arX

iv:1

312.

6471

v1 [

stat

.ME

] 2

3 D

ec 2

013

Statistical Science

2013, Vol. 28, No. 4, 564–585DOI: 10.1214/13-STS445c© Institute of Mathematical Statistics, 2013

Wind Energy: Forecasting Challenges forIts Operational ManagementPierre Pinson

Abstract. Renewable energy sources, especially wind energy, are toplay a larger role in providing electricity to industrial and domesticconsumers. This is already the case today for a number of Europeancountries, closely followed by the US and high growth countries, forexample, Brazil, India and China. There exist a number of techno-logical, environmental and political challenges linked to supplement-ing existing electricity generation capacities with wind energy. Here,mathematicians and statisticians could make a substantial contribu-tion at the interface of meteorology and decision-making, in connec-tion with the generation of forecasts tailored to the various operationaldecision problems involved. Indeed, while wind energy may be seen asan environmentally friendly source of energy, full benefits from its us-age can only be obtained if one is able to accommodate its variabilityand limited predictability. Based on a short presentation of its phys-ical basics, the importance of considering wind power generation asa stochastic process is motivated. After describing representative op-erational decision-making problems for both market participants andsystem operators, it is underlined that forecasts should be issued in aprobabilistic framework. Even though, eventually, the forecaster mayonly communicate single-valued predictions. The existing approachesto wind power forecasting are subsequently described, with focus onsingle-valued predictions, predictive marginal densities and space–timetrajectories. Upcoming challenges related to generating improved andnew types of forecasts, as well as their verification and value to forecastusers, are finally discussed.

Key words and phrases: Decision-making, electricity markets, forecastverification, Gaussian copula, linear and nonlinear regression, quantileregression, power systems operations, parametric and nonparametricpredictive densities, renewable energy, space–time trajectories, stochas-tic optimization.

Pierre Pinson is Professor in the Modeling of

Electricity Markets, Centre for Electric Power and

Energy, Department of Electrical Engineering, Technical

University of Denmark, Elektrovej 325(058), 2800 Kgs.

Lyngby, Denmark e-mail: ppin@dtu.dk.

This is an electronic reprint of the original articlepublished by the Institute of Mathematical Statistics inStatistical Science, 2013, Vol. 28, No. 4, 564–585. Thisreprint differs from the original in pagination andtypographic detail.

1. INTRODUCTION

Increased concerns related to climate evolutionand energetic independence have supported the nec-essary technological and regulatory developments tobroaden the energy mix all around the world, witha particular emphasis placed on renewable energysources (Letcher, 2008). Among the various candi-dates, wind energy showed the most rapid and con-sistent deployment of power generating capacities.By June 2012, the cumulative installed wind powercapacity worldwide had reached 254 GW, and it

1

2 P. PINSON

is still increasing at a rapid pace [WWEA (WorldWind Energy Association) (2012)]. Besides all themathematical and statistical challenges in the de-velopment of turbines (aerodynamics, materials andstructures, etc.), and in the deployment of wind en-ergy capacities (e.g., wind resource estimation, logis-tics optimization), those relating to power systemsoperations and electricity markets have attractedsubstantial and growing interest over the last 3 dec-ades. This is since, in contrast with conventionalgeneration means, wind power generation cannotbe scheduled at will, except maybe by curtailingthe output of the wind turbines. Wind power isproduced as the wind blows: the dynamics of windpower generation are the result of a nonlinear con-version and filtering of wind dynamics through theturbines’ rotor and electric generator. It makes thatthe traditional operational methods used for con-ventional generators cannot directly apply to windenergy. For that reason, of the various renewableenergy sources, wind, solar, wave and tidal energyare often referred to as stochastic power generation,owing to their inherent variability and uncertainty.Wind energy is by far the renewable energy source

that has attracted the most attention of researchersand practitioners. It is clear, however, that a numberof operational and economic issues will be the samefor the other forms of renewable energy sources. Inpractice, such challenges require the modeling andforecasting of the wind power generation process atvarious temporal and spatial scales, to be subse-quently used as input to decision-making. Our ob-jective here is to give an overview of these forecastsand of challenges stemming from their generationand verification. It is to be noted that forecasting isonly one aspect of better accommodating renewableenergy generation, such as that from the wind intoexisting power systems and electricity markets. Forinstance, from a more general perspective of invest-ment, regulation and policy, even the way wind en-ergy should be compared to conventional technolo-gies challenges traditional practice (Joskow, 2011).Similarly, when assessing resource adequacy (i.e.,making sure that the overall generating capacity issufficient to meet demand at all times) and compe-tition in electricity markets, it is argued that theimpact of renewable energy sources on market dy-namics ought to be accounted for (Wolak, 2013).The most classical statistical problem involving

wind energy is that of resource assessment, thatis, focusing on unconditional distributions of wind

speed and the corresponding potential power gener-ation. In practice, this is based on estimating margin-al wind distributions given a (potentially limited)sample of wind measurements on site and/or in thevicinity of the sites of interest. Even though thesemarginal distributions are highly valuable for theoptimal siting and design of wind farms, they havenearly no value for the operational management ofwind power generation: they give an unconditionalpicture only, hence, they do not give information onthe volatile and conditional characteristics of windand power dynamics at the relevant spatial and tem-poral scales. A succession of two papers published inthe Journal of Applied Meteorology in 1984 is a sym-bol of the transition from models for limiting distri-butions only to dynamic models. There, the seminalwork of Conradsen, Nielsen and Prahm (1984) on fit-ting Weibull distributions to samples of wind speedmeasurements of various lengths is literally followedby that of Brown, Katz and Murphy (1984), whichcertainly was the first paper looking at dynamic (lin-ear time-series) models for the prediction of windspeed and corresponding power generation. Not solong after, Haslett and Raftery (1989) bridged thegap between the two by focusing on the dynamicspatio-temporal structure of wind speed over Ire-land and its implications for the wind energy re-source. Since then, ample research was performedon stochastic dynamic models for the prediction ofwind power generation at lead times between a fewminutes and up to several days ahead, accounting ornot for spatial effects. For an exhaustive review ofthe state of the art in that field, the reader is referredto Giebel et al. (2011), while a solid introductionto the physical concepts involved can be found inLange and Focken (2006). Our state of knowledge to-day is that optimal decision-making involving windpower generation calls for predictions generated ina probabilistic framework. These should inform ofuncertainties through predictive marginal densities,but also potentially of spatio-temporal dependenciesthrough trajectories, which are known as scenariosin the operations research literature. As a very re-cent example of how forecasts in their most sim-ple deterministic form, or as space–time trajectories,may be used as input to operational problems, thereader is referred to Papavasiliou and Oren (2013),focusing on a unit commitment problem (i.e., theleast-cost dispatch of available generation units) un-der transmission network constraints.

WIND ENERGY: FORECASTING CHALLENGES 3

Wind power generation is first introduced in Sec-tion 2 as a stochastic process observed at discretepoints in space and in time. Subsequently, in orderto underline the importance of probabilistic fore-casts (in contrast to deterministic, single-valued fore-casts), Section 3 describes representative decisionproblems involving wind energy in power systemsoperations and its participation in liberalized elec-tricity markets. Section 4 then covers the varioustypes of forecasts used today and to be employed inthe future for optimal decision-making. The paperends in Section 5 with a discussion that covers (i) thecurrent and foreseen challenges for forecast improve-ment, (ii) the proposal of thorough and appropriateverification frameworks, and (iii) the importance ofbridging the gap between forecast quality and value.

2. WIND POWER GENERATION AS A

STOCHASTIC PROCESS

Some of the early works on dynamic modeling andforecasting of wind power generation were cast in aphysical deterministic framework, as, for instance,Landberg and Watson (1994) on local wind condi-tions, and similarly for the follow-up study (Land-berg, 1999) on power generation. Today however,there is a broad consensus that wind power gen-eration should be modeled as a stochastic process,whatever the spatial and temporal scales involved.A part of uncertainty comes from our lack of knowl-edge of all the physical processes involved, combinedto our limited ability to account for all of themin mathematical and statistical models. There mayalso be some inherent uncertainty in the data gen-erating process. The choice for appropriate distribu-tions may not be straightforward.The physical basics of wind power generation are

presented in Section 2.1. Definitions and notationare introduced subsequently in Section 2.2. Finally,the Western Denmark data set is described in Sec-tion 2.3. It will be used for illustrating the differentforms of forecasts that will be described throughoutthe paper.

2.1 Physical Basics of Wind Power Generation

The generation of electric power from the windrelies on atmospheric processes. The power outputof a single wind turbine is a direct function of thestrength of the wind over the rotor swept area.Coarsely simplifying the meteorological aspects in-volved, winds originate from the movement of air

masses from high to low pressure areas: the largerthe difference in pressure, the stronger the result-ing winds. On top of that come the boundary layereffects, complexifying wind behavior due to natu-ral obstacles, friction effects, the nature of the sur-face itself, temperature gradients, etc. The bound-ary layer is formally defined as the lower part ofthe atmosphere where wind speed is affected by thesurface. The resulting level of complexity makes thatthe characteristic features of wind variability may bebetter described in the frequency domain (Mur Amadaand Bayod Rujula, 2010). Our state of the knowl-edge on wind dynamics in the boundary layer, and,more generally, mesoscale meteorology, is today stilllimited: resulting models of wind characteristics havesystematic and random errors.Wind speed exhibits fluctuations over a wide range

of frequencies. Those in the order of days are in-duced by the movement of synoptic weather pat-terns, that is, by general changes in weather sit-uations. These are modeled within global weathermodels such as those run at the European Centrefor Medium-range Weather Forecasts (ECMWF, inthe UK) and at the National Centers for Environ-mental Prediction (NCEP, in the US), among oth-ers. Those models encompass well-known dynam-ics of state variables for the global weather, whilewind components are a by-product derived from theevolution of these state variables. In terms of fore-casting, several directions are thought of today forimproving the estimation of the initial state of theAtmosphere and also to better account for potentialuncertainties in the model and its parametrization(Palmer, 2012).Fluctuations referred to as diurnal and semi-diurnal

cycles (with periods of 24 and 12 hours) are mainlyinduced by thermal exchanges between the surface(land or sea) and the Atmosphere. Their magnitudevaries as a function of local climate and seasons.At these time scales, the phenomena involved arefairly well known, though certain aspects like theirimpact on wind profiles (i.e., the way wind evolveswith height) still are a subject of active research, forexample, Pena Diaz, Gryning and Mann (2010). Atfrequencies in the order of minutes to hours, localeffects potentially including the presence of cumu-lus clouds, convective cells, precipitation, waves (foroffshore sites), etc. are the drivers of wind speedvariations. Here, the physical and mathematical as-pects may become more challenging owing to thecombination of a substantial number of interacting

