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Probabilistic Short-Term Wind 

Forecasting Based on Pinball 

Loss Optimization

Preprint 

Mucun Sun,1 Cong Feng,1 Jie Zhang,

1

Erol Kevin Chartan,2 and Bri-Mathias Hodge2

1 University of Texas at Dallas 

2 National Renewable Energy Laboratory 

Presented at the 2018 IEEE International Conference on 

Probabilistic Methods Applied to Power Systems (PMAPS)

Boise, Idaho

June 24–28, 2018

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Probabilistic Short-Term Wind Forecasting Based

on Pinball Loss Optimization

Mucun Sun, Cong Feng, Jie Zhang

The University of Texas at Dallas

Richardson, TX 75080, USA

Erol Kevin Chartan, Bri-Mathias Hodge

National Renewable Energy Laboratory

Golden, CO 80401, USA

Abstract—Probabilistic wind power forecasts quantify uncertainty and have the potential to aid in the economic grid

integration of wind power at large penetration levels. In this

paper, a probabilistic wind forecasting approach based on pinball

loss optimization is proposed, in conjunction with a multimodel machine learning-based ensemble deterministic forecasting

framework. By assuming the point-forecasted value as the mean

at each point, one unknown parameter (i.e., standard deviation)

of a predictive distribution at each forecasting point is determined

by minimizing the pinball loss. A surrogate model is developed

to represent the unknown distribution parameter as a function

of deterministic forecasts. This surrogate model can be used

together with deterministic forecasts to predict the unknown distribution parameter and thereby generate probabilistic forecasts.

Numerical results of case studies show that the proposed method

has improved the pinball loss by up to 35% compared to a

baseline quantile regression forecasting model.

Index Terms—probabilistic wind forecasting, optimization, surrogate model, machine learning, pinball loss.

I. INTRODUCTION

The uncertain and variable nature of wind imposes challenges to integrate wind power, particularly at large penetration

levels. Improved wind forecasts are needed to assist power

system planning and operations.

A number of wind forecasting technologies have been

developed in the literature and have also been applied to a

variety of power system operation and planning problems. For

example, Lee et al. [1] used improved wind power forecasts

to reduce the cost of system ancillary services and to conduct

a system risk analysis. Botterud et al. [2] applied wind

power forecasts in unit commitment and economic dispatch

decision-making to provide dynamic operating reserves, which

benefits system operators and electricity traders. In electricity

markets for energy, conventional deterministic forecasts are

not sufficient to describe the inherent unpredictability of wind

power, which is compensated by operating reserves. As a

result, probabilistic forecasts that provide quantitative uncertainty information associated with wind power are expected

to better assist power system operations better. Methods of

constructing predictive distributions can be mainly classified

into parametric and nonparametric approaches in terms of

distribution shape assumptions [3]. A prior assumption of the

predictive distribution shape is made via parametric methods.

Once an analytical form of the predictive distribution is

defined, parameters describing this distribution can be determined from data, which generally requires low computational

cost. Distribution parameters can be estimated through different methods, and nonlinear time series is one of the most

popular. For example, Pinson et al. [4] proposed a conditional

parametric autoregression model to estimate the parameters of

a Generalized Logit-Normal (GL-normal) distribution, which

is a discrete-continuous mixture of GL-normal distribution and

two probability masses.

For distribution-free nonparametric approaches, the predictive distribution is estimated through a finite number of

observations. Quantile regression (QR) and kernel density

estimation (KDE) are traditional nonparametric probabilistic

forecasting methods [5]. Haben et al. [6] proposed a nonparametric hybrid method, which combines the KDE and

QR to generate probabilistic load forecasts. Ordiano et al.

[7] conducted probabilistic solar power forecasting using a

nearest-neighbor-based nonparametric method.

