The lack of efficient methods for de-trending of wind speed resource data may lead to erroneous wind turbine fatigue and ultimate load predictions. The present paper presents two models, which quantify the effect of an assumed linear trend on wind speed standard deviations as based on available statistical data only.
The first model is a pure time series analysis approach, which quantifies the effect of non-stationary characteristics of ensemble mean wind speeds on the estimated wind speed standard deviations as based on mean wind speed statistics only. This model is applicable to statistics of arbitrary types of time series.
The second model uses the full set of information and includes thus additionally observed wind speed standard deviations to estimate the effect of ensemble mean non-stationarities on wind speed standard deviations. This model takes advantage of a simple physical relationship between first-order and second-order statistical moments of wind speeds in the atmospheric boundary layer and is therefore dedicated to wind speed time series but is not applicable to time series in general.
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Wind speed statistics are usually denoted as resource data, and such statistics comprise information on mean wind speed, wind speed standard deviation and maximal/minimal wind speed realizations within each of the recording periods defining the measuring campaign. In the present paper, the wind speed standard deviation—or in its normalized form, the turbulence intensity—associated with such resource data is in focus.
Wind speed fluctuations in the atmospheric boundary layer (ABL) contain virtually all scales and are traditional classified in micro-scales (turbulence), convective scales (buoyancy-driven diurnal variations) and macro-scales (weather systems and planetary waves). Real (finite length) wind speed time series are therefore indeed non-stationary, and the concept of stationary wind speed time series is therefore an abstraction. However, for practical reasons, the wind speed standard deviation is nevertheless traditionally computed as based on assumptions of ergodicity and stationarity of the wind speed process in question. Likewise, turbulent wind fields are traditionally described and modelled as a stationary zero-mean multi-dimensional stochastic process imposed on a time-invariant mean wind speed field using Reynolds de-composition. From a mathematical point of view, this formulation has the advantage of being significantly more tractable compared with a non-stationary formulation.
The assumptions of ergodicity and stationarity basically mean that the ideally requested ensemble statistics are approximated by the statistics of sampled time traces of the process. As a consequence, a possible time-varying (ensemble) mean is replaced by its time average over the time span considered; and a time-varying (ensemble) variance is replaced by the sum of the time-averaged (ensemble) variance over the time span considered and the variance of the time-varying (ensemble) mean over the same time span. Thus, analysing non-stationary time series uncritically, pretending stationarity, eventually leads to erroneous results; e.g. distortion of the ‘true’ turbulence spectra, which in turn may be detrimental for prediction of turbulent-driven fatigue and ultimate loading of structures exposed to fluctuating ABL wind fields.
In the general non-stationary case, all statistical characteristics are time dependent. However, an often used simplifying approach is to consider the high frequency fluctuating part of the time series as stationary ‘around’ a trend, which essentially means that the inherent non-stationarity is assumed to be attributed to a time variation in the (ensemble) mean value only, whereas the (ensemble) variability around this mean value is assumed time independent. This in turn means that the directly derived variance, apart from the true (micro-scale) turbulence variance, contains a macro-scale contribution originating from the assumed mean wind speed being off the true ensemble mean value. With this setting, a proper analysis of micro-scale turbulence characteristics—e.g. in relation to short-term wind gust magnitude prediction or assessment of wind turbine fatigue loading—requires elimination of possible macro-scale contributions in the measured time series characteristics.
For simulation of fatigue loading of wind turbines erected at a given site, it is particularly important to separate the directly measured wind speed variability in a micro-scale contribution and a macro-scale contribution, respectively.[2, 3] This is because the ‘total’ turbulence standard deviation (i.e. the directly measured standard deviation including both the micro-scale and macro-scale variabilities), when used as input to turbulence generators that assume stationary turbulence, will be distributed over the entire frequency range according to the presumed turbulence spectrum. However, in reality, only the stationary part of the turbulence (i.e. the micro-scale part) displays the prescribed frequency behaviour, whereas the contribution originating from mean level changes (i.e. the macro-scale variability) affects the low-frequency fluctuations only.[2, 3] Neglecting de-trending may consequently result in significant errors in the prediction of fatigue lifetime consumption.
An illustrative example is the computation of fatigue damage of a wind turbine blade. Fatigue damage, and thereby fatigue lifetime, is a highly non-linear process dictated by material stress cycles imposed by the external fluctuating loading. For blade flapwise fatigue loading, atmospheric turbulence is by far the dominating load driver. Fatigue modelling associated with such stochastic loading is a challenging task, and failure criteria as well as fatigue degradation assessment models are in general purely empirical. The most widely used approach for fatigue damage estimation is the linear Palmgren–Miner damage accumulation approach, combined with an S–N curve characterizing the material fatigue performance. This is a crude approach, basically neglecting effects of load cycle sequencing and assuming that partial damages, each characterized by a specific load cycle amplitude, can be linearly accumulated.[3, 4] Applying this approach for fatigue damage estimation, the blade fatigue damage is roughly proportional to the turbulence intensity raised to the power m, where m denotes the Wöhler exponent characterizing the relevant S–N curve. For blade composite materials, m may easily exceed 10. With m equal to 10 and a realistic relative trend contribution to the true micro-scale turbulence intensity of the order of 10%, the trend induced increase in turbulence intensity causes the blade flapwise fatigue loading to increase by a factor of approximately 2.5, corresponding to a reduction of the blade fatigue lifetime of the order of 60%.
