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Contract No. DE-AC36-08GO28308

Wind Power Forecasting Error 

Distributions over Multiple 

Timescales 

Preprint 

Bri-Mathias Hodge and Michael Milligan

To be presented at the Power & Energy Society General Meeting

Detroit, Michigan

July 24-29, 2011

Conference Paper

NREL/CP-5500-50614

March 2011
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1 

Abstract — Wind forecasting is an important consideration in 

integrating large amounts of wind power into the electricity grid. 

The wind power forecast error distribution assumed can have a 

large impact on the confidence intervals produced in wind power 

forecasting. In this work we examine the shape of the persistence 

model error distribution for ten different wind plants in the 

Electric Reliability Council of Texas (ERCOT) system over 

multiple timescales. Comparisons are made between the 

experimental distribution shape and that of the normal 

distribution. The shape of the distribution is found to change 

significantly with the length of the forecasting timescale. The 

Cauchy distribution is proposed as a model distribution for the 

forecast errors and model parameters are fitted. Finally, the 

differences in confidence intervals obtained using the Cauchy 

distribution and the normal distribution are compared.

Index Terms—Wind power generation, wind energy,

stochastic systems, error probability, forecasting

I. INTRODUCTION

ith the increasing amounts of wind generation being 

added into the electricity system, the importance of 

being able to accurately forecast the wind power output for 

future time stages is also increasing. More accurate wind 

power forecasts can lead to economic efficiency in the unit 

commitment and dispatch process, as fewer reserves must be 

kept, or deployed, in order to compensate for changes in the 

wind power output. Although most wind power forecasts are 

currently given as point forecasts, a more useful approach 

involves producing interval forecasts. Interval forecasts can 

complement the point forecast by providing bounds on the 

expected future value with associated probabilities. This can 

be a very important consideration in the commitment and 

dispatching of generating units. For example, Fabbri et al. 

reported on the cost associated with supplemental regulation 

reserves required because of forecasting errors in the Spanish 

electricity market [1]. They found that the total annual cost of 

wind forecast errors for a single wind plant ranged from 

€15,000-18,000 per MW of installed capacity, depending on 

the forecasting time frame. They also observed that these 

costs decline with increasing installed capacity, demonstrating 

the benefits of geographical diversity in wind plant placement.

 As a further illustration of the importance of wind forecasting 

errors, consider a system in which there is expected to be 1000 

MW of total load with a point forecast of 100 MW of wind 

 The authors are with the National Renewable Energy Laboratory, Golden, 

CO 8041 USA (email: bri-mathias.hodge@nrel.gov, 

michael.milligan@nrel.gov) 

power for the next time period. Now we assume in this 

situation that the marginal producer is a 50 MW gas turbine 

producing at only 40% of capacity; that is 20 MW. Using 

only the point forecast we do not know whether the 30 MW of 

capacity available in the marginal gas turbine will be sufficient 

if the wind power forecast is inaccurate. If we use an interval 

forecast that specifies the 95% confidence lower bound for the 

wind power output to be 85 MW, the spinning reserve 

capacity in the marginal gas turbine should be sufficient to 

handle any up regulation. However, if the lower 95% 

confidence interval for the forecast is only 40 MW, the system 

operator would mostly likely bring more reserves online to 

handle a lower wind output situation. 

The production of these forecast confidence intervals is 

often calculated using an assumed error distribution on the 

point forecast. The errors are often assumed to follow a 

normal distribution [2-5], though Weibull [6] and Beta [7] 

distributions have also been utilized. Lange studied the 

distribution of wind power forecast errors for timescales 

between 6 and 48 hours ahead with a particular focus on the 

additional errors created from converting wind speed forecasts

created by numerical weather prediction to wind power output 

[8]. The study demonstrates that while the NWP errors are 

well represented by a Gaussian curve, the power forecast error 

distributions exhibit both skewness and excess kurtosis. 

