn) is the vector of model coefficients and θ0 = 1.

2.3. 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 Φi and θj are the autoregressive and moving average coefficients of the ARMA model.

2.4. 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 Φiθj and λk are the autoregressive, moving average and exogenous coefficients of the ARMAX model, and et is the exogenous input term.

2.5. 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 non-stationary 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 − q−1 and Φm(q) is a stationary and invertible AR(m) operator; xt, ωt, Φi and θj 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.

2.6. Autoregressive fractionally integrated moving average (ARFIMA)

The autoregressive fractionally integrated moving average (ARFIMA) model is used for long-memory 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 − L)d is the fractional differencing operator.

2.7. 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 et develops an ARIMAX process of order (m, d, n, p). The ARIMAX (m, d, n, p) model is represented by [3]:


where ωt is the white noise. Φi, θj and λk are the coefficients of the autoregressive, moving average and exogenous inputs, respectively.

2.8. 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 xt and v are k × 1 vectors of variables and constants, respectively. L is the maximum lag in the VAR model, Ai is a k × k matrix of lag order parameters, and ωt = (ω1t,  …, ωkt) is the vector of white noise [6, 7].

2.9. 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 xt is the observed time series values; εt is the error; σt is the conditional standard deviation; and a0 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.

3. 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.

3.1. MSE

Mean square error (MSE) is calculated by:


3.2. NMSE

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


3.3. RMSE

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


3.4. NRMSE

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


3.5. MAE

Mean absolute error is calculated by:


3.6. NMAE

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


3.7. MRE

Mean relative error (MRE) is calculated by:


3.8. MBE

Mean bias error (MBE) is calculated by:


3.9. MAPE

Mean absolute percentage error is calculated by:


3.10. MASE

Mean absolute scaled error is calculated by:


3.11. MSPE

Mean square percentage error is calculated by:


4. 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 non-stationarity 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.

4.1. 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.