Similar results were obtained in Hansen using a simple heuristic fatigue load model, where only the fatigue loading caused by the stochastic part of the wind field was considered. This analysis demonstrated reductions in estimated tower fatigue life consumption of the order of 40% when using de-trended turbulence characteristics computed from measured full-scale high sampled time series. As no existing fatigue models are able to account for non-stationary stochastic load processes, de-trending is a mandatory pre-processing step when using fatigue simulation approaches based on a stationarity assumption.
In the present context, we adopt the simplification of stationary* (ensemble) variance and further define the trend as composed of the sum of all Fourier modes present in the signal with wavelengths longer than the time span of the time signal considered. This is equivalent with the trend definition proposed in Kaimal and Finnigan.
In case high sampled data of the time series are available, the type of trend within each particular recording period can in principle be identified using a Fourier de-composition technique including Fourier modes with wavelengths longer than the observation period and therefore consistent with the trend definition or, alternatively, by using more classical de-trending approaches like first differencing approaches (e.g. autoregressive integrated moving average models), higher order differencing approaches, curve-fitting approaches in which the trend is described as a deterministic function in time (e.g. obtained from a least-squares fit to a parameterized curve or from physical considerations) and digital filtering in which the trend is described as a suitable linearly filtered version of the original time series. Based on the identified trend, the raw standard deviation estimate, as computed based on the time-averaged ensemble mean, is easily corrected to give the true micro-scale ensemble turbulence standard deviation. In practice, a linear variation (trend) of the ensemble mean wind speed over each considered recording period is often assumed and estimated using a least-squares approach.
In case high sampled time series data are not available, traditional de-trending approaches as described earlier are not applicable. De-trending of resource data therefore calls for alternatives, which is the topic of the present paper. This type of de-trending has previously been treated in Hansen using a heuristic approach involving an empirically based calibration to certain terrain types. Contrary to the approach in Hansen, the present contribution deals with models, which quantify the effect of de-trending on estimated wind speed standard deviation as based on the available statistical data only.
Two models are presented. The first model quantifies the effect of non-stationary characteristics of the ensemble mean wind speed on the estimated wind speed standard deviation as based on observed mean wind speeds only. The second model uses the full set of information and includes, in addition, estimated (directly computed) raw wind speed standard deviations to estimate the effect of ensemble mean non-stationary characteristics on the estimated wind speed standard deviations. The developed techniques allow in principle for identification of both linear and non-linear trends, as well as for quantification of the associated consistent estimates of the de-trended micro-scale turbulence standard deviations.
The capabilities of the two models are analysed by comparing their predictions with predictions obtained from a traditional linear de-trending of high sampled time series data. A huge amount of wind speed time series data, extracted from ‘Database on Wind Characteristics’, is used for this comparative analysis, and the selected data represent a broad variety of different terrain types.
2 STATEMENT OF PROBLEM
We consider a sequence of consecutive time series recordings of equal length T and assume that only the statistics (i.e. mean value and variance) of these are known. We seek a procedure for de-trending of such data based on the available statistical information only. The proposed algorithms will be based on the conjecture that de-trending of a variance measure, associated with a given time series, can be performed using only the statistics of the particular time series in question together with the statistics of the two ‘neighbouring’ time series (i.e. the two time series that in time ‘surround/embed’ the time series in question). This assumption is motivated by the fact that the wind speed autocovariance function predominantly is a decreasing function of the involved time shift.
2.1 Definition of trend
It is characteristic that the problem in hand usually does not offer any detailed information on the nature of a possible trend in available time series. For this reason, we will consider trend as the special kind of non-stationary behaviour, where the mean value exhibits a linear trend, whereas the variance of the stochastic process in question, with respect to this linear varying mean value, is assumed constant in time. With this setting, the stochastic process in question is a stationary process when referring to the linear varying ensemble mean value.
We thus consider the total (unknown) signal as composed by a sum of a non-stationary (i.e. time dependent) ensemble mean value and a zero mean stationary stochastic process. We will assume the stationary part of the signal to be uncorrelated with the non-stationary part (which may be caused by a frontal passage, extremely large turbulence eddies, etc.), which is justified by the presence of a ‘spectral gap’ in ABL wind speed spectra separating the micro-scale turbulence from the synoptic scale peaks. The spectral gap is at scales comparable with the characteristic recording length for resource data (i.e. of the order 10 min)—in fact the location of the spectral gap has motivated the chosen recording length of resource data. With the adopted assumptions, the variance of the total signal is thus evaluated as the sum of the variance of the stationary and non-stationary parts, respectively.
