2013 46th Hawaii International Conference on System Sciences Standardized Software for Wind Load Forecast Error Analyses and Predictions Based on Wavelet-ARIMA Models Applications at Multiple Geographically Distributed Wind Farms Zhangshuan Hou zhangshuan.hou@pnl.gov Yuri V. Makarov yuri.makarov@pnnl.gov Nader A Samaan nader.samaan@pnnl.gov Pavel V Etingov pavel.etingov@pnnl.gov Abstract This study introduces a standardized forecast error analysis and prediction tool that can be implemented into a software package that predicts the uncertainty range for generation resources involved in the power grid balancing service. The tool is reusable and transferrable to power generation systems of different variability. Given the multi-scale variability and uncertainty of wind generation and forecast errors, a natural choice is to use time frequency representation (TFR) as a view of the corresponding time series represented over both time and frequency. Here we use wavelet transform (WT) to expand the signal in terms of wavelet functions that are localized in both time and frequency. Each WT component is more stationary and has a consistent auto-correlation pattern. We combined wavelet analyses with time-series forecast approaches such as the autoregressive integrated moving average (ARIMA) model and tested the approach at five different wind farms with similar or different field and weather conditions. The prediction capability is satisfactory the day-ahead prediction of errors matches the original error values very well, including the patterns. The observations are well located within the predictive intervals. Integrating our wavelet-arima ( stochastic ) model with the weather forecast model ( deterministic ) will improve our ability significantly to predict wind power generation and reduce predictive uncertainty. 1. Introduction Software product line engineering is an industrially validated methodology for developing software products and software-intensive systems and services faster, at lower costs, and with better quality and higher end-user satisfaction. In addition to the computational efficiency and accuracy of a software platform and its components, the developers also consider the transferability, maintenance, strategic reuse of the platform, and exploiting the variability to be built into the platform. In this study, we focus on power generation systems. The goal is to develop a standardized forecast error analysis and prediction tool that can be implemented into a software package that predicts the uncertainty range for generation resources involved in the power grid balancing service, including their required capacity, ramp, and ramp duration requirements in the near future. All sources of uncertainty and variability are to be exploited, including system load, wind and solar generation, and others. The uncertainty prediction tools are expected to be efficient, accurate, and applicable to various power generation systems in a standardized manner. We developed a tool beginning with analyzing wind power generation. Wind power generation is directly linked to weather conditions, and thus the first aspect of wind power forecasting is the prediction of future values of the necessary weather variables at the level of the wind farm. This is done by using numerical weather prediction (NWP) models. Such models are based on equations governing the motions and forces affecting motion of air masses. From the knowledge of the actual state of the atmosphere, the system of equations allows one to estimate what the evolution of state variables e.g., temperature, velocity, humidity, and pressure will be at a series of 3-dimensional grid points. The meteorological variables needed as input for wind power prediction obviously include wind speed and direction but also possibly temperature, pressure, and humidity [1-5]. The wind load forecast, however, is subject to great uncertainty because of various sources of uncertainty associated with these inputs and modeling assumptions. One can check the accuracy of a model forecast by looking at the average misfit or other measures, such as sum-squares of forecast errors, and check the consistency and reliability by looking at 1530-1605 2012 U.S. Government Work Not Protected by U.S. Copyright DOI 10.1109/HICSS.2013.495 5003 5005
