An investigation of load extrapolation according to IEC Ed. 3

Transcription

1 A ivestigatio of load extrapolatio accordig to IEC 64- Ed. Rolf Gez, Siemes Wid Power A/S, Kristia Bedix Nielse, DNV Global Wid Eergy, Peter Hauge Madse, Siemes Wid Power A/S, Abstract: The preset versio of the IEC 64- Ed stadard suggests a method to extrapolate ultimate stregth loads to a 5 year occurrece period based o loads calculated durig productio. Sice load calculatios for productio states are ormally performed for a limited time period the extreme loads are estimated by extrapolatio, usig a probabilistic approach. The curret paper presets results for the load extrapolatio, for a multi megawatt Siemes pitch regulated, variable speed turbie, where the cotrol shows a strog iteractio with the loads. The ivestigatio focuses o the threshold used i the Peak-over-Threshold method suggested i IEC 64- Ed ad the ifluece of choice of threshold o the extrapolated loads. The choice of probability desity fuctios (PDF) for the idividual exceedace fits is also ivestigated by comparig results usig various distributios. The ivestigatio icludes loads from several sesors, with mai focus o flap wise bedig momet. Results show that for a pitch regulated variable speed turbie the turbie cotrol system has a limitig ifluece o the extreme loads. For this reaso a fit to all extremes above the suggested threshold (IEC 64- Ed) of.4 times the stadard deviatio is quite poor for the ivestigated distributios ad yields overly coservative results. If the PDFs are oly fitted to the tail of the data (e.g. the maximum values) the extrapolated loads are reduced. To reliably fit to the tail of the data a larger amout of seeds is required. The ivestigatio shows that the required amout of seeds is o the order of seeds additioal desig load case must be cosidered. The load case covers the extrapolatio of simulated loads to a 5-year extreme load. I the the stadard a iformative aex F is give, which describes a method to extrapolate from simulated loads. With this aex as a basis a program has bee implemeted i Matlab to carry out the aalysis automatically. However durig the programmig phase several possibilities were discovered to divert from the method outlied i the aex. This gave rise to the ivestigatios preseted i the paper. To give a striget presetatio the idividual issues have bee addressed as they appear i the program flow. This ca be illustrated by the flow chart preseted i Figure. All decisios that have bee addressed i this paper have bee marked i red color. The preseted simulatio results are obtaied by a ew geeratio aeroelastic code, BHawC (Bous Eergy Horizotal axis wid turbie Code), developed at Bous Eergy A/S ad Siemes Wid Power A/S durig the recet four years. The mai aim with this developmet is to esure a detailed represetatio of geometrical o-liearities for all desig critical compoets. The code represets the structure by a geometrically o-liear fiite elemet model, ad the aerodyamic load calculatio is based o the BEM method with extesios aimig at takig dyamic flow behaviour ito accout [6]. Further the results show that usig a extrapolatio algorithm based o wid speed bis yield overly coservative results. This is caused by the high, limited loads are preset at oly a few wid speeds, ad the lower loads at other wid speeds thus domiate the extrapolatio. The results from the extrapolatio of loads from simulatios are compared with results from measuremets o the actual turbie. Itroductio With the ew editio of the IEC 64- stadard: Wid turbie geerator systems, Part : Safety requiremets, a Figure : Structure of program.

