Skill frecasting frm different wind pwer ensemble predictin methds Pierre Pinsn 1, Henrik Aa. Nielsen 1, Henrik Madsen 1 and Gerge Karinitakis 2 1 Technical University f Denmark, Infrmatics and Mathematical Mdelling, 2900 Kgs. Lyngby, Denmark 2 Ecle des Mines de Paris, Center fr Energy and Prcesses, 06904 Sphia Antiplis, France E-mail: pp@imm.dtu.dk Abstract. This paper presents an investigatin n alternative appraches t the prviding f uncertainty estimates assciated t pint predictins f wind generatin. Fcus is given t skill frecasts in the frm f predictin risk indices, aiming at giving a cmprehensive signal n the expected level f frecast uncertainty. Ensemble predictins f wind generatin are used as input. A prpsal fr the definitin f predictin risk indices is given. Such skill frecasts are based n the dispersin f ensemble members fr a single predictin hrizn, r ver a set f successive lk-ahead times. It is shwn n the test case f a Danish ffshre wind farm hw predictin risk indices may be related t several levels f frecast uncertainty (and energy imbalances). Wind pwer ensemble predictins are derived frm the transfrmatin f ECMWF and NCEP ensembles f meterlgical variables t pwer, as well as by a lagged average apprach alternative. The ability f risk indices calculated frm the varius types f ensembles frecasts t reslve amng situatins with different levels f uncertainty is discussed. 1. Intrductin The large scale integratin f wind generatin capacities induces difficulties in the management f a pwer system. An additinal challenge is t cnciliate this deplyment with the n-ging deregulatin f the Eurpean electricity markets. Increasing the value f wind generatin thrugh the imprvement f predictin systems perfrmance is ne f the pririties in wind energy research needs fr the cming years [1]. Relevant predictin hrizns are up t 72- hur ahead, justifying the chice fr this frecast length in the present paper. A state f the art n wind pwer frecasting has been published by Giebel et al [2]. A large part f recent research wrks in wind pwer frecasting has cncentrated n assciating uncertainty estimates t pint frecasts. Pinsn and Karinitakis [3] have described tw cmplementary appraches that cnsist in prviding frecast users with skill frecasts (cmmnly in the frm f risk indices) r alternatively with prbabilistic frecasts. Fr a thrugh discussin n the prbabilistic frecasting alternative, we refer t Pinsn et al [4] and references therein. Here fcus is given t the frmer type f uncertainty indicatrs. It appears that lw quality frecasts f wind generatin are partly due t pwer predictin mdels, and partly t Numerical Weather Predictin (NWP) systems. Indeed, during sme perids weather dynamics can be relatively mre predictable, while at sme ther pint in time they may prve t be unpredictable, and this regardless f the frecasting methd emplyed. c 2007 Ltd 1
Since pwer predictins are derived frm nnlinear transfrmatins f wind speed frecasts, the level f uncertainty in meterlgical predictins may be amplified r dampened thrugh this transfrmatin. Prviding frecast users with an a priri warning n the expected level f predictin uncertainty may allw them t develp alternative (and mre r less risk averse) strategies. In an peratinal cntext, a skill frecast assciated t a pint predictin may be mre easily understd than prbabilistic frecasts. Als, skill frecasts are nt directly related t a given predictin methd: since they relate t an assessment f the inherent predictability f the weather dynamics, they are expected t prvide an a priri warning whatever the cnsidered single-valued frecast. Certain wrks in that directin are based n the definitin f weather dynamics indicatrs [5]. Mre precisely, methds frm synptic climatlgy take advantage f measurements f wind speed and directin, as well as pressure, in rder t classify lcal weather cnditins. Cnsequently, [5] relates such typical weather patterns t different levels f frecast uncertainty. In cntrast, ensemble frecasts f wind generatin are used here as input fr deriving skill frecasts. Such type f frecasts cnsists f a set f alternative predictin scenaris ver a perid f interest. Ensemble frecasts f meterlgical variables are cmmnly prduced by integrating the uncertainty in the cmputatin f NWPs [6]. Different types