Focken et al. investigated the smoothing of forecast errors for 

multiple wind plants at timescales from 6 to 48 hours ahead 

[9]. They determined that geographical diversity of wind 

plants and ensemble forecasts could reduce the forecast errors, 

with increased benefits for increased wind plant distance. 

Bludszuweit et al. examined the forecast error distribution of 

the persistence model and find that the distribution is “fattailed” and should not be modeled using the normal 

distribution [7]. They measure variable kurtosis values 

between different time scales (with a minimum of ten-minute 

averaged output data), and model the error distributions using 

a beta function. This function was then applied to the sizing 

of an energy storage system that will act to smooth wind 

power output.

Previous studies have examined the distribution of wind 

power forecast errors, but have tended to look at timescales of 

six hours or more. In this work we examine the forecast error 

distributions for a number of different timescales, ranging 

from one minute to three hours, demonstrating that the normal 

distribution is unsuitable to represent the error distributions at

these smaller timescales. We then examine the differences in 

distribution shape with regard to timescale. A number of 

Wind Power Forecasting Error Distributions 

over Multiple Timescales

Bri-Mathias Hodge, Member, IEEE and Michael Milligan, Senior Member, IEEE

W
2 

different distribution types are then fitted to sample time series 

and a new distribution is found to outperform those reported in 

the literature. Finally this new distribution is applied to the 

problem of generating forecast error confidence intervals. 

The remainder of the paper is organized as follows. In 

Section II the methods and data used in this study are detailed. 

Section III reports on the results of analyzing the forecast error 

over a number of timescales, and demonstrates the 

effectiveness of modeling the observed forecast errors with a 

Cauchy distribution. Conclusions are then drawn and future 

areas for examination outlined in Section IV. 

II. METHODS AND DATA

In this section we describe some of the important methods 

utilized in the study. Section II-A contains information on the 

datasets analyzed. The persistence model is described in 

Section II-B, and Section II-C discusses the differences 

between a pair of methods for data aggregation. Background 

information on the statistical distributions discussed in the 

work is given in Section II-D. 

A. Data Utilized 

In this study we have utilized data from the Electric 

Reliability Council of Texas (ERCOT) interconnection area in 

the United States. The dataset used is from the year 2009 and 

contains one minute average power output for ten different 

wind plants ranging in size from 30 MW to 215 MW. In 

addition, we have also included a series composed of the 

combined output of the ten wind plants with a total capacity of 

approximately 940 MW. This aggregated ERCOT-wide time 

series is very useful for helping to identify trends that may not 

be apparent in the individual wind plant data. Additionally, 

this time series is more representative of the patterns noticed 

at the Independent System Operator (ISO) level where the 

geographic diversity of a number of wind plants tends to 

smooth the data. Two main subsets of the data were used: a 

winter period and a summer period. The winter period 

consists of the months of January until April and the summer 

period is comprised of June until September. 

B. Persistence Model Forecasting 

There have been a number of statistical techniques 

developed for the forecasting of time series data, but none are 

as widely applicable, and surprisingly effective, as the 

persistence model. While a simple approach, the persistence 

model is the baseline against which all other forecasting 

methods must be compared. Equation (1) shows the 

persistence model, where Pˆ is the forecast power output, 

tP )( is the measured power output at time t , and k is the 

forecast delay. 

)()|( )1( ˆ + = tPtktP

One issue with the use of the persistence model is the choice 

of the time delay between the observation and forecast. For 

longer forecast time periods, it is reasonable to assume that the 

previous point can be measured and relayed with sufficient 

time allowed to make the forecast for the next period. 

However, as the forecast period decreases the importance of 

this market closure delay increases. However, in this work we 

have assumed no delay in the forecast and thus every forecast 

is simply the output at the previous time point. It is important 

to note that the forecast errors have been standardized as a 

fraction of the wind plant installed capacity, creating forecast 

errors contained within the interval [-1,1]. 