2 The turbie used for simulatios ad measuremets is the Siemes.6 MW variable speed, variable pitch turbie. Specifically, the turbie used i this paper is the.6 MW turbie placed at the Høvsøre test site o the west coast of Jutlad, Demark. The turbie has a rotor diameter of 7 m ad is fitted with Siemes B5 blades. Selectio of threshold I Ed. of the IEC 64- stadard a threshold is suggested, which is give as µ +. 4σ () However, this threshold is ot robust for o-gaussia sigals but is oly applicable for sigals that are Gaussia distributed or with oly slight deviatios from a Gaussia distributio. Ufortuately ot all the simulated loads are Gaussia distributed. This is i particular the case for loads i the wid speed rage, where the turbie achieves rated power. Drops i wid speed at this poit may result i drastic short term chages i the load. This is clearly illustrated i Figure, which shows the simulated shaft torsio momet at m/s. Probability distributio type The choice of probability distributio is the most critical poit i determiig the 5-year extreme load. For this reaso a umber of probability distributios have bee tested. The distributios tested are Gumbel -parameter Weibull Geeralized Extreme Value (GEV) -parameter Log-ormal The tests of the idividual distributios are based o the goodess-of-fit to the data.. Goodess-of-fit criterio The goodess-of-fit criterio employed i this paper is a stadard Aderso-Darlig goodess-of-fit test []. The advatage of this criterio over the Kolmogorov-Smirov ad Chi-squared tests is that the Aderso-Darlig test is more sesitive to deviatios i the tail of the data. Furthermore the Aderso-Darlig test ca be modified to accout for small sample sizes, which is a importat property for the data sets cosidered i this paper. The Aderso-Darlig criterio is defied as: A = ( ) ( i ) [ l( ) + l( )] i= w i w i+ (4) Figure : Shaft torsio momet, My, m/s From the figure it is apparet that usig the threshold suggested by IEC 64- will result i o extremes at all beig foud by the peak-over-threshold method. It is therefore suggested that a alterative threshold be used istead. This threshold is suggested to be of the form ( ( Time ) ) µ + C max series µ () The costat C is determied by requirig that the two thresholds should be idetical for a distributed Gaussia sigal. Assumig a maximum of the Gaussia sigal with 5 data poits of.95σ ad combiig the two equatios gives a coefficiet C of.5. Istead of the threshold preseted i IEC 64- Ed. it is therefore suggested that the followig threshold is used. ( max( Time ) ) µ +.5 series µ () where is the sample size ad w i is the value of the cosidered CDF at poit i. This is valid for all the cosidered distributios i this paper. However, i order to take ito accout the effect of small sample sizes, it is ecessary to correct the A -values i differet ways. For the Weibull, Gumbel ad GEV distributios the followig correctio should be used ( +. ) A m = A whereas for the logormal distributio this correctio should be used istead ( ) A m = A + For the correctio factors differet critical values also exist for the distributios. For the Weibull, Gumbel ad GEV distributios the critical value at a cofidece level of α =.5 is.757, while it is.75 for the logormal distributio. It should be oted here that the results from the goodessof-fit tests are iheretly ucertai due to the low umber of data poits. For a reasoable result a sample size of at least should be used. Sice this is ot possible here, ad sice the goodess-of-fit test is merely used to fid the best possible fit it is reasoable to use the goodess-of-fit as a simple compariso tool. (5) (6)

3 . Choice of distributio The goodess-of-fit criterio chose here is ot i itself sufficiet to evaluate what type of distributio gives the best fit to the data. Although the Aderso-Darlig criterio is more sesitive to deviatios i the tail, it is equally sesitive to deviatios i the tail at both eds of the distributio. However, for the case of load extrapolatio it is of much greater importace how the distributio fits to the high ed of the data poits. It will be show i sectio 4 that the data are domiated by several pheomea, ad that fittig to the high ed of the data is essetial i achievig accurate results. Sice a miimum of 5 poits per wid speed is required it is possible to get data sets that cotai data from two differet pheomea, thus uderliig the ecessity that the fit to the tail of the data is the best possible. For these reasos it is chose to also employ a visual compariso of the fits before choosig the type of distributio best suited for modelig extreme load respose.. Cumulative distributio fuctios This sectio briefly describes the used cumulative distributio fuctios (CDF) ad the estimatio of parameters i the idividual CDF s For the -parameter Weibull distributio the CDF is defied as: F x γ = exp α β where α is the scale parameter, β is the shape parameter ad γ is the locatio parameter. The parameters i the distributio are determied by usig the method of momets described i []. The three first momets are determied by s m~ x = i= x i = ( x x) i= = ( x i x) i= i (7) (8) (9) () The parameters i the distributio ca the be determied by solvig the system of equatios: f ( α) ˆ λ : = Γ : = Γ ˆ γ : = x Γ ˆ λ ( α) Γ ( α) Γ ( α) +Γ ( α) [ Γ ( α) Γ ( α) ] ( ˆ α) Γ ( ˆ α) s ( ˆ α) m~ [ s ] = () F ( x γ ) β exp α = x γ exp exp α β,, forβ forβ = > α α forx β + γ β + γ β () ( wheβ ) ad x < ( whe > ) Note that for β =, the GEV distributio reduces to the Gumbel distributio. For both the CDF ad parameter estimatio i the GEV distributio the statistical software package WAFO [4] has bee used. The parameters are determied usig a estimatio based o probabilityweighted momets (PWM) [5]. The probability weighted momet of a radom variable ca be defied by the exceedace probability q = (-p) as: a = m q m x ( q) dq The ubiased estimates of a m ca be determied as: aˆ m () i = xi = m m i (4) Hoskig [5] defied certai liear combiatios of PWMs, referred to as L-momets, which are aalogue to ordiary momets. The first three L-momets are defied as: λ = aˆ λ = aˆ aˆ λ = 6aˆ 6aˆ + aˆ τ = λ λ (5) The parameter τ is the L-skewess. Based o these three L- momets the three parameters i the GEV distributio ca be estimated. l z = + τ l β = 7.859z z α = Γ λ β β ( + k)( ) α γ = λ ( Γ( + k) ) β For the logormal distributio the CDF is []: F l =.5 +.5erf σ µ (6) (7) For the GEV distributio the CDF is give as []:

4 where µ is the estimated mea ad σ the estimated stadard deviatio of the data x ad erf() is the error fuctio defied as: erf = x t e dt π (8) The parameters are estimated usig a stadard MLE estimate, which ca be determied by maximizig the partial derivatives of the log-likelihood fuctio Λ. Λ = l µ l φ (9) σx i σ.4 Aalysis results Before treatig the results a few poits o the simulatios should be metioed here. The turbie modeled is the.6 MW Siemes turbie. It is desiged for IEC class IA. This defies the parameters of the wid speed distributios as A =. m/s, k =.. The turbulece level is set at 5-9% for the etire wid speed rage. I.e. simulatios at each wid speed is are ru for 5, 7 ad 9% turbulece. The reaso for this turbulece modelig, which is far from the IEC64- defiitios, is that the turbulece levels match the turbulece at the Høvsøre site. This eables a direct compariso betwee measured values ad simulated values. From this plot it is clear that although the logormal ad Gumbel distributios give the highest goodess-of-fit values they do ot fit well to the highest data poit from the data set. The ature of the fits is such that the CDF of the two distributios predicts higher probabilities for the high load data tha are actually simulated. This meas that the CDF reaches a probability of earlier tha idicated by the data, which leads to lower calculated extrapolatio values tha idicated by the simulatios. The highest data poit should therefore be weighted heavier tha the lower data poits. Based o this visual compariso the -parameter Weibull distributio with a cetral momet based estimatio of parameters is chose for the further ivestigatios. 4 Number of data poits The threshold foud above esures that a sufficiet umber of data poits are foud whe performig the peak-overthreshold aalysis. However, fidig ay give umber of poits does ot isure good distributio fits ad therefore ot accurate results. The problem with fittig to the data is that the data may be domiated by several differet pheomea. This gives differet slopes whe plottig the raked data o a double log-scale (i.e. A Gumbel-plot), as see i Figure. I the table below the average fit quality is give for several chaels for the cosidered distributios. Tower bottom bedig Tower top torsio Flapwise blade bedig Edgewise blade bedig Gumbel Weibull GEV LogNorm Based o the table, the average fit quality is lowest for the geeralized extreme value distributio, wherefore we have chose ot to cotiue with this distributio. However, for the remaiig three distributio types it is ecessary to make a visual compariso of the fits to be able to determie the appropriate distributio type. Below the results for the distributio types are plotted alog with the extracted data poits for the blade root flap momet at 5 m/s. Cumulative probability Flapwise blade bedig 5 m/s Normalized load [-] Peaks Weibull LogNormal Gumbel Figure : Fit for the differet distributio types Figure : Blade root momet, rak plot, 5 poits per wid speed From the figure it is clear that the wid turbie acts as a o-liear filter, ad that the turbie shows differet operatioal behaviour related to the differet parts of the curve. This is most likely caused by the turbie cotrol system. It is also clear that if too may of the poits o the lower part of the curve are used for fittig it will ted to lower the high tail of the data. This will result i the overestimatig the extrapolated loads beyod the values implied by the data poits plotted i Figure. For this reaso it is ecessary to fid the appropriate umber of data poits to use for each wid speed ad each load sigal. To determie what the appropriate umber of poits will be the goodess-of-fit criterio is agai used. The goodess-offit is ivestigated for a rage of umber of poits. The rage covered is 5,, 5,, 5 ad poits. For each of the data sets i the rage the average goodess-of-fit for the cosidered chaels is foud. They have bee plotted i Figure 4.