4 P. PINSON

Fig. 1. Power curve of the Vestas V44 (600 kW) wind tur-bines installed at the Klim wind farm, for an air density of1.225 kg/m3.

physical processes. Higher frequencies (seconds to afew minutes, not considered in the present paper)see a dominance of turbulence effects, which are aparticular concern for the structural design of tur-bines, fatigues studies and, potentially, control. Fi-nally, at the other end of the spectrum, very lowfrequencies also seen as long-term wind trends, haveattracted increased attention recently since humanactivity and climate evolution may potentially im-pact surface winds at these time scales; see Vautardet al. (2010), for instance. In the following, empha-sis is placed on time scales in the order of minutesto days, where existing meteorological challenges in-clude the better understanding of the physical pro-cesses and their interaction, as well as their model-ing.Wind speed is the meteorological variable of most

relevance to power generation. The process of theconversion of wind to electric power for a single windturbine is described by its power curve. It is alsoinfluenced by air density (being a function of tem-perature, pressure and humidity) to a minor extent.Power curves for different turbines roughly have thesame shape for all manufacturers and turbine types.In order to discuss and illustrate what manufac-turer’s (i.e., theoretical) and observed power curvesmay look like, let us take the example of the Klimwind farm in Western Denmark. It is composed of35 Vestas V44 wind turbines having a capacity of600 kW each, yielding a nominal capacity of 21 MW.The nominal capacity of a wind turbine or of a windfarm is the power output it generates within therange of wind conditions over which it was designedto operate, ideally. Figure 1 depicts the power curvefor a V44 turbine. The power production is null be-

low the cut-in wind speed (4 m/s), then sharply aug-ments between the cut-in and rated wind speeds(16 m/s). At rated speed, it reaches its nominalpower Pn. The power production is nearly constantbetween rated and cut-off wind speeds (here 25 m/s).At cut-off speed, the turbine stops for security rea-sons. This power curve example is for a fairly oldwind turbine model, since this wind farm started op-erating in 1996. Various technological improvementshave been permitted to lower cut-in and rated windspeeds, which are today between 2 and 4 m/s for theformer one and between 12 and 15 m/s for the latterone. Moreover, cut-off wind speeds may reach up to34 m/s. In a general manner there may also be adifference between the maximum (peak) and nomi-nal power values (up to 10–20%). Most importantly,the nominal capacity of today’s wind turbines is upto 7–8 MW.A power curve such as in Figure 1 is a theoretical

one, since it gives the power output of a single tur-bine exposed to ideal wind conditions as if in a windtunnel (i.e., not altered by obstacles, without turbu-lence and for the turbine always perfectly facing thewind), for a given air density. In practice, however,wind turbines are almost always gathered in windfarms with potentially a mix of different turbinetypes. The combination of these individual powercurves will not be the same as that of any of theindividual turbine types. Besides, depending uponthe prevailing wind direction, some of the turbineswithin a wind farm may mask the others—the so-called shadowing effect, therefore reducing the windseen by these turbines. This combines with addi-tional surrounding topographic and orographic ef-fects (i.e., hills, forest, etc.), making that the variousturbines within a wind farm are constantly seeingdifferent wind conditions, which also are differentfrom the free-stream wind at a reasonable distanceaway from the wind farm. Consequently, the result-ing wind farm power curve has features far morecomplex than the theoretical power curves providedby the manufacturers for individual wind turbines.Figure 2 depicts the empirical power curve of the

Klim wind farm based on hourly wind speed (at10 m above ground level) and power measurementscollected over the first 6 months of 2002. For bothtypes of measurements, a record for a given pointin time corresponds to the average value over thepreceding hour. Measurement errors in power andwind speed observations certainly contribute to thescatter of data observed. However, the main reason

WIND ENERGY: FORECASTING CHALLENGES 5

Fig. 2. Example empirical power curve for the Klim windfarm over a 6-month period in 2002, based on hourly mea-surements of wind speed and corresponding power output.Marginal distributions of wind speed and power are also repre-sented above, and, respectively, right of, the power curve itself.

for that scatter is the impact of other meteorolog-ical variables, as, for example, wind direction andair density, on the power generation from the windfarm. For measured wind speeds of 5 m/s the ob-served power output of the wind farm varies between0 and 7 MW, while for wind speeds of 10 m/s, thatsame power output may be between 6 and 15 MW.Other reasons for these variations include naturalchanges in the environment of the wind farm, agingof turbine components, etc. At the turbine, windfarm or portfolio (i.e., a group of geographically dis-tributed wind farms, though jointly operated) level,all empirical power curves exhibit characteristics dif-fering from those of theoretical ones, also with asubstantial scatter of observations. Other interest-ing empirical power curves for wind farms in Crete,as well as challenges related to their modeling, wererecently discussed by Jeon and Taylor (2012).

2.2 Preliminaries and Definitions

Owing to the combination of complex physicalprocesses, and since we may not have a perfect un-derstanding of all these processes anyway, it is ac-knowledged that one should account for a randomuncertainty component in the modeling of energygeneration from wind turbines. Wind power is there-fore considered as a discrete stochastic process, thatis, as a set of random variables Ys,t observed at dis-crete points in time t and in space s. Depending

upon the practical setup, it may reduce to a tempo-ral process with a set of random variables Yt for suc-cessive times, for instance, if concentrating on a sin-gle wind farm or on a fixed (geographically spread)portfolio, or to a spatial process with a set of ran-dom variables Ys for a given time but for variouslocations, for instance, if looking at maps of windenergy resource over a region. The correspondingrealizations of the process are denoted by ys,t in themore general spatio-temporal case, or, more simply,by yt and ys in the temporal and spatial cases, re-spectively. The notation f and F are used for proba-bility density and cumulative distribution functions(abbreviated p.d.f. and c.d.f.) of the random vari-ables involved, with appropriate indices.Wind power generation as a stochastic process ex-

hibits features that can be seen as fairly unique, eventhough relevant parallels with stochastic processesfor other renewable energy sources, in meteorologyand hydrology or in economics and finance, exist.Some of these characteristic features come from thevery nature of wind, while some others are directlylinked to the process of converting the energy inthe wind to electric power. First of all, wind com-ponents and resulting wind speed have a combina-tion of dynamic and seasonal features, which mayvary depending on local wind climates and regionsof the world. Besides, when focusing on spatial andtemporal scales of relevance to power systems op-erations and electricity markets, the various mete-orological phenomena involved induce switches inthe dynamic behavior of wind fluctuations and intheir predictability, yielding a nonstationary process[see the discussion by Vincent et al. (2010), for in-stance]. Inspired by models developed in the econo-metrics literature, the existence of successive periodswith different levels of predictability of wind speedswas first captured with a Generalised Auto Regres-sive Conditional Heteroscedastic (GARCH) modelby Tol (1997), though focusing on coarser daily windrecords.In parallel, the conversion of the energy in the

wind to electric power acts as a nonlinear transferfunction (as represented in Figure 2) making windpower generation a nonlinear and bounded stochas-tic process. There may even be smooth temporalchanges in this nonlinear transfer function owing to,for example, aging of equipment, changes in exter-nal environment, etc. The transfer function shapesthe predictability of wind power generation. Conse-quently, conditional densities of wind power genera-

6 P. PINSON

tion should be seen as non-Gaussian, with their mo-ments of order greater than 1 directly influenced bytheir mean (Lange, 2005; Bludszuweit, Domınguez-Navarro and Llombart, 2008). Truncated Gaussian,Censored Gaussian and Generalized Logit–Normaldistributions were proposed as relevant candidatesfor the modeling of conditional densities of windpower generation (Pinson, 2012). In terms of stochas-tic differential equations, this would translate to hav-ing a state-dependent diffusion component. The flatparts of the transfer function also yield concentra-tion of probability mass at the boundaries, poten-tially requiring to consider wind power generationas a discrete-continuous mixture, similar to precipi-tation, for instance.After proposing a suitable model structure, and

estimating its parameters, such a model may be em-ployed to simulate time-series of wind power gener-ation for one or several locations, for instance, as in-put to power systems and market-related analysis.In most cases, however, forecasting is the final ap-plication. Predictions fed into operational decisionproblems always are for future points in time andrarely for new locations at which no observationsare available. Consequently, even though spatial as-pects are of crucial interest, the problem at hand ismainly seen as a temporal forecasting problem. Theset of m locations is denoted by

s= s1, s2, . . . , sm.(2.1)

In parallel, the set of n lead times is

t+ k= t+ 1, t+ 2, . . . , t+ n,(2.2)

where n is the forecast length. Lead times are spacedregularly and with a temporal resolution equal tothe sampling time of the process observations. Sincethe power generation process is bounded, it can bemarginally normalized, so that

ys,t+k ∈ [0,1]mn.(2.3)

At time t the aim is to predict some of the char-acteristics of

Ys,t+k

(2.4)= Ys,t+k; s= s1, . . . , sm, k = 1, . . . , n,

a multivariate random variable of dimension m× n

in the complete spatio-temporal case, or of

Yt+k = Yt+k;k = 1, . . . , n,(2.5)

a multivariate random variable of dimension n, inthe simpler setup where spatial considerations aredisregarded.In the most general case, the forecaster issues at

time t for the set of lead times t+ k a probabilis-tic forecast Fs,t+k|t (here a predictive c.d.f.) describ-ing as faithfully as possible the c.d.f. Fs,t+k of therandom variable Ys,t+k, given the information avail-able up to time t. It hence translates to a full de-scription of marginal densities for every location andlead time, as well as spatio-temporal dependenciesamong the set of m locations and n lead times. Thisclearly comprises a difficult problem, both in termsof generating such forecasts and also for their veri-fication. Consequently, since degenerate versions ofthat problem may be more tractable, a number ofthem have been dealt with in the literature, for in-stance, for the forecasting of marginal densities foreach location and lead time individually, or even byforecasting some summary statistics (more precisely,mean and quantiles) of these marginal densities only.The combination of all uncertainties, related to

physical aspects to be accounted for in the models,but also in connection with the data-generating pro-cess, obviously is going to impact the quality of theresulting forecasts. In Section 4 some of the mostcommon approaches to forecasting will be reviewed.They all tend to disregard the specific contributionsof physical and data-generating processes to forecastquality. Alternative proposals in a robust forecastingframework could therefore be beneficial.