C. Mean Values versus Point Values in Data Aggregation 

One of the goals of the current study is to examine the 

changes in forecast errors of the same data set over a number 

of different timescales. Since the basic wind power data is at 

the minute timescale, we need to aggregate the data set in 

order to examine forecast errors for five minute, fifteen 

minute, one hour and three hour time periods. There are two 

basic ways that one could turn the minute data into data a 

longer timescale; either using point values and simply 

removing values not coincident with the desired time steps, or 

use the mean of the values for the previous time period. As 

may be seen from Fig. 1 the point method produces Root 

Mean Squared Forecast Error (RMSFE) values that are 

substantially higher at all of the timescales under 

consideration for all of the individual wind plants as well as 

the aggregated output. 

Fig. 1. Increase in RMSFE for point periods as a percentage of the RMSSFE 

found using the mean value period with the persistence model without a 

forecast delay. These values were obtained for the four month winter period. 

Fig. 1 represents the values obtained using the persistence 

method with no time lag over a four month period starting at 

the beginning of the year. For this reason we have used the 

averaged minutely values over the previous time period in the 

subsequent analysis. For the sake of comparison the RMSFE 

values for each wind plant using the mean values at each time 

period length are shown in Fig. 2. 
3 

Fig. 2. RMSFE found using the mean values for each wind plant during the 

summer period. Note the log-log scale of the plot. 

D. Statistical Distributions 

The probability density function is used to describe the 

range of values that a random variable can obtain, and the 

likelihood that a sample falls in a particular interval. In this 

work the primary focus is on two particular distributions: the 

normal (or Gaussian) distribution and the Cauchy-Lorentz 

Distribution. The beta distribution family, where the 

distribution can vary along the interval (0,1) based on the 

values of two shape parameters α and β, will also be discussed 

along with the Weibull distribution; which is a continuous 

distribution with two parameters relating to shape and scale, 

denoted by k and λ respectively. Besides the standard 

deviation and mean, we will apply two other statistical 

measures to the characterization of the forecast error 

distributions. Skewness is a measure of the asymmetry of the 

probability distribution. This is demonstrated in Fig. 3 by the 

Weibull distribution that is slightly positively skewed. 

Skewness is the third standardized moment, as shown in (2). 

)2(

3





















 



 − = σ

μ

γ

X

E

where γ is the skewness, X is the random variable, σ is the 

standard deviation and μ is the mean. 

Kurtosis is a measure of the magnitude of the peak of the 

distribution, or conversely how fat-tailed the distribution is, 

and is the fourth standardized moment, defined in (3). 

)3( 4

) 4 (

σ

ε

κ

E =

where κ is the kurtosis and ε is the normalized forecast error. 

 A distribution with a high kurtosis is known as leptokurtic 

and it describes a distribution where more of the variance is 

due to a lesser number of large deviations rather than the very 

frequent small deviations. The Cauchy distribution in Fig. 3 

displays high kurtosis compared to the normal distribution, 

possessing a more pronounced peak, slimmer shoulders and 

longer tails. The difference between the kurtosis of a sample 

distribution and that of the normal distribution is known as the 

excess kurtosis. In the subsequent analysis the term kurtosis 

will be treated synonymously with excess kurtosis since we 

use the normal distribution as a baseline for the comparison of 

distributions. 

Fig. 3. Illustration of the different probability distributions on the x-interval 

[0,1]. Blue is the normal distribution with mean = 0.5 and standard deviation 

= 0.2. Red is the beta distribution with shape parameters α = β =5. The 

Weibull distribution is shown in green with k = 5 and λ = 0.5. The Cauchy 

distribution is in black with xo = 0.5 and γ = 0.05. 

III. RESULTS

Having established the importance of wind power 

forecasting error distributions and described the relevant 

statistical background, we now characterize the distributions 

for the datasets under consideration. In Sections III-A and IIIB we examine the forecast error distribution shapes and use a 

pair of graphical techniques to compare the resulting 

distributions to the normal distribution. In Sections III-C and 

III-D we use a pair of numerical techniques to examine the 

forecast error distribution shapes. After ruling out the normal 

distribution as a valid model for the sample distributions, 

Section III-E endeavors to identify a more accurate model 

distribution. Finally, in Section III-F we apply the results of 

the new model to the problem of generating wind forecast 

error confidence intervals. 