In the proposed approach, we will approximate the non-stationary ensemble mean part by a time-dependent polynomial expression, 〈vti(t)〉, yielding the following expression for the measured signal vti(t)
where 〈*〉 denotes the ensemble mean operator, vsi(t) denotes the stochastic component associated with the ith measurement and N is the number of consecutive recording intervals of length T. In order to provide the link between the polynomial ensemble mean description and a conventional linear trend description, we will define the requested linear trend in terms of that particular linear mean value slope, which results in the least possible variance of the total measured signal when corrected by a zero mean linear trend in the ensemble mean value. The trend-corrected signal, vci(t), is thus given as
and we thus seek the trend slope (gradient), hi, that yields the lowest possible variance of vci(t). The stochastic part is uncorrelated with the non-stationary part of the signal as well as with the introduced linear trend. The variance of the trend-corrected signal defined by equation (2) is thus expressed as the sum of the variance of the stationary stochastic part and the variance of the trend-corrected non-stationary part. As the variance of the stochastic part in this context is a constant, the sought linear trend is therefore the particular trend, which minimizes the variance of the trend-corrected non-stationary part. The variance of the trend-corrected non-stationary part, which is known to have the same mean value as the original signal, is given by
where subscript (ci)n refers to linear trend correction of the ith non-stationary part and denotes the measured mean associated with the ith sampling interval. Note that the linear trend introduced in equation (2) is the only class of linear relationships consistent with the known mean value . The sought slope is thus obtained from the following equation:
In the proposed models, we will operate with polynomial estimates of 〈vti(t)〉 of up to fourth order (i.e. fourth-order polynomial estimates of 〈vti(t)〉 in the first model, and third-order polynomial estimates of 〈vti(t)〉 in the second model, respectively).
Introducing a fourth-order polynomial estimate for the non-stationary ensemble mean value, 〈vti(t)〉, embedded in equation (4), we obtain the following relation between the optimal linear slope and the polynomial coefficients introduced in equation (6) (cf. APPENDIX A, equation (A.7))
The corresponding generic expression for a third-order polynomial representation of the ensemble mean is obtained directly from equation (5) by putting ai equal to zero. As seen, the optimal slope is linear in the polynomial coefficients as expected.
With the introduced definitions in place, we turn to formulation of the specific models in the next sections. Two algorithms are proposed. The first of these is based only on available information on mean values, whereas the second algorithm in addition utilizes the available variance information.
3 MODEL 1
We consider consecutive time intervals of the form [iT;(i + 1)T]. As mentioned, trends in time series express a non-stationary (i.e. time dependent) behaviour of ensemble means. To describe this non-stationary behaviour, we will assume that the ensemble mean wind speed within the ith time interval, 〈vi(t)〉, can be approximated by a fourth-order polynomial expression as
To determine the polynomial coefficients, we exploit the available mean value information and require a ‘smooth’ transition from one polynomial to the next (i.e. a smooth transition of 〈vti(t)〉 from one time interval to the succeeding time interval). This gives us the following linear system of equations in the polynomial coefficients:
where (*)’ denotes time differentiation. Introducing equation (6) into equation (7), we obtain
The derived spline-like polynomial interpolation can be applied on a statistical sequence consisting of the mean values from an arbitrary number, N, of consecutive recording intervals. However, inspection of the system of equations defined in (7) reveals that additional four equations are needed to ‘close’ the system. This is also in analogy with conventional spline interpolation and is caused by ‘end effects’, as the conditions on smooth transition between successive polynomial approximations cannot be applied there. As the required supplement, we will adopt the following ‘boundary conditions’ at the ends
Considering the polynomial approximations as deflection shapes, the boundary conditions expressed in (9) have a mechanical analogy. Considering a uniform beam, simple beam theory (i.e. Bernoulli–Euler) yields a direct proportionality between the second derivative of the deflection shape and the cross-sectional moment, and a direct proportionality between the third derivative of the deflection shape and the cross-sectional shear force. In this analogy, equation (9) prescribes zero moments and zero shear forces at the beam ends, which is specified to avoid end effects to influence the deflection shapes (i.e. the spline expressions). Although the derived ‘curve-fitting’ is applicable for an arbitrary number of observation points, we will, as mentioned, for the trend analysis restrict us to the special case N = 3.
To conclude the description, we shall briefly motivate the selected fitting expression (6) or rather the order of the selected polynomial. Basically, we aim at an order as low as possible to limit the number of parameters to be estimated. In a conventional (natural) cubic-spline approximation, the polynomial order is 3. In this framework, the polynomials are defined as extending between available known point values, and the fit is required to pass through these points. In addition, the fit is required smooth by prescribing identical first and second derivatives of successive polynomials at the points of intersection. In this case, the ‘end effects’ cause additional two conditions† to be specified to close the system.
In the present case, we do not have information on specific measuring points but have information on average quantities over some intervals. Therefore, it makes no sense to ‘force’ the fit to pass through certain points. For each considered interval, this recognition results in a replacement of two ‘passing-through-conditions’ with one integral requirement—the mean value requirement. As a consequence, additional smoothness conditions must be specified at the polynomial intersections. If a third-order polynomial has been selected for the present fitting, conditions ensuring continuity up to (and including) the second derivative must have been imposed. This gives three smoothness conditions at each polynomial intersection, which in turn means that a total of three additional ‘boundary conditions’ would have to be specified at the end points. As 3 is an odd number, the ‘symmetry’ of the end point conditions would be violated, as it makes no sense to introduce one condition linking the behaviour at the two ends. Therefore, besides constraints on the two second derivatives at the ends, we would be forced to introduce a constraint on the first derivative or on the signal itself at one of the ends. This is somewhat more restrictive—especially when no obvious conditions of this kind are available from the character of the problem. By selecting a fourth-order polynomial for the fitting, these problems are circumvented.
4 MODEL 2
Contrary to the previous algorithm described in Section 3, the present algorithm takes advantage of all the available information—i.e. includes both the mean value information and the variance information in the fitting. The expense of including the variance information in the fitting is that (second-order) nonlinearities are introduced in the system of equations defining the best possible polynomial coefficients.