f meterlgical ensemble predictins are cnsidered, prvided either by the Eurpean Centre fr Medium-range Weather Frecasts (ECMWF, 51 members) [7, 8] r by the Natinal Centre fr Envirnmental Predictin (NCEP, 11 members) [9]. They are cnverted t ensemble predictins f wind pwer fllwing the methd described by Nielsen et al [10]. They will be referred as ECMWF-EPW and NCEP-EPW, respectively. A specificity f the cnversin mdel in cmparisn with mre classical pwer curve mdels is that it frces pwer frecasts t span the whle range f ptential pwer values. Finally, lagged average ensembles, which cnsist in a set f frecasts with cmmn lead time, but issued at different time rigins, are used as a benchmark. They are here btained by lagging wind pwer predictins prduced with ECMWF cntrl frecasts, and include 5 members. A full descriptin f the varius ensemble types is available in [11, Ch. 5]. The paper is structured as fllwing. In a first stage, the methdlgy fr skill frecasting is described, with emphasis n the definitin f predictin risk indices and the way they shuld be related t frecast uncertainty. Then, the ability f the varius risk indices t infrm n expected uncertainty is evaluated and discussed by using the test case f a Danish ffshre wind farm, ver a perid f 10 mnths. Cncluding remarks end the paper. 2. Skill frecasts based n wind pwer ensembles The methdlgy fr skill frecasting dedicated t the wind pwer frecasting applicatin is develpped here. First, a definitin f predictin risk indices is prpsed. It is based n the dispersin f wind pwer ensembles ver a single predictin hrizn, r ver a set f successive lk-ahead times. It is then explained hw such predictin risk indices may be used as skill frecasts, i.e. frecasts f the distributins f expected predictin errrs. The relatin between predictin risk indices and the level f predictin errr is described with cnditinal prbability diagrams. 2.1. Definitin f predictin risk indices Owing t the relatin between the spread f ensemble members and the standard deviatin f the errrs described in e.g. [11], it is prpsed here t define predictin risk indices as a measure f this ensemble spread. This measure is a cntinuus ne, in cntrast t sme categrical measures 1 intrduced in the meterlgical literature, such as mde ppulatin [12] r ensemble statistical entrpy [13]. Such chice is mtivated by the cnclusins frm Grimit [14], stating 1 The basic idea f categrical measures f ensemble spread cnsists in dividing the range f pssible frecast values in several bins, and t cunt the numbers f ensemble members falling in each bin. 2
that cntinuus measures f ensemble spread are mre apprpriate if frecast s users have a cntinuus utility functin 2. We assume that this is case r users f wind pwer predictins, either fr the management r trading f wind generatin. Ensemble predictins f wind pwer issued at time t cnsist in a set f J alternative predictins ˆp (j) t+k/t (j = 1,...,J) fr any lead time t + k. The weighted standard deviatin σ t,k f ensemble members is used as a measure f spread fr that lk-ahead time. σ t,k is given by σ t,k = J J 1 J j=1 ( ) w j ˆp (j) 2 t+k/t pj t+k/t 1 2 (1) such that the sum f the weights w j ttals 1, and with p J t+k/t predictins fr that lead time, that is the mean f the J alternative p J t+k/t = 1 J J j=1 ˆp (j) t+k/t (2) In equatin (1), the weights may reflect the ability f the ensemble members t give an assessment f predictability. If cnsidering fr instance an algrithm that derives a best-guess frecast as a weighted average f the ensemble members, these weights culd be directly used in the calculatin f σ t,k. A similar remark is valid if cnsidering lagged average ensembles, fr which the weights in the ptimal cmbinatin f the alternative predictins are a functin f their age. Ensemble frecasts and pwer measures have different tempral reslutin. Tempral interplatin is thus used in rder fr bth f them t have a 15-minute reslutin. Even thugh, the actual tempral reslutin f ECMWF and NCEP meterlgical ensembles is f six hurs. In additin, weather predictability des nt have an instantaneus nature: it is very unlikely that wind generatin wuld be easily predictable fr a given lk-ahead time, and then highly unpredictable fr the fllwing ne. Hence, it is envisaged here t estimate predictability ver a time perid. The use f the weighted standard deviatin is generalized by cmputing the average f σ t,k ver a set f cnsecutive hrizns, frm lk-ahead time k 1 t k 2. This average weighted standard deviatin defines a Nrmalized Predictin Risk Index, abbreviated NPRI, calculated as NPRI(k 1,k 2 ) := 1 k 2 k 1 + 1 k 2 i=k 1 σ t,i (3) with σ t,i given by equatin (1). In the fllwing, NPRI h dentes the predictin risk index calculated n a per-hrizn basis (i.e. such that k 1 = k 2 ) while NPRI d stands fr perids f 24 hurs (i.e. day 1, day 2, etc). 