A. Histograms 

In order to examine the distributions of the persistence 

forecast errors, histograms of the data were first created and 

analyzed. The number of bins used exceeded the number 

needed according to Scott’s rule [10] in all cases and a value 

of n = 300 was found to work well over all of the timescales. 

An examination of the sample histograms provided in Fig. 4 

and Fig. 5 show examples of leptokurtic distributions at two 

0.0 0.2 0.4 0.6 0.8 1.0

0123456

x

Density
4 

different timescales. The sample distributions have more 

pronounced peaks, steeper shoulders and fatter tails than the 

normal distributions generated using the sample means and 

standard deviations. Distributions are shown from both 

summer and winter four-month time periods. 

Fig. 4. Histogram of five minute persistence forecast error for wind plant #3 

during the four month winter period. γ = -2.14, κ = 151.25. The blue line 

represents a normal distribution with the same mean and standard deviation. 

The black line is an approximate fit to the binned data. 

Fig. 5. Histogram of one hour persistence forecast error for wind plant #8 

during the four month summer period. γ = -0.02, κ = 6.18. The blue line 

represents a normal distribution with the same mean and standard deviation. 

The black line is an approximate fit to the binned data. 

B. Normal Quantile-Quantile Plots 

The quantile-quantile (Q-Q) plot is a means by which two 

distributions can be graphically compared. Though the 

histograms shown in the previous section seem to indicate that 

the forecast error distributions are not normal over the 

timescales considered, the use of normal Q-Q plots can 

provide additional assurance. When the quantiles of the 

sampled quantity are compared with the quantiles of a normal 

distribution they should follow a linear pattern and 

approximately lie on the line y = x if the two distributions are 

similar. 

The normal Q-Q plot in Fig. 6 shows the one minute 

forecast error distributions for wind plant #1. The non-normal 

nature of the distribution is immediately apparent from the 

non-linear nature of the quantile comparison, with 

significantly arched patterns suggesting a high degree of 

kurtosis, as confirmed by numerical calculation. 

Fig. 6. Normal Quantile-Quantile plot for wind plant #1 – one minute 

persistence forecast error for the four month summer period. γ = -0.45, κ = 

1717.81. 

Fig. 7 shows a normal Q-Q plot of the one hour forecast 

error for wind plant #6. The plotted values are much more 

linear than those found in the previous plot, but still exhibit 

noticeable deviations. The distribution is also leptokurtic, as 

evidenced by the kurtosis value calculated. 

Fig. 7. Normal Quantile-Quantile plot for the wind plant #6 – one hour 

average persistence forecast error for the four month summer period. 

γ = -0.08, κ = 8.04. 

Finally, Fig. 8 is an example of a normal Q-Q plot where 

the distribution is close to normal. The plotted points are very 
5 

close to linear and the small kurtosis value confirms the visual 

assessment. 

Fig. 8. Normal Quantile-Quantile plot for the combined ERCOT data – three 

hour average persistence forecast error for the four month summer period. 

γ = 0.02, κ = 0.82. 

C. Kurtosis of the Forecast Errors 

One of the more interesting findings to come from 

examining the kurtosis values for the distribution of 

persistence forecasting errors over different timescales is the 

trend of decreasing kurtosis with larger timescales. As may be 

seen in Table I and Fig. 9, the kurtosis values have a strong 

decreasing trend with increasing timescale. 