Although possible, we will not formulate the present algorithm for a fit to an arbitrary number of statistical sets, each consisting of mean and variance, but rather restrict the formulation to three consecutive sets in agreement with the conjecture stated in Section 2. This restriction is partly due to notational convenience, partly motivated by the conviction that more consecutive sets than three are inferior to the suggested de-trending methodology, because the correlation between ensemble means will decrease for increasing time separation and eventually vanish for very large separations in time. However, for a spline-like formulation of the complete course of a mean value non-stationarity, a fit to an arbitrary number of statistical sets is relevant because of continuity requirements.
Driven by general lack of information on the atmospheric stability, we make the simplification that the flow, from which the available statistical information originates, is associated with neutral atmospheric conditions, whereby Monin–Obukhov scaling dictates the standard deviation of the velocity fluctuations, σ, to be proportional to the friction velocity, u*, and therefore invariant through the boundary layer. Thus,
with α being the proportionality constant. With buoyancy neglected, the logarithmic law for the vertical mean wind field further applies whereby
where z is the measuring altitude, z0 is the roughness length and κ denotes the von Kármán constant (κ ≈ 0.4). Consequences of the assumed neutral stratification are addressed in Section 6.
At this stage, we need to distinguish between on-shore and offshore upstream conditions. In the present context, the major difference between these two classes of upstream conditions is that, contrary to on-shore upstream conditions, the roughness length (and thereby the friction velocity) depends on the mean wind speed for offshore conditions, because the wave field responsible for the surface roughness depends on the mean wind speed. First, the algorithm is formulated in case of on-shore upstream conditions, and subsequently, the required modifications associated with offshore conditions are dealt with.
4.1 On-shore conditions
In case of on-shore upstream conditions, we assume identical (i.e. directionally independent) upstream roughness conditions associated with three consecutive statistical sets. This is expressed by a constant roughness length z0 for such sets. With this assumption, the turbulence intensity, C(z), is identical for three consecutive statistical sets. Combining equations (10) and (11), we obtain the following expression for the turbulence intensity:
For a particular sensor, the turbulence intensity, C(z), is seen to simplify to a constant, C, for each of the considered three statistical sets, as these relate to a common specific measuring height. This result is based on assumed constant ensemble mean values over the respective recording intervals, which is evidently not fulfilled in the present context. However, for high Reynolds numbers, invariance of turbulence intensity with mean wind speed further follows directly as a consequence of Navier–Stokes equations. Note, however, that for fast frontal passages in inhomogeneous terrain conditions, the presumed velocity invariance of C might be problematic because of the most likely associated fast change in the wind direction. Further, fast frontal behaviours are often related to cold fronts, and the turbulence in the transition zone of such fronts may, in addition to the mechanically driven turbulence effects described in equation (10), potentially be substantially affected by thermal effects. Nevertheless, we will stick to the assumption of velocity invariance of C within consecutive three statistical sets because of lack of reasonable alternatives and also because fast frontal passages are relatively rare phenomena.
For three consecutive statistical sets, we approximate the non-stationary (ensemble) mean behaviour by a third-order polynomial as
To determine the polynomial coefficients, we exploit the available mean and variance information and require a ‘smooth’ transition from one polynomial to the next. This gives us the following system of equations in the polynomial coefficients
where the first and second right-hand terms in the last relation of equation (14) are the true variance obtained from equation (12) (i.e. the variance excluding any possible trend and thus referring to a stationary stochastic wind speed process with mean ) and the variance contribution arising from the possible linear ensemble mean trend, respectively. The latter term is explained in detail in APPENDIX B.
For given three consecutive time series, the system of equations defined by (14) contains 13 unknowns—i.e. 12 polynomial coefficients and the constant C—and 12 equations provided by three mean value conditions, three variance conditions and six conditions securing the smooth polynomial transitions. Thus, at least one additional condition is needed to close the system of equations. Given the relatively restrictive assumptions leading to the system of equation (14), we aim at an over-determined system by introducing additional two conditions—one at each ‘end’ of the total time span defined by the considered three consecutive time series. This is primary to retain symmetry in the conditions constraining the sought polynomial. Guided by the considerations leading to equation (9), we adopt the following additional constraints:
The combination of relations (14) and (15) thus provides a non-linear system with one supernumerary. Introducing equation (13) in the system of equations in (14), we finally obtain
where the trend gradients, hi, have been expressed in terms of the polynomial coefficients using (A.8) and sign(*) denotes the sign operator. We see that, except for C, relation (17) is a linear system in the unknowns, although with slightly implicit characteristics due to the sign(hi) term. Note also that the right-hand side of the second equation in (17) never becomes imaginary, as the total variance always exceeds the trend-corrected variance.
