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Contract No. DE-AC36-08GO28308

Wind Power Forecasting Error 

Frequency Analyses for 

Operational Power System 

Studies

Preprint

A. Florita, B.-M. Hodge, and M. Milligan

To be presented at the 11th Annual International Workshop on 

Large-Scale Integration of Wind Power into Power Systems 

as well as on Transmission Networks for Offshore Wind Power 

Plants Conference

Lisbon, Portugal

November 13–15, 2012

Conference Paper

NREL/CP-5500-56086

August 2012
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1 

Wind Power Forecasting Error Frequency Analyses for 

Operational Power System Studies 

Anthony R. Florita, Bri-Mathias Hodge, and Michael Milligan

National Renewable Energy Laboratory

Golden, CO 80401; USA

Abstract—The examination of wind power forecasting errors is 

crucial for optimal unit commitment and economic dispatch of 

power systems with significant wind power penetrations. This 

scheduling process includes both renewable and nonrenewable 

generators, and the incorporation of wind power forecasts will 

become increasingly important as wind fleets constitute a larger 

portion of generation portfolios. This research considers the 

Western Wind and Solar Integration Study database of wind power 

forecasts and numerical actualizations. This database comprises 

more than 30,000 locations spread throughout the western United 

States, with a total wind power capacity of 960 GW. Error analyses 

for individual sites and for specific balancing areas are performed 

using the database, quantifying the fit to theoretical distributions 

through goodness-of-fit metrics. Insights into wind-power 

forecasting error distributions are established for various levels of 

temporal and spatial resolution, contrasts made among the 

frequency distribution alternatives, and recommendations put forth 

for harnessing the results. Empirical data are used to produce more 

realistic site-level forecasts than previously employed, such that 

higher resolution operational studies are possible. This research 

feeds into a larger work of renewable integration through the links 

wind power forecasting has with various operational issues, such as 

stochastic unit commitment and flexible reserve level 

determination.

Keywords—wind forecasting, error frequency, hyperbolic 

distribution, operational power system

I. INTRODUCTION

The wind power supplying energy in power systems has 

increased greatly throughout the previous decade. Unlike 

conventional thermal units, the variability and uncertainty of 

wind power has led to concerns about how wind power is utilized 

in power system operations. In the United States, there have been 

a number of studies of the impact of larger penetrations of wind 

power on system operations [1]–[7]. One of the results of these 

studies has been the recognition that wind power forecasting is an 

important technology enabling greater wind power penetration 

because it reduces the uncertainty of the wind power output. 

Because wind power forecasting plays a critical role in these 

integration studies, a proper statistical characterization of wind 

power forecasting errors assumed in operational studies is 

necessary to ensure accurate results.

The wind power forecasts used in operational studies are 

often divided into two forecast periods: day ahead and hour 

ahead. These correspond to the decision time frames in the 

production cost models used to simulate system operations, and 

are modeled on a generalization of the Unit Commitment and 

Economic Dispatch (UCED) problem. In this paradigm, variable 

values, such as load and wind power, are forecast the day before 

to ensure that slow starting thermal units are available to meet the 

anticipated load. One hour before the realization, the variable 

values are forecast again so that dispatch decisions may be made. 

Generally speaking, commitment decisions determine whether a 

unit will be online, whereas dispatch decisions fine tune the 

output level of units that will be online. However, some faststarting units can be committed in the dispatch time frame, if 

necessary. Forecast errors in the unit commitment stage can have 

substantial economic consequences, if they are large enough that 

they cause a different commitment than would have been 

performed with an optimal forecast. For example, if the wind 

power is over-forecast by 500 MW, a cheaper, slow-starting coal 

unit, which would have been started if the forecast were more 

accurate, might need to be replaced by a more expensive faststarting natural gas unit. However, if the forecast error is only 1 

MW, the natural flexibility in the system will be able to make up 

the difference. For these reasons, most concern is placed on large 

forecast errors in the day-ahead time frame—i.e., the tails of the 

forecast error distribution have the greatest economic impact and 

there is more uncertainty in day-ahead forecasts. 

This research seeks to first ascertain a methodology for fitting 

wind power forecasting errors at various levels of temporal and 

spatial resolution, then to harness empirical data for updating 

existing (numerical) site-level forecasts to increase their fidelity 

to measured phenomena. Both topics strive toward aiding higher 

resolution operational power system studies. The forecasting 

errors are examined in the sense of frequency and are not 

examined time sequentially; however, the updated forecasts are 

inherently a time series while maintaining distribution moments 

informed by the empirical data. The data for wind power 

forecasts and numerical actualizations come from the Western 

Wind and Solar Integration Study (WWSIS), with more than

30,000 locations spread throughout the western United States and 

a total wind power capacity of 960 GW.