2.2. Relating NPRI and predictin errrs 2.2.1. Cnsidering predictin errrs as energy imbalances Predictin errrs are expressed in the frm f energy imbalances since it is aimed at shwing that NPRI can be used fr infrming n the level f expected predictin errr ver a certain perid f time. Energy imbalances are defined here as the difference, in abslute value, between predicted and measured amunts f energy ver a perid f interest. Bth measured and predicted amunts f energy are nrmalized 2 The utility functin fr a frecast s user is intrduced and further discussed by Pinsn et al [15]. 3
quantities, since pwer frecasts and measures are nrmalized values (by the nminal capacity P n ). The energy imbalance d t+k 2 t+k 1 between lead time t + k 1 and lead time t + k 2 is d t+k 2 t+k 1 = E t+k 2 t+k 1 Êt+k 2 t+k 1 = t r k 2 i=k 1 p t+i ˆp t+i/t (4) where Êt+k 2 t+k1 and Et+k 2 t+k 1 are the predicted and measured quantities f energy ver this perid f time, while t r is the tempral reslutin f wind pwer predictins. By calculating energy imbalances in abslute value, prductin surplus and shrtage are similarly accunted fr. Predictin risk indices are meant fr estimating the expected level f uncertainty, but cannt give the sign f frecast errrs. If cnsidering a single lk-ahead time, the nrmalized imbalance equals the predictin errr in abslute value. And, fr successive hrizns, it is equivalent t the average abslute errr ver this time interval. Predictin risk indices shuld prvide infrmatin n the expected level f frecast uncertainty whatever the pint predictin methd cnsidered. Therefre, we d nt cncentrate hereafter n the use f the best available pint frecast f wind generatin that can be derived frm ensembles, i.e. given by the ensemble mean (r the weighted mean fr lagged average ensembles) [11]. Instead, pint predictins are prduced as it is cmmnly dne tday, that is by applying a statistical pwer curve mdel t the cntrl frecasts prvided by meterlgical ffices, here frm ECMWF r NCEP. 2.2.2. Cnditinal prbability diagrams fr relating NPRI t the level f expected predictin errr Fllwing discussin in [14], the relatinship between predictin risk indices and level f predictin errr is drawn frm a prbabilistic perspective. This prpsal ges against the traditinal apprach cnsisting in fitting a linear regressr between measures quantifying the ensemble spread and predictr s skill, assciated with a crrelatin cefficient assessing the strength f this relatin (see [16, 17, 18] amng thers). The incnsistency f using the crrelatin cefficient has been discussed by Grimit and Mass [17]: cnsidering it fr measuring the strength f the relatinship between the ensemble spread and the predictr s skill implicitly assumes a linear relatin between these tw variables, which is nt true in practice 3. A pssibility fr expressing the relatin between NPRI and related predictin errr in a prbabilistic manner is t use cntingency tables, which give the prbabilities f events defined by the ccurrence f NPRI-range/errr-range pairs. Such idea has been prpsed first by Hutekamer [19] and cnsequently applied by Whitaker and Lughe [18]. Thugh, ur chice ges fr cnditinal prbability diagrams similar t thse used by Mre and Kleeman [20], which easily give a visual infrmatin n the relatin between NPRI and predictin uncertainty levels. Cnditinal prbability diagrams summarize distributins f energy imbalances given NPRI values. The range f NPRI values is divided int categries defined as equally ppulated classes. This fllws frm the idea that it is nt the value f NPRI by itself that tells if the situatin is mre r less uncertain, but mre where this value is lcated in the climatlgical distributin f NPRI values [13, 18]. Als, cnsidering equally ppulated classes