TABLE I 

KURTOSIS VALUES FOR PERSISTENCE FORECAST ERROR WITH NO TIME LAG 

OVER A FOUR MONTH SUMMER PERIOD AT DIFFERENT TIME SCALES

 1 Minute 5 Minute 15 Minute 1 Hour 3 Hour 

#1 1,717.89 328.43 163.59 39.50 25.07 

#2 210.06 62.61 34.09 8.24 3.42 

#3 46.61 42.17 29.05 9.33 2.34 

#4 122.11 51.94 30.63 7.61 2.23 

#5 595.13 69.88 31.87 9.30 2.65 

#6 900.03 137.24 34.84 8.04 2.16 

#7 79.91 27.79 19.69 5.16 1.72 

#8 50.72 51.31 28.08 6.18 1.13 

#9 119.07 37.23 18.41 6.23 1.99 

#10 328.18 81.80 38.93 5.14 1.39 

ERCOT 149.12 16.32 7.85 2.51 0.82 

It must be noted that the increasing timescale also 

corresponds to an increasing number of values used in the 

calculation of the time series values, due to the mean values 

used in the analysis. This trend can be interpreted to be a 

result of the central limit theorem because as the timescale 

increases so do the number of factors that may act on the wind 

power output values. This would also helps to explain why 

the aggregated ERCOT data’s kurtosis values are far lower 

than simply the mean of the values for all of the individual 

wind plants in the summer data. However, the opposite effect 

is observed in the winter data, with the ERCOT forecast error 

distribution kurtosis values consistently larger than the mean 

values for the individual wind plants, as seen in the distinct 

purple line in Fig. 9. 

Fig. 9. Kurtosis values for persistence forecast error distributions for the 

winter time period. Note the log – log scale of the plot. 

D. Skewness of the Forecast Errors 

In addition to the strength of the peak, the symmetry of the 

distributions is another very important aspect of the forecast 

error distribution characterization. The skewness values of the 

forecast error distributions for the various wind plants and 

timescales are shown in Table II. The trends in the data with 

respect to timescale are much weaker than in the kurtosis 

values; however, there is a weak trend toward more symmetric 

distributions with increasing time. In general, most of the 

skewness values are relatively minor, suggesting that the 

distribution of errors is close to symmetric. Similar to the 

kurtosis values for the summer period, the combined ERCOT 

data shows more Gaussian behavior than the individual wind 

plant mean skewness values alone would suggest. 

TABLE II 

SKEWNESS VALUES FOR PERSISTENCE FORECAST ERROR WITH NO TIME LAG 

OVER A FOUR MONTH SUMMER PERIOD AT DIFFERENT TIME SCALES

 1 Minute 5 Minute 15 Minute 1 Hour 3 Hour 

#1 -0.44 -0.48 -0.11 0.35 1.13 

#2 -0.75 -0.08 0.94 0.63 0.44 

#3 0.00 -0.40 0.06 0.06 0.16 

#4 -0.40 1.56 1.48 0.56 0.18 

#5 4.21 1.67 1.03 0.50 0.00 

#6 -4.26 -0.92 0.06 -0.08 -0.12 

#7 -0.15 0.48 0.61 0.13 -0.03 

#8 0.14 -0.06 -0.48 -0.02 0.02 

#9 -0.25 -0.20 0.07 -0.08 -0.17 

#10 -1.07 0.10 0.26 -0.24 0.00 

ERCOT 0.25 0.01 -0.07 0.05 0.02 

E. Distribution Fitting 

Having established that the normal distribution is a poor fit 

for the persistence forecast error distributions at the timescales 

1 2 5 10 20 50 100 200

1e-01 1e+01 1e+03 1e+05

Kurtosis of Persistence Forecast Error Distributions

Time Period (Min)

Kurtosis
6 

under study, the next step in generating better forecasting 

intervals is to find a model that can more accurately represent 

the observed distributions. To accomplish this goal we have 

chosen three different model distributions that resemble those 

observed in the data. The Beta [7] and Weibull [6] 

distributions have been previously suggested in the literature, 

however, the Cauchy distribution appeared to be the most 

natural fit to the authors. In order to accommodate the fitting 

of the Beta and Weibull distributions, which are supported 

only on the intervals [0,1] and [0,∞) respectively, the forecast 

errors were converted from the [-1,1] interval to lie on the 

(0,1) interval. The distributions models were fit to the 

observed data using a maximum-likelihood optimization 

routine fitdist from the MASS package in the R statistical 

software environment [11]. 