II. METHODS AND DATA

In this section, some background is provided on the primary 

statistics used to examine the forecast error distributions studied 

in this research. Additionally, the empirical data sets examined in 
2 

this work are described. The analysis was performed using the R 

statistical computing environment [8], using additional pertinent 

packages [9], [10].

A. Some Statistical Background

Most work in wind power forecasting uses the first two 

standard moments, mean and variance, to describe the observed 

error distribution. Although these two metrics provide some 

important information about the distribution, further information 

is available from the third and fourth statistical moments: 

skewness and kurtosis. Skewness is a measure of the symmetry 

of the distribution; whereas kurtosis describes the relationship 

between the peak and tails of the distribution. A negative skew 

indicates a long left tail, with the bulk of the distribution on the 

right side. A leptokurtic distribution is one with a high kurtosis 

value; whereas a platykurtic distribution has a low kurtosis value. 

Observed kurtosis values are often compared to the kurtosis value 

of the normal distribution, which can be fully described by the 

first two statistical moments. This value is known as the excess 

kurtosis, and in what follows, the term excess kurtosis is used 

synonymously with kurtosis. Because particular interest is placed 

on the tails of the distribution, the analysis of these two moments 

will help enable a more accurate characterization of the error 

distribution.

Recent research into the aforementioned wind power 

forecasting and related concerns [11]–[13] has shown the 

inadequacy of the Gaussian distribution for fitting errors of 

interest. It has also hinted at the applicability of the hyperbolic 

distribution. Nonetheless, for the data examined herein, 

numerous theoretical distributions were considered, including: 

Cauchy, Laplace, Beta, the Generalized Lambda, as well as many 

appropriate piecewise distributions. In the interest of brevity, 

only major results and insights were included.

B. Investigated Data

The primary data set examined in this work was produced by 

3TIER [14], [15], as part of the WWSIS Phase 1 [6]. Wind 

speeds at 100 m were simulated at 10-min intervals for the years 

2004, 2005, and 2006, at more than 30,000 locations in the 

Western United States, using a Numerical Weather Prediction 

model (NWP). These values were then converted to a wind 

power output value, using the SCORE-lite statistical correction 

methodology [15], [16], based on the assumption that each 

location contained 10 3-MW turbines. The 10-min values were

averaged into hourly wind power output from each of the sites; 

henceforth these values will be referred to as the “actuals.” An 

additional run of the NWP was performed with different 

boundary conditions to produce day-ahead wind power forecasts 

for each of the sites selected. These values will be referred to as 

“forecasts” in what follows. It is noted that these forecasts were 

not subject to the same site-specific statistical processing 

normally applied to operational systems [14].

Although the simulated “actuals” and “forecasts” were 

obtained from state-of-the-art modeling techniques, it was 

desirable to bring empirical data into consideration. Thus, data 

have been analyzed from three distinct regions of the United 

States and results later made to bear on the “forecasts” for 

enhanced fidelity to reality; these more realistic forecasts are 

termed “updated forecasts.” The updated forecasts will be used in 

future operational studies addressing the UCED problem from 

various perspectives. The following describes the empirical data 

utilized for these concerns.

The first data set comprises aggregated wind power output 

and forecasts from all of the active wind power plants in the 

Electric Reliability Council of Texas (ERCOT) interconnection 

during a 13-month period. The total wind capacity included in 

this data set is approximately 9,000 MW. This data set includes

only day-ahead forecasts, made once a day at 16:00 the previous 

day. It is important to note that the day-ahead forecast is not at a 

consistent timescale and includes forecasts between 8 and 32 

hours in advance. The second data set considered consists of a 

year’s worth of day-ahead forecasts from the California 

Independent System Operator (CAISO) region. These forecasts 

are made at 05:30 and are valid for the following midnight-tomidnight time frame, thus they represent periods 18 to 42 hours 

in advance. They include forecasts for 16 different wind plants 

with a total capacity of approximately 940 MW. The third data

set used consists of the forecasts and output from a wind plant in 

the Xcel Energy Colorado territory. The wind plant studied has 

an approximate nameplate capacity of 300 MW. This data set 

includes three months of data from the summer and fall seasons 

with hourly forecasts produced every 15 minutes for the next 72 

hours.

