Fatigue Estimation Methods Comparison for Wind Turbine Control
FFATIGUE ESTIMATION METHODS COMPARISONFOR WIND TURBINE CONTROL
J.J. BARRADAS BERGLIND AND RAFAEL WISNIEWSKI
Automation & Control, Department of Electronic SystemsAalborg University, Aalborg East 9220, DK. e-mail: { jjb,raf } @es.aau.dk Abstract.
Fatigue is a critical factor in structures as wind turbines exposedto harsh operating conditions, both in the design stage and control during theiroperation. In the present paper the most recognized approaches to estimatethe damage caused by fatigue are discussed and compared, with special focuson their applicability for wind turbine control. The aim of this paper is toserve as a guide among the vast literature on fatigue and shed some light onthe underlying relationships between these methods. Introduction and Motivation
Fatigue has been widely and exhaustively studied from different perspectives, andthe literature is vast and approached from different perspectives; thus, incorporatingfatigue or wear in components of a wind turbine in a control problem may seem asa daunting task. Fatigue is regarded as a critical factor in structures such as windturbines, where it is necessary to ensure a certain life span under normal operatingconditions in a turbulent environment. These environmental conditions lead toirregular loadings, which is also the case for waves and uneven roads. The mainfocus of the present is on fatigue estimation methods for wind turbine control, andas such the most widely used methods are described, with special emphasis in theapplicability of these techniques for control.In general, fatigue can be understood as the weakening or breakdown of a materialsubject to stress, especially a repeated series of stresses. From a materials perspec-tive, it can be also thought of as elastoplastic deformations causing damage on acertain material or structure, compromising its integrity.Fatigue is a phenomenon that occurs in a microscopic scale, manifesting itself asdeterioration or damage. Consequently, it has been of interest in different fields andhas been studied extensively with different perspectives; a very detailed history offatigue can be found in [1]. It could be argued that two major turning points inthe history of fatigue came firstly with the contributions of W¨ohler, who suggesteddesign for finite fatigue life in the 1860’s [2] and the so-called W¨ohler curve (orS-N curve stress versus number of cycles to failure) which still sets the basis for
Key words and phrases.
Fatigue Estimation, Load Estimation, Fatigue for Wind Turbine Con-trol, Rainflow Counting, Spectral Methods, Hysteresis Operator.
Pre-print submitted to Wind Energy. a r X i v : . [ m a t h . O C ] N ov J.J. BARRADAS BERGLIND AND RAFAEL WISNIEWSKI theoretical damage estimation; and secondly with the linear damage accumulationrule by Palmgren [3] and Miner [4], still under use nowadays.2.
Fatigue Estimation for Wind Turbine Control
Perhaps the most recognized and used measure for fatigue damage estimation isthe so-called rainflow counting (RFC) method, which is used in combination withthe Palmgren-Miner rule. In the wind turbine context, the impact on fatigue froma load can be described by an equivalent damage load (EDL); basically, the EDLis calculated using the Palmgren-Miner rule to determine a single, constant-ratefatigue load that will produce equivalent damage [5].Load or fatigue reduction techniques for wind turbines can be roughly divided inactive and passive. The former makes use of the controller, e.g., by changing thepitching angle or the generator torque, while the latter entails the design of thestructure. In [6], both strategies are combined to reduce loads in the blades. In thewind turbine control context, the control algorithm may have substantial effectson the wind turbine components; for example, controlling the pitching angle maylead to thrust load changes, which consequently affects the loads on the towerand blades [7]. In [7], [8], [9] reductions in loading are achieved by controllingthe pitch of each blade independently; the damage of different control strategies isassessed by EDL, using S-N curves. In [10], [11] a load reduction control strategiesare proposed, where the damage is evaluated using the RFC algorithm. Modelpredictive control (MPC) strategies using wind preview have been proposed in [12],[13] to reduce loads, evaluated via EDL. In [14], control strategies were designed, byapproximating fatigue load by an analytical function based on spectral moments.The Aeolus project [15] has a simulation platform, which considers the fatigue loadof wind farm for optimization as a post-processing method.A large amount of the current control methods rely on the calculation of the dam-age either by EDL or RFC, which can be only used as post-processing tools; othermethods are based on minimization of some norms of the stress on different com-ponents of the wind turbine, which are hoped to reduce fatigue, but they are nota reliable characterization of the damage [12], [16]. Thus, in this paper we will in-troduce and compare the most recognized fatigue estimation methods, and exploredifferent alternatives with a focus on whether they can be incorporated in controlloops and thus be used in the controller synthesis directly.3.