2.3 The Western Denmark Data Set

A data set for the Western Denmark area is usedas a basis for illustration. It consists of wind powermeasurements as collected by Energinet.dk, the trans-mission system operator in Denmark. This regionhas one of the highest wind power penetrations (i.e.,the share of wind power in meeting the electric en-ergy demand) in the world, consistently between 25and 30% over the last few years.Wind power measurements are originally available

at more than 400 geographically distributed grid-connection points. Observations have an hourly res-olution over a period between 1 January 2006 and 24October 2007. They represent average hourly powervalues. For operational purposes, these are gatheredin 15 so-called control zones depicted in Figure 3along with their identification numbers. The totalnominal capacity slightly evolved during this periodthough generally being around 2.5 GW. In order to

WIND ENERGY: FORECASTING CHALLENGES 7

Agg. zone Orig. zones % of capacity

1 1, 2, 3 312 5, 6, 7 183 4, 8, 9 174 10, 11, 14, 15 235 12, 13 10

Fig. 3. The Western Denmark data set: original locations for which measurements are available, 15 control zones definedby Energinet.dk, as well as the 5 aggregated zones. The total nominal capacity for Western Denmark was 2.5 GW over theperiod covered by this data set.

additionally simplify this case-study, the original 15control zones are aggregated into 5 zones only (seeFigure 3), each having a different share of the overallwind power capacity for that region. All power mea-surements are normalized by the respective nominalcapacities of the 5 aggregated zones. This aggrega-tion is for the sake of example only and could beseen as wind power generation portfolios operatedby a set of power producers in that region. Work-ing at such a coarse spatial resolution certainly issufficient for some decision problems, also simplify-ing modeling and estimation challenges. However,it may be that for some applications the statisti-cian and forecaster has to work with the original400-location data set, so that he has to finely an-alyze and model the observed spatio-temporal dy-namics; see Girard and Allard (2013), for instance.This would be the case if all the owners/operatorsof these individual wind farms ask for predictionsin order to design market offering strategies or forthe network operator to perform very detailed sys-tem simulations based on the impact of spatiallydistributed wind power generation.Some of the features of this data at such temporal

and spatial scales can be observed from the exampleepisode with 24 hours of hourly wind power mea-surements in Figure 4, for the 5 aggregated zones ofWestern Denmark. The spatio-temporal interdepen-dence structure of the wind power generation pro-cess, as induced by the inertia in weather phenom-ena and resulting winds, especially results in smoothtemporal variations at each zone, individually, as

well as in similarities in the patterns observed atthe various zones. These spatio-temporal dependen-cies are necessarily strengthened by the aggregationprocedure employed. For instance, the drop in powergeneration observed in zone 4 on 19 March 2007 at8:00 UTC (i.e., the 20th time step) is also visiblefor zone 5, at the same time and with a similarmagnitude, while it may potentially be related toa drop of lesser magnitude observed in zones 2 and3 around the same time. Note that UTC (for Coor-dinated Universal Time) is the most common stan-dard for referring to time in the meteorological andwind energy communities.

3. SOME REPRESENTATIVE OPERATIONAL

DECISION-MAKING PROBLEMS INVOLVING

WIND ENERGY

Some of the representative operational decisionproblems are described here, while a more exten-sive overview of such problems may be found inAckermann (2012). The side of power producers istaken first, by considering the issue of designing op-timal offering strategies in electricity markets. Sub-sequently taking the side of the system operator in-stead (like Energinet.dk, the transmission systemoperator for Western Denmark), an issue of risingimportance is that of quantifying the necessary back-up generation to accommodate variability and lim-ited predictability of wind power generation. Thesetwo decision-making problems are somehow interre-lated, since the quantification of necessary backup

8 P. PINSON

Fig. 4. Example episode with normalized wind power measurements for the 5 zones of the Western Denmark data set over24 hours, starting from the 18 March 2007 at 12 UTC.

capacities is performed in a dynamic way, condi-tional on the clearing of the electricity market. Forboth types of problems, forecasts for other quantitiesthan wind power generation may be necessary, likeload and prices. There exists substantial literatureon the statistical modeling and forecasting of thesemarket variables. The interested reader is referredto Weron (2006) for an overview.

3.1 Participation of Wind Energy in Electricity

Markets

In a number of countries with significant windpower generation, electricity markets are organizedas electricity pools, gathering production and con-sumption offers in order to dynamically find thequantities and prices for electricity generation andconsumption that permit to maximize social wel-fare. These electricity pools typically have two ma-jor stages which are the day-ahead and the balancingmarkets. The electricity pool for Scandinavia, usedas an example here, is commonly known as the NordPool. For an overview of some the European elec-tricity markets and of the way they deal with windpower generation, see Morthorst (2003). A paralleloverview for the case of US electricity markets canbe found in Botterud et al. (2010).Electricity exchanges first take place in the day-

ahead market for the next delivery period, that is,the next day. Production offers and consumption

bids are to be placed for every time unit beforegate closure, occurring 12 hours before delivery inthe Nord Pool, where market time units are hourly.At the time t of gate closure, wind power produc-ers shall propose energy offers based on forecastswith lead times t+ k, k ∈ 13,14, . . . ,37. The mar-ket clearing is there to match production offers andconsumption bids through a single auction process,yielding the system price πc

t+k and the program ofthe market participants, that is, a set of energy blocksyct+k to be delivered by wind power producers,1 forevery market time unit. The superscript c indicatesthat this combination of energy quantity and pricedefines a contract. Power producers are financiallyresponsible for any deviation from this contract. In-deed, in a second stage, the balancing market man-aged by the system operator ensures the real-timebalance between generation and load, while trans-lating to financial penalties for those who deviatefrom their contracted schedule. The prices for buy-ing and selling on the balancing market are denotedby πb

t+k and πst+k, respectively. They are generally

less advantageous than those in the day-ahead mar-ket, fairly volatile and substantially different from

1Note that the notation yct+k is used abusively for simplifi-

cation. This is since the energy block for hour t+ k is neces-sarily equal to the average power production value yc

t+k overthat one-hour period.

WIND ENERGY: FORECASTING CHALLENGES 9

one another in a two-price settlement system likethat of the Nord Pool. The combination of the in-herent uncertainty of wind power predictions and ofthe asymmetry of balancing prices encourages mar-ket participants to be more strategic when designingoffering strategies (Skytte, 1999).Simplifying the decision problem for clarity, po-

tential dependencies among time units and in spacethroughout the network are disregarded. A windpower producer is seen as participating with a globalportfolio of wind power generation in the electricitymarket. The overall market revenue Rt+k is a ran-dom variable, which, given the decision variable yct+k

and the random variable Yt+k, can be defined as

Rt+k = st+k(yct+k) +Bt+k(Yt+k, y

ct+k),(3.1)

where the first part corresponds to the revenue fromthe day-ahead market, st+k(y

ct+k) = πc

t+kyct+k, while

the second is that from the balancing market, tobe detailed below. Following Pinson, Chevallier andKariniotakis (2007) (among others), this revenue canbe reformulated as a combination of revenues andcosts in a way that the decision variable appears inthe balancing market term only

Rt+k = St+k(Yt+k)− Bt+k(Yt+k, yct+k),(3.2)

that is, as the sum of a stochastic, though fatalsince out of the control of the decision-maker, com-ponent St+k from selling of the energy actually pro-duced through the day-ahead market, minus anotherstochastic component Bt+k, whose characteristicsmay be altered through the choice of a contract yct+k.The imbalance is also a random variable, given byYt+k−yct+k, yielding the following definition for Bt+k:

Bt+k(Yt+k, yct+k)

(3.3)

=

π↓t+k(Yt+k − yct+k), Yt+k − yct+k ≥ 0,

−π↑t+k(Yt+k − yct+k), Yt+k − yct+k < 0,

where π↓t+k and π

↑t+k are referred to as the regula-

tion unit costs for downward and upward balancing,respectively. They are readily given by

π↓t+k = πc

t+k − πst+k,(3.4)

π↑t+k = πb

t+k − πct+k.(3.5)

For most electricity markets regulation unit costsare always positive, making Bt+k ≥ 0, while the over-all market revenue has an upper bound obtainedwhen placing an offer corresponding to a perfect

point prediction, yct+k = yt+k|t = yt+k. As this is notrealistic, and accounting for the uncertainty in windpower forecasts, optimal offering strategies are to bederived in a stochastic optimization framework. As-suming that the wind power producer is rational,his objective is to maximize the expected value ofhis revenue for every single market time unit, sincethis permits to maximize revenues in the long run.Additionally considering the market participant as aprice-taker (i.e., not influencing the market outcomeby his own decision), and having access to forecasts

of the regulation unit costs (π↓t+k|t and π

↑t+k|t), the

optimal production offer y∗t+k at the day-ahead mar-ket is given by

y∗t+k = argminyct+k

E[Bt+k(Yt+k, yct+k)].(3.6)

This stochastic optimization problem has a closed-form solution, as first described by Bremnes (2004),that is, for any market time unit t+ k, the optimalwind power production offer is given by

y∗t+k = F−1t+k|t

(

π↓t+k|t

π↓t+k|t + π

↑t+k|t

)

,(3.7)

where Ft+k|t is the predictive c.d.f. issued at time t

(the decision instant) for time t+ k. In other words,the optimal offer corresponds to a specific quantileof predictive densities, whose nominal level α is a di-rect function of the predicted regulation unit costsfor this market time unit. That problem is a variantof the well-known linear terminal loss problem, alsocalled the newsvendor problem (Raiffa and Schaifer,1961). It was recently revisited by Gneiting (2010),who showed that for a more general class of costfunctions [i.e., generalizing that in (3.3)] optimalpoint forecasts are quantiles of predictive densitieswith nominal levels readily determined from the util-ity function itself, analytically or numerically. Notethat appropriate forecasts of regulation unit costsare also needed here. It was shown by Zugno, Jonssonand Pinson (2013) and the references therein thatthese may be obtained from variants of exponentialsmoothing (in its basic form or as a conditional para-metric generalization) and then directly embeddedin offering strategies such as those given by (3.7).In their simplest form, market participation prob-

lems involving wind energy rely on a family of piece-wise linear and convex loss functions, for which op-timal offering strategies are obtained in a straight-forward manner, as in the above. These only re-quire quantile forecasts for a given nominal level or

10 P. PINSON

maybe predictive densities of wind power genera-tion for each lead time individually. However, whencomplexifying the decision problem by adding de-pendencies in space (e.g., spatial correlation in windpower generation, network considerations) and intime (e.g., accounting for the temporal structure offorecast errors), it then requires a full descriptionof Ys,t+k (ideally in the form of trajectories), in-stead of marginal densities for the whole portfolioand for each lead time individually. The same goesfor alternative strategies of the decision-makers, forinstance, if one aims to account for risk aversion.The resulting mathematical problems do not rely onstudying specific families of cost functions, but in-stead translate to formulating large scenario-basedoptimization problems, in a classical operations re-search framework. Some of the resulting stochasticoptimization problems may be found in Conejo, Car-rion and Morales (2010). The price-taker assump-tion is also to be relaxed to a more general stochas-tic optimization framework, where wind-market de-pendencies are to be described and accounted for(Zugno et al., 2013).