f NPRI values allws us t cmpare skill frecasts made frm ECMWF-EPW, NCEP-EPW r lagged average ensembles as input, independently f the range f their ensemble spread values. A similar reasning applies fr energy imbalances, which are nrmalized by their climatlgical value depending n the lk-ahead perid. This climatlgical value crrespnds t the average imbalance ver the 3 Actually, it has been shwn fr an ideal ensemble f infinite size that the spread-errr crrelatin can be written analytically, as a functin f the tempral variability f the ensemble spread [19]. In this mdel, the predictin errr is in abslute value. Fr an infinite spread variability, this spread-errr crrelatin asympttes t 0.8 [18]. 4
10-mnth evaluatin perid fr each lk-ahead perid. When mentining imbalance levels, they will indeed be relative and expressed in percentage f their climatlgical value. Thus, we will study hw NPRI has the ability t tell if these imbalances are lwer r higher than usual, independently f the glbal perfrmance f the cnsidered pint predictin methd. 3. Results In this Sectin is evaluated the ability f predictin risk indices t differentiate between situatins with lw and high uncertainty depending n the use f ECMWF-EPW, NCEP-EPW, r lagged average ensembles as input. This study is fr the test case f the Tunø Knb wind farm, lcated few kilmeters ff the east cast f Jutland in Denmark, with a nminal capacity P n f 5MW. The perid fr which bth meterlgical and pwer data are avalaible cvers the first 10 mnths f 2003. The tempral reslutin f 15 minutes fr pwer measurements is chsen as the tempral reslutin fr the study. 3.1. Pintwise estimatin f expected uncertainty In a first stage, the ability f NPRI t infrm n the level f predictin uncertainty when calculated fr each lk-ahead time is evaluated, fr the three sets f wind pwer ensemble predictins. As explained in paragraph 2.1, NPRI h crrespnds t the weighted standard deviatin f the ensemble members fr a given lk-ahead time. The weights fr its calculatin are set t 1/J fr ECMWF-EPW and NCEP-EPW (where J the number f ensemble members equals 11 and 51, respectively). Alternatively, the weights given in table 1 are used fr the case f the lagged averaged ensembles, fllwing [11]. Owing t the limited amunt f data available (nly 300 series f wind pwer predictins ver a 10-mnth evaluatin perid), and als fr cmparisn with results f the secnd part f the study, NPRI and energy imbalance values are gathered fr each day ahead. Table 1. Weights used fr calculating the NPRI when cnsidering the lagged average ensembles. These weights are thse that wuld als be used fr cmbinatin f ensemble members in rder t btain an ptimal single-valued predictin. Frecast age [hurs] 24 48 72 96 Weight 0.36 0.26 0.21 0.17 3.1.1. The NPRI h ability t infrm n the expected imbalance level Figure 1 gives the example f a cnditinal prbability diagram fr ECMWF-EPW fr day 3 (i.e. fr lk-ahead times between 48- and 72-hur ahead). It takes the frm f a set f bxplts, summarizing distributins f energy imbalances given the class f NPRI h values. Bxplts are centred n the average NPRI h values fr each class. Five different NPRI classes are cnsidered, fr which the related empirical distributins f imbalances are cmpsed by 1440 items each. Evlutin f mean values gives the general trend between NPRI h and level f predictin errr. There is a steady (and quasi linear) increase in the mean imbalance level when ging frm lwest t highest NPRI h class. When NPRI h values belng t the NPRI h class 1, the average imbalance level equals 30% f the climatlgical ne. Thugh, fr class 5, this average imbalance is mre than 5 times larger, reaching 155% f the climatlgical value. Using NPRI h with ECMWF- EPW prves t be a pssibility fr reslving between situatins with varius levels f expected imbalances. The mst interesting infrmatin cmes frm the quted quantiles f cnditinal prbability distributins given the NPRI h class, since it infrms n a lwer and an upper bund fr expected 5