All three data sets provide useful information. The day-ahead 

ERCOT and CAISO forecasts are important when performing the 

day-ahead unit commitment process. The Xcel forecasts provide 

useful information on how forecasts improve with decreasing 

forecasting horizon. The Xcel forecasts also show the forecast 

errors that can be expected during smaller operational timescales, 

though they lack the smoothing of a geographically diverse data

set. Additionally, the Xcel data set consists of a single season, 

instead of at least one year for the other two data sets. It is 

important to note that all of the data sets use methodologically 

similar forecasting systems, based on NWP models with 

statistical post-processing, but the forecasts come from two 

different forecast providers. For the concerns of this article, as 

explained in a later section with greater detail, only statistical 

moments from the error distributions of the three empirical data

sets were considered directly (through an interpolation approach) 

for producing the updated forecasts.

An overview of the analysis of the wind power forecasting 

data set and its updating for the needs of WWSIS Phase 2 is 

provided in Fig. 1. This paper provides a summary of the crucial 

aspects of the research methodology and some final results; a 

forthcoming report provides greater detail.
3 

Fig. 1. Overview of wind power forecasting analysis and production of updated forecasts that consider empirical data

III. CHARACTERIZATION OF FORECASTING ERRORS

To understand the implications of wind power forecasting 

errors in wind integration studies, a statistical characterization of 

the errors is helpful. This characterization can identify the range 

of errors expected, with their corresponding frequency. Such a 

characterization can be particularly useful for determining the 

operating reserves necessary to compensate for the forecasting 

errors. Here, two different levels of the wind forecasting data 

were examined: the forecasts at individual sites and forecasts 

aggregated throughout balancing areas.

A. Individual Sites

The forecasts that are most commonly used by wind power 

developers and plant operators are those created for individual 

wind plants. These forecasts may be used by operators to bid into 

the day-ahead market in areas where wind may participate in 

such a market. Here, the forecast errors for a typical 30-MW 

wind plant in the WWSIS data set were examined. Fig. 2 

illustrates the typical error frequency behavior from one of the 

30,000 sites, with mean = -1.139, var = 60.4, skew = -0.206, and 

kurt = 2.16. The distribution is leptokurtic, and its negative bias 

underestimates the actual production by ~3.8% on average; the 

negative skew compensates in terms of the whole distribution. 

The assumption of normality is obviously erroneous, and the best 

fit from the myriad of theoretical distributions considered came

from the hyperbolic distribution. The hyperbolic distribution is 

common in financial modeling and applications. This may be 

intuitive to some readers because any forecasting can be thought 

of as actively chasing realizations of future measurements: with a 

physically driven, somewhat predictable process, there will be 

many relatively small errors and considerably fewer relatively 

large errors. 

B. Aggregation of Balancing Areas 

Although the forecasts at individual wind plants are important for 

wind plant operators, the aggregate forecast for all of the wind 

plants is an important consideration for the Independent System 

Operator (ISO) or Balancing Authority (BA). Here, all of the 

wind plant outputs and forecasts for each of the balancing areas 

were aggregated according to the high-wind scenario of WWSIS 

Phase 2. Aggregation throughout a wide geographic domain 

typically creates distributions that are more conventional than 

from individual plants, as more atmospheric phenomena are 

involved, and large forecasting errors become less correlated with 

increasing distance. Fig. 3 provides a typical example from a BA, 

with mean = 137.7, var = 5.84*10^5, skew = 0.122, and kurt = 

2.05. The distribution is leptokurtic, and its positive bias 

overestimates the actual production by ~2.6% on average; the 

positive skew compensates in terms of the whole distribution. It 

should be noted the error bias is less than that of the individual 

site, which would be expected according to the spatial and 

temporal smoothing effects previously noted. In addition to better 

forecasts at the larger scale because of aggregation and 

smoothing effects from averaging, there is a visually, and more 

importantly numerically, better fit for the tails of the distribution. 

This is fundamentally important because these extreme events 

drive the economics of the UCED problem and are of great 

interest in operational power system studies.
4 

Fig. 2. The site-level forecasting errors and pdf fitting

Fig. 3. The BA-level forecasting errors and pdf fitting

C. Goodness-of-Fit Evaluation

From simple visual inspection, the normal distribution is not 

helpful for the concerns of this research. In the interest of space, 

the numerous theoretical distributions considered for fitting the 

forecast errors were not considered here.