3.2 Quantification of Necessary Power Systems’

Reserves

On the other side, the electric network operatorhas the responsibility to ensure a constant matchof electricity generation and consumption, outsideof the market framework described before. It in-volves the quantification of so-called reserve capac-ities, prior to actual operations, to be readily avail-able if needed. This may be either for supplement-ing generation lacking in the system, for example,in case of asset outages, general loss of productionand unforeseen increase in electricity demand, or,alternatively, for lowering the overall level of gener-ation in the system when demand is less than pro-duction. For an overview, see Doherty and O’Malley(2005).For simplicity and clarity, the timeline here is the

same as for the market participation problem de-scribed earlier. Potential dependencies among timeunits and in space throughout the network (as in-duced by potential network congestion) are disre-garded. The system operator has to make a decisionat time t (market gate closure) for all time unitst+ k of the following day. Reserves are to be quan-

tified as two numbers q↓t+k and q

↑t+k for the whole

power system, for downward (when consumption is

less than production) and upward (conversely) bal-ancing, respectively. The choice of optimal reservelevels is linked to a random variable Ot+k describ-ing all potential deviations from the chosen dispatch(consisting in the reference values for generation andconsumption at every time t+k). This random vari-able is commonly referred to as the system genera-tion margin.Ot+k can be defined as a sum of random vari-

ables representing all uncertainties involved. Theseinclude (i) potential forecast errors εL for the elec-tric load, (ii) the probability of generation lossthrough asset outages (assets being conventional gen-erators, transmission lines and other equipment),and (iii) potential forecast errors εY for the variousforms of stochastic power generation. For simplicity,we only consider wind power here, corresponding tothe operational situation where, as in most coun-tries, wind power is the prominent form of stochas-tic power generation. In a more general setup thecombination of uncertainties with, for example, so-lar and wave energy, should also be accounted for.These various uncertainties are fully characterizedby probabilistic forecasts available at time t: f εL

t+k|t

for the load, fGt+k|t for generation losses, and f

εYt+k|t

for wind power generation. This means that, be-sides the wind generation forecasts discussed in thispaper, additional predictions of potential genera-tion losses (e.g., the probability of failure of variousequipment) are to be issued, for instance, based onreliability models in the spirit of Billinton and Allan(1984). Forecasts for the electric load can in additionbe obtained from one of the numerous methods re-cently surveyed by Hahn, Meyer-Nieberg and Pickl(2009), though very few of them look at full predic-tive densities.Assuming independence of the various random vari-

ables, the overall uncertainty, represented by a pre-dictive marginal density fO

t+k|t, is obtained through

convolution,

fOt+k|t = f

εLt+k|t ∗ f

Gt+k|t ∗ f

εYt+k|t.(3.8)

This predictive density is split into its positive andnegative parts, yielding fO+

t+k|t and fO−

t+k|t, since de-

cisions about downward and upward reserve capac-ities are to be made separately.After such a description of system-wide uncertain-

ties, the system operator can plug this density intoa cost-loss analysis (Matos and Bessa, 2010), similar

WIND ENERGY: FORECASTING CHALLENGES 11

in essence to the market participation problem pre-sented in the above. Based on cost functions g↓ andg↑ for the downward and upward cases, the optimalamounts of reserve capacities (in an expected utilitymaximization sense) are the solution of stochasticoptimization problems of the form

q↑∗t+k = argmin

q↑t+k

E[g↑(O−t+k, q

↑t+k)](3.9)

and

q↓∗t+k = argmin

q↓t+k

E[g↓(O+t+k, q

↓t+k)],(3.10)

which may be solved analytically or numerically,depending upon the complexity of the cost func-tions. Here the optimal reserve levels relate to spe-cific quantiles of the predictive densities for the sys-tem margin Ot+k. However, it would be difficult tolink the optimal reserve levels to specific quantilesof the input predictive densities of wind power gen-eration.In its more complex form the reserve quantifica-

tion problem requires accounting for dependenciesin space and in time, similar to the trading prob-lems, with many more considerations relating to op-erational constraints, for example, unit characteris-tics (capability to increase or decrease power outputover a predefined period of time—so-called rampingcharacteristics, nonconvexities in costs, etc.), and,potentially, risk aversion. The resulting stochasticoptimization problems take the general form of thosedescribed and analyzed in Ortega-Vazquez andKirschen (2009). They require space–time trajecto-ries for all input variables.

4. MODELING AND FORECASTING WIND

POWER IN A PROBABILISTIC FRAMEWORK

Decision-making problems relating to an optimalmanagement of wind power generation in power sys-tems and electricity markets require different typesof forecasts as input. The lead forecast range consid-ered in the above is between 13 and 37 hours ahead,with an hourly temporal resolution for the forecasts.In practice, various forecast ranges, spatial and tem-poral resolutions, are of relevance depending uponthe decision problem. For instance, the shorter leadtimes, say, between 10 minutes and 2 hours ahead,are also crucial for a number of dispatch and controlproblems. Below are presented the leading forms offorecasts for wind power generation, as well as ex-ample approaches to generate them.

4.1 Point Predictions

The traditional deterministic view of a large num-ber of power system operators translates to prefer-ring single-valued forecasts. These so-called pointpredictions are seen as easier to appraise and handleat the time of making decisions.When describing at time t the random variable

Ys,t+k of a set of locations s and lead times k,point forecasts comprise a summary value for eachand every marginal distribution of Ys,t+k in timeand in space. Typically, if one aims at minimizing aquadratic criterion (i.e., in a Least Squares sense),a point forecast for location s and lead time k cor-responds to the conditional expectation for Ys,t+k

given the information set available up to time t, thechosen model and estimated parameters. With re-spect to a predictive density fs,t+k|t for location s

and lead time k, that point forecast therefore corre-sponds to

ys,t+k|t =

∫ 1

0yfs,t+k|t(y)dy.(4.1)

Integration is between 0 and 1 since one is dealingwith power values normalized by the nominal ca-pacity of the wind farm or group of wind farms ofinterest.To issue point predictions at time t, the forecaster

utilizes an information set Ωt, a set containing mea-surements Ωo

t (including observations of power andof relevant meteorological variables, the notation “o”meaning “observation”) over the area covered, as

well as meteorological forecasts Ωft (with “f” for

“forecast”) for these relevant variables, Ωt ⊆ Ωot ∪

Ωft . Based on this wealth of available information,

different types of models of the general form

Ys,t+k = h(Ωt) + εs,t+k(4.2)

were proposed, where εs,t+k is a noise term with zeromean and finite variance.Indeed, when focusing on a single location (a wind

farm), it may be that point forecasts can be issuedin an inexpensive way based on local measurementsonly, and in a linear time-series framework. The firstproposal in the literature is that of Brown, Katzand Murphy (1984), using Auto-Regressive MovingAverage (ARMA) models for wind speed observa-tions and for lead times between a few hours anda few days. When focusing on wind power directlyfor very short range (say, for lead times less than

12 P. PINSON

2 hours), even simpler Auto-Regressive models oforder p, that is,

Ys,t+k = θ0 +∑

i∈L

θiYs,t−i+1 + σεs,t+k,(4.3)

are difficult to outperform, possibly after data trans-formation (Pinson, 2012). In the above, L ⊂ N

+ isa set of lag values of dimension p, while εs,t+k isa standard Gaussian noise term, scaled by a stan-dard deviation value σ. In addition, k = 1 if the ARmodel is for 1-step ahead prediction only, or to beused in an iterative fashion for k-step ahead predic-tion, while k > 1 if one uses the AR model for directk-step ahead forecasting.These models were generalized by a few authors

by accounting for off-site observations and/or by ac-counting for regime-switching dynamics of the time-series. A regime-switching version of the modelin (4.3) assumes different dynamic behaviors in thevarious regimes, as expressed by

Ys,t+k = θ(rt)0 +

∑

i∈L

θ(rt)i Ys,t−i+1 + σ(rt)εt+k,(4.4)

where rt is a realization at time t of a regime se-quence defined by discrete random variables, withrt ∈ 1,2, . . . ,R, ∀t, andR is the number of regimes.The number of lags and the noise variance may dif-fer from one regime to another. The regime sequencecan be defined based on an observable process, likewind direction at time t or a previous wind powermeasurement, yielding models of the Threshold Auto-Regressive (TAR) family, which are common in econo-metrics (Tong, 2011). As an example for wind speedmodeling and forecasting, Reikard (2008) proposedto consider temperature as driving the regime-switch-ing behavior in wind dynamics. In contrast, the classof Markov-Switching Auto-Regressive (MSAR) mod-els, also popular in econometrics since the work ofHamilton (1989), assumes that the regime sequencerelies on an unobservable process. MSAR modelswere shown to be able to mimic the observable switch-ing in wind power dynamics, especially offshore, thatcannot be explained by available meteorological mea-surements (Pinson and Madsen, 2012).Incorporating off-site information in regime-switch-

ing models of the form of (4.4) was proposed byGneiting et al. (2006) and subsequently in a moregeneral form by Hering and Genton (2010), when fo-cusing on a data set for the Columbia Basin of east-ern Washington and Oregon in the US. The modelin the regime-switching space–time (RST) approach

originally proposed by Gneiting et al. (2006) can beformulated as

Ys,t+k = θ(rt)0 +

∑

i∈L

θ(rt)i Ys,t−i+1

(4.5)+

∑

sj∈S

∑

i∈Lj

ν(rt)j,i Ysj ,t−i+1 + σ(rt)εt+k,

that is, in the form of a TARX model (TAR withexogenous variables), where a set of terms is addedto the regime-switching model of (4.4), for obser-vations at off-site locations sj ∈ SX and for a set oflagged values i ∈Lj at this location. Such models al-low considering advection and diffusion of upstreaminformation, but require extensive expert knowledgefor optimizing the model structure.Conditional parametric AR (CP–AR) models are

another natural generalization of regime-switchingmodels,

Ys,t+k = θ0(xt) +∑

i∈L

θi(xt)Ys,t−i+1

(4.6)+ σ(xt)εs,t+k,

where instead of considering various regimes withtheir own dynamics, the AR coefficients are replacedby smooth functions of a vector (of low dimension,say, less than 3) of an exogenous variable x, for in-stance, wind direction only in Pinson (2012). Thenoise variance can be seen as a function of x, or as aconstant, for simplicity. CP–AR models are relevantwhen switches between dynamic behaviors are notthat clear. Meanwhile, they also require fairly largedata sets for estimation, which are more and moreavailable today. Their use is motivated by empiricalinvestigations at various wind farms, where it wasobserved that specific meteorological variables (e.g.,wind direction, atmospheric stability) can substan-tially impact power generation dynamics and pre-dictability in a smooth manner.Other forms of conditional parametric models were

proposed for further lead times, also requiring addi-tional meteorological forecasts as input. As an ex-ample, a simplified version of the CP–ARX model(CP–AR with exogenous variables) of Nielsen (2002)writes