The solution to (17) is obtained in an iterative manner starting with initial guesses of C and sign(hi). Analysing successive series of statistical sets, the initial guess of C may conveniently be selected as the final solution to C from the previously analysed three consecutive sets—otherwise, a reasonable value of the order of 0.95 times the average of the relevant three measured turbulence intensities may be specified as initial guess. Having specified C and sign(hi), (17) is an over-determined (linear) system consisting of 14 relations between the 12 polynomial coefficients. We denote the resulting system matrix by A, the vector of the unknown polynomial coefficients in the jth iterative step by k(j) and the corresponding jth right-hand side by b(j). With the introduced notation equation, (17) simplifies to
We are looking for a least-squares optimal solution to equation (18) in the sense that the Euclidean norm of (Ak − b) is minimized. Such a solution is readily obtained using the Moore–Penrose pseudo-inverse of the system matrix, A+, as
Because of the over-determined character of the system, the jth solution, given by equation (19), most likely results in three different new estimates of C, as the relations constituting the system of equations (18) are not necessarily identically satisfied in each iteration. The new estimates of C can be derived from (16) as
where subscript (j) indicates estimates resulting from the jth iteration. The characteristic value of C, associated with the jth iteration step, is finally defined by
where a suitable relaxation, defined in terms of the factor k, has been introduced to stabilize the iteration process. Selecting k = 0.99 has proven to work satisfactorily for the investigated data (cf. Section 6).
To establish a criterion for stopping the iterative procedure defined by equations (19)-(22), we need an error measure. Four different error measures are defined and described in APPENDIX C, and these have subsequently been tested against available data, with the threshold for the error measures selected as εt = 10− 7. Only insignificant differences in performance among the criteria have been identified when measured in terms of size of the Euclidean norm of (Ak − b) at convergence, and as criterion (C.1) seems marginally superior to the other measures in terms of robustness, we recommend adopting this criterion.
Finally, it should be noted that the convergence criterion have been supplemented by a stop criterion, in case the number of iterations exceeds 199. This stop criterion is, however, almost never activated, indicating a satisfactory convergence in the large majority of the investigated cases.
4.2 Offshore conditions
We now shift from on-shore to offshore upstream conditions where, as mentioned, the roughness length depends on the mean wind speed. This dependence is often quantified in terms of the simple Charnock relationship
with g being the acceleration of gravity and kC denoting the Charnock constant.
Introducing equation (23) into equation (11) yields
Because the roughness relationship expressed in equation (23) is primarily associated with small‡ surface waves with time scales of the order of a few minutes, we assume equation (24) to adapt, in a statistical sense, instantaneously§ to a change in the mean wind speed .
For a given recording (i.e. sensor), the height above terrain, z, is constant, and equation (24) may be solved numerically for u* using an iterative type of scheme. However, considering the approximate character of the Charnock relation, we consider an approximate solution to (24) as sufficient for the present purpose.¶ In this respect, we take advantage of the following reformulation:
and because u* is usually in the range [0.5; 1] (cf. Panofsky and Dutton), [u* − 1] is considered a small quantity. The Taylor expansion to second order of equation (25) yields
Introducing equations (25) and (26) into (24) yields the following approximate third-order algebraic equation in u*
The sought closed-form solution to equation (27) can be expressed as
with q and r defined as
In arriving at equation (28), it has been utilized that we seek real roots only and further that kC << 1.
Based on equations (10) and (28), we finally recognize that for offshore upstream conditions, the last three relations of the system of equation (14) must be replaced by
whereby, compared with the on-shore case, α just replaces C and replaces . The remaining part of the model complex is unaffected by the introduced wind speed dependent roughness in the offshore case.
For the iterative solution of the modified version of the system of equation in (14), two fundamentally different strategies exist. One strategy is initially to fix the values of once and for all based on a priori known values of the Charnock constant, the von Kármán constant and the measurement height. The resulting iterative scheme is thus analogous to the one applied for the on-shore case, however, in this case defining an iterative determination of the parameter α. This parameter is, however, known a priori with good accuracy for flat terrain conditions (Panofsky and Dutton, p. 160), which is presumed comparable with offshore conditions in this respect. With this approximation we have for the longitudinal turbulence component α = 2.39, for the transversal turbulence component α = 1.92 and for the vertical turbulence component α = 1.25. For the horizontal turbulence component, as e.g. obtained from a cup anemometer, α may be approximated by (cf. APPENDIX D)
More uncertainty is associated with the Charnock constant. This is the reason for formulating a second alternative scheme. In this scheme, the basic idea is to consider α as known a priori (possibly via equation (31)) and subsequently determine the optimal value of the Charnock constant, kC, in an iterative manner. We note that this parameter is considered constant for the considered three consecutive statistical sets. However, because of the over-determined character of the resulting system of equations, the iterative scheme will in general result in three different values of kC (one for each of the investigated three statistical sets) in analogy with the iterative scheme for C in the on-shore case. Isolating the gradient hi from equation (30), we obtain the following analogy to the system of equations (17) for the offshore case
with defined in equation (28). In general, αi is constant and thus independent of ‘i’. However, in case of cup anemometer recordings, αi varies among the considered three consecutive statistical sets through its dependence on the mean wind speed.
In analogy with the on-shore case, we see that, except for , relation (32) is a linear system in the unknowns, although with slightly implicit characteristics due to the sign(hi) term. Note again that the right-hand side of the second equation in (32) never becomes imaginary, as the total variance always exceeds the trend-corrected variance.