Focus was placed on ensuring the hyperbolic distribution met 

standards associated with a goodness-of-fit statistic for all BAs 

and a subset of selected sites of interest to the WWSIS Phase 2 

high-wind scenario, after “best fits” were determined. All 

implementations of distribution fitting relied on a maximum 

likelihood routine using the Nelder-Mead simplex method for the 

appropriate initialization [9]. To ensure the robustness of the fit 

parameters, the routine was evaluated 1,000 times (for each BA 

and selected site) as part of a bootstrapping approach 

programmed by the authors, the final result being the best 

estimates of the four hyperbolic parameters for every BA and 

selected site.

The goodness-of-fit statistic utilized [17] a Cramer-von

Mises statistic as follows:

( ) nn

i zW

n

i

i 12

1

2

12 2

1

2  + 

 



 − ∑ −= =

, (1)

It has a null hypothesis that the sample comes from the

hyperbolic distribution, which is accepted/rejected after four

distinct calculation steps:

1) The maximum likelihood estimates are obtained; 

2) The Cramer-von Mises statistic is calculated; 

3) Tabular values of W^2 are consulted at significance level 

alpha and the null hypothesis is accepted or rejected; and

4) The p-value of the test can be calculated by interpolation,

if necessary.

The null hypothesis was accepted at the alpha = 0.05 significance

level for all BAs and each selected site. This validated the 

applicability of the hyperbolic distribution for the forecast error 

data under investigation for WWSIS Phase 2. It should also be 

noted that the hyperbolic distribution provided a superior fit at 

the “bus” and “plant” aggregation levels of Fig. 1 than all the 

theoretical distributions previously discussed and considered as 

part of this research.

IV. COMPARISON WITH OPERATIONAL

To fully understand the implications of the simulated wind power 

forecasts used in wind integration studies, an understanding of 

how they compare with the errors observed in real operational 

forecasting systems is necessary; the empirical data was 

previously described in Section II-B. Understanding where the 

simulated forecast errors differ allows the identification of their 

limitations for use in wind integration studies. Furthermore, the 

empirical data can be harnessed to create more realistic forecast 

error distributions that help increase model fidelity of wind 

integration studies; the updated and more realistic forecasts will 

be termed the “updated forecasts,” and are necessary only at the 

site level because of the topology and required inputs of such 

power system studies. Therefore, the methodology illustrated in 

Fig. 1 relied on extracting crucial information (i.e., statistical 

moments) from the empirical operational forecasting system data, 
5 

then used it to produce the updated forecasts superior to those 

from WWSIS Phase 1 [6]. The updated forecasts are currently 

being used as part of various ongoing UCED investigations 

involving wind integration.

One of the fundamental problems with wind integration 

studies is that researchers are evaluating a future that does not yet 

exist. One of the implications of this reality is that although

actual wind power forecasting errors for all of the 960 GW of 

installed wind capacity considered in the WWSIS would be 

preferred, we must create these errors, keeping in mind important 

features that should be represented as close to the current state as 

possible. With only limited empirical operational forecasting 

information available, it is therefore necessary to interpolate data 

to infer reasonable values for modifying the forecasts to produce 

the updated forecasts. To be able to numerically compare the 

important features of the error distributions, the first four 

statistical moments were chosen.

In the process of updating the forecasts to a more faithful 

representation of reality (i.e., still erroneous but not containing 

statistical anomalies), a key assumption made was that the bias 

and skew of the updated forecast error distribution should be 

equal to zero. That is, there was no physical reason the forecasts 

should, on average, over- or under-forecast, and there was no 

reason to expect a higher frequency of over- or under-forecast 

events. Therefore, in the conversion process, primary focus was 

placed on obtaining realistic magnitudes of variance and kurtosis, 

as these were the main statistical features of interest from an error 

frequency viewpoint. To produce the updated forecasts, the 

partitioning of the errors in a time series fashion was a function 

of the original forecast errors. This was meant to maintain the 

majority of the site-to-site correlations that were a product of the 

NWP forecast generation process. An overview of the conversion 

process follows in the next section.

V. CONVERSION PROCESS

A visual representation of the updating process for modifying 

the original WWSIS Phase 1 forecasts to produce the updated 

forecasts is shown near the bottom of Fig. 1. The goal was to 

produce more realistic site-level forecasts, updating the original 

forecasts through the reliance on operational system data (i.e., 

statistical moments) and the standardization/extraction of 

statistical anomalies. A step-by-step process description follows.

The first step was to obtain the best hyperbolic fit to the BAlevel data; examples and reasoning for this at both the BA and 

site levels were discussed above. However, an additional driver 

was the manner in which the forecasts would be used in the 

WWSIS Phase 2 study. The study uses a zonal transmission 

model, meaning that forecasts are essentially aggregated at the 

BA level. With the distribution fit, the four moments were saved 

from each BA.