Ys,t+k

= θc0(xt) cos

(

2πht+k

24

)

+ θs0(xt) sin

(

2πht+k

24

)

(4.7)+ θ1(xt)Ys,t + θ2(xt)g(ut+k|t, vt+k|t, k)

+ σεs,t+k,

WIND ENERGY: FORECASTING CHALLENGES 13

Fig. 5. Example episode with point forecasts for the 5 aggregated zones of Western Denmark, as issued on 16 March 2007at 06 UTC. Corresponding power measurements, obtained a posteriori, are also shown.

where ut+k|t and vt+k|t are forecasts of the windcomponents (defining wind speed and direction) atthe level of the wind farm of interest. The vector xt

includes wind direction and lead time. In addition,g is used for a nonlinear conversion of the informa-tion provided by meteorological forecasts to powergeneration, for instance, modeled with nonparamet-ric nonlinear regression (local polynomial or spline-based). The model in (4.7) finally includes diurnalFourier harmonics for the correction of periodic ef-fects that may not be present in the meteorologi-cal forecasts, with ht+k the hour of the day at leadtime k.Besides (4.7), a number of alternative approaches

were introduced in the past few years for predictingwind power generation up to 2–3 days ahead basedon both measurements and meteorological forecasts.Notably, neural networks and other machine learn-ing approaches became popular after the originalproposal of Kariniotakis, Stavrakakis and Nogaret(1996) and more recently with the representativework of Sideratos and Hatziargyriou (2007). For allof these models, parameters are commonly estimatedwith Least Squares (LS) and Maximum Likelihood

(ML) approaches (and a Gaussian assumption forthe residuals εs,t+k), potentially made adaptive andrecursive so as to allow for smooth changes in themodel parameters (accepting some form of nonsta-tionarity), while reducing computational costs. Itwas recently argued that employing entropy-basedcriteria for parameter estimation may be beneficial,as in Bessa, Miranda and Gama (2009), since theydo not rely on any assumption for the residual dis-tributions. A more extensive review of alternativesstatistical approaches to point prediction of windspeed and power can be found in Zhu and Genton(2012).As an illustration, Figure 5 depicts example point

forecasts for the 5 aggregated zones of Western Den-mark, issued on 16 March 2007 at 06 UTC based onthe methodology described by Nielsen (2002). Thesehave an hourly resolution up to 43 hours ahead, inline with operational decision-making requirements.The well-captured pattern for the first lead timesoriginates from the combination of the trend givenby meteorological forecasts with the autoregressivecomponent based on local observational data. Forthe further lead times, the dynamic wind power gen-

14 P. PINSON

eration pattern is mainly driven by the meteorologi-cal forecasts, though nonlinearly converted to powerand recalibrated to the specific conditions at thesevarious aggregated zones.In contrast with the introductory part of this sec-

tion, where it was mentioned that point forecastscorresponded to conditional expectation estimates,Gneiting (2010, and references therein) discussedthe more general case of quantiles being optimalpoint forecasts in a decision-theoretic framework.Indeed, in view of the operational decision-makingproblems described in Section 3, it is the case that ifone accounts for the utility function of the decision-makers at the time of issuing predictions, such fore-casts would then become specific quantiles,

ys,t+k|t = F−1s,t+k|t(α),(4.8)

whose nominal level α is determined from the util-ity function and the structure of the problem itself.The information set and models to be used for issu-ing quantile forecasts are similar in essence to thosefor point predictions in the form of conditional ex-pectations. The estimation of model parameters isthen performed based on the check function criterionof Koenker and Bassett (1978) or any general scor-ing rules for quantiles (Gneiting and Raftery, 2007),instead of quadratic and likelihood-based criteria.An example approach to point forecasting of windpower generation where point forecasts actually arequantiles of predictive densities is that of Møller,Nielsen and Madsen (2008), based on time-adaptivequantile regression.

4.2 Predictive Marginal Densities

Point forecasts in the form of conditional expecta-tions are somewhat “just the mean of whatever mayhappen.” These are not optimal inputs to a largeclass of decision problems. Since the nominal levelof quantile forecasts to be used instead may varyin time while depending upon the problem itself,or might be even unknown, issuing predictive den-sities certainly is more relevant. Given the randomvariable Ys,t+k whose characteristics are to be pre-dicted, these actually are predictive marginal densi-ties fs,t+k|t for all locations and lead times involved,

individually, with Fs,t+k|t the corresponding predic-tive c.d.f.s.Today such a type of wind power forecast is issued

in both parametric and nonparametric frameworks.In the former case, based on an assumption for the

shape of predictive marginal densities (for instance,motivated by an empirical investigation), one has

fs,t+k|t = f(ys,t+k; θs,t+k|t),(4.9)

where f is the density function for power to be gen-erated at location s and time t+ k, for the chosenprobability distribution, for example, truncated/

censored Gaussian or Beta. In (4.9), θs,t+k|t is thepredicted value for the vector of parameters fullycharacterizing that distribution, for instance, a vec-tor of parameters consisting of location and scale pa-rameters for the truncated/censored Gaussian andBeta distributions. For these classes of distributionscharacterized by such limited sets of parameters only,point forecasts as conditional expectations, comple-mented by a variance estimator, for example, us-ing exponential smoothing or based on an ARCH/GARCH process, permit to directly obtain locationand scale parameters of predictive marginal densi-ties. This reliance on a limited number of parametersmay be seen as desirable since it eases subsequentestimation and related computational cost.Models for the density parameters take a general

form similar to that in (4.2) (and subsequent modelsin Section 4.1), that is, based on linear or nonlinearmodels with input a subset Ωt from the informa-tion set at time t. Example parametric approachesinclude the RST approach of Gneiting et al. (2006)for predicting wind speed with truncated Gaussiandistributions and that of Pinson (2012) using Gen-eralized Logit–Normal distributions for wind power,also compared with censored Gaussian and Beta as-sumptions. Similarly, Lau and McSharry (2010) pro-posed employing Logit–Normal distributions for ag-gregated wind power generation for the whole Re-public of Ireland.In contrast, nonparametric approaches, since they

do not rely on any assumption for the shape of pre-dictive densities, translate to focusing on sets ofquantile forecasts defining predictive c.d.f.s. Theseare conveniently summarized by such sets of quan-tile forecasts,

Fs,t+k|t = q(αi)s,t+k|t; 0≤ α1 < · · ·< αi < · · ·

(4.10)< αl ≤ 1,

with nominal levels αi spread over the unit interval,though, in practice, Fs,t+k|t is obtained by interpo-lation through these sets of quantile forecasts. Actu-ally, nonparametric approaches to quantile forecasts

WIND ENERGY: FORECASTING CHALLENGES 15

may suffer from a limited number of relevant obser-vations for the very low and high nominal levels α,say, α,1 − α < 0.05, therefore potentially compro-mising the quality of resulting forecasts. This wasobserved by Manganelli and Engle (2004) when fo-cusing on risk quantification approaches in finance,and more particularly on dynamic quantile regres-sion models for very low and high levels. Even thoughthe application of interest here is different, the nu-merical aspects of estimating models for quantiles ofwind power generation for very low and high levelsare similar. It may therefore be advantageous un-der certain conditions to define nonparametric pre-dictive densities for their most central part, say,α,1−α > 0.05, while parametric assumptions couldbe employed for the tails.A number of approaches for issuing nonparamet-

ric probabilistic forecasts of wind power were pro-posed and benchmarked over the last decade. In themost standard case, these are obtained from alreadygenerated point predictions and, potentially, associ-ated meteorological forecasts. Maybe the most well-documented and widely applied methods are thesimple approach of Pinson and Kariniotakis (2010)consisting in dressing the available point forecastswith predictive densities of forecast errors made insimilar conditions, as well as the local quantile re-gression of Bremnes (2004) and the time-adaptivequantile regression of Møller, Nielsen and Madsen(2008), to be used for each of the defining quan-tile forecasts. The approach of Møller, Nielsen andMadsen (2008) comprises an upgraded version ofthe original proposal of Nielsen, Madsen and Nielsen(2006), where quantile forecasts of wind power gen-eration are conditional to previously issued pointforecasts and to input wind direction forecasts. Asfor point predictions, neural network and machinelearning techniques became increasingly popular overthe last few years for generating nonparametric prob-abilistic predictions based on a set of quantiles(Sideratos and Hatziargyriou, 2012). In contrast tothese methods using single-valued forecasts of windpower and meteorological variables as input, a rel-evant alternative relies on meteorological ensemblepredictions, that is, sets of multivariate space–timetrajectories for meteorological variables as issued bymeteorological institutes [see Leutbecher and Palmer(2008) and the references therein], which are thentransformed to the wind power space. Ensemble fore-casts attempt at dynamically representing uncer-tainties in meteorological forecasts (as well as spa-tial, temporal and inter-variable dependencies), by

jointly accounting for misestimation in the initialstate of the Atmosphere and for parameter uncer-tainty in the model dynamics. To obtain probabilis-tic forecasts of wind power generation from suchmeteorological ensembles, conventional approachescombine nonlinear regression and kernel dressing ofthe ensemble trajectories, as in the alternative pro-posals of Roulston et al. (2003); Taylor, McSharryand Buizza (1999) and Pinson and Madsen (2009).In a similar vain, a general method for the con-version of probabilistic forecasts of wind speed topower based on stochastic power curves, thus ac-counting for additional uncertainties in the wind-to-power conversion process in a Bayesian framework,was recently described by Jeon and Taylor (2012).Example nonparametric forecasts are shown in Fig-

ure 6 for the same period as in Figure 5, as obtainedby applying the method of Pinson and Kariniotakis(2010) to the already issued point predictions andtheir input meteorological forecast information. Thecharacteristics of these predictive densities smoothlyevolve as a function of a number of factors, for ex-ample, lead time, geographical location, time of theyear and level of power generation (since nonlin-ear and bounded power curves shape forecast uncer-tainty). By construction, and through adaptive es-timation, these predictive densities are probabilisti-cally calibrated, meaning that observed levels for thedefining quantile forecasts correspond to the nom-inal ones. This is a crucial property of probabilis-tic forecasts to be used as input to decision prob-lems such as those of Section 3, since a probabilisticbias in the forecasts would yield suboptimality of re-sulting operational decisions. Actually, in addition,probabilistic calibration is also a prerequisite for ap-plication of the methods described in the followingin order to generate trajectories.