5 A^-value for fits No. of data poits [-] Tower bottom bedig Flapwise blade bedig Tower top torsio Edgewise blade bedig Figure 4: Average goodess-of-fit for selected chaels as a fuctio of the umber of data poits. From the figure it is apparet that overall the higher the umber of poits used the better the achieved fit is. As with the selectio of distributio type it is ot sufficiet to merely use the goodess-of-fit as a criterio. This ca be see i Figure. However at oly 5 data poits the statistical ucertaity is large. With a ever icreasig umber of poits the statistical ucertaity would be reduced. This is ufortuately at the cost of a ever icreasig error i estimatio of the extrapolated loads, which is caused by the fact that more ad more poits i the lower tail of the data is icluded. This dilemma is illustrated whe comparig Figure ad 5. peaks are colored by the wid speed ad plotted i the same rak plot as show i figure. Figure 6: Blade root momet, rak plot, poits per wid speed. From this figure it ca be see that the poits that represet the tail of the data, belogs to oly 4 wid speeds. These 4 wid speeds are i the rage of -4 m/s. If the loads are extrapolated for wid speeds i the rage of -5 m/s the results are as show i Figure 7. From the plot it ca be see that the domiatig wid speeds are -5 m/s. If we compare this with a rak plot of the peaks, as show i Figure 8, it is clear that the loads domiatig the extrapolatio are ot the highest peaks, which are at - m/s, but istead are those below the kik i the curve. Whe comparig the two plots the umber of poits beyod the kik i the data poits is see to icrease oly slightly with a icreasig umber of poits, whereas the umber of poits below the kik icreases more tha liearly with the umber of poits. Figure 7: Extrapolated loads for the flapwise blade bedig momet. Wid speed rage -5 m/s. Figure 5: Blade root momet, rak plot, poits per wid speed. Based o these plots it is suggested that usig poits per wid speed would be a reasoable compromise betwee a sufficiet umber of data poits for lower statistical ucertaity ad poor fits to high ed data for a high umber of data poits. The approach to fittig to the data based o this ca be questioed. It seems that for turbies usig pitch regulatio ad variable speed techology the simulatio peaks ca ot be fitted by a simple distributio, because the system exhibits o-liear behavior. These results correspod fairly well to the results foud i [7]. I figure 6 the idividual Figure 8: Blade root momet, rak plot, poits per wid speed bi.

6 Although some poits above the kik i the curve are from the wid speeds domiatig the extrapolatio these do ot determie the distributio fit for the cosidered wid speed. This is show i Figure 9 for 4 m/s. Std. deviatio factor C [-] Number of seeds [-] Tower bottom bedig Flapwise blade bedig Tower top torsio Edgewise blade bedig Figure : The factor C as a fuctio of the umber of seeds. Figure 9: Distributio fit for flapwise blade bedig momet at 4 m/s. From this plot it is clear that the data are ot distributed i a way that the cosidered distributios are able to fit. Therefore the estimated load will be overly coservative. This is a geeral problem with the method of extrapolatio. That although there may be distict kik i the data the method used is ot able to pick it up, because the distributios which are fitted to the lower ed of the data poits, domiate the probability sum. Therefore it is suggested that further work is doe to determie a algorithm, which is capable of extrapolatio usig the etire set of data poits from all wid speeds. 5 Number of seeds The umber of seeds required is determied solely o basis of the robustess of the extreme load extrapolatio, i.e. the umber of seeds at which the extrapolatio has coverged towards a result. The covergece is based o the stadard deviatio of the cosidered data poits. If we take the mea of the cosidered data poits ad add a factor C times the stadard deviatio, the extrapolated load ca be foud. The factor C is used as a measure of covergece. The umber of seeds required for the factor C to coverge is used as the umber of seeds at which the extrapolatio achieves covergece. A total of 5 seeds have bee simulated for all wid speed bis. Ad the covergece has bee determied by extrapolatig for seeds i blocks of three. The factor C has bee plotted for chaels of iterest i Figure. From this plot it is clear that covergece of the estimated extrapolated loads is achieved at aroud 9 seeds, depedig o the cosidered chael. Based o this the further ivestigatios are carried out for 9 seeds. 6 Compariso with measuremets The paper so far has bee based o simulatios made i BHawC. The pheomea ecoutered i the simulatios, should be preset i the measuremets i order to validate the approach take for extrapolatio of extreme loads. Furthermore the distributio of peaks should be the same for both measuremets ad simulatios; otherwise the results of the extrapolatio for the simulatios would ot be reproducible by extrapolatio of the measuremets, which would idicate that the approach take for extrapolatio of loads is ot correct for the actual loads. The measuremets are take from a period coverig through The measuremets are recorded as te-miute time series with a sample frequecy of 5 Hz, which is the same time step as i the simulatios. The measuremets are recorded usig the Siemes i-house PLM measuremet system. For the.6 MW turbie at Høvsøre the measuremet system cosists of chaels. Of these chaels there are 5 structural chaels coverig blade bedig, shaft bedig ad torsio ad tower bedig ad torsio. The umber of measuremets at high wid speeds is limited. For this reaso it has oly bee possible to perform the extrapolatio at 5 seeds per wid speed bi. This will to some extets ifluece the results, sice it has bee show i sectio 5 that it takes 9 seeds for the extrapolatio to achieve covergece. Sice the purpose of this sectio is solely to compare measuremets ad simulatios this is acceptable. The results may ot be accurate, but for compariso purposes 5 seeds of simulatios ad 5 seeds of measuremets, should give the same results, if the data are distributed i the same way. For the flapwise blade bedig momet, the measuremets are distributed like show i figure.