Fatigue Estimation Methods
Some of the most recognized approaches to estimate the damage caused by fatiguewill be discussed and compared in the sequel. From a materials perspective, anextensive survey for homogeneous materials was done in [17]. In the wind turbinecontext, [5] goes through the counting and spectral techniques used for wind turbinedesign. The perspective taken here is from a control point of view and as such wecategorize the fatigue estimation methods as follows:(1) Counting methods(2) Frequency domain or spectral methods
ATIGUE ESTIMATION METHODS COMPARISON FOR WIND TURBINE CONTROL 3 (3) Stochastic methods(4) Hysteresis operatorIn all cases, we assume that the input signal is obtained from time history of theloading parameter of interest, such as force, torque, stress, strain, acceleration, ordeflection [18].3.1.
Counting Methods.
Cycle counting methods are algorithms that identifyfatigue cycles by combining and extrapolating information from extrema (maximaand minima) in a time series. These algorithms are used together with damageaccumulation rules, which calculate the total damage as a summation of incre-ments. The most popular method among the counting methods is the so-calledrainflow counting (RFC) method, jointly with the Palmgren-Miner rule of lineardamage accumulation to calculate the expected damage. The Palmgren-Miner ruleis the most popular due to its simplicity; however, by applying it one assumes afixed-load, neglecting interaction and sequence effects that might have a significantcontribution to the damage, e.g., [19] for tests with random loading.Other cycle counting methods include: peak-valley counting (PVC), level-crossingcounting (LCC), range counting (RC), and range-pairs counting (RPC); for moredetails see [20] and [21]. Here, we will focus on the RFC method, which is the mostwidely used and the most accurate in identifying the damaging effects caused bycomplex loadings, [22]. The rainflow counting method, first introduced by Endo[23], has a complex sequential and nonlinear structure in order to decompose arbi-trary sequences of loads into cycles, and its name comes from an analogy with roofscollecting rainwater to explain the algorithm, sometimes also referred to as pagodaroof. A figure depicting the described procedure is shown below in Figure 1.
StructuralModel TimeHistory RainflowCount StressRangeHistogram PalmgrenMinerRule
FatigueLife
Figure 1.
Rainflow counting damage estimation procedure.For many materials there is an explicit relation between number of cycles to failureand cycle amplitude, which is known as S-N or W¨ohler curves, given as a line in alog-log scale as s k N = K, (3.1)where k and K are material specific parameters and N is the number of cycles tofailure at a given stress amplitude s . Then, for a time history, the total damageunder the linear accumulation damage (Palmgren-Miner) rule is given as J.J. BARRADAS BERGLIND AND RAFAEL WISNIEWSKI D ( T ) = N ( T ) (cid:88) i =1 ∆ D i = N ( T ) (cid:88) i =1 N i , (3.2)for damage increments ∆ D i associated to each counted cycle, N i the number ofcycles to failure associated to stress amplitude s i , and the number of all countedcycles N ( T ). Taking the S-N curve relationship in (3.1), we can rewrite (3.2) as D ( T ) = N ( T ) (cid:88) i =1 s ki K . (3.3)Different RFC algorithms have been proposed such as [24] and [25], with differentrules but providing the same results. A way to implement the RFC algorithm isusing the Rainflow toolbox introduced in [26]. An example is presented below,using the wind turbine model from the standard NREL 5MW wind turbine [27],running is closed-loop with standard pitch and torque controllers. The input usedfor the comparison is a time series of the tower bending moment extracted afterthe simulation of 600 seconds. The results are presented on Figure 2. On thetop the input stress is shown, and in the bottom part the instantaneous damageand the accumulated damage are shown. For our example, we will let k = 4 and K = 6 . × as in [28], where the value of k is adequate for steel structures. Forthis example, the instantaneous damage was extrapolated to its causing time, suchthat it can be plotted in the right time scale instead of the reduced turning-pointscale. M t [ N m ] Time [s]Fatigue Damage Estimation using RFC0 100 200 300 400 500 60001234 x 10 −8 D i [ − ] Time [s] 0 100 200 300 400 500 60001234 x 10 −7 D a c [ − ] Time [s]
Figure 2.