The solution to (32) is obtained in an iterative manner starting with initial guesses of kC and sign(hi) along the lines described for the on-shore case. The initial guess of kC may be selected as kC = 0.0167 as proposed in ESDU International. For the jth iteration, initial values of , , are estimated from equation (30) using the gradient results, hi(j − 1), obtained from the (j − 1)th iteration. The initial values are subsequently used to determine new values of the parameters kC,i(j)/(gz) from equations (27) and (28) as
In analogy with equation (22), a characteristic value, kC,(j)/(gz), defined by
is introduced and subsequently used to define from equation (28). The iteration process is continued until 1/ε(j), as defined in terms of one of the measures outlined in APPENDIX C, exceeds a suitable a priori selected threshold 1/εt. Note that if one of the measure (C.1), (C.2) or (C.3) is applied, the parameter C related to on-shore conditions should be replaced by the parameter kC related to offshore conditions.
5 TREND-CORRECTED VARIANCE
Having obtained a trend estimate based on either Model 1 or Model 2, the trend-corrected variance measure can be determined according to the following expression:
The detailed derivation of this expression can be found in APPENDIX B.
In case Model 2 is applied, there exists a simple alternative based on equations (10) and (12) for offshore and on-shore cases, respectively. For the on-shore case, the trend-corrected variance is determined as the square of the product of the resulting C(j) (i.e. the turbulence intensity associated with the idealized atmospheric flow situation) and the respective mean wind speed. Thus,
For the offshore case, the trend-corrected variance is determined as the square of the product of αi and the resulting friction velocity . Thus,
6 RESULTS AND DISCUSSION
The capabilities of the two proposed models are assessed and discussed by comparing model predictions with variance characteristics obtained from traditional linear de-trending of measured wind speed time series recorded in the ABL. A variety of sites, representing different terrain types and wind climates, have been selected for this comparison. The data material originates from Database on Wind Characteristics.
For each site, successive 10 min measurements, within a 1 year period, have been extracted for the analysis. The averaged results, associated with the verification of Model 1, are summarized in Table 1, where CRaw, CC and CM1 denote the raw turbulence intensity (i.e. turbulence intensity including trend contribution), the turbulence intensity obtained by conventional de-trending and the turbulence intensity de-trended using Model 1, respectively. For a particular site, the average is formed over all time series represented in the investigated 1 year period. The resulting turbulence intensities are therefore to be interpreted as site characteristics. In addition to the three turbulence intensity measures, the averaged relative trend contributions, ∆CC = (CRaw − CC)/CRaw and ∆CM1 = (CRaw − CM1)/CRaw, have been specified along with a measure of the goodness-of-fit of the model prediction, CM1/CC. A goodness measure of 1 indicates that the model trend prediction and the traditional linear trend identification, which in the present context is assumed to be the true trend value, are identical.
Table 1. Comparison of de-trended turbulence intensities obtained from Model 1 and from traditional linear de-trending of time series data, respectively.
De-trending with Model 1
Oak Creek, USA
Horns Rev, Denmark
Oak Creek, USA
A general observation is that the importance of de-trending varies significantly among the investigated sites, with the relative trend contribution, (CRaw − CC)/CRaw, ranging from 3% to 14%. Coming to the performance of Model 1, it is characteristic that the model significantly under-predicts the trend contribution. The spline-like approach in Model 1 thus has a tendency of producing a smoother mean wind speed ‘trajectory’ than observed in the time series data.
To further investigate the performance of Model 1, a more detailed analysis has been performed using the data from the Norwegian Sletringen site (referred to as ‘Sle’ in Table 1), which represents a very significant trend contribution. Based on all available 10 min time series, the distribution, in terms of the probability density functions (PDFs), of the three turbulence intensities introduced in Table 1 has been determined. The results are given in Figure 1. Also evaluated is the distribution of the difference between the true de-trended turbulence intensity (CC) and the de-trended turbulence intensity predicted by the model (CM1).
It is characteristic that the site turbulence intensity varies considerably within the subset of available recordings. Figure 1 shows that the performance of Model 1 apparently depends significantly on the level of the observed turbulence intensity. It is moreover seen that Model 1 accounts directly for only a limited part (i.e. approximately 60%) of the trend contribution contained in the individual recordings from the Sletringen site. However, more serious is that the model fails to predict the correct mean of the site trend contribution.
As for the capability of Model 2, analogous results are presented in Table 2 and Figure 2, respectively. Referring to Table 2, we observe that Model 2 performs very well and identifies almost exactly the true (average) trend as determined from conventional linear de-trending of the time series.
Table 2. Comparison of de-trended turbulence intensities obtained from Model 2 and from traditional linear de-trending of time series data, respectively.
De-trending with Model 2
Oak Creek, USA
Horns Rev, Denmark
Oak Creek, USA
Having validated Model 2 in terms of average turbulence intensities, we turn to a detailed analysis of the distribution turbulence intensities at a particular site. To facilitate comparison with the performance of Model 1, we have again selected the Sletringen site, and the results appear in Figure 2.
It is seen that the quality of the Model 2 predictions is independent of the size of the observed turbulence intensity; the Model 2 PDF prediction matches very well the PDF associated with the conventionally linear de-trending algorithm for the whole range of turbulence intensities present in the data material. It is therefore concluded that Model 2 performs satisfactorily in terms of statistical measures (i.e. average and PDF).