Fig. 10. Illustration of the fitted probability distributions on the x-interval 

[0,1] for wind plant #8 using hour average data for the winter time period. 

Blue is the beta distribution with shape parameters α = 25.2414 and β = 

25.2692. The Weibull distribution is shown in green with k = 7.1940 and λ = 

0.5285. The Cauchy distribution is in red with xo = 0.4996 and γ = 0.0215. 

The log-likelihood values for the fitted distributions indicate 

that the Cauchy distribution is a better fit for the forecast error 

distributions for a majority of the wind plants at all timescales. 

The Cauchy distribution fit was better than the Beta and 

Weibull distributions in 89% and 95% of the 55 cases, 

respectively. One such example is shown in Fig. 10, where 

the model distributions are compared to the histogram of the 

forecast error distribution for wind plant #8 at the hour 

timescale. In this case the Cauchy distribution represents a 

20.24 % and 16.12% improvement over the Weibull and Beta 

distributions, respectively, in terms of the optimized loglikelihood values. For the ERCOT case at the 15 minute 

timescale the Cauchy distribution represents a 54.39 % and 

36.19% improvement over the Weibull and Beta distributions, 

respectively, in terms of the optimized log-likelihood values. 

Fig. 11. Illustration of the fitted probability distributions on the x-interval 

[0,1] for combined ERCOT data using 15 minute average data for the winter 

time period. Blue is the beta distribution with shape parameters α = 82.0833 

and β = 82.1398. The Weibull distribution is shown in green with k = 9.4390 

and λ = 0.5160. The Cauchy distribution is in red with xo = 0.4999 and γ = 

0.0069. 

F. Forecast Probability Intervals 

Thus far we have characterized the distribution of forecast 

errors for the persistence model over a number of timescales. 

We have also identified a distribution model that can represent 

the distribution of errors more accurately than the Gaussian 

distribution, the beta distribution and the Weibull distribution. 

But what are the practical implications of these findings on 

wind forecasting? 

Fig. 12 shows the ERCOT data power forecast at 15 minute 

intervals along with 90% and 95% confidence intervals 

created using both a Cauchy distribution fit to the data and a 

normal distribution based on the data’s mean and standard 

deviation values. The confidence intervals were created 

through 10,000 draws from each distribution. It is interesting 

to see that the Cauchy distribution produces a tighter interval 

for the 90% confidence values than the normal distribution. 

However, while the difference between the 90% and 95% 

confidence intervals for the normal distribution is relatively 

minor, there is a large difference for the Cauchy distribution. 

In fact, the 95% confidence interval for the Cauchy 

distribution is broader than for the normal distribution. This 

interesting result can be explained by examining the shapes of 

the respective distributions. While the Cauchy distribution 

produces more instances close to the mean than the normal 

distribution, it also has fat tails that indicate large deviations at 

the ends of the distribution, leading to narrow bounds for 

lower percentage confidence intervals and very wide bounds 

at high percentage confidence intervals. For the ERCOT 15 

minute series shown in Figure 12, the 90% confidence interval 

band for the Cauchy distribution is 18.62% of total capacity, 

while for the normal distribution it is 25.32% of total capacity. 

Wind Farm #8 - Hour Forecast Error

x

Density

0.0 0.2 0.4 0.6 0.8 1.0

0 10 20 30 40

ERCOT Data - 15 Minute Forecast Error

x

Density

0.0 0.2 0.4 0.6 0.8 1.0

0 10 20 30 40
7 

The 95% confidence interval spans 30.42% of the capacity for 

the normal distribution and 38.35% for the Cauchy 

distribution. 