Pinball loss is one of most popular metrics for evaluating

the performance of probabilistic forecasting [8]. In this paper,

a novel two-step probabilistic wind forecasting method is developed based on pinball loss optimization. First, deterministic

forecasts are generated (with any deterministic forecasting

methods). Second, a set of unknown parameters in the predictive distribution are optimized determined by minimizing the

pinball loss. The optimal prediction distribution parameters are

first determined in the training data set. A surrogate model is

developed to represent the unknown distribution parameter as

a function of deterministic forecasts. At the forecasting state,

the surrogate model is then used with deterministic forecasts

to predict the unknown distribution parameter and thereby

generate probabilistic forecasts.

The remainder of the paper is organized as follows. Section

II describes the proposed probabilistic forecasting method,

including a multi-distribution model and pinball loss-based

optimization process and a deterministic forecasting method.

Section III applies the developed pinball loss-based probabilistic forecasting method to multiple wind data set and compares

the forecasting performance with benchmark models. Concluding remarks and future work are discussed in Section IV.

II. METHODOLOGY

An optimal pinball loss-based short-term probabilistic forecasting method is developed in this paper, and the overall

framework is illustrated in Fig. 1. This is a two-step probabilistic forecasting method, consisting of deterministic fore1
Gaussian

FN x μ, σ

2

Gamma

FG x k, θ , μ = kθ

Noncentral T

FT x ν, λ , 

μ =

ν

2

Γ(

ν − 1

2

)

Γ(

ν

2

)

λ

Laplace

FL x μ, σ

Gaussian Distribution

qm = FN

−1

(

m

100)

Gamma Distribution

qm = FG

−1

(

m

100)

Noncentral t 

Distribution

qm = FT

−1

(

m

100)

Laplace Distribution

qm = FL

−1

(

m

100)

Cumulative Distribution Quantile Function

Historical Wind 

Data

Neural Network 

(NN)

Support Vector 

Machine (SVM)

Gradient Boosting 

Machine (GBM)

Random Forest 

(RF)

Deterministic 

Wind Forecasts

Ensemble 

Algorithm

Deterministic 

Forecasts = 

Mean (μ)

Pinball Loss: L qm, xi = ൞

1 −

m

100

× qm − xi xi < qm

m

100

xi − qm xi ≥ qm

Optimal standard 

deviation set

argmin

σ ෍

i=1

n

෍

m=1

99

L(qm, xi)

Probabilistic Wind 

Forecasts

Reliability and 

σ Sharpness Analysis

Deterministic Wind Forecasting Wind Forecasting Kernel Selection Pinball Loss Model Optimization Probabilistic Wind Forecasting

Pinball Loss Analysis

σො = f(xp)

Surrogate model of 

optimal standard 

deviations

Estimated σ of test data

Fig. 1: Overall framework of the pinball loss-based probabilistic wind forecasting framework

casts generation and predictive distribution (type and parameters) determination. A machine learning-based multi-model

forecasting framework (MMFF) is first adopted to generate

short-term deterministic wind forecasts (i.e., 1-hour-ahead).

To generate probabilistic forecasts, deterministic forecasts are

considered as a means of predictive distributions, as illustrated

in Fig. 2, and unknown parameters of the predictive distributions are solved by minimizing pinball loss. The distribution

with the minimum pinball loss in conjuction with a surrogate

model are used to generate probabilistic forecasts.

Time (h)

Deterministic Wind Speed Forecasts (m/s

)

Predictive Distribution

0 2 4 6 8 10 12 14 16 18 20 22

0 2 4 6 8 10 12 14 16 18

Fig. 2: Deterministic wind speed forecasts and predictive

distribution

A. MMFF deterministic forecasting

The proposed pinball loss-based probabilistic forecasting

methodology is a two-step method, which can be applied with

any deterministic wind forecasts. In this paper, an MMFF

system consisting of an ensemble of four single machine

learning algorithms with various kernels is adopted to generate

deterministic forecasts. Details of the MMFF method can be

found in [9].

