Probabilistic Forecasts of Wind Power Generation by Stochastic Differential Equation Models

• Author: Henrik Madsen,Marco Zugno,Jan Kloppenborg Møller
• Date: 01 April 2016
• Published date: 01 April 2016
• DOI: http://doi.org/10.1002/for.2367

Journal of Forecasting,J. Forecast. 35, 189–205 (2016)

Published online 1 December 2015 in Wiley Online Library (wileyonlinelibrary.com)DOI: 10.1002/for.2367

Probabilistic Forecasts of Wind Power Generation by Stochastic

Differential Equation Models

JAN KLOPPENBORG MØLLER,MARCO ZUGNO AND HENRIK MADSEN

DTU Compute, Matematiktorvet, Technical University of Denmark, DK-2800 Lyngby, Denmark

ABSTRACT

The increasing penetration of wind power has resulted in larger shares of volatile sources of supply in power

systems worldwide. In order to operate such systems efficiently, methods for reliable probabilistic forecasts of future

wind power production are essential. It is well known that the conditional density of wind power production is highly

dependent on the level of predicted wind power and prediction horizon. This paper describes a new approach for

wind power forecasting based on logistic-type stochastic differential equations (SDEs). The SDE formulation allows

us to calculate both state-dependent conditional uncertainties as well as correlation structures. Model estimation is

performed by maximizing the likelihood of a multidimensional random vector while accounting for the correlation

structure defined by the SDE formulation. We use non-parametric modelling to explore conditional correlation

structures, and skewness of the predictive distributions as a function of explanatoryvariables. Copyright © 2015 John

Wiley & Sons, Ltd.

KEY WORDS non-linear forecasting; state space models; stochastic differential equations; wind power;

probabilistic forecasting

INTRODUCTION

In 2013 wind power accounted for more than 33% of the total demand for electricity in Denmark, and in December

(2013) about 53% of the electricity load was covered by wind power. Integrating this world-leading amount of wind

power into the power systems calls for state-of-the-art methods for wind power forecasting. Methods for generating

point forecasts of wind power for operational decision-making have been used since 1995 (Madsen et al., 1995).

However, the solution of the decision-making problems in electricity markets typically requires the full predictive

density of future wind power production, and in some cases also its correlation structure (Morales et al., 2014).

A literature review with a focus on methods for point predictions and with some focus also on the meteorological

aspects can be found in Giebel et al. (2011) and Costa et al. (2008). In practice, forecasts are needed for real-time

monitoring, operation scheduling, production and maintenance planning, and energy trading (cf. Morales et al., 2014).

The corresponding lead times range from the very short term (say, 5 minutes) and up to the medium range (say, up to

5–7 days).

Still, even after about 20 years of research, wind power forecasting remains a challenge from a statistical point

of view. This is firstly due to the nonlinear and double-bounded nature of wind power production (Pinson, 2012).

Nonlinearity is a result of the sigmoidal dependence between wind power production and wind speed as well as,

possibly, direction. In addition, nonlinearities may be present when describing both the conditional mean and the

conditional variance (Trombe et al., 2012). Finally, the process should generally be seen as non-stationary, with

successive periods exhibiting different dynamics. In a probabilistic framework, conditional predictive densities may

be generated in a parametric framework, for instance based on (censored) Gaussian, Beta, or mixtures of distributions

(Pinson, 2012; Trombe et al., 2012), or alternatively in a non-parametric framework, e.g. using quantile regression

for modelling and predicting a set of quantiles with varying nominal proportions. Because of the complexity of the

predictive distribution for long look-ahead times, the non-parametric frameworkis often preferred for lead times from

hours to a few days ahead, e.g. the quantile regression approach of Nielsen et al. (2006) and time-adaptive quantile

regression of Møller et al. (2008).

To further describe the interdependence between marginal predictive densities for a set of future lead times in

a multivariate framework,Pinson et al. (2009) proposed to issue trajectories of wind powergeneration where the inter-

dependence structure is modelled with a Gaussian copula. Such modelling then simplifies to the tracking of a time-

varying covariance matrix. Alternatively, trajectories of wind power generation can be obtained as in Nielsen et al.