4.3 Spatio-Temporal Trajectories

Both point forecasts and predictive densities aresuboptimal inputs to decision-making when spatialand temporal dependencies are involved. It is thenrequired to fully describe the density of the spatio-temporal process Ys,t+k. Following a proposal by(Pinson et al. (2009)) for wind power and, morerecently, by Moller, Lenkoski and Thorarinsdottir(2013) for multiple meteorological variables, the prob-

abilistic forecast Fs,t+k|t can be fully characterizedunder a Gaussian copula by the predictive marginalc.d.f.s Fs,t+k|t, ∀s, k, and by a space–time covariance

matrix Ct linking all locations and lead times. In

16 P. PINSON

Fig. 6. Example episode with probabilistic forecasts for the 5 aggregated zones of Western Denmark (and correspondingmeasurements obtained a posteriori), as issued on 16 March 2007 at 06 UTC. They take the form of so-called river-of-bloodfan charts [termed coined after Wallis (2003)], represented by a set of central prediction intervals with increasing nominalcoverage rates (from 10% to 90%).

that case, using notation similar to that of Moller,Lenkoski and Thorarinsdottir (2013),

Fs,t+k|t(ys,t+k|Ct)(4.11)

= Φmn(Φ−1(Fs,t+k|t(y))s,k|Ct),

where ys,t+k was defined in (2.3) and Φ is the c.d.f.of a standard Gaussian variable, while Φmn is thatfor a multivariate Gaussian of dimension m × n.Going beyond the Gaussian copula simplification,one could envisage employing more refined copulas,though at the expense of additional complexity. Theinterested reader may find an extensive introductionto copulas in Nelsen (2006).This type of construction of multivariate proba-

bilistic forecasts for wind power generation in spaceand in time has clear advantages. Indeed, given thatall predictive densities forming the marginal densi-ties are calibrated, it may be assumed that one dealswith a latent space–time Gaussian process consist-ing of successive multivariate random variables Zt

(each of dimension m × n) with realizations given

by

zt = Φ−1(Fs,t+k|t(ys,t+k));(4.12)

s= s1, s2, . . . , sm, k = 1,2, . . . , n.

Consequently, this latent Gaussian process can beused for identifying and estimating a suitable para-metric space–time structure or, alternatively, if m×n is low and the sample size large, for the trackingof the nonparametric (sample) covariance structure,for instance, using exponential smoothing.Similarly, one of the advantages of this construc-

tion of multivariate probabilistic forecasts based ona Gaussian copula is that it is fairly straightfor-ward to issue space–time trajectories. Rememberthat these are the prime input to a large class ofstochastic optimization approaches, such as the ad-vanced version of the problems presented in Sec-tions 3.1 and 3.2, where representation of space–time interdependencies is required. Such trajectoriesalso are a convenient way to visualize the complexinformation conveyed by these multivariate proba-bilistic forecasts, as hinted by Jorda and Marcellino

WIND ENERGY: FORECASTING CHALLENGES 17

Fig. 7. Set of 12 space–time trajectories of wind power generation for the 5 aggregated zones of Western Denmark, issuedon the 16 March 2007 at 06 UTC.

(2010), among others. Let us define by

y(j)s,t+k|t = y

(j)s,t+k|t; s= s1, s2, . . . , sm,

k = 1,2, . . . , n,(4.13)

j = 1,2, . . . , J,

a set of J space–time trajectories issued at time t.As an illustrative example, Figure 7 gathers a set ofJ = 12 space–time trajectories of wind power gen-eration for the same episode as in Figures 5 and 6.The covariance structure Ct used to fully specify thespace–time interdependence structure is obtained byexponential smoothing of the sample covariance ofthe latent Gaussian process. The trajectories arethen obtained by first randomly sampling from amultivariate Gaussian variable with the most up-to-date estimate of the space–time covariance struc-

ture. Denote by z(j)t the jth sample, whose compo-

nents z(j)s,t+k will directly relate to a location s and

a lead time k in the following. These multivariate

Gaussian samples are converted to wind power gen-eration by a transformation which is the inverse ofthat in (4.12). This yields

y(j)s,t+k = F−1

s,t+k|t(Φ(z(j)s,t+k)) ∀s, k, j.(4.14)

This type of visualization allows to appraise thetemporal correlation in wind power generation andpotential forecast errors through predictive densi-ties, giving an extra level of information if comparedto the probabilistic forecasts of Figure 6. There areobvious limitations stemming from the dimensional-ity of the random variable of interest. For instance,here, the spatial interdependence structure, thoughserving to link these trajectories, is nearly impossi-ble to appreciate.

5. DISCUSSION: UPCOMING CHALLENGES

Three decades of research in modeling and fore-casting of power generation from the wind have ledto a solid understanding of the whole chain from tak-ing advantage of available meteorological and power

18 P. PINSON

measurements, as well as meteorological forecasts,all the way to using forecasts as input to decision-making. Today, methodologies are further developedin a probabilistic framework, even though forecastusers may still prefer to be provided with single-valued predictions. Some important challenges arecurrently under investigation or identified as partic-ularly relevant for the short to medium term. Theseare presented and discussed below, with emphasisplaced on new and better forecasts, and forecast ver-ification, as well as bridging the gap between fore-cast quality and value.

5.1 Improved Wind Power Forecasts: Extracting

More out of the Data

Improving the quality of wind power forecasts is aconstant challenge, with strong expectations linkedto the increased commitment of the meteorologi-cal community to issue better forecasts of relevantweather variables, mainly surface wind components.This will come, among other things, from a betterdescription of the physical phenomena involved, es-pecially in the boundary layer, as well as from anincreased resolution of the numerical schemes usedto solve the systems of partial differential equations.Meanwhile, for statisticians, there are paths to-

ward forecast improvement that involve a better uti-lization of available measurement data, combiningmeasurements available on site and additional obser-vations spatially distributed around that site. Windforecasts used to issue power forecasts over a re-gion seldom capture fully the spatio-temporal dy-namics of power generation owing to, for example,a too coarse resolution (spatial and temporal) andtiming errors with respect to passages of weatherfronts. However, all distributed meteorological sta-tions and wind turbines may serve as sensors in or-der to palliate for these deficiencies. For the exam-ple of the Western Denmark data set, Girard andAllard (2013) explored the spatio-temporal charac-teristics of residuals after capturing local dynamicsat all individual sites, hinting at the role of pre-vailing weather conditions on the space–time struc-ture. For the same data set, Lau (2011) investigatedan anisotropic space–time covariance model basedon a Lagrangian approach, conditional to prevailingwind direction over the region. Based on such anal-ysis, it is required to propose nonlinear and non-stationary spatio-temporal models for wind powergeneration, for instance, using covariance structuresconditional to prevailing weather conditions, in the

spirit of Huang and Hsu (2004). An advantage willbe that, instead of having to identify and estimatemodels for every single site of interest (more than400 for the Western Denmark data set), and at var-ious spatial and temporal resolutions of relevanceto forecast users, a single model would fit all pur-poses at once. Even though more complex and po-tentially more costly in terms of parameter estima-tion, they could lead to a substantial overall reduc-tion in computational time and expert knowledgenecessary to set up and maintain all individual mod-els. Alternatively, approaches relying on stochasticpartial differential equations ought to be consideredowing to appealing features and recent advances intheir linkage to spatio-temporal covariance struc-tures, as well as improved computational solving(Lindgren, Rue and Lindstrom, 2011). Challengesthere, however, relate to the complexity of the sto-chastic processes involved, requiring to account forthe state-dependent diffusion part, and also forchanges in the very dynamics of wind components,as induced by a number of weather phenomena. It isnot clear how all these aspects could be accountedfor in a compact set of stochastic differential equa-tions, which could be solved with existing numericalintegration schemes.The increasing availability of high-dimensional data

sets, with a large number of relevant meteorologicaland power systems variables, possibly at high spatialand temporal resolutions, gives rise to a number ofchallenges and opportunities related to data aggre-gation. These challenges have already been identi-fied in other fields, for example, econometrics, whereaggregation has shown its interest and potential lim-itations. A relevant example work is that of Giaco-mini and Granger (2004), which looks at the prob-lem of pooling and forecasting spatially correlateddata sets. On the one hand, considering differentlevels of aggregation for the wind power forecastingproblem can permit to ease the modeling task, byidentifying groups of turbines with similar dynamicbehavior which could be modeled jointly. On theother hand, this would lower the computational bur-den by reducing model size and complexity. Propos-als related to aggregation should, however, fully con-sider the meteorological aspects at different tempo-ral and spatial scales, which may dynamically condi-tion how aggregate models would be representativeof geographically distributed wind farms. One couldbuild on the classical Space–Time Auto-Regressive

WIND ENERGY: FORECASTING CHALLENGES 19

(STAR) model of Cliff and Ord (1984) by enhanc-ing it to having dynamic and conditional space–timecovariance structures. In a similar vain, dynamicmodels for spatio-temporal data such as those in-troduced by Stroud, Muller and Sanso (2001) andfollow-up papers are appealing, since they providean alternative approach to data aggregation by see-ing the overall spatial processes as a linear combina-tion of a limited number of local (polynomial) spa-tial processes in the neighborhood of appropriatelychosen locations. Overall, various relevant directionsto space–time modeling could be explored, based onthe substantial literature existing for other processesand in other fields.