B. Multi-distribution model

A multi-distribution database is formulated to model the

possible shapes of the predictive distribution. Four distribution

types are considered: Gaussian, Gamma, Laplace, and noncentral t distributions. Probability density functions (PDFs) of

the four distributions are listed in [10]. Parameters used to

describe the four predictive PDFs are all related to their mean

and standard deviation values. Therefore, all of the PDFs can

be represented in the form of mean μ and standard deviation σ,

namely f(x|μ, σ). A cumulative distribution function (CDF),

F(x|μ, σ), can be deduced through the integration of a PDF.

C. Pinball loss-based optimization

Pinball loss is one of the most popular metrics for evaluating

probabilistic forecasts [8], and it is a function of observations

and quantiles of a forecast distribution. A smaller pinball loss

value indicates a better probabilistic forecast.

L(qm, xi) =







(1 −

m

100

) × (qm − xi), xi < qm

m

100

× (xi − qm), xi ≥ qm

(1)

2
where xi represents the ith hour observation, m represents

a quantile percentage from 1 to 99, and qm represents the

predicted quantile. For a given m percentage, the quantile qm

represents the value of a random variable whose accumulated

probability density (i.e., CDF) is m percentage. The quantiles

of different distributions types are represented by a standard

deviation σ. In this paper, the optimal standard deviation σ is

determined by minimizing the sum of the pinball loss function

L(·), by considering the appropriate constraints. A genetic

algorithm is used to solve this optimization problem. A genetic

algorithm is a widely used heuristic method for solving both

constrained and unconstrained optimization problems [11]. In

this study, the maximum number of iterations is set to 100, and

the iteration stops if the improvement is less than 0.001. At

each deterministic forecasting time point, an optimal standard

deviation that minimizes the pinball loss of this single point

is found accordingly. The optimization problem is formulated

as follows:

min

σ

X99

m=1

L(qm(σ), xi)

subject to (2)

σl ≤ σ ≤ σu

where σl and σu represent the lower and upper bounds of

the unknown standard deviation, respectively. In this paper,

the lower and upper bounds of the standard deviations are set

to be 0 m/s and 10 m/s, respectively [12]. The distribution

with the minimum pinball loss is selected as the predictive

distribution shape. The optimal σ’s estimated using the training

data are used to construct a surrogate model to be used in the

forecasting stage.

D. Surrogate model

To generate probabilistic forecasts, an optimal standard

deviation value is needed at every forecasting time point. To

obtain this optimal standard deviation value, a surrogate model

is developed to represent the optimal standard deviation as

a function of deterministic forecasting value based on the

training data, which is expressed by:

σˆ = f(xp) (3)

where xp is a point forecast, and f(·) is a surrogate model of

the optimal standard deviation of the predictive distribution.

A support vector regression method is used in this paper to

construct the surrogate model. This surrogate model is used to

estimate the standard deviation of the predictive distribution in

the forecasting stage, thereby generating the final probabilistic

forecasts.

III. CASE STUDIES AND RESULTS

A. Data Summary

The proposed pinball loss-based probabilistic forecasting

approach is applied to 7 locations for wind speed forecasts.

The wind speed data are collected near hub height with a

1-hour resolution [13]. The duration of the collected data is

summarized in Table I. For all locations, the first 2/3 of data

are used as training data. The last 1/12 of the training data

is used to build a surrogate model between optimal standard

deviation and deterministic forecasts. The effectiveness of the

forecasts is evaluated by the remaining 1/3 of data. Although

the proposed method is capable of generating forecasts at

multiple forecasting timescales, only 1-hour-ahead forecasts

are generated in this study.