(2006) by nonlinear transformation of ensemble forecasts of relevant meteorological variables (see Leutbecher and

Palmer, 2008, for an overview of ensemble forecasting), hence accounting for the dynamic interdependence structure

Correspondence to: Jan Kloppenborg Møller, DTU Compute, Matematiktorvet, Technical University of Denmark Building 303B, DK-2800

Lyngby,Denmark. E-mail: jkmo@dtu.dk

Copyright © 2015 John Wiley & Sons, Ltd

190 J. K. Møller, M. Zugno and H. Madsen

in the inherent uncertainty of meteorological forecasts used as input to wind power forecasting. These approaches

are based on either static models or approaches with a simple tracking of the parameters, and hence they give a fairly

static view of the interdependence structure.

In this paper, which is a further elaboration of the work presented in Møller et al. (2013) (proofs and comparison

with benchmark models are included in the present work), a new method based on stochastic differential equations

(SDEs) is suggested. The advantage of this method is that by selecting a proper specification and parametrization

of the SDE, basically all the required properties, like nonlinearities, time variation, non-stationarity, double-bounded

variations and varying quantiles, can be described. Furthermore, we demonstrate that this approach also offers a

flexible description of the variation of interdependence structure of the predictions errors. Compared to the competing

approaches described above, our method requires a significantly smaller number of parameters.

In our view, SDEs are a more suitable framework to describe the physics of wind power production than classical

time series approaches. Indeed, the mechanics of dynamical systems are usually formalized as systems of differential

equations. In the specific setting of wind power forecast, this implies that wind speed and wind direction can be

included based on a mechanistic understanding of the problem. The results presented in this paper indicate that the

combination of SDE models and multi-horizon forecasting has great potential.

Following this introductory section, the general framework is described in the next section, while the dataset used

for the empirical investigation is introduced in the third section. Multivariatepredictive densities issued in a Gaussian

framework (possibly after some transformation), and with various modelling of the covariancestructure, are presented

in the fourth section. These Gaussian predictive densities are used as benchmark models. The fifth section gives a

very short description of the general SDE framework used in the present work. The new method, which is the key

contribution in the present work (in particular, Theorems 1 and 2), based on SDEs is presented in the sixth section,

together with a number of candidates for the parametrization of the SDEs. Finally, numerical results are presented

and compared in the seventh section, and concluding remarks are given in the eighth section.

GENERAL FRAMEWORK

We consider the problem of wind powerforecasting with a look-ahead time of up to 2 days. As further described in the

next section, a 48-hour prediction of wind power production is issued every 6th hour. Within this setup the observed

wind power production (average of 1 hour) is considered as a realization of a 48-dimensional random variable, and

hence the density of the observation is calculated in a multivariate setting.

After normalizing with the installed capacity, the ‘true’ underlying distribution will be contained in the hypercube

Œ0; 1n,wherenis the dimension of the variable. Here we consider forecast horizons hD¹1; : : : ; 48º,i.e.nD48.

Rather than testing different parametric distributions, the scope of this paper is to demonstrate the improvements

that can be achieved by considering SDEs. This implies that we will not apply transformation to the original data, even

though this might be beneficial. Hence we assume that the general distribution can be approximated by a multivariate

normal distribution:

YiN. O

Yi;†i/(1)

where the index irefers to a realization of the 48-dimensional variable under some conditions (determined by the

specific conditions, e.g. wind speed, and hence the index i), O

Yiand †iare functions of meteorological forecasts

and look-ahead time. In this setting we use point predictions provided by the widely used wind power prediction

tool (WPPT) (see Madsen et al., 1998; WPPT, 2012), which uses meteorological forecasts and local data as input

for predicting wind power production, but provide only a point forecast in terms conditional expectation. The focus

of this work is on describing the structure of the residuals. The 48-dimensional WPPT forecast issued for series iis

denoted Q

piDŒQpi;1;:::; Qpi;48, and a natural first approach would be to select

O

YiDQ

piI†iDF.Q

pi;h/(2)

where hDŒ1; : : : ; 48Tdenotes the forecast horizons and Fis a matrix function (with the natural restrictions

of positive definiteness). We use these types of models as benchmark models, since the structure is similar to the

SDE models presented in the fifth section. In the fourth section the covariance structure is obtained by a direct

parametrization of F, while Fand O

Yare obtained as the solution to SDEs in the fifth and sixth sections.

DATA

The data for this study consist of hourly averages of wind power production from the Klim wind power plant located

in the northern part of Denmark. The dataset covers the period March 2001 through April 2003, and the total number

Copyright © 2015 John Wiley & Sons, Ltd J. Forecast. 35, 189–205 (2016)