error distributions to see whether the wind loads are systematically under- or over-estimated and whether the errors have non-normal or multi-modal distributions. The summary statistics can be used to provide confidence intervals for quantifying uncertainty associated with model forecasts [6]. Further analyses can be performed to capture the patterns in the variability of the errors. The patterns can be represented quantitatively as the deterministic or quasi-deterministic components reflecting some biases and periodic processes contained in the error curves. These underlying variations result in nonstationary stochastic error processes that are the processes with changing statistical characteristics. The deterministic components can be subtracted from the original process, creating a stationary stochastic residual. By predicting the deterministic curves and subtracting them from the forecast obtained in a traditional way, we can improve the quality of forecasts. A model with integrated stochastic and deterministic components is expected to significantly reduce the model forecast errors and their uncertainty ranges (i.e., yielding narrower confidence intervals). Variability of wind generation can be regarded at various time scales, including seasonal variations, daily cycles, and/or fluctuations at short-term scale(s). The variations are not of the same order for different time scales and usually have different auto-correlation patterns, which limits the use of traditional time-series forecast approaches such as the autoregressive integrated moving average (ARIMA) model. The multi-scale variations of the wind load forecast errors warrant the approaches (e.g., wavelet decomposition) that can separate the signals at the different time scales and extract the major contributor(s) in the mixture of errors [7,8]. Wavelet transform (WT) has several advantages for time-series forecast and uncertainty evaluation. Each of the WT decomposed components is more stationary and has a much more consistent auto-correlation pattern, which make the statistical forecast models (such as ARIMA) more applicable. Moreover, if a signal is a mixture of components varying at different time scales (frequencies), one should be aware that higher frequencies are better resolved in time, and lower frequencies are better resolved in frequency. This means that a certain high-frequency component can be located better in time (with less relative error) than a low-frequency component. On the contrary, a low-frequency component can be located better in frequency compared to high-frequency component. Therefore, it is better to evaluate the uncertainty for each component separately instead of using a simple measure such as standard deviation. This is intuitively true when the forecast errors have strong multi-modal distributions. One more advantage of WT is its unique ability to identify anomalous/extreme events (outliers) with localized time and frequency resolution [9-12]. 2. Methodology Given the multi-scale variability of weather conditions, wind generation, and forecast errors, it is a natural choice to use time frequency representation (TFR) as a view of the corresponding time series represented over both time and frequency. Here we use WT to expand the signal in terms of wavelet functions that are localized in both time and frequency. Wavelet transform can be defined for different classes of functions. The intention in this transformation is to address some of the shortcomings of the Fourier transform. Instead of fixing the time and the frequency resolutions, one can let both resolutions vary in the time frequency plane to obtain a multiresolution analysis. This variation can be carried out without violating the Heisenberg inequality. One can design a set of basis functions (wavelets) by choosing a proper basic wavelet (mother wavelet) and then use a delayed and scaled version of the mother wavelet. The wavelet at scale (, at time t, and time delay is obtained from the following equation [10]: (1) Based on function expansion theory, the forward and inverse continuous wavelet transform (CWT) is as follows: Forward CWT: (2) Inverse CWT: (3) W is wavelet coefficients at scale and time delay. is the reconstructed signals through inverse WT of coefficients W. The forward transformation separates the original signals into multiple components. At each step, the signal is decomposed simultaneously using a high-pass filter h and a low-pass filter g, resulting in detail coefficients and approximation coefficients (the remaining mixed signal), respectively (Figure 1). Figure 1. Synthesis of wavelet decomposition. 5004 5006