TABLE I: Data duration at selected sites

Site Data duration

Boulder NWTC 2009-01-02 to 2012-12-31

Bovina50 2010-10-10 to 2012-10-08

Bovina100 2010-03-03 to 2012-03-01

CapeMay 2007-09-26 to 2009-09-24

CedarCreek H06 2009-01-02 to 2012-12-31

Goodnoe Hills 2007-01-01 to 2009-12-31

Megler 2010-11-03 to 2012-11-01

B. Pinball loss optimization results

Pinball loss values with different predctive distributions are

listed in Table II. The sum of pinball loss is averaged over all

quantiles from 1% to 99% and normalized by the maximum

wind speed at each site. A lower loss score indicates a better

probabilistic forecast. The Laplace distribution with MMFF

has the smallest pinball loss value at all locations except

CedarCreek H06. The lower pinball loss in CedarCreek H06

using the Laplace distribution with persistence method is

mainly because the persistence deterministic forecasts perform

better than the MMFF forecasts. A quantile regression method,

persistence-Laplace (PS Laplace) method with pinball loss

optimization, and an MMFF-Laplace method without pinball

loss optimization are used as baselines in case studies. The

MMFF-Laplace forecasts have improved the pinball loss by up

to 35% compared to the three benchmark models. Therefore,

the Laplace distribution is finally chosen to generate probabilistic wind speed forecasts. Note also that the methods of

MMFF-Gaussian, MMFF-Gamma, and MMFF-Laplace perform similarly, which indicates that the optimization can help

achieve a better accuracy with different predictive distribution

types. For the baseline method of MMFF-Laplace without

pinball loss optimization, a random standard deviation value is

selected from the range between the minimum and maximum

values of the optimal σ. We repeat this process 30 times to

obtain an average sum of pinball loss without optimization.

The training time of the MMFF-laplace method ranges from

1 to 2 hours, and the forecasting time ranges from 2 to 5

miniutes.

C. Deterministic forecasting results

Standard metrics of root mean squared error (RMSE), mean

absolute error (MAE), and their corresponding normalized

indices, i.e., NMAE and NRMSE, are adopted to evaluate

the deterministic forecasting performance. For these metrics,

a smaller value indicates better performance. Deterministic

forecasting errors using MMFF at the selected locations are

3
TABLE II: Normalized optimal sum of pinball loss

Site Boulder NWTC Megler CedarCreek H06 Goodnoe Hills Bovina50 Bovina100 CapeMay

QR 2.22 1.76 2.03 1.96 2.56 2.44 1.95

MMFF Gaussian 1.74 1.26 1.44 1.35 1.86 1.69 1.27

MMFF Gamma 1.74 1.26 1.43 1.35 1.87 1.69 1.27

MMFF Laplace 1.72 1.25 1.43 1.35 1.85 1.68 1.26

MMFF noncentral t 1.74 1.81 2.20 2.21 2.68 3.41 2.56

MMFF Laplace (without opt) 2.94 2.93 2.40 2.39 3.53 3.08 2.45

PS Laplace 1.81 1.29 1.34 1.38 1.92 1.69 1.32

Note: The smallest normalized optimal sum of pinball loss at each location is in boldface.

summarized in Table III. It is shown that the one-hour-ahead

NMAE and NRMSE are in the range of 3%-5% and 4%-

7%, respectively. An example of the forecasts at the Megler

site from 2012-02-01 to 2012-02-04 are shown in Fig. 3. The

persistence method is used as a baseline, and the forecasting

errors are also summarized in Table III. Overall, the accuracies

of the MMFF deterministic forecasts are better than those of

persistence forecasts, except CedarCreek H06.

D. Probabilistic forecasting results

With estimated scale parameters through pinball loss minimization and surrogate modeling, predictive wind speed distributions are determined, and the quantiles q1, q2, ..., q99 can

be calculated. To better visualize probabilistic forecasts, the

99 quantiles are converted into nine predictive intervals Iβ

(β=10, ..., 90) in a 10% increment. Fig. 3(a) shows an example

of probabilistic wind speed forecasts at the Megler site from

2012-02-01 to 2012-02-04. The width of the predictive interval

varies with the level of wind speed fluctuation. When the wind

speed fluctuates significantly, the predictive interval tends to be

wider, i.e., the uncertainty in wind speed forecasts is relatively

higher. Fig. 3(b) shows probabilistic forecasts generated from

the baseline quantile regression method at the same site and

time period. The predictive intervals of the proposed MMFFLaplace with pinball loss optimization method are narrower

than those of the quantile regression method. Thus, there is

less uncertainty in the proposed probabilistic forecasts.