7 Figure : Distributio of measured flapwise bedig momets. The distributio of data is very similar to those for the simulatios, which are show i figure. This idicates that the simulatios determie the actios of the turbie sufficietly well. This has also bee validated through the structural load verificatio, which has bee carried out for the turbie type approval. Figure 4: Extrapolatio chart for the extrapolatio of simulated flapwise blade bedig momet. The differece betwee the cotributios from differet wid speeds, ca be accouted for by mior differeces betwee wid speed distributio ad turbulece levels of the measuremets ad simulatios. The measuremets are domiated by loads betwee 8-5 m/s, whereas the simulatios are domiated by -5 m/s as well as 8-5 m/s. It ca however also be see that if there were small differeces i turbulece ad wid speed the slope of the 8-5 m/s curves i the simulatios would quickly lead to these becomig the domiatig wid speeds. 7 Coclusio Figure : Distributio of simulated flapwise blade bedig momets. If the extrapolatio is carried out for both simulatios ad measuremets the results are as show i figures ad 4. Figure : Extrapolatio chart for the extrapolatio of measured flapwise blade bedig momet The mai focus of this paper has bee the ivestigatio of the extrapolatio method preseted i IEC 64- Ed. Aex F. Based o this method the threshold, type of distributio ad required umber of seeds have bee determied for a Siemes.6 MW pitch regulated, variable speed wid turbie. The mai fidigs cocerig the implemetatio of the outlied algorithm are that a -parameter Weibull distributio gives the most robust results, whe used oly for the highest load values i each wid speed bi. Oly the highest values ca be used for extrapolatio due to the limitig ifluece o the load from the turbie cotrol system. At least 9 seeds were ecessary to achieve robust results of the extrapolatio. The followig poits may also be draw based o this paper Neither the Gumbel or the GEV distributio seems to fit the high ed tail of the data as well as the Weibull distributio. Depedig o the data poits used the use of these distributios may lead to both uder- or overestimatio of the extreme desig loads. The threshold suggested i IEC 64- is ot robust for pitch regulated, variable speed turbies. The algorithm suggested i IEC64- may result i over-estimatio of the extreme desig loads, sice cotributios from the highest loads are igored whe extrapolatig based o wid speed bis.

8 It is suggested that further work is carried out o the followig subjects i extesio of this paper. Developmet of a algorithm that allows for extrapolatio o all peaks from all wid speeds at the same time, to allow for oly the loads limited by the cotrol system to be used i extrapolatio A compariso with the extreme loads from IEC64- Ed. to determie the actual turbie desig loads ad check whether a icrease i loads is foud ad whether this is justified based o the turbie characteristics. Refereces []. D Agostio, R.B. ad Stephes, M.A., editors, Goodess-of-fit Techiques, Marcel Dekker, Ic., 986. []. Cohe, A.C. ad Whitte, B.J., Modified Maximum Likelihood ad Modified Momet Estimators for the Three-Parameter Weibull Distributio, Commuicatios i statistics, A, Theory ad methods, Vol., No., P , 98 []. Johso, N. L., Kotz, S. ad Balakrisha, N., Cotiuous Uivariate Distributios, Vol., Wiley, 994 [4]. WAFO, Wave aalysis for Fatigue ad oceaography ver...., dowload: 6--4, [5]. Hoskig, J.R.M., Wallis, J.R. ad Wood, E.F., Estimatio of the geeralized extreme-value distributio by the method of probability-weighted momets, Techometrics (7), pp. 5-6, 985 [6]. Rubak, R. ad Peterse, J.T., Moopile as Part of Aeroelastic Wid Turbie Simulatio Code, Proceedigs, Copehage Offshore Wid, 5 [7]. Moriarty, P.J., Holley, W.E., Butterfield, S.P., Extrapolatio of Extreme ad Fatigue Loads Usig Probabilistic Methods. NREL/TP Golde, CO: Natioal Reewable Eergy Laboratory, 4.