However, when it comes to the performance on individual time series, deviations between CM2 and CC are observed, and Model 2 explains the trend perfectly for only approximately 60% of the analysed time series. This is comparable with the performance of Model 1 on individual time series. However, as the agreement between the PDF of CM2 and CC is good even in the distribution tail regime, where only relatively few observations are available, it can implicitly be concluded that the performance of Model 2 most likely will be satisfactory even when tested on the basis of averages containing only a limited number of individual time series. This conjecture is supported by the fact that the goodness-of-fit of the model prediction, CM2/CC, seems uncorrelated with the amount of data available (cf. Table 2)—at least for recording time spans larger than 150 hours. Finally, we observe that the PDF of (CC − CM2) is almost symmetric around zero, which explains the very satisfactory behaviour of Model 2 when measured in terms of average turbulence intensities (cf. Table 2).
A prerequisite for the Model 2 formulation is that neutral atmospheric conditions prevail. However, as this model is characterized by being an over-determined system, deviations from the neutral atmospheric assumption do not necessarily imply significant deviations between model predictions (CM2) and the true de-trended turbulence intensity (CC). This is because the model predictions result from a balance between mean wind speed information and variance information. To investigate the sensitivity of Model 2 with respect to the atmospheric stability conditions, we have selected two sites for a detailed analysis. The two sites are Sletringen and Oak Creek, respectively.
Sletringen is situated on the Norwegian west coast, and stable atmospheric conditions dominate—most probably as a result of warm air coming from the Atlantic Ocean (heated by the Gulf Stream) to the coastal regime with a relatively cold surface due to limited sun exposure. Oak Creek, on the other hand, is dominated by unstable atmospheric conditions due to significant surface heating by the sun.
The thermal effects affect the turbulence production, however, depending on the mean wind speed. For neutral atmospheric conditions, the turbulence intensity is independent of the mean wind speed (cf. equation (12)). For unstable atmospheric conditions, the thermally driven turbulence production adds to the mechanical turbulence production, although more significantly in the low mean wind speed regime. For stable atmospheric conditions, the thermally driven buoyancy effect tends to suppress the mechanical turbulence production, although more pronounced in the low mean wind speed regime. This is illustrated in the left-hand figures of Figures 3 and 4, respectively, where the turbulence intensity at the two sites has been plotted as function of the mean wind speed.
From the right-hand figure in Figure 3, we observe that Model 2 tends to under-estimate the trend contribution for the Sletringen site, in agreement with the model interpreting the (negative) thermal turbulence production as a trend contribution. From the right-hand figure in Figure 4, we observe that Model 2 tends to over-estimate the trend contribution for the Oak Creek site by interpreting the (positive) thermal turbulence production as a trend contribution. Although the investigated thermal effects are pronounced—especially for the Oak Creek site—the predictions from Model 2 are only moderately affected, indicating that the over-determined character of Model 2 introduces sufficient flexibility in the model to assure some robustness against violation of the neutral stability assumption.
Under high wind conditions, mechanical generation of turbulence prevails, and the atmospheric stability conditions therefore tend to approach neutral conditions. We should therefore expect Model 2 to perform better under such conditions. This is also reflected in the right-hand figures in Figures 3 and 4, respectively, where the goodness measure, CM2/CC, is seen to approach 1 for increasing mean wind speed.
7 SUMMARY AND CONCLUSION
We have developed two models that allow for quantification of the effect of an assumed linear trend, originating from macro-scale changes in mean wind speed levels, on measured micro-scale turbulence standard deviations. The model input is standard statistical data from consecutive time series of equal but otherwise arbitrarily extends T. We operate with successive fits to ‘segments’ consisting of only three consecutive recording time spans rather than a ‘full’ fit involving all available recording time spans. This procedure has been chosen to save computer time and, because we consider the effects from times, separated more than T from the time span of interest, as marginal with respect to estimation of the linear trend gradient over the time span in focus. Whereas the first model predicts the trend based solely on mean wind speed observations, the second model includes in addition information on the raw wind speed standard deviation.
The inclusion of wind speed standard deviation in the modelling framework requires a relation that links mean wind speed information with information on wind speed standard deviation. To obtain a link, which can be supported by mean wind speed and wind speed standard deviation only, requires thermal turbulence effects to be excluded from the modelling and thus implicitly an assumption of neutral atmospheric conditions. This will to some extent potentially restrict the use of the second model. However, practical experience shows that the second model is relatively robust to violation of the assumed neutral stratification, which in turn means that this model can be successfully applied also for thermally affected data.
The developed models have been validated in an average sense by comparing their predictions with true trend contributions obtained from conventional linear de-trending of high sampled wind speed time series. Depending on the actual site characteristics, de-trended turbulence intensities are in average (i.e. averaged over all available results associated with a particular site) reduced in the range 3–14% compared with raw turbulence intensities.
For the selected sites, Model 1 seems to significantly under-estimate the true trend contribution, whereas Model 2 offers a significantly more precise estimate of the trend contribution. When it comes to prediction of trend contribution in individual time series, some deviations should be expected for both models. However, the analysis indicates that only a limited number of time series results should be averaged to ensure an almost correct estimate from Model 2. This is notable, because one of the important applications of de-trending is to characterize site micro-scale turbulence, e.g. associated with estimation of fatigue loading of wind turbines.
Based on the analysis, it can be concluded that the embedded constraint in Model 2, introduced by the physical link between the first and second statistical moments, has proven very efficient in the present context, and it is consequently recommended to use Model 2 for de-trending of statistical data. In a future perspective, the turbulence description in Model 2 shall be extended to include also thermal turbulence production in an attempt to further refine the model prediction. However, this extension inevitably requires availability of temperature difference statistics, which in turn limits application of the extended method to sites where such information is available.