1) Reliability: Reliability (RE) stands for the correctness of

a probabilistic forecast that matches the observation frequencies [14]:

RE =



ξ

(1−α)

N

− (1 − α)



× 100% (4)

where N is the number of test samples, and ξ

(1−α)

is the

number of times that the actual test samples lie wthin the

αth prediction interval. With measured emperical coverage,

a reliability diagram can be plotted to describe the quantile

forecast series with different nominal proportions. A reliability

plot shows whether a given method tends to systematically

underestimate or overestimate the uncertainty. In this study,

the nominal coverage rates range from 10% to 90% with

a 10% increment. Fig. 4 shows the reliability curves of the

probabilistic forecasts at the CapeMay, Megler, and Bovina50

sites. A forecast presents better reliability when the curve is

closer to the diagonal. Overall the quantile regression has

0 10 20 30 40 50 60 70 80 90

Time (h)

0

2

4

6

8

10

12

Wind Speed (m/s)

90% intervel

80% intervel

70% intervel

60% intervel

50% intervel

40% intervel

30% intervel

20% intervel

10% intervel

Prediction

Actual

(a) MMFF-laplace with pinball loss optimization

0 10 20 30 40 50 60 70 80 90

Time (h)

0

2

4

6

8

10

12

14

Wind Speed (m/s)

90% intervel

80% intervel

70% intervel

60% intervel

50% intervel

40% intervel

30% intervel

20% intervel

10% intervel

Prediction

Actual

(b) Quantile regression

Fig. 3: MMFF-Laplace and quantile regression forecasts at the

Megler site

better reliability performance because the confidence band

of the quantile regression is much wider than that of the

proposed MMFF-Laplace method. A wider confidence band

indicates that the results take more errors into consideration;

however, note that the reliability over the 90th confidence

intervals is similar between the proposed method and the

baseline quantile regression method, which is generally more

4
TABLE III: Deterministic forecasting results using MMFF and PS

Method Site Boulder NWTC Megler CedarCreek H06 Goodnoe Hills Bovina50 Bovina100 CapeMay

MMFF NMAE(%) 4.71 3.36 3.86 3.72 5.00 4.56 3.39

NRMSE(%) 6.87 4.65 5.43 5.09 6.64 6.28 4.74

PS NMAE(%) 4.97 3.51 3.69 3.85 5.27 4.68 3.59

NRMSE(%) 7.23 4.87 5.08 5.27 7.04 6.46 4.98

important in probabilistic forecast applications in power system operations. Also, overall the MMFF-Laplace with pinball

loss optimization has much better reliability than that of

the MMFF-Laplace without pinball loss optimization, which

indicates the effectiveness of the pinball loss optimization.

2) Sharpness: Sharpness indicates the capacity of a forecasting system to forecast extreme probabilities [15]. This

criterion evaluates the predictions independently of the observations, which gives an indication of the level of usefulness

of the predictions. For example, a system that provides only

uniformly distributed predictions is less useful for decisionmaking under uncertainty. Predictions with perfect sharpness

are discrete predictions with a probability of one (i.e., deterministic predictions). The sharpness is measured by the average size of the predictive intervals. The sharpness of the proposed pinball loss-based MMFF-Laplace forecasts; quantile

regression; pinball loss-based MMFF with other distribution

types, and MMFF-Laplace without pinball loss optimization

at the CapeMay, Megler, and Bovina50 site are compared

in Fig. 5. The sharpness of pinball loss based forecasts are

better than that of the baseline QR method. Also, the expected

intervals size increases with increasing nominal coverage rate.