Rainflow counting algorithm example, using the tool-box from [26].Other outputs provided by the toolbox in [26] are amplitude and cycle mean his-tograms, as well as the so-called rainflow matrix (RFM), from which the number
ATIGUE ESTIMATION METHODS COMPARISON FOR WIND TURBINE CONTROL 5 of counted cycles with a given amplitude and mean value are obtained from thegiven stress history. Since the RFM will play a role further on this paper, we willelaborate on its construction. Load signals can be discretized to a certain number oflevels, allowing an efficient storage of the cycles in a so-called rainflow matrix, whichis an upper triangular matrix by definition. Consequently, cycle amplitudes andmean values can be grouped in bins, such that the cycle count can be summarizedas a matrix (for details see [29], and Chapter 2 in [21]); sometimes this matrix isshown transposed. The rainflow matrix for the aforementioned example is depictedon Figure 3 for 10 bins, where cycle mean is on the y − axis, cycle amplitude in the x − axis and number of cycles in the z − axis. Cycle Amplitude C yc l e M ean Rainflow Matrix 1 2 3 4 5 6 7 8 9 1012345678910 24681012
No. of cycles
Figure 3.
Rainflow Matrix, using the toolbox from [26].Lastly, NREL has a an estimator of fatigue-life called
MLife (currently in alphaversion, an improvement on
MCrunch [30]), which runs the RFC algorithm of [26].
MLife calculates fatigue life for one or several time series, incorporating the Good-man correction to the damage calculation (to account and correct for the fixed-loadassumption). These calculations include short-term damage equivalent loads anddamage rates, lifetime results based on time series, accumulated lifetime damage,and time until failure [31].3.2.
Spectral Methods.
An alternative to counting methods are the so-calledspectral or frequency domain methods [32], which assume narrow band processesand calculate the lifetime estimate by using an empirical formula that uses thespectral moments of the input signal; the aim of these methods is to approximate therainflow density of the RFC algorithm. This procedure is depicted on Figure 4. It isworth mentioning that some of these methods are based on empiric formulas, beingessentially black-box and may be restricted to Gaussian histories. A comparison ofdifferent spectral methods was carried out in [33].
J.J. BARRADAS BERGLIND AND RAFAEL WISNIEWSKI
StructuralModel Fatigue ProbabilityDensityFunction
FatigueLife
PowerSpectralDensity
BLACKBOX
Modeller
Figure 4.
Spectral methods damage estimation procedure.Spectral methods are based on statistical information of the signal of interest, i.e.,its spectral moments. Following from [32] and [14], the m th spectral moment of theprocess x ( t ) is defined as λ xm = 1 π ∞ (cid:90) f m · S x ( f ) df, (3.4)where S x ( f ) is the power density (PSD) of the process, with the following properties λ x = σ x , λ x = σ x and λ x = σ x . (3.5)In other words, the variance of the process is given by λ x , the variance of the process’first derivative is then given by the second moment, and lastly the variance of theprocess’ second derivative is given by the fourth moment. Consequently, followingthe results in [34] and [35] the damage rate for narrow-banded Gaussian stresshistories is given by d (cid:102) = 12 π (cid:114) λ λ K (cid:16) (cid:112) λ (cid:17) k Γ (cid:18) k (cid:19) , (3.6)where Γ( · ) corresponds to the gamma distribution, and k , K are the S-N parametersused in the RFC case. In [34], the authors proposed an estimate of the expectedfatigue damage rate given as the narrow-band approximation augmented with acorrection factor to account for the process not necessarily being narrow-band E [ d ] ≈ d (cid:102) · (cid:0) b + (1 − b ) α k +12 (cid:1) (3.7)with b = ( α − α ) (cid:2) .