relative imbalance [%] 0 100 200 300 10 20 30 40 NPRI [%] h Figure 1. Cnditinal prbability diagram giving the relatin between NPRI h and level f energy imbalance. NPRI h values are calculated frm ECMWF-EPW. Results are fr day 3 (predictin hrizns frm 48 t 72-hur ahead). Empirical distributins are made up with 1440 elements. Bxplts give the 10% and 90% quantiles (lwer and upper tips), the lwer and upper quartiles (bx bunds), the median (central line) and finally the mean (). imbalances. In figure 1, ne sees fr instance that if NPRI h lies in the first class, then 90% f imbalances are belw 90% f the climatlgical imbalance level (fr the cnsidered lk-ahead time), while there is still a 10% prbability that the level f imbalance exceeds 340% f the climatlgical imbalance value if the NPRI h value belngs t class 5. The imbalance distributins becme much wider when NPRI h values are larger: the 10% quantiles are still clse t zer, but the 90% nes get much higher. This upper bund n the expected imbalance level infrms n the risk f relying n the prvided wind pwer pint predictin. Frm a risk aversin pint f view, it wuld be preferable t make cnservative decisins if NPRI h values belng t class 5. Nte that here, imbalances are relative t their climatlgical level in rder t better illustrate the fact that predictin uncertainty is lwer r higher than usual. Thugh, in an peratinal cntext, expected levels f imbalance culd be expressed with physical units (e.g. in MWh r in percentage f the maximum pssible generatin ver the time range). The example f day 3 has been chsen fr describing the relatinship between NPRI h calculated frm ECMWF-EPW and level f imbalance in a prbabilistic manner. The same kind f relatin can be witnessed fr the ther days, r fr the ther types f ensembles. Thugh, that relatinship may exhibit slightly different characteristics, which are studied in the fllwing paragraph. 3.1.2. Cmparing the cases fr which NPRI h is calculated with ECMWF-EPW, NCEP-EPW and the lagged average ensembles The relatin between NPRI h (fr ECMWF-EPW) and imbalance 6
n a per-lk-ahead time basis has been described abve. Here, a cmparisn is made between the infrmatin cntent f the three types f ensemble predictins f wind generatin cnsidered. This cmparisn is pssible fr the first 3 days ahead. The specific case f day 1 is irrelevant since ECMWF-EPW are nly available at the end f this first day in an peratinal cntext. Fcus is given t day 2 (lk-ahead times between 24 and 48-hur ahead) fr highlighting the differences between the varius wind pwer ensembles. Similar analyses were carried ut fr day 3, and the fllwing cmments are representative fr the whle study. Even if the NPRI h values are nt scattered in the same manner when cnsidering NCEP- EPW, ECMWF-EPW, r lagged average ensembles, using classes f NPRI h values enables t study the inherent ability f the varius ensemble appraches t reslve between situatins with lwer and higher uncertainty. These categries f NPRI h values permit t leave aside the prblem f their distributins and t assess hw their variatins may infrm n expected imbalance level. Therefre, when cmparing the varius appraches, we d nt mentin ranges f NPRI h values, but nly the NPRI h class, numbered frm 1 t 5. Table 2 gathers sme f the quantiles (r (α), with α the prprtin) and the mean µ f the cnditinal prbability distributins f imbalances, given the NPRI h class, fr the three types f ensemble predictins. Table 2. Characteristics f cnditinal imbalance distributins given the NPRI h class. r (α) dentes the quantile with prprtin α, while µ relates t the mean. Bth NPRI h and imbalance values are gathered ver day 2 (lk-ahead times between 24 and 48-hur ahead). Results are fr ECMWF-EPW, NCEP-EPW, and the lagged average ensembles respectively. Empirical distributins are made up with 1440 elements. Imbalance values crrespnd t relative imbalances (in % f their climatlgical level). (a) - ECMWF-EPW NPRI h class r (0.1) r (0.25) r (0.5) r (0.75) r (0.9) µ 1 1.3 2.9 10.2 28.7 68.9 27.7 2 4.2 15.2 42.7 89.6 176.7 70.5 3 10.8 30.1 78.4 157.7 277.7 113.7 4 15.2 44.3 113.5 202.7 298.3 137.4 5 16.0 47.5 116.4 223.5 333.6 150.7 (b) - NCEP-EPW NPRI h class r (0.1) r (0.25) r (0.5) r (0.75) r (0.9) µ 1 1.4 4.7 10.8 29.3 84.8 30.4 2 10.5 22.4 40.3 91.3 192.6 75.9 3 17.2 39.8 76.2 146.1 243.4 108.7 4 24.9 56.5 103.3 176.5 275.7 130.93 5 26.8 69.4 135.2 218.7 305.1 154.0 (c) - lagged average ensembles NPRI h class r (0.1) r (0.25) r (0.5) r (0.75) r (0.9) µ 1 1.4 3.7 13.0 40.2 99.8 40.0 2 5.1 20.0 57.3 117.0 223.5 89.1 3 9.2 28.9 88.4 171.6 264.4 114.9 4 11.2 35.3 101.8 192.6 290.2 127.6 5 8.7 27.1 76.7 187.8 323.5 128.3 The variability f the mean imbalance can be seen as a criterin fr evaluating the ability 7