Also, the MMFF-Laplace with pinball loss optimization has

much better sharpness than that of the MMFF-Laplace without

pinball loss optimization. The interval size of the pinball

loss-based MMFF-Laplace forecasts ranges from 2% to 18%,

which indicates low sharpness.

IV. CONCLUSION

In this paper, an optimal pinball loss based probabilistic

wind forecasting method was developed, in conjunction with

a multi-model deterministic forecasting framework. Different

shapes of predictive distributions are tested and compared,

including Gaussian, Gamma, Laplace, and noncentral t distributions. The optimal shape parameter of the predictive

distribution is determined by minimizing the sum of pinball

loss using training data. This optimal shape parameter is used

in the forecasting stage through surrogate modeling. We found

that the laplace distribution presents the best pinball loss.

Results showed that the proposed probabilistic forecasting

method could reduce the pinball loss by up to 35% compared

to the baseline methods. The relationship between the accuracy

of deterministic and probabilistic forecasts will be explored in

future work.

V. ACKNOWLEDGEMENT

This work was authored in part by Alliance for Sustainable

Energy, LLC, the manager and operator of the National

Renewable Energy Laboratory for the U.S. Department of

Energy (DOE) under Contract No. DE-AC36-08GO28308.

Funding provided by U.S. Department of Energy Office of

Energy Efficiency and Renewable Energy Wind Water Power

Technologies Office. The views expressed in the article do

not necessarily represent the views of the DOE or the U.S.

Government. The U.S. Government retains and the publisher,

by accepting the article for publication, acknowledges that the

U.S. Government retains a nonexclusive, paid-up, irrevocable,

worldwide license to publish or reproduce the published form

of this work, or allow others to do so, for U.S. Government

purposes.

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5
0 20 40 60 80 100

Nominal Proportion [%]

0

10

20

30

40

50

60

70

80

90

100

Estimated Coverage [%]

QR

MMFF-Gamma

MMFF-Laplace

MMFF-Gaussian

MMFF-t

MMFF-withoutopt

PS-Laplace

Ideal

(a) CapeMay

0 20 40 60 80 100

Nominal Proportion [%]

0

10

20

30

40

50

60

70

80

90

100

Estimated Coverage [%]

QR

MMFF-Gamma

MMFF-Laplace

MMFF-Gaussian

MMFF-t

MMFF-withoutopt

PS-Laplace

Ideal

(b) Megler

0 20 40 60 80 100

Nominal Proportion [%]

0

10

20

30

40

50

60

70

80

90

100

Estimated Coverage [%]

QR

MMFF-Gamma

MMFF-Laplace

MMFF-Gaussian

MMFF-t

MMFF-withoutopt

PS-Laplace

Ideal

(c) Bovina50

Fig. 4: Reliability of probabilistic forecasts on selected sites

10 20 30 40 50 60 70 80 90

Nominal Proportion [%]

0

10

20

30

40

50

60

70

80

δ [%]

QR

MMFF-Gamma

MMFF-Laplace

MMFF-Gaussian

MMFF-t

MMFF-withoutopt

PS-Laplace

(a) CapeMay

10 20 30 40 50 60 70 80 90

Nominal Proportion [%]

0

10

20

30

40

50

60

70

80

90

100

δ [%]

QR

MMFF-Gamma

MMFF-Laplace

MMFF-Gaussian

MMFF-t

MMFF-withoutopt

PS-Laplace

(b) Megler

10 20 30 40 50 60 70 80 90

Nominal Proportion [%]

0

20

40

60

80

100

120

δ [%]

QR

MMFF-Gamma

MMFF-Laplace

MMFF-Gaussian

MMFF-t

MMFF-withoutopt

PS-Laplace

(c) Bovina50

Fig. 5: Sharpness of probabilistic forecasts on selected sites

6