112 (1 + α α − ( α + α )) e . α + ( α − α ) (cid:3) ( α − (3.8)and ATIGUE ESTIMATION METHODS COMPARISON FOR WIND TURBINE CONTROL 7 α = λ √ λ λ , α = λ √ λ λ . (3.9)In [14] and [28], the numerical integration of the spectral density as in (3.5) isavoided, since the spectral moments are computed by means of polynomial eval-uation and differentiation, involving a logarithm and an inverse tangent function.This allowed the method to be incorporated in the control loop.In order to compare the spectral method with the example presented in the previoussection, the spectral moments λ = ( λ , λ , λ , λ ) of the time series were calculatedusing the WAFO toolbox [36] (through integration) λ = { . E , − . E , . E , . E } , (3.10)and then the damage was computed using the Benasciutti approximation, using the Matlab script in Appendix B.3. of [28], such that d B = 4 . E − , (3.11)which is a little off compared to the RFC case; this can be explained by the fact thatwe need to scale the damage rate according to the geometry of the system, whichis generally unknown. However, the obtained damage rate can be normalized to beused for control purposes, for details see [14]. In [37] the RFC method is comparedwith the spectral method using Dirlik’s formula (which approximates the rainflowdensity, see [38]) for fatigue analysis of several components of wind turbines, whereit is concluded that spectral methods work very well in some cases, but ratherpoorly in others due to the narrow band assumption. However, spectral methodsdo have the advantage of conveniently relying on spectral information that is easierto estimate from limited data.3.3. Stochastic Methods.
In [39], a thorough survey of stochastic methods for fa-tigue estimation in materials is presented, including reliability-inspired approaches,evolutionary probabilistic approaches and models for random fatigue crack growth.Modeling fatigue as a stochastic process makes sense due to the random nature offatigue, which becomes more obvious under time-varying random loading.Due to the broadness of this class of methods, we will focus on one example of theevolutionary approach. Following [39], by introducing the hypothesis that the pro-cess is Markovian, such that future outcomes only depend on present information,disregarding the past. This way, we will have a random process with only forwardtransitions, E → E → · · · → E k → E k +1 · · · → E n = E ∗ , (3.12)where E denotes a damage-free state and E ∗ characterizes the ultimate damageor destruction. Letting P k ( t ) be the probability that the specimen at time t is on J.J. BARRADAS BERGLIND AND RAFAEL WISNIEWSKI state E k (notice that the state transitions are discrete, while the time evolution iscontinuous), then we obtain the following system of differential equations dP ( t ) dt = q P ( t ) dP k ( t ) dt = q k P k ( t ) + q k − P k − ( t ) , k ≥ , (3.13)or in shorter notation dP k ( t ) dt = QP k ( t ) , k ≥ , (3.14)which corresponds to a Markov chain (MC) with intensity or transition matrix Q .Markov chains are well studied and have been successfully used in control settings;however, a shortcoming of this approach is that it is assumed that the intensitymatrix Q is not generally known. It could be assumed that the intensities areobtained from physical experiments, but this would correspond to a certain load;so, if the load changes, the parameters will change as well. However, the elements of Q could be identified, using for instance recursive maximum likelihood identificationmethods, in order to capture the shifts in the load introduced by the controller.In the present, for the sake of comparison, we will make use of the equivalencein [29], where a method to convert between rainflow matrix to a Markov matrix ispresented. As an example, we take the rainflow matrix depicted in Figure 3 and usethe WAFO toolbox to convert it into a Markov matrix, and obtain its correspondingintensity matrix Q . Additionally, the MC is simulated for as many steps as thelength of turning points of the RFC algorithm, such that the instantaneous damagecan be reconstructed in the appropriate time instances. The simulation of the MCis presented on Figure 5, where the size of the MC corresponds to the number ofbins of the RFM.