Fig.12. Plot of the wind power forecasts and associated forecast confidence 

intervals for the combined ERCOT output during a section of the winter 

period. The black line represents the power forecast. The normal distribution 

confidence intervals at 90% and 95% are shown in light blue and blue 

respectively. The Cauchy distribution confidence intervals at 90% and 95% 

are shown in red and dark red. 

It is a slightly different story for the wind plant #5 hour 

series shown in Fig. 13, the result of a more platykurtic 

Cauchy distribution. The 90% confidence interval for the 

Cauchy distribution is nearly identical to the 95% confidence 

interval for the normal distribution. Once again the difference 

between the 90% and 95% confidence intervals for the Cauchy 

distribution is much wider than for the normal distribution due 

to the heavy tails. The Cauchy distribution 90% confidence 

interval is 36.86% of total capacity, while for the normal 

distribution it is 30.56% of total capacity. The 95% 

confidence interval for the normal distribution spans 36.39% 

of the capacity and 76.36% for the Cauchy distribution. 

The results indicate that the normal distribution can 

overestimate the range of values for lower percentage 

confidence intervals, while underestimating the range for 

higher percentage confidence intervals. This difference can 

have important consequences in the determination of operating 

reserve levels necessary in systems with high wind 

penetration. More accurate modeling of the true wind power 

output confidence intervals with a Cauchy, or other, 

distribution can lead to economic savings through more 

efficient operating reserve allotment and operation. The 

confidence intervals chosen during operations may differ both 

by the balancing authority in question and the time of day. 

The current state of the system will guide the choice of 

confidence intervals and knowing the large marginal 

differences in the Cauchy distribution for confidence intervals 

above 90% can help inform the decision making process to 

ensure economic and reliable system operation. 

Fig. 13. Plot of the wind power forecasts and associated forecast confidence 

intervals for wind plant #5 output during a section of the winter period. The 

black line represents the power forecast. The normal distribution confidence 

intervals at 90% and 95% are shown in light blue and blue respectively. The 

Cauchy distribution confidence intervals at 90% and 95% are shown in red 

and dark red. 

IV. CONCLUSION

In this work we have examined the shape of wind power 

forecast error distributions through a statistical analysis. The 

distributions were found to differ greatly from the commonly 

assumed normal distribution, and the kurtosis of the 

distribution was found to vary with timescale. The Cauchy 

distribution has been proposed as a means of representing the 

forecast error distributions and was compared with some of 

the other distributions used in the literature. Finally, the effect 

that the new forecast error distribution model has on the 

calculation of wind power forecast confidence intervals was 

illustrated. 

The current work suggests a number of directions for 

further examination. The analysis can be expanded to include 

more sources of wind power output data, as well as data at 

even smaller time scales. One possible caveat to the current 

work is that the wind power data sets are taken from the bus 

level and so they may include instances of wind curtailment 

that may skew the ends of the error distributions. The 

examination of the forecast error distributions at differing 

geographic scales would also be an interesting extension of the 

current work. The effect of forecasting lag within the 

persistence model is another area for further study. The 

examination of forecast error distributions for more 

sophisticated forecasting models, such as ARMA and ARIMA 

models, is also planned. 

The current work also has applicability beyond the field of 

wind forecasting. The analysis found in this study could be 

used in the development of methods for the scaling down of 

wind power data. The quality of wind power output data is an 

important concern in wind integration studies. In cases where 

0 10 20 30 40 50

0 200 400 600 800 1000

ERCOT Power Forecasts

Time Period (15 Minute Intervals)

Power (MW)

0 5 10 15 20 25

0 50 100 150

Wind Farm #5 Power Forecasts

Time Period (Hour Intervals)

Power (MW)
8 

the wind data available is at a higher timescale than desired for 

the study, conditioning of the data is necessary in order to 

obtain data at the correct time scale. The findings from using 

the persistence model are especially relevant in this case 

because it represents what can be expected in the next time 

frame without any statistical transformation. The use of the 

Cauchy distribution to create data for the smaller scale time 

points in a scaling down effort could lead to the use of more 

representative wind power data in these studies.