In the frequency domain, WT tells which components are important; in the time domain, we know when the component contributes the most. These provide guidance on how to adjust a wind power forecast based on error analyses, while the inverse transformation can be used to reconstruct the time series or to provide short- or long-term forecasts if combined with forecasting approaches such as ARIMA. In time-series analysis, an ARIMA model is a generalization of an autoregressive moving average (ARMA) model. These models are fitted to time-series data either to better understand the data or to predict future points in the series (forecasting). The model is generally referred to as an ARIMA(p,d,q) model in which p, d, and q are non-negative integers that refer to the order of the autoregressive, integrated (differencing), and moving average parts of the model, respectively [13,14]. In this study, we apply ARIMA(p,d,q) to each WTdecomposed component of wind load forecast errors. The parameters p, d, and q are trained for each component. Generally, higher-frequency components need an ARIMA model with a larger autoregressive parameter p, while lower-frequency signals require a larger differencing parameter d and moving average parameter q. 3. Case studies wavelet power spectrum analyses In this section, we summarize the wavelet decomposition and wavelet-arima forecast results for several case studies. Three wind farms located close to each other are studied. As a comparison, forecast errors of wind farms that are located within a large geographical area are used. The wavelet decomposition results are shown in Figures 2-4. Note that the wind generation forecast error time series originally is expressed in megawatts, but the units are normalized in the figures and become dimensionless ( ). Figure 2. Wavelet analyses of load forecast errors, Windfarm #1. a) Original time series of load error time series consisting of three main components: bi-weekly, seasonal (quarterly), and half-weekly; d) high contribution of the bi-weekly component i around t = 7000, 8200, and 3600 hours. 3.1. Three nearby wind farms in California The three wind farms are closely located; therefore, the corresponding forecast errors share similar variability at various time scales. The biweekly component dominates all three error time series and should be included in the stochastic component model. There also exist some differences: at Windfarm #1, the dominance of the biweekly component is much more obvious than at Windfarm #2 and Windfarm #3, while at Windfarm #3, the variability at the different time scales is comparable, although one can still see three major components (half-weekly, bi-weekly, and quarter-annually). This information provides some guidance as to the component(s) on which we should focus at each wind farm. Figure 3. Wavelet analyses of load forecast errors, Windfarm #2. a) Original time series of load error time series consists of three main components: bi- 5005 5007
weekly, seasonal (quarterly), and half-weekly; d) high contribution of the bi-weekly component around t = 7000, 8200, and 3600 hours. Figure 4. Wavelet analyses of load forecast errors, Windfarm #3. a) Original time series of load error time series consists of three main components: biweekly, seasonal (quarterly), and half-weekly; d) high contribution of the bi-weekly component around t = 7000, 8200, and 3600 hours. Figure 5. Wavelet analyses of load forecast errors, Windfarm #4. a) Original time series of load error time series dominated by the monthly component, followed by the seasonal (quarterly) component; d) high contribution of the bi-weekly component around t = 3000 to 4500 hours and t = 8200 hours. 3.2. Wind farms distant from each other Forecast errors at two other wind farms (Windfarm #4 and Windfarm #5) located far away from Windfarm #1 also were studied. Their corresponding forecast errors behave very differently. At Windfarm #4, the monthly component dominates, while at Windfarm #5, the quarterly signal is the most important (Figures 5 and 6). Therefore, at each wind farm of unique geographical attributes, the error analyses should be performed, which help identify the major component to be included into the stochastic error forecast model. Figure 6. Wavelet analyses of load forecast errors, Windfarm #5. a) Original time series of load error time series dominated by low-frequency (quarterly) components followed by the monthly and the bi-weekly components; d) high contribution of the biweekly component around t = 1000 to 2000 hours and t = 8200 hours. 5006 5008
4. Case studies wavelet-arima reconstruction and forecast Wavelet-decomposed components tend to have more consistent autoregressive patterns, which make the use of the ARIMA(p,d,q) forecast more reasonable. For the high-frequency component, a larger p is generally preferred; while for low-frequency signals (trends), a differencing term is needed to capture the trend. Therefore, it is unreasonable to directly apply ARIMA tools to a mixture time series such as wind generation forecast errors. Fortunately, a combination of wavelet and ARIMA deals nicely with the multiscale variability; appropriate ARIMA parameters for each scale can be chosen adaptively. As demonstrated in Section 3, the tool can be standardized such that the time series are decomposed into similar multi-scales by observing several major common components across wind farms. Here we demonstrate the capability of wavelet