The second step was to use the empirical operational data to 

interpolate what the moments should be (from empirical data 

and a curve fit) as a function of a given BA’s capacity; the 

moments from the original forecast errors and the empirical 

interpolation were roughly similar. Analysis of the data revealed 

that the variance and kurtosis values of both the operational and 

original WWSIS forecast data was strongly correlated with the 

wind capacity considered, an indirect measure of the geographic 

diversity.

The third step involved determining the values of the 

hyperbolic parameters, with the location (mean) and asymmetry 

(skew) parameters assumed null, as discussed above, such that 

the empirical-derived operational moments could be matched in 

an optimal fashion. The optimization process involved a particle 

swarm routine programmed by the authors to evaluate a moment 

matching subroutine until convergence on the optimal 

hyperbolic parameters. The result was that the character of the 

original forecast errors (with hyperbolic distribution) was 

maintained, minus the statistical anomalies, and the empirical 

data concerning moments was imparted on the updated forecast 

error distribution. Because of histogram binning, the updated 

forecast error density showed only minor visual differences from 

the original of Fig. 3, and was therefore not provided here; 

however, there were distinct numerical differences. Figs. 4 and 5 

illustrate the numerical differences between the WWSIS Phase 1 

forecast errors and the updated theoretical ones, which took

empirical data into account and were not biased or skewed. 

From the figures, typical site-level examples of minor and 

significant differences were noted, respectively. Fig. 4 shows a 

case where the updated forecast errors (i.e., the theoretical 

distribution signified by the line) were only slightly different 

than the original WWSIS forecast errors—mostly in the tails, 

where there were less than a dozen samples/hours per year; the 

update was minimal. Fig. 5 shows a case were the updates were

significantly different than the original WWSIS forecast 

errors—the original forecast error data was hyperbolic but 

anomalous because of one sample year of hourly data that was a 

function of one forecasting instance, and was not indicative of 

realistic and quality site-level forecasts. The latter was the 

reason for updated forecasts, as well as time/computation limits 

on such minor stochastic considerations.

The fourth step involved determining the error at each hour of 

the year according to its quantile. This was achieved by stepping 

through the original WWSIS forecasts, determining its quantile 

at that hour, and mapping that to the quantile of the theoretical 

distribution to allow the determination of the updated error at 

each hour. The process is illustrated in Fig. 6 for one hour in the 

95% quantile.

The fifth and final step was to take the 8,760 hourly errors 

and partition them to the site level to produce the updated 

forecasts. This could be accomplished in one of two ways, both 

of which were executed: (1) The hourly error could be uniformly 

distributed to each site, and/or (2) The hourly error could be

distributed to each site according to its relative error, i.e., the 

site’s hourly error as a proportion of total BA error from the 
6 

original WWSIS error forecasts. The sign of the error was

maintained from the original to the updated forecasts.

Fig. 4. Quantile-quantile plot of updated (theo.) forecast errors 

and the original WWSIS data: minor differences

Fig. 5. Quantile-quantile plot of updated (theo.) forecast errors 

and the original WWSIS data: significant differences

Fig. 6. The BA-level forecast errors mapped (according to 

quantile and sign) from the original WWSIS distribution to the 

updated forecast distribution

VI. CONCLUSIONS AND FUTURE WORK

In this work, we compared the forecast error distributions from 

the WWSIS data set with those taken from operational 

forecasting systems. The results of the analysis enabled the 

modification of the forecasts from the original WWSIS data set 

to more accurately reflect the current state in wind power 

forecasting. This is important especially because there were 

significant differences in the tails of the error distributions, i.e., 

the largest forecasting errors that have the most economic 

impact. Sensitivity studies are planned in the WWSIS Phase 2 to 

quantify the economic impacts of the improved wind forecasts. 

The modified forecasts will be made publicly available 

alongside the original WWSIS data. In a broader context, this 

work highlights the need for accurate portrayal of wind power 

forecasting errors in high wind penetration integration studies. 

This is an often overlooked aspect of these studies that can 

significantly influence the results of production cost simulations. 

Future work will look to further disaggregate the wind power 

forecasting errors into different categories where similar results 

can be expected, e.g., based on weather conditions. This will 

further increase the fidelity of the wind power forecasts used in 

wind integration studies.

VII. ACKNOWLEDGMENTS

The authors would like to thank Debra Lew and Greg 

Brinkman for their many conversations on forecasting errors in 

production cost models. 

This work was supported by the U.S. Department of Energy 

under Contract No. DE-AC36-08-GO28308 with the National 

Renewable Energy Laboratory.
7 

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