f the different appraches fr dissciating between several levels f frecast uncertainty. That variability can be quantified by the rati between the mean imbalances fr NPRI h class 5 and 1. This rati is equal t 5.4, 5.1 and 3.2 fr ECMWF-EPW, NCEP-EPW and the lagged average ensembles, respectively. ECMWF-EPW and NCEP-EPW have a higher differentiatin ability (with a slight advantage fr ECMWF-EPW), far better than that f the lagged average ensembles. Then, fcus is given t the quantiles f cnditinal imbalance distributins. The increase in the spread 4 f these distributins when ging frm the first t the fifth class is mre significant fr ECMWF-EPW, fllwed by NCEP-EPW and the lagged average ensembles. This can als be seen as anther criterin fr stating that ECMWF-based ensembles better reslve amng situatins, since the variatins in the range f expected imbalances are mre prnunced. If lking separately at lwer (r (0.1) and r (0.25) ) and upper (r (0.75) and r (0.9) ) quantiles, ne sees that lwer quantiles are mre variable fr NCEP-EPW while upper quantiles are mre variable fr ECMWF-EPW. The first ne better reslves the lw part f cnditinal imbalance distributins while the secnd better differentiates the upper part f these distributins. Therefre, if having a risk aversin pint f view, NPRI h used with ECMWF-EPW gives a mre valuable infrmatin n the risk ne may face when relying n the prvided pint predictin. The increase in the mean imbalance depending n NPRI h class is nt as steady fr the lagged average ensembles than fr the thers. In additin, the fur lwest quantiles decrease between class 4 and 5. If the mean imbalance is higher fr NPRI h class 5, it is nly because this NPRI h class cntains very large predictin errrs. But, it als cntains mre lw predictin errrs than the furth class. Nte that this lagged averaging apprach, even if less infrmative fr skill frecasting n a per-step ahead basis, has the great advantage f being a gratis and easily applicable alternative t the use f ECMWF-based and NCEP-based ensemble predictins. 3.2. Estimatin f the uncertainty fr a lk-ahead perid In a secnd stage, the pssibility f infrm n expected uncertainty fr a lk-ahead perid is cnsidered, by calculating NPRI ver a set f successive lk-ahead times. The benefits f this tempral integratin f uncertainty estimatin is assessed by lking at the relatin between NPRI d classes and energy imbalances. Bth quantities are calculated ver 24 hurs (thus fr 96 cnsecutive lk-ahead times). While Möhrlen [21] discussed the benefits f cnsidering a larger area when assessing the spread-skill relatinship f wind pwer ensembles, the aim here is t shw hw skill frecasting can benefit frm tempral averaging f spread and skill. Als, adding this tempral cmpnent wuld be relevant fr real-wrld applicatins, since NPRI d wuld be related t levels f energy imbalance ver the lk-ahead perid cnsidered. Nte that current methds fr uncertainty estimatin f wind pwer predictins nly fcus n prviding pintwise uncertainty estimates (cf. discussin in Pinsn et al [4]). The 300 series f ensemble predictins ver the 10-mnth evaluatin perid are cnsidered. Bth NPRI d values and energy imbalances are calculated fr lk-ahead times between 0 and 24-hur ahead (day 1), 24 and 48-hur ahead (day 2), etc. NPRI d values are srted in 5 equally ppulated classes. T each f these classes are assciated the empirical distributins f related energy imbalances, which cntain 60 items each. The same quantities than in the abve paragraph are used fr summarizing the characteristics f cnditinal prbability distributins (i.e. mean, median, quartiles and 10 and 90% quantiles). 3.2.1. The NPRI d ability t infrm n the expected imbalance level Fcus is given first t the same example than that cnsidered in paragraph 3.1, which relates t the use f NPRI d with 4 Here, the spread f the imbalance distributins can be quantified by the inter quartile range, r alternatively by the distance between the quantiles with prprtin 0.1 and 0.9. In general, the inter quartile range is preferred, since it cnsists a mre rbust measure f the spread f an empirical distributin. 8