V. ACKNOWLEDGMENT

The authors gratefully acknowledge the contributions of

Yih-huei Wan for supplying the data used in the study.

VI. REFERENCES

[1] A. Fabbri, et al., "Assessment of the Cost Associated With Wind 

Generation Prediction Errors in a Liberalized Electricity Market," 

IEEE Transactions on Power Systems, vol. 20, August 2005.

[2] K. Methaprayoon, et al., "An integration of ANN wind power 

estimation into UC considering the forecasting uncertainty," in 

IEEE Industrial and Commercial Power Systems Technical 

Conference, Saratoga Springs, NY, 2005.

[3] V. Pappala, et al., "A Stochastic Model for the Optimal Operation 

of a Wind-Thermal Power System," IEEE Transactions on Power 

Systems, vol. 24, pp. 940 - 950, May 2009.

[4] E. Castronuovo and J. Lopes, "On the Optimization of the Daily 

Operation of a Wind-Hydro Power Plant," IEEE Transactions on 

Power Systems, vol. 19, pp. 1599 - 1606, August 2004.

[5] R. Doherty and M. O'Malley, "A New Approach to Quantify 

Reserve Demand in Systems With Significant Installed Wind 

Capacity," IEEE Transactions on Power Systems, vol. 20, pp. 587 - 

595, May 2005.

[6] K. Dietrich, et al., "Stochastic unit commitment considering 

uncertain wind production in an isolated system," in 4th 

Conference on Energy Economics and Technology, Dresden, 

Germany, 2009.

[7] H. Bludzuweit, et al., "Statistical Analysis of Wind Power Forecast 

Error," IEEE Transactions on Power Systems, vol. 23, pp. 983 - 

991, August 2008.

[8] M. Lange, "On the Uncertainty of Wind Power Predictions - 

Analysis of the Forecast Accuracy and Statistical Distribution of 

Errors," Journal of Solar Energy Engineering, vol. 127, pp. 177-

184, 2005.

[9] U. Focken, et al., "Short-term prediction of the aggregated power 

output of wind farms - a statistical analysis of the reduction of the 

prediction error by spatial smoothing effects," Journal of Wind 

Engineering and Industrial Aerodynamics, vol. 90, pp. 231 - 246, 

2002.

[10] D. Scott, "On optimal and data-based histograms," Biometrika, vol. 

66, pp. 605-610, 1979.

[11] "R: A Language and Environment for Statistical Computing," ed. 

Vienna, Austria: R Foundation for Statistical Computing, 2010.

VII. BIOGRAPHIES

Bri-Mathias Hodge (M’10) received the B.S. degree in chemical engineering 

from Carnegie Mellon University in 2004. He received a M.S. from the 

Process Design and Systems Engineering Laboratory of Åbo Akademi, Turku, 

Finland in 2005, and completed the Ph.D. in chemical engineering at Purdue 

University in 2010.

He is currently a post-doctoral researcher in the grid integration team for 

the National Renewable Energy Laboratory (NREL). His research interests 

include energy systems modeling, simulation and optimization.

Michael Milligan (M’98, SM’10) received the B.A. degree from Albion 

College, Albion, MI, and the M.A. and Ph.D. degrees from the University of 

Colorado, Boulder.

He is part of the grid integration team for the National Renewable Energy 

Laboratory (NREL), where he has worked on various aspects of electricity 

supply related to wind energy since 1992. He has worked on issues such as the 

ancillary service impacts of wind generation, the value of accurate wind 

forecasting, optimal selection of geographically dispersed wind power plants, 

modeling wind plant variability, and reliability contribution of wind power 

plants. He has authored or coauthored more than 90 reports and book chapters 

and has served on numerous technical review committees for wind integration 

studies around the United States.
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Wind Power Forecasting Error Distributions over Multiple 

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In this paper, we examine the shape of the persistence model error distribution for ten different wind plants in the 

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Wind power generation; wind energy; stochastic systems; error probability; forecasting

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