5.2 New Forecast Methodologies and Forecast

Products

As a result of these efforts, new types of fore-casts will be available to decision-makers in the formof continuous surfaces and trajectories, from whichpredictions with any spatial and temporal resolutioncould be dynamically extracted. Similar to the de-velopment of meteorological forecasting, the need forlarger computational facilities might call for central-izing efforts in generating and issuing wind powerpredictions. Actually, in the opposite direction, ashare of practitioners request predictions of lowercomplexity that could be better appraised by abroader audience and more easily integrated into ex-isting operational processes. For instance, since ac-commodating the variability of power fluctuationswith successive periods of fast-increasing and fast-decreasing power generation is seen as an issue bysome system operators in the US and in Australia,methodologies were proposed for the prediction ofso-called ramp events, where the definition of these“ramp events” is based on the very need of thedecision-maker (Bossavy, Girard and Kariniotakis,2013; Gallego et al., 2013).Besides, even though alternative parametric as-

sumptions for predictive marginal densities havebeen analyzed and benchmarked, for example, Beta(Bludszuweit, Domınguez-Navarro and Llombart,2008), truncated and censored Gaussian, and Gen-eralized Logit–Normal (Pinson, 2012), there is noclear superiority of one over the others, for all po-tential lead times, level of aggregation and wind dy-namics themselves. This certainly originates fromthe nonlinear and bounded curves representing theconversion of wind to power, known to shape pre-dictive densities. Such curves may additionally betime-varying, uncertain and conditional on various

external factors. This is why future work should con-sider these curves as stochastic power curves, alsodescribed by multivariate distributions, as a gener-alization of the proposal of Jeon and Taylor (2012).Their impact on the shape of predictive densitiesought to be better understood. Then, combined withprobabilistic forecasts of relevant explanatory vari-ables, for instance, from recalibrated meteorologicalensembles, stochastic power curves would naturallyyield probabilistic predictions of wind power genera-tion, in a Bayesian framework. This is since stochas-tic power curves comprise models of the joint den-sity of meteorological variables and of correspond-ing wind power generation. Predictive densities ofwind power generation would then be obtained byapplying Bayes rule, that is, by passing probabilisticforecasts of meteorological variables through suchstochastic power curve models.To broaden up, and since operational decision-

making problems are based on interdependent vari-ables (power generation from different renewable en-ergy sources, electric load and potentially marketvariables), multivariate probabilistic forecasts for rel-evant pairing, or for all of them together, shouldbe issued in the future, with the weather as thecommon driver. Similar to the proposal of Moller,Lenkoski and Thorarinsdottir (2013) for multivari-ate probabilistic forecasts of meteorological variables,one could generalize the space–time trajectories ofSection 4.3 to a multivariate setup. Alternativesshould be thought of, allowing to directly obtainsuch spatio-temporal and multivariate predictions,instead of having to go through predictive marginaldensities first.

5.3 Verifying Probabilistic Forecasts of

Ever-Increasing Dimensionality

Forecast verification is a subtle exercise alreadyfor the most simple case of dealing with point fore-casts, to be based on the joint distribution of fore-casts and observations (Murphy and Winkler, 1987).Focus is today on verifying forecasts in a probabilis-tic framework, for instance, following the paradigmof Gneiting, Balabdaoui and Raftery (2007) orig-inally introduced for the univariate case, based oncalibration and sharpness of predictive marginal den-sities. The nonlinear and double-bounded nature ofthe wind power stochastic process (possibly also adiscrete-continuous mixture) renders the verificationof probabilistic forecasts more complex, especiallyfor their calibration. It generally calls for an exten-sive reliability assessment conditional on variables

20 P. PINSON

known to impact the shape of predictive densities:level of power, wind direction, etc. In addition, thebenchmarking and comparison of forecasting meth-ods ought to account for sample size and correla-tion issues, since evaluation sets often are of lim-ited size (though of increasing length now that somewind farms have been operating for a long time),while correlation in forecast errors and other cri-teria (skill score values, probability integral trans-form) is necessarily present for forecasts with leadtimes further than one step ahead. Verifying high-dimensional forecasts, like space–time trajectories inthe most extreme case, based on small samples willnecessarily yield score values that may not fully re-flect actual forecast quality even though the scoreused is proper. Indeed, the deviations from the ex-pected score value, which could be estimated betterwith larger samples, would be substantial. Corre-lation issues may only magnify this problem, sincethey somewhat reduce the effective sample size forestimation. An illustration of the combined effects ofsampling and correlation on the verification of prob-abilistic forecasts can be found in Pinson, McSharryand Madsen (2010).Going from univariate to multivariate aspects,

Gneiting et al. (2008) explained how the previouslyintroduced paradigm can be readily generalized formultivariate probabilistic predictions, yielding anevaluation framework including skill scores and di-agnostic tools. An application to the verification oftemporal trajectories of wind power generation inPinson and Girard (2012) illustrated its potentiallimitations stemming from the high-dimensionality(there, n= 43 lead times) of the underlying randomvariables. Following the discussion in Section 5.1,it is clear that new views on forecast verificationought to be introduced and evaluated as dimension-ality increases. For instance, recent work by Heringand Genton (2011) showed how to compare spatialpredictions in a framework inspired by the Diebold–Mariano test and with limited assumptions on thespatial processes themselves, thus permitting to dealwith high-dimensional predictions by focusing ontheir spatial structure.

5.4 Bridging the Gap Between Forecast Quality

and Value

Murphy (1993) introduced 3 types of goodnessfor weather forecasts, also valid and relevant forother types of predictions like for wind power. Outof these 3, quality and value play a particular role:(i) quality relates to the objective assessment of how

well forecasts describe the stochastic process of in-terest (and its realizations), regardless of how theforecasts may be used subsequently, while (ii) valuecorresponds to the economic/operational gain fromconsidering forecasts at the decision-making stage.Through the introduction of representative opera-tional decision problems in Section 3, it was shownthat optimal forecasts as input to decision-making ina stochastic optimization framework take the formof quantiles, predictive marginal densities or, finally,trajectories describing the full spatio-temporal pro-cess. However, it is not clear today how improv-ing the quality of these forecasts, for instance, interms of reduced skill score values or increased prob-abilistic calibration, may lead to added value forthe decision-makers, especially when they might usethese forecasts sub-optimally. In practice, this willcall for more analytic work in a decision-theoreticframework, by better linking skill scores of the fore-casters and utility of the decision-makers, as well asfor a number of simulation studies in order to simu-late the usage of forecasts of varying quality as inputto a wide range of relevant operational problems.Full benefits from integrating wind power genera-tion into existing power systems and through elec-tricity markets will only be obtained by optimallyintegrating forecasts in decision-making.

ACKNOWLEDGMENTS

The author was supported by the EU Commis-sion through the project SafeWind (ENK7-CT2008-213740) and the Danish Public Service Obligation(PSO) fund under RadarSea (contract no. 2009-1-0226), as well as the Danish Strategic ResearchCouncil under ‘5s’—Future Electricity Markets (12-132636/DSF), which are hereby acknowledged. Theauthor is also grateful to ENFOR, DONG Energy,Vattenfall Denmark and Energinet.dk for providingthe data used in this paper. Acknowledgments arefinally due to Tilmann Gneiting (Heidelberg Uni-versity), Adrian Raftery (University of Washington)and Patrick McSharry (University of Oxford) forrows of inspiring discussions on probabilistic fore-casting and forecast verification, as well as JulijaTastu, Pierre-Julien Trombe, Jan K. Møller and Hen-rik Madsen, among others, at the Technical Univer-sity of Denmark, for contributing to broadening ourunderstanding of spatio-temporal dynamics and un-certainties in wind power modeling and forecasting.Acknowledgments are finally due to two reviewersand a guest editor for their comments and sugges-tions.

WIND ENERGY: FORECASTING CHALLENGES 21

REFERENCES

Ackermann, T. (2012). Wind Power in Power Systems, 2nded. Wiley, Chichester.

Bessa, R. J., Miranda, V. and Gama, J. (2009). Entropyand correntropy against minimum square error in offlineand online three-day ahead wind power forecasting. IEEET. Power Syst. 24 1657–1666.

Billinton, R. and Allan, R. N. (1984). Reliability Evalua-tion of Power Systems. Plenum Press, London.

Bludszuweit, H., Domınguez-Navarro, J. A. and Llom-

bart, A. (2008). Statistical analysis of wind power forecasterrors. IEEE T. Power Syst. 23 983–991.

Bossavy, A., Girard, R. and Kariniotakis, G. (2013).Forecasting ramps of wind power production with numeri-cal weather prediction ensembles. Wind Energy 16 51–63.

Botterud, A., Wang, J., Miranda, V. and Bessa, R. J.

(2010). Wind power forecasting in US electricity markets.The Electricity Journal 23 71–82.

Bremnes, J. B. (2004). Probabilistic wind power forecastsusing local quantile regression. Wind Energy 7 47–54.

Brown, B. G., Katz, R. W. and Murphy, A. M. (1984).Time series models to simulate and forecast wind speedand wind power. J. Appl. Meteor. 23 1184–1195.

Cliff, A. D. and Ord, J. K. (1984). Space–time modelingwith applications to regional forecasting. Trans. Inst. Br.Geogr. 66 119–128.

Conejo, A., Carrion, M. and Morales, J. M. (2010).Decision-Making under Uncertainty in Electricity Markets.Springer, New York.

Conradsen, K., Nielsen, L. B. and Prahm, L. P. (1984).Review of Weibull statistics for estimation of wind speeddistributions. J. Appl. Meteor. 23 1173–1183.

Doherty, R. and O’Malley, M. (2005). A new approachto quantify reserve demand in systems with significant in-stalled wind capacity. IEEE T. Power Syst. 20 587–595.

Gallego, C., Costa, A., Cuerva, A., Landberg, L.,Greaves, B. and Collins, J. (2013). A wavelet-based ap-proach for large wind power ramp characterisation. WindEnergy 16 257–278.

Giacomini, R. and Granger, C. W. J. (2004). Aggrega-tion of space–time processes. J. Econometrics 118 7–26.MR2030964

Giebel, G., Brownsword, R., Kariniotakis, G., Den-

hard, M. and Draxl, C. (2011). The state-of-the-art inshort-term prediction of wind power: A literature overview,2nd ed. Technical report. Available at orbit.dtu.dk.

Girard, R. and Allard, D. (2013). Spatio-temporal prop-agation of wind power forecast errors. Wind Energy. Toappear.

Gneiting, T. (2010). Quantiles as optimal point forecasts.Int. J. Forecasting 27 197–207.

Gneiting, T., Balabdaoui, F. and Raftery, A. E. (2007).Probabilistic forecasts, calibration and sharpness. J. R.Stat. Soc. Ser. B Stat. Methodol. 69 243–268. MR2325275

Gneiting, T. and Raftery, A. E. (2007). Strictly properscoring rules, prediction, and estimation. J. Amer. Statist.Assoc. 102 359–378. MR2345548

Gneiting, T., Larson, K., Westrick, K., Genton, M. G.

and Aldrich, E. (2006). Calibrated probabilistic fore-casting at the stateline wind energy center: The regime-

switching space–time method. J. Amer. Statist. Assoc. 101968–979. MR2324108

Gneiting, T., Stanberry, L. I., Grimit, E. P., Held, L.

and Johnson, N. A. (2008). Assessing probabilistic fore-casts of multivariate quantities, with an application to en-semble predictions of surface winds. TEST 17 211–235.MR2434318

Hahn, H., Meyer-Nieberg, S. and Pickl, S. (2009). Elec-tric load forecasting methods: Tools for decision making.European J. Oper. Res. 199 902–907.