ARIMA forecast of the error time series at three wind farms (Windfarm #1, Windfarm #4, and Windfarm #5) with different geographical attributes. Equation (3) is used to reconstruct the error time series (Figure 7b) through inverse wavelet transform of the WT-decomposed components (e.g., the black lines in Figure 8 for time < 8736 hours). When ARIMA prediction is applied to each component beyond time = 8750 hours, the predicted values can also be combined through inverse wavelet transform for the overall prediction of the forecast errors (see Figure 8; the red lines are the predictions, the blue lines are 95% predictive bounds). Error Value (MW) Error Value (MW) Error Value (MW) 200 0-200 200 0-200 200 0-200 Original data 1000 2000 3000 4000 5000 6000 7000 8000 9000 Time (hours) Reconstructed data 1000 2000 3000 4000 5000 6000 7000 8000 9000 Time (hours) residuals 1000 2000 3000 4000 5000 6000 7000 8000 9000 Time (hours) Figure 7. The original wind power forecast error time series, the wavelet-reconstructed data, and the differences between the original and reconstructed, at Windfarm #1. As shown in Figure 8, each of the decomposed components has consistent frequency and autocorrelation patterns, which make the use of ARIMA more appropriate and reliable. For high-frequency components, usually more AR terms are needed, while for low-frequency components, differencing terms are necessary for trending patterns. When the predictions for all the components are summed, the overall prediction of error values is as shown in Figure 9. The last 24-hour wind forecast errors are used to validate the wavelet ARIMA predictions with dayahead predictions. As can be seen in Figures 9, 10, and 11, the day-ahead predictions from the wavelet ARIMA combination are very accurate. The predicted pattern matches the original data very well. As a summary, wavelet transform successfully decomposes the wind generation forecast errors at various wind farms into the same set of multi-scales and helps visualize the several major common components. This then enables a standardized wavelet ARIMA prediction tool for predicting the forecast errors and the associated uncertainties, which help improve forecast accuracy and decision making in the power grid balancing service. -20 0 10 20-15 -5 5 15-10 -5 0 5 10 Figure 8. ARIMA day-ahead prediction (and 95% predictive intervals) of the error time series (normalized, dimensionless) for different components of the original error time series, at Windfarm #1. The figures show only data beyond t = 8500 hours to highlight the predicted error values. The predictions (red lines) and predictive bounds (blue lines) are for t > 8736 hours. -20 0 10 20-15 -5 0 5 10-5 0 5 5007 5009
(MW) -200 0 100 original data (MW) (MW) -1500-500 500 1500 original data (MW) -200 0 100 reconstruction and day-ahead prediction (MW) (MW) -1500-500 500 1500 reconstruction and day-ahead prediction Figure 9. Wavelet-ARIMA day-ahead prediction (and 95% predictive intervals) of wind power forecast error at Windfarm #1. (MW) (MW) -300-100 100 300-300 0 200 original data reconstruction and day-ahead prediction Figure 10. Wavelet-ARIMA day-ahead prediction (and 95% predictive intervals) of wind power forecast error at Windfarm #4. Figure 11. Wavelet-ARIMA day-ahead prediction (and 95% predictive intervals) of wind power forecast error at Windfarm #5. 5. Summary In this study, we introduced an effort to develop a software product for power generation forecast error analyses and prediction with better quality and generality. The wavelet ARIMA data analyses framework for wind load forecast error analyses, prediction, and uncertainty reduction was demonstrated. Given the multi-scale variations and complex combinations of the multi-scale components in the error time series, wavelet decomposition provides a nice means to separate the signals. It can tell us which decomposed component(s) is important and therefore should be included in the stochastic component of a wind load forecast. It also tells when a specific component plays a more important role or has higher contributions in the overall errors. When other information such as geographical or topographical information is available, one can use the wavelet spectrum information to analyze the underlying mechanisms of why and when these components play important roles. The wavelet analyses can be combined with timeseries forecast approaches such as ARIMA. We tested the wavelet ARIMA model at three different wind farms located far away from each other. The prediction capability was satisfactory the day-ahead prediction of errors matched the original error values very well, including the patterns. The original values were located well within the predictive intervals. 5008 5010
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