ReferencesForecast variableForecast horizonError metricTime series methodDataFinding
[9]5, 15, 30, and 60 min averaged global horizontal irradiance (GHI)5 min to several hoursMAPERegressions in logs, ARIMA, and hybrid (ARIMA and ANN)4 years of hourly GHI data for three locations in USAARIMA can obtain better results if used in logs with time varying coefficients
[10]Daily GHI1 dayRMSE, NRMSE, MAE, and MBEAR, ARMA19 years of daily GHI from the metrological station of Ajaccio, FranceAR and ANN models perform better than other prediction methods (ARMA, k-Nearest Neighbors, Markov Chains, etc.), if the time-series data is not pre-processed
[11]Hourly GHI1 hMBE and RMSEARIMAMeteorological data including GHI, diffuse horizontal irradiance (DHI), direct normal irradiance (DNI) and cloud cover from two weather stations in USA (Miami and Orlando)Cloud cover information yields to more accurate forecasting
[12]Half daily values of GHIUp to 3 daysNRMSEARHourly GHI measurements from stations of the Spanish National Radiometric NetworkNeural network models obtain better results for almost all stations except for Lerida station where the clearness index-based models outperform
[13]Hourly solar irradiance1 hRMSE, and NRMSENaive, ARMA144 months of hourly solar irradiance of the Paris suburb of AlfortvilleARMA model has competitive results as compared to similarity method (SIM), support vector machine (SVM) and neural network (NN)
[14]Hourly solar radiation1 hRMSE, and NRMSEHybrid (ARMA and time delay neural network (TDNN))10 months of solar radiation data from the observation station in Nanyang Technological UniversityThe combination of the ARMA and TDNN provides more accurate results than each individual forecaster
[15]Daily average of solar irradiance1–15 hMAPEARIMASolar irradiance data from a 4.0 kW PV panel in the city of Awali, Kingdom of BahrainARIMA models are proved to effectively capture the auto-correlative structure of the solar irradiance
[16]Daily solar irradiance and surface air temperature1 dayNAARIMASolar irradiance and surface air temperature data from 10 meteorological stations in Europe and 4 stations in AsiaVarious climate time series are dependent on long-range variability of solar irradiance
[17]Hourly solar power from PV systems1 h up to 36 hRMSEAR, AR with exogenous input (ARX), RX (regressive model with no endogenous variables)1 year of solar power observations from 21 PV systems in DenmarkARX model with both solar power observations and numerical weather predictions (NWPs) as the input outperforms the AR model for forecast horizons longer than 2 h ahead
[18]Hourly GHI, DHI and DNI1 hRMSE, and MBEAR5 min GHI data from Jeddah, Saudi Arabia for a five-year intervalUsing sunshine duration, relative humidity and air temperature as the inputs result in the most accurate forecast by the developed adaptive model
[19]Monthly average solar radiation1 monthRMSELinear regression (LR)Daily GHI and meteorological data in Darwin, Australia from 2000 to 2011LR obtains the best predictions compared to Angstrom-Prescott-Page (APP) and ANN models
[20]Hourly PV power1 and 2 hMAE, MBE, RMSE, and NRMSEARIMAHourly average power of a 1 MW PV power plant located in Merced, California collected between November 2009 and August 2011ANN-based forecasting models including the ANN and GA-optimized ANN obtain better predictions than Persistent, ARIMA and k-NN models
[21]Hourly GHI1 hNRMSEHybrid (ARMA and ANN)6 years of hourly solar radiation and meteorological data from five locations in the Mediterranean area in FranceCombining ARMA and ANN enhances the forecast accuracy
[22]Hourly solar irradiation24 hNRMSEARMA2 years of meteorological data from Ajaccio, FranceANN outperforms the ARMA by 1.3 points reduction in the error estimate
[23]Daily GHI1 dayRMSE, NRMSE, MAE, and MBEAR, ARIMA30 min global solar radiation data in Corsica Island, France from January 1998 to December 2007An ANN with exogenous and endogenous data outperforms univariate forecasters such as ARMA models
[24]Solar irradiance12 hRMSE, and MASEHybrid (ARIMA-Back Propagation)Hourly solar irradiance observations from National Solar Radiation Data Base (NSRDB) site between 2008 and 2009The hybrid ARIMA-BP does not outperform ARIMA
[25]Solar power1 minMAE, MSE, and MAXEHybrid (Wavelet, ARMA, and Nonlinear Autoregressive model with exogenous variables (NARX))1 min solar power data from the solar panel at UCLA for nearly 200,000 observationsCapability of the ARMA process to model the linear features of the data and the NARX advantage to compensate the error of Wavelet-ARMA enhances the forecast accuracy of the hybrid Wavelet-ARMA-NARX method
[26]Solar generation1–5 hMAE, and MSEARMA14 years of hourly solar radiation data from SolarAnywhereARMA outperforms the persistence model for short and medium term solar predictions
[27]Hourly solar irradiance1 h and 3 hRMAEHybrid (non-linear regression and PR)Solar radiation data from sensors, and National Digital Forecast Database, as well as the meteorological measurements from local airports in Los Angeles regionThe hybrid method excels the benchmark methods including the regression, ARIMA and ANN by 40% and 33.33% for 1-h and 3-h ahead, respectively

Table 1.

Summary of the articles with time series methods (individual or hybrid) for solar radiation/power forecasting.

4.2. 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. Table 2 provides the summary.