Hamilton, J. D. (1989). A new approach to the economicanalysis of nonstationary time series and the business cycle.Econometrica 57 357–384. MR0996941

Haslett, J. and Raftery, A. E. (1989). Space–time mod-elling with long-memory dependence: Assessing Ireland’swind power resource (with discussion). J. R. Stat. Soc. Ser.C Appl. Stat. 38 1–50.

Hering, A. S. and Genton, M. G. (2010). Powering up withspace–time wind forecasting. J. Amer. Statist. Assoc. 10592–104. MR2757195

Hering, A. S. and Genton, M. G. (2011). Comparing spa-tial predictions. Technometrics 53 414–425. MR2850473

Huang, H. C. and Hsu, N. J. (2004). Modeling transporteffects on ground-level ozone using a non-stationary space–time model. Environmetrics 15 251–268.

Jeon, J. and Taylor, J. W. (2012). Using conditional ker-nel density estimation for wind power density forecasting.J. Amer. Statist. Assoc. 107 66–79. MR2949342

Jorda, O. and Marcellino, M. (2010). Path forecast eval-uation. J. Appl. Econometrics 25 635–662. MR2758077

Joskow, P. L. (2011). Comparing the costs of intermittentand dispatchable electricity generating technologies. Amer.Econ. Rev. 100 238–241.

Kariniotakis, G. N., Stavrakakis, G. S. and Noga-

ret, E. F. (1996). Wind power forecasting using advancedneural networks models. IEEE T. Energy Conver. 11 762–767.

Koenker, R. and Bassett, G. Jr. (1978). Regression quan-tiles. Econometrica 46 33–50. MR0474644

Landberg, L. (1999). Short-term prediction of the powerproduction from wind farms. J. Wind Eng. Ind. Aerodyn.80 207–220.

Landberg, L. and Watson, S. J. (1994). Short-term pre-diction of local wind conditions. Bound.-Layer Meteor. 70171–195.

Lange, M. (2005). On the uncertainty of wind powerpredictions—Analysis of the forecast accuracy and statis-tical distribution of errors. J. Solar Energ.-T. ASME 127

177–184.Lange, M. and Focken, U. (2006). Physical Approach to

Short-Term Wind Power Prediction. Springer, Berlin.Lau, A. (2011). Probabilistic wind power forecasts: From ag-

gregated approach to spatiotemporal models. Ph.D. thesis,Univ. Oxford, Oxford, UK.

Lau, A. and McSharry, P. (2010). Approaches for multi-step density forecasts with application to aggregated windpower. Ann. Appl. Stat. 4 1311–1341. MR2758330

Letcher, T. M. (2008). Future Energy: Improved, Sustain-able and Clean Options for Our Planet. Elsevier, Amster-dam.

22 P. PINSON

Leutbecher, M. and Palmer, T. N. (2008). Ensemble fore-casting. J. Comput. Phys. 227 3515–3539. MR2400226

Lindgren, F., Rue, H. and Lindstrom, J. (2011). An ex-plicit link between Gaussian fields and Gaussian Markovrandom fields: The stochastic partial differential equationapproach. J. R. Stat. Soc. Ser. B Stat. Methodol. 73 423–498. MR2853727

Manganelli, S. and Engle, R. F. (2004). A comparison ofvalue-at-risk models in finance. In Risk Measures for the21st Century (G. Szego, ed.) 123–143. Wiley, Chichester.

Matos, M. A. and Bessa, R. J. (2010). Setting the operatingreserve using probabilistic wind power forecasts. IEEE T.Power Syst. 26 594–603.

Moller, A., Lenkoski, A. and Thorarinsdottir, T. L.

(2013). Multivariate probabilistic forecasting using ensem-ble Bayesian model averaging and copulas. Q. J. RoyalMet. Soc. 139 982–991.

Møller, J. K., Nielsen, H. A. and Madsen, H. (2008).Time-adaptive quantile regression. Comput. Statist. DataAnal. 52 1292–1303. MR2418566

Morthorst, P. E. (2003). Wind power and the conditionsat a liberalized power market. Wind Energy 6 297–308.

Mur Amada, J. and Bayod Rujula, A. (2010). Variabil-ity of wind and wind power. In Wind Power (S. M. Muy-

een Vukovar, ed.) 289–320. Intech Open, Rijeka, Croatia.Murphy, A. H. (1993). What is a good forecast? An essay

on the nature of goodness in weather forecasting. Wea.Forecasting 8 281–293.

Murphy, A. H. and Winkler, R. L. (1987). A generalframework for forecast verification. Mon. Wea. Rev. 115

1330–1338.Nelsen, R. B. (2006). An Introduction to Copulas, 2nd ed.

Springer, New York. MR2197664Nielsen, T. S. (2002). Online prediction and control in non-

linear stochastic systems. Ph.D. thesis, Dept. Informaticsand Mathematical Modelling, Technical Univ. Denmark.

Nielsen, H. A., Madsen, H. and Nielsen, T. S. (2006).Using quantile regression to extend an existing wind powerforecasting system with probabilistic forecasts. Wind En-ergy 9 95–108.

Ortega-Vazquez, M. A. and Kirschen, D. S. (2009). Esti-mating the spinning reserve requirements in systems withsignificant wind power generation penetration. IEEE T.Power Syst. 24 114–124.

Palmer, T. N. (2012). Towards the probabilistic Earth-system simulator: A vision for the future of climate andweather prediction. Quart. J. Royal Met. Soc. 665 841–861.

Papavasiliou, A. andOren, S. S. (2013). Multiarea stochas-tic unit commitment for high wind penetration in atransmission constrained network. Oper. Res. 61 578–592.MR3079732

Pena Diaz, A., Gryning, S. E. and Mann, J. (2010). Onthe length-scale of the wind profile. Quart. J. Royal Met.Soc. 136 2119–2131.

Pinson, P. (2012). Very-short-term probabilistic forecastingof wind power with generalized logit–normal distributions.J. R. Stat. Soc. Ser. C Appl. Stat. 61 555–576. MR2960738

Pinson, P., Chevallier, C. and Kariniotakis, G. (2007).Trading wind generation from short-term probabilistic

forecasts of wind power. IEEE T. Power Syst. 22 1148–1156.

Pinson, P. and Girard, R. (2012). Evaluating the qualityof scenarios of short-term wind power generation. Appl.Energ. 96 12–20.

Pinson, P. and Kariniotakis, G. (2010). Conditional pre-diction intervals of wind power generation. IEEE T. PowerSyst. 25 1845–1856.

Pinson, P. and Madsen, H. (2009). Ensemble-based proba-bilistic forecasting at Horns Rev. Wind Energy 12 137–155.

Pinson, P. and Madsen, H. (2012). Adaptive modellingand forecasting of offshore wind power fluctuations withMarkov-switching autoregressive models. J. Forecast. 31

281–313. MR2924797Pinson, P., McSharry, P. E. and Madsen, H. (2010). Reli-

ability diagrams for nonparametric density forecasts of con-tinuous variables: Accounting for serial correlation. Quart.J. Royal Met. Soc. 136 77–90.

Pinson, P., Nielsen, H. A., Madsen, H., Pa-

paefthymiou, G. and Klockl, B. (2009). Fromprobabilistic forecasts to statistical scenarios of short-termwind power production. Wind Energy 12 51–62.

Raiffa, H. and Schaifer, R. (1961). Applied StatisticalDecision Theory. Division of Research, Harvard BusinessSchool, Boston.

Reikard, G. (2008). Using temperature and state transitionsto forecast wind speed. Wind Energy 11 431–443.

Roulston, M. S., Kaplan, D. T., Hardenberg, J. andSmith, L. A. (2003). Using medium-range weather fore-casts to improve the value of wind energy production. Re-new. Energ. 28 585–602.

Sideratos, G. and Hatziargyriou, N. D. (2007). An ad-vanced statistical method for wind power forecasting. IEEET. Power Syst. 22 258–265.

Sideratos, G. and Hatziargyriou, N. D. (2012). Proba-bilistic wind power forecasting using radial basis functionneural networks. IEEE T. Power Syst. 27 1788–1796.

Skytte, K. (1999). The regulating power market on theNordic power exchange Nord Pool: An econometric analy-sis. Energ. Econ. 21 295–308.

Stroud, J. R., Muller, P. and Sanso, B. (2001). Dynamicmodels for spatiotemporal data. J. R. Stat. Soc. Ser. BStat. Methodol. 63 673–689. MR1872059

Taylor, J. W., McSharry, P. E. and Buizza, R. (1999).Wind power density forecasting using ensemble predictionsand time series models. IEEE T. Energy Conver. 24 775–782.

Tol, R. S. J. (1997). Autoregressive conditional het-eroscedasticity in daily wind speed measurements. Theo.Appl. Clim. 56 113–122.

Tong, H. (2011). Threshold models in time series analysis—30 years on. Stat. Interface 4 107–118. MR2812802

Vautard, R., Cattiaux, J., Yiou, P., Thepaut, J. N. andCiais, P. (2010). Northern Hemisphere atmospheric still-ing partly attributed to an increase in surface roughness.Nat. Geosci. 3 756–761.

Vincent, C. L., Giebel, G., Pinson, P. and Madsen, H.

(2010). Resolving nonstationary spectral information inwind speed time series using the Hilbert–Huang transform.J. Appl. Meteor. Clim. 49 253–267.

WIND ENERGY: FORECASTING CHALLENGES 23

Wallis, K. F. (2003). Chi-squared tests of interval and den-sity forecasts, and the bank of England’s fan charts. Int. J.Forecasting 19 165–175.

Weron, R. (2006). Modeling and Forecasting ElectricityLoads and Prices: A Statistical Approach. Wiley, New York.

Wolak, F. A. (2013). Regulating competition in wholesaleelectricity supply. In Economic Regulation and Its Reform:What Have We Learned? (N. L. Rose, ed.). Univ. ChicagoPress. To appear.

WWEA (World Wind Energy Association) (2012). 2012—Half-year report. Technical report. Available at http://www.wwindea.org/webimages/Half-year report 2012.pdf.

Zhu, X. and Genton, M. G. (2012). Short-term wind speedforecasting for power system operations. Int. Stat. Rev. 802–23. MR2990340

Zugno, M., Jonsson, T. and Pinson, P. (2013). Tradingwind energy based on probabilistic forecasts of both windgeneration and market quantities. Wind Energy. 16 909–926.

Zugno, M., Morales, J. M., Pinson, P. and Madsen, H.

(2013). Pool strategy of a price-maker wind power pro-ducer. IEEE T. Power Syst. 28 3440–3450.