ECMWF-EPW fr hrizns between 48 and 72-hur ahead. Related cnditinal prbability diagram is depicted in figure 2. As an interpretatin f these cnditinal distributins, ne sees fr instance that the relative imbalance ver day 3 is between 55 and 295% f its climatlgical level when a NPRI d value lies in the fifth class. Fr cmparisn, this same relative imbalance ranges frm 5 t 85% f the climatlgical value nly when NPRI d belngs t the first class. relative imbalance [%] 0 50 100 200 300 10 20 30 NPRI [%] d Figure 2. Cnditinal distributins f energy imbalance given NPRI d classes. Bth imbalances and NPRI values are calculated ver day 3 (predictin hrizns frm 48 t 72-hur ahead). Empirical distributins are made up with 60 elements. Bxplts give the 10% and 90% quantiles (lwer and upper tips), the lwer and upper quartiles (bx bunds), the median (central line) and finally the mean (). Similarly t the analysis carried ut in paragraph 3.1, the increase f the mean energy imbalance with the NPRI d class is steady and quasi linear, with mean imbalance levels ranging frm 35 t 150% f their climatlgical value. Therefre, the ability f NPRI d t be an indicatr f expected uncertainty is still valid when cnsidering tempral averaging. In additin, imbalance distributins fr every NPRI d class appear t be sharper than thse btained when using NPRI h as an uncertainty indicatr (cf. figure 1). Fr instance, the inter quartile range fr NPRI class 4 equals 165% in the latter case, while it is nly f 105% fr the frmer ne. These distributins are sharper first because upper quartiles are at lwer level and als because lwer quartiles are at a higher level (the same remark is valid fr the 10% and 90% quantiles). Tempral averaging f skill smthes ut differences between lw and large predictin errrs. Frm a skill frecasting pint f view, sharper distributins f expected imbalance are beneficial, since they give mre cnfidence in the estimatin f frecast uncertainty. 3.2.2. Cmparing the cases fr which NPRI d is calculated with ECMWF-EPW, NCEP-EPW and the lagged average ensembles The cmparisn between the three types f ensemble predictins is again carried ut fr day 2. Table 3 gathers sme quantiles and the mean f cnditin 9
imbalance distributins given the NPRI d class. If n particular mentin, the fllwing remarks are als valid fr day 1 and day 3. Table 3. Characteristics f cnditinal imbalance distributins given the NPRI d class. r (α) dentes the quantile with prprtin α, while µ relates t the mean. Bth NPRI d and imbalance values are gathered ver day 2 (lk-ahead times between 24 and 48-hur ahead). Results are fr ECMWF-EPW, NCEP-EPW, and the lagged average ensembles respectively. Empirical distributins are made up with 60 elements. Imbalance values crrespnd t relative imbalances (in % f their climatlgical level). (a) - ECMWF-EPW NPRI d class r (0.1) r (0.25) r (0.5) r (0.75) r (0.9) µ 1 3.6 8.6 24.4 48.8 88.3 34.9 2 23.5 41.2 61.4 100.8 171.1 76.9 3 29.7 60.9 94.7 140.3 192.7 112.3 4 67.3 91.1 115.0 164.1 203.0 130.1 5 60.1 102.5 125.6 176.4 222.6 144.6 (b) - NCEP-EPW NPRI d class r (0.1) r (0.25) r (0.5) r (0.75) r (0.9) µ 1 3.5 10.9 24.0 36.9 71.9 35.8 2 23.8 41.0 80.6 110.4 140.7 81.6 3 53.6 70.9 92.4 140.3 189.7 108.9 4 70.5 87.2 118.8 158.0 198.1 129.1 5 61.8 95.6 134.3 181.0 234.5 143.5 (c) - lagged average ensembles NPRI d class r (0.1) r (0.25) r (0.5) r (0.75) r (0.9) µ 1 7.1 12.9 30.4 49.1 97.9 41.9 2 8.1 44.3 75.8 118.4 170.1 88.9 3 47.4 77.9 96.8 153.2 229.1 116.3 4 29.9 77.4 115.6 154.8 194.9 117.8 5 55.1 88.9 120.9 176.3 195.0 135.5 The ratis between mean imbalances fr NPRI d class 5 and NPRI d class 1 equal 4.2, 4 and 3.2, fr ECMWF-EPW, NCEP-EPW and lagged average ensembles, respectively. The value f the rati fr lagged averaging ensembles is similar t that calculates when fcusing n NPRI h, while it is significantly lwer fr the tw thers. This decrease is mainly due t the smthing f skill, nt t a diminutin in the ensembles ability t reslve amng situatins. In a general manner, ECMWF-EPW and NCEP-EPW are still mre skilful fr indicating the expected imbalance level. Thugh, lagged average ensembles gain frm the cnsideratin f a tempral cmpnent fr uncertainty estimatin. Imbalance distributins when cnsidering NPRI d are much sharper than when cnsidering NPRI h fr the three types f ensembles. The inter quartile range is here between 26 (fr NCEP- EPW) and 40% (fr ECMWF-EPW) fr the first class f NPRI d values. These values are slightly higher than thse in table 3. But fr the ther NPRI d classes, it is actually the inverse: the inter quantile range is much lwer when imbalance values are srted depending n NPRI d values. The reductin f the inter quartile range is up t 50%. Therefre, in terms f skill frecasting, NPRI d appears t be a better indicatr, wing t these sharper distributins. 10