ReferencesForecast variableForecast horizonError metricTime series methodDataFinding
[28]Hourly average wind Speed1 hNAARMA2 years of wind speed data from Quetta in PakistanARMA is more appropriate for prediction intervals and probability forecasts
[29]Wind power density1–10 daysMAE, and RMSEAR-GARCH, ARFI-GARCHDaily midday wind speed measurements from 1995 to 2004, as well as weather ensemble predictions from 1997 to 2004 for five wind farms in UKWeather ensemble-based forecasters are shown to perform better than time series models and atmospheric-based models
[30]Mean hourly wind speed1 hRMSEAR, and ARIMA744 hours of wind speed measurements in Odigitria of the Greek island of Crete in March 1996The neural logic-based models perform better than the time series methods
[15]Daily average of wind speed1–15 hMAPEARIMAWind speed data from a 1.7 kW wind turbine in the city of Awali, Kingdom of BahrainARIMA models are proved to effectively capture the auto-correlative structure of the wind speed
[31]Wind speed3 hRMSEARWind speed data measured every 3-h in three Mediterranean sites in CorsicaAR is sufficient to simulate 3-h wind speeds
[32]Wind speed1, 2 and 3-step(s)MAE, MAPE and MSEHybrid (Wavelet Packet-ARIMA-BFGS (Broyden-Fletcher-Goldfarb-Shanno))Half-hourly wind speed measurements from 20 December 2011 to 5 January 2012 in Chinese Qinghai wind farmThe ARIMA models have better time performance than the ANN models in approximating wind speed time series while providing a little lower accuracy
[33]Hourly mean wind speed and direction1 hMAEARMA, and VARHourly average wind data from May 1 to October 21, 2002 in two wind sites in North Dakota, USAARMA forecasts the wind speed better than the component model whereas the opposite is observed for wind direction forecasting
[34]Wind power3 hMAPE, and NMAEARIMAWind power data in PortugalThe ARIMA model is used as a benchmark to evaluate the performance of the proposed hybrid Wavelet-PSO-ANFIS forecasting method
[35]Wind speed1–24 hMAE, and RMSEAR, ARX, ARX-GARCH, Hybrid (ARX-TN (truncated normal), ARX-GARCH-TN)3 years of hourly wind speed observations from a meteorological station in Denmark, as well as wind speed predictions based on a NWP model from the Danish Meteorological InstituteThe time series models are used as benchmark methods to evaluate the performance of the developed stochastic differential equation for probabilistic wind speed forecasting
[36]Wind speed/power1–24 hMAE, MBE, RMSE, MASE NMBE, NMAE, and NRMSEAR, ARMA, and ARIMAWind speed, wind direction, humidity, solar radiation, temperature, atmospheric pressure, and heat radiation data from two anemometric measuring towers in La Ventosa, MexicoResults show that the developed method based on support vector regression is more accurate than the persistence and autoregressive models
[37]Wind speed/power1 and 2 day(s)Daily mean error (DME)fractional-ARIMA (f-ARIMA)4 weeks of hourly average wind speed data from four wind monitoring sites in North DakotaThe proposed f-ARIMA is more accurate than the persistence method
[38]Average hourly wind speed1 hME, MSE, and MAEHybrid (ARIMA-ANN)1 month of wind speed measurements in three regions of MexicoThe combination of ARIMA and ANN predicts the wind speed with more accuracy than the individual ARIMA and ANN
[39]Wind speed1 dayMAPEHybrid (KF-ANN model based on ARIMA)Daily wind speed observations from two meteorological stations in Mosul, Iraq and Johor, MalaysiaThe ARIMA model provides inaccurate wind speed forecasts due to its limitation to capture the nonlinearity of the wind speed patterns
[40]Wind speed1, 2 and 3-step(s)MAE, MSE, and MAPEHybrid (ARIMA-ANN and ARIMA-Kalman)Hourly wind speed measurements from a stationBoth hybrid methods can obtain accurate forecasts and are appropriate for non-stationary wind speed datasets
[41]Wind speed1 hNAARMA-GARCH7 years of hourly wind speed data from an observation site in Colorado, USAThe ARMA-GARCH model is proved efficient in capturing the trend change of wind speed mean and volatility over time
[42]Hourly average wind speed1 h up to 10 hRMSEARMA9 years of hourly wind speed data of five locations in Navarre, SpainFor longer term forecasting, the ARMA models with transformed and standardized data perform better than the persistence model
[43]Wind speed1 monthMSE, MAE, and MAPEARIMA7 years of wind speed measurements from the South Coast of Oaxaca, MexicoARIAM models provide more sensitivity than the ANN methods to the adjustment and prediction of the wind speed
[44]Win speed1–6 min(s), and 1–6 hour(s)MAE, and MAPEHybrid (Empirical mode decomposition (EMD)-Least squares support vector machines (LSSVM)-AR)1 year of wind speed data measurements in Beloit, Kansas, USAThe proposed hybrid approach is proved more accurate than the existing forecasting approaches
[45]Wind speed/power generation1 hMAE, and RMSEHybrid (ARIMA-ANN/SVR)2 years of hourly wind data from a 1.5 MW wind turbine in North Dakota, USAThe hybrid approaches are practical for both wind speed and power forecasting but not the best for all the forecasting time horizons
[46]Wind speed15 minMAPE, MSPE, and MAEUnivariate and multivariable ARIMAWind speed data from the Wind Engineering Research Field Laboratory (WERFL) at five different heights at Texas Tech UniversityMultivariate models are more accurate than the univariate models and they are both less accurate than the recurrent neural network models

Table 2.

Summary of the articles with time series methods (individual or hybrid) for wind speed/power forecasting.

5. 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.

© 2017 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Mahmoud Ghofrani and Musaad Alolayan (December 20th 2017). Time Series and Renewable Energy Forecasting, Time Series Analysis and Applications, Nawaz Mohamudally, IntechOpen, DOI: 10.5772/intechopen.70845. Available from:

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