20 40 60 80 100 120 140 160 18012345678910
Steps, k
Markov Chain Simulation E i ( k ) Figure 5.
Markov Chain simulation, using the WAFO toolbox [36].
ATIGUE ESTIMATION METHODS COMPARISON FOR WIND TURBINE CONTROL 9
Then, the damage evolution is scaled according to the RFM amplitudes, and after-wards the Palmgren-Miner rule is used. One of the possible realizations is comparedagainst the RFC method on Figure 6. Note that many realizations for the damageevolution are possible, since the MC in (3.14) is governed by probabilities. −8 Damage, D i R F C m e t hod Time[s] 0 200 400 60001234 x 10 −7 Accumulated Damage, D ac Time[s]0 200 400 60000.511.52 x 10 −8 M a r k o v C ha i n Time[s] 0 200 400 60001234 x 10 −7 Time[s]
Figure 6.
RFC versus Markov chain method damage comparison.3.4.
Hysteresis Operator.
As mentioned in [24] and [18], the purpose of the RFCmethod is to identify the closed hysteresis loops in the stress and strain signals. In[40], an incremental method for the calculation of dissipated energy under randomloading is presented, where the dissipated hysteresis energy to failure is used as thefatigue life parameter; the physical interpretation is that as some of the energy isdissipated, certain damage is introduced to a material or structure.In [41] an equivalence between symmetric RFC and a Preisach hysteresis operator isprovided. This is a very useful result, since it gives the opportunity to incorporatethe fatigue estimation online in the control loop. Additionally, this method isstrongly related to the physical behavior of the damaging process as explained in[42]. If one associates values to individual cycles or hysteresis loops, it is beingassumed that the underlying process is rate independent, thus meaning that onlythe loops themselves are important, but not the speed with which they are traversed;in other words, what causes the damage is the cycle amplitude and not how fastit occurs. Rate independent processes are mathematically formalized as hysteresisoperators, see [43], [44] [41].The aforementioned equivalence in [41] between symmetric rainflow counting (RFC)and a type of Preisach operator, is given as D ac ( s ) = (cid:88) µ<τ c ( s )( µ, τ ) N ( µ, τ ) = Var( W ( s )) . (3.15)where the left-hand side corresponds to the damage given by the RFC with c ( s )( µ, τ )being the rainflow count associated with a fixed string s = ( v , · · · , v N ), countingbetween the values of µ and τ , and N ( µ, τ ) denotes the number of times a repetitionof the input cycle ( µ, τ ) leads to failure.The right-hand side of (3.15) is the variation of a special hysteresis operator, namelythe Preisach operator defined as, W ( s ) = (cid:90) µ<τ ρ ( µ, τ ) R µ,τ ( s ) dµdτ. (3.16)with density function ρ ( µ, τ ), interpreted as a gain that changes with the differentvalues of µ and τ , being a function of N ( µ, τ ). To interpret the right-hand side of(3.15) we will need to introduce the relay operator R µ,τ ( s ) = R µ,τ ( v , · · · , v N ) =( w , · · · , w N ), where its output is given by w i = , v i ≥ τ, , v i ≤ µ,w i − , µ < v i < τ. (3.17)with µ < τ and w − ∈ { , } given. The relevant threshold values for the relays R µ,τ in the Preisach operator W ( s ) then lie within the triangle P = (cid:8) ( µ, τ ) ∈ R , − M ≤ µ ≤ τ ≤ M (cid:9) . (3.18)known as the Preisach plane. The variation operator Var( · ) is a counting elementdefined as Var( s ) = N − (cid:88) i =0 | v i +1 − v i | (3.19)for an arbitrary input sequence s = ( v , · · · , v N ); so essentially, Var( W ( s )) rep-resents the counting between the thresholds µ and τ , weighted by certain gain ρ .Notice as well, that the limit under the integral defining the Preisach operator in(3.16) is congruent with the RFM being upper triangular.In order to apply this fatigue estimation method to the previous example, thePreisach operator W ( s ) was approximated as a parallel connection of three relayoperators ATIGUE ESTIMATION METHODS COMPARISON FOR WIND TURBINE CONTROL 11 H ( s ) = (cid:88) i ν ( µ i , τ i ) R µ i ,τ i ( s ) , (3.20)for i = { , , } . The thresholds were set to ( µ , τ ) = ( − . M, . M ), ( µ , τ ) =(0 . M, . M ) and ( µ , τ ) = ( − . M, − . M ) corresponding to uniform dis-cretization, where M is the bound for the Preisach plane in (3.18) calculated as M = max { min { s } , max { s }} . The initial conditions of the relays were given ac-cording to the following condition: w − ( µ i , τ i ) = (cid:26) , µ i + τ i < , , µ i + τ i ≥ . (3.21)Lastly, since the Preisach density function ρ ( µ, τ ), captured by the weightings oneach relay ν ( µ i , τ i ) is unknown, the individual weightings of each relay were normal-ized such that ν = α , ν = α , ν = α for ν + ν + ν = 1. Thus the accumulateddamage can be written in closed form as D ac ( s ) = Var ( H ( s )) , (3.22)where we let the input signal s be the tower bending moment from the previousexamples.A comparison between the RFC, using the procedure described before, and the hys-teresis method obtained by (3.22) is shown in Figure 7. Even though the magnitudein the damage given by the hysteresis method is off scale, this could be resolvedby identifying the Preisach density, see [45] for an identification procedure and asummary of other identification methods.It is worth mentioning that the results in (3.15) apply to symmetric RFC. Asmentioned in [42] not all RFC methods are symmetric; however, for symmetricRFC the so-called Madelung rules apply, i.e., deletion pairs commute, meaningthat it does not matter the order in which the sequences are deleted. However, ifthe primal concern is to apply this technique online, no deletion is actually possiblesince the estimation is done directly on measurements.3.5. Crack Growth approaches.
Another alternative for fatigue estimation isthe crack growth approach, which can be both addressed from a deterministic view-point using Paris’ law ([46]), or a stochastic perspective using for example jumpprocesses, diffusion processes or stochastic differential equations (SDEs). However,in the crack growth approach a microscopic scale perspective is taken, thus makingit difficult to transport to system level. We refer the interested readers to [47], [17]and the references therein. −8 Damage, D i R F C m e t hod Time[s] 0 200 400 60000.511.522.533.5 x 10 −7 Accumulated Damage, D ac Time[s]0 200 400 60000.20.40.60.81 H ys t e r e s i s M e t hod Time[s] 0 200 400 6000246810 Time[s]
Figure 7.
RFC versus Hysteresis method damage comparison.4.