4. Cnclusins The present investigatin n the use f wind pwer ensemble predictins has revealed their ptential fr assciating skill frecasts t pint predictins f wind generatin. Fcus has been given t the pssibility f cmmunicating differently n expected level f predictin uncertainty. Skill frecasts take the frm f predictin risk indices. They may be seen as estimates f shrt-term predictability f wind generatin. They thus cmprise a cmprehensive signal n the cnfidence frecast users may have in the pwer predictins prvided regardless f the methd emplyed. The predictin risk index NPRI that has been intrduced reflects the spread f ensemble members fr a single r a set f successive lk-ahead times. Varius types f wind pwer ensemble frecasts have been cnsidered: lagged average ensembles (btained by lagging the ECMWF cntrl frecasts, 5 members), as well as wind pwer ensembles derived frm ECMWF (51 members) and NCEP (11 members) ensemble predictins f meterlgical variables. The investigatin has been carried ut n the case-study f the Tunø Knb wind farm, ver a perid f 10 mnths. The methdlgy develped has cnsisted in cnsidering varius equally ppulated classes f NPRI values (mre precisely 5 classes), and in establishing their prbabilistic relatin with energy imbalance levels. It has been shwn that fr all different types f wind pwer ensembles cnsidered, NPRI culd prvide a useful infrmatin n expected level f frecast uncertainty. An imprtant pint relates t the pssibility and interest f defining predictin risk indices fr a lk-ahead perid. They then permit t infrm n the level f expected energy imbalance ver the perid cnsidered. This cntrasts with the cmmn prviding f pintwise uncertainty estimates. Mrever, an imprtant cnclusin is that the gratis alternative f making up wind pwer ensembles by lagging available pint predictins prved t be valuable fr estimating the level f expected predictin uncertainty. Cnsidering NCEP-based r ECMWF-based ensembles f wind generatin is justified by their better ability f reslving between lw and high predictability situatins. Perspectives regarding fllw-up studies include: (i) a validatin f the results n varius types f test-cases lcated in znes with different meterlgical characteristics (fr which predictability may be mre r less easily estimated); (ii) further investigatin n ther pssibilities fr estimating the disagreement between ensemble members e.g. with categrical measures (mde ppulatin and ensemble entrpy); (iii) a study f ther ensemble predictin systems, which may be mre apprpriate fr shrt-range applicatins than thse cnsidered here; (iv) the use f such predictin risk indices in frecast cmbinatin r regime-switching methds in rder t dampen the risk f large predictin errrs. Finally, a last perspective cncerns the real-wrld utilizatin f predictin risk indices by end-users f wind pwer frecasts. As a first step, they can be cmmunicated as a cmplement t pint predictins. This way, frecast users will get used t that infrmatin, as a signal n the cnfidence they may have n the frecasts prvided. Then, a secnd step will be t define hw t make alternative decisins (mre r less cnservative depending n the risk aversin f frecast users) depending n the value f the predictin risk index, and t demnstrate the resulting peratinal benefits. Acknwledgments The results presented have been generated as part f tw prjects: Wind Pwer Ensemble Frecasting supprted by Danish PSO funds (ORDRE-101295/FU2101), and Anems partly funded by the Eurpean Cmmissin (ENK5-CT2002-00665), which are hereby greatly acknwledged. The authrs als gratefully acknwledge Elsam Engineering A/S fr prviding the pwer data, ECMWF and NCEP fr prviding the meterlgical ensemble data. 11
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