Methods Comparison and Discussion
The aforementioned fatigue estimation methods share certain relations betweeneach other. Firstly, there is an equivalence between the rainflow matrix and theMarkov matrix or intensity of the Markov chain. Moreover, both have zeros be-low the diagonal, which is also the case for the Preisach plane P in the Hysteresismethod. The Spectral methods are related to RFC, since their intention is to ap-proximate the rainflow density by spectral formulas, and they also relate to thestochastic methods in that their goal is to approximate certain density function.The hysteresis method is strongly related to the RFC, since the RFC actually iden-tifies the closed hysteresis loops by counting cycles. A sketch of these relationshipsis depicted on Figure 8.Furthermore, a method comparison summary is shown on Table 1, where advantagesand disadvantages are presented for each method previously introduced.For the next comparison part we will focus just on the MC instead of the whole sto-chastic methods class, which is quite broad. The accumulated damage provided bythe RFC, MC and Hysteresis methods are compared in Figure 9. The damage givenby the hysteresis was normalized, such that it matches the accumulated damage ofthe RFC. The spectral method example could not be included, since the methoddelivers the damage rate itself and not instantaneous measurements. For the RFCand the MC method presented here, the instantaneous damage is given every timean extrema occurs and zero elsewhere, which is exactly what the hysteresis does, ATIGUE ESTIMATION METHODS COMPARISON FOR WIND TURBINE CONTROL 13 M ean Amplitude
Hysteresis Rainflow Counting Markov Chain
Preisach Plane Rainflow Matrix Intensity / Markov Matrix
Spectral methods
Amplitude N o . cyc l e s Density, A m p li t ude , Cumulated cycles, A m p li t ude , Probability Density Function
Figure 8.
Relationship between the compared methods.
Method Advantages Disadvantages
Rainflow Active Standard (ASTM E1049) Post-processingCounting Widely used Relies on linear accum. hypothesisAlgorithmic, very non-linearSpectral Can be used for control Black-boxBased on statistical measures Narrow-band approximationStochastic Account for random loading Parameters generally unknownMethods Could be used for prediction May involve PDEs, SDEsVery abstract formulationHysteresis Online estimation Typically hard control problemStrong physical interpretation Density generally unknownClose mathematical form Approximation may be needed
Table 1.
Methods advantages and disadvantages.i.e., hold the value between certain thresholds. All these techniques can be used aspost-processing tools, however not all of them can be used in the control loop. Abrief summary is presented on Table 2, where it is reported if the methods can beimplemented directly online or indirectly, i.e., not using measurements. The spec-tral methods are included indirectly, since they were included in the loop throughtransfer functions and not based on measurements. The Markov chain could beincluded online if the intensity matrix is parametrized with respect to the controls,which may not be realizable. 5.
Conclusions
The literature regarding fatigue estimation methods is vast, since fatigue is an entirediscipline by itself. The aim of the present paper is to provide a guide to the most
Method Online Indirect Comments
RFC - - Only Post-processingSpectral - X Moments obtained by transfer functionHysteresis X - Approximation may be needed
Table 2.
Methods applicability for control. −7 Time[s]Normalized Accumulated Damage D a c [ − ] RFCHysteresisMarkov Chain
Figure 9.
Normalized accumulated damage for different estima-tion methods.recognized methods, which were assembled in four groups. These methods werepresented and compared, from a control perspective in a Wind Turbine settingby estimating the damage from a tower bending moment time-series. A chartdescribing their advantages and disadvantages is presented on Table 1 and theirapplicability to control in Table 2. We also attempted to shed some light on theunderlying relations between them.Summarizing, the most widely used and standardized method is the RFC, butits algorithmic nature restricts its usage primarily as a post-processing tool. Thespectral methods provide an alternative by trying to emulate the rainflow densityfunction, they are based on statistical measures that are easier to calculate, butthey are black-box and restricted (mainly) to narrow-band processes. The stochas-tic methods can accommodate the randomness of fatigue, but their constructionis abstract and complicated, often involving stochastic or partial differential equa-tions, and their parameters may need identification. The hysteresis method can beimplemented online, acting on instantaneous measurements, but its complex and
ATIGUE ESTIMATION METHODS COMPARISON FOR WIND TURBINE CONTROL 15 non-linear nature results in hard control problems. In general, one could say thatthe controller will influence the loading in the wind turbine components, and thusfor implementing any of these techniques in the control loop, variable load shouldbe considered by the estimation method in some sense.6.
ACKNOWLEDGEMENT
This work was partially supported by the Danish Council for Strategic Research(contract no. 11-116843) within the ‘Programme Sustainable Energy and Envi-ronment’, under the “EDGE” (Efficient Distribution of Green Energy) researchproject.
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