This work was partly sponsored by the Danish Energy Agency, contract no. 33032-0085, and partly by EU, contract no. 045813. Further, the analysis has benefited from measurements downloaded from the Internet database, Database of Wind Characteristics (http://www.winddata.com/) located at DTU, Denmark. Wind field time series from the following sites have been applied: Horns Rev (ELSAM, Denmark); Middelgrunden, Oak Creek, Tobøl and Lammefjord (Risø DTU, Denmark); Skipheia and Sletringen (Norwegian University of Science and Technology, Norway); and Hanford (NREL, USA). Finally, careful reading and comments from the anonymous reviewers are acknowledged.
The optimal linear trend derived from a polynomial representation of a mean value non-stationarity
In Section 2.1, the optimal linear trend was defined as the trend resulting in the least possible variance of the total de-trended signal. This was in turn shown to be equivalent to the particular trend minimizing the variance of the trend-corrected mean value non-stationary given by (cf. equation (3))
The optimal trend slope is thus obtained from the equation
The gradient expression (A.2) can be reformulated as
Introducing equation (A.3) into equation (A.2), we obtain the following equation for the optimal trend gradient:
We now introduce the following general polynomial expression for the non-stationary ensemble mean value
where K denotes the order of the polynomial. Introducing equation (A.5) into equation (A.4) yields
For K = 4, which is applied in Model 1, equation (A.6) simplifies to
For K = 3, which is applied in Model 2, equation (A.6) simplifies to
A simple check of the derived expressions is to let the non-stationary ensemble mean be described by a first-order polynomial. With this assumption, equation (A.6) reduces to
which is evidently correct.
Correction of variance by subtraction of the contribution caused by a linear ensemble mean trend
Following the considerations in Section 2.1, the signal corrected for a linear trend, vci(t), is expressed as (cf. equation (2))
The variance of the trend-corrected signal, which is restricted to have the same mean value as the total/measured signal, is therefore given by
has been used to obtain the last identity in equation (B.4). In analogy with the considerations in Section 2.1, the optimal linear trend is defined by the equation
Now, finally introducing equation (B.6) into equation (B.4) yields
Note that the trend slope occasionally is defined in terms of a slope normalised with the mean wind speed as
In this case, the trend-corrected variance is expressed as
which is evidently correct.
To establish a criterion for stopping the iterative procedure defined by equations (19)-(22), an error measure is needed. Four possible relative measures are suggested: (i) a measure based on the relative error on unknowns squared in an aggregated sense; (ii) an alternative fragmental approach relying on a sum of the relative errors squared associated with each of the respective 13 unknowns; (iii) an approach based on a sum of relative errors squared associated with basic, directly physical interpretable, quantities only; and (iv) a measure based on the relative differences in residuals squared in an aggregated sense.
As for the aggregated approach based on the unknowns, the error measure is defined as
with ‖ * ‖2 denoting the Euclidean norm and being the vector of polynomial coefficients, k(j), extended with the parameter C(j) defined in equation (22).
As for the fragmental approach, the error measure is defined as
with kl(j) denoting the lth polynomial coefficient associated with the jth iteration. Whereas the aggregated approach may potentially be completely dominated by the numerically largest parameter, the fragmental approach essentially presents an error measure with all parameters equally weighted.
The third approach has a mathematical structure similar to the fragmental approach but considers only relevant parameters with a direct physical interpretation, i.e. the four parameters Cj and hi(j). This error criterion reads
The error criterion defined by equation (C.3) can be interpreted as a weighted variety of the error measure defined by equation (C.2), with the weights on the polynomial coefficients defined by equation (21).
Finally, for the residual-based approach, the error measure is defined by
Independent of the chosen error measure, the iteration procedure is concluded as soon as 1/ε(j) exceeds a suitable a priori selected threshold 1/εt. The selected threshold may, however, depend on the chosen error measure. Note that except for criterion (C.1), the suggested error measures may fail in case one or more of the involved parameters equal zero in iteration j. Because of the over-determined character of the system in question, this event is, however, not very likely to happen with criterion (C.4).
Approximate expressions for standard deviations of cup anemometer recordings
Cup anemometers are widely used for wind speed recordings. This sensor does, however, not resolve the conventional wind speed components but rather the magnitude of the (instantaneous) horizontal wind vector. Therefore, the relationship between the friction velocity and the observed standard deviation (cf. Equation (10)) must be adjusted accordingly.
Denoting the mean wind speed by , the longitudinal turbulence component by vl and the transversal turbulence component by vt, the cup anemometer recording, Vcup, is given by
The second-order Taylor approximation of this relation is
The mean cup anemometer wind speed recording may thus be approximated by
with and denoting the variances of the longitudinal and transversal turbulence components, respectively. The variance of Vcup is thus in turn expressed as
where E[*] denotes the mean value operator.
To reduce expression (D.4) further, we introduce an arbitrary zero mean multivariate PDF and denote it by flt(vl,vt). We thus have the following general expression for (mixed) higher order moments
Utilizing that longitudinal and transversal turbulence components, in a given spatial point, are uncorrelated, equation (D.5) simplifies to
where fl(vl) and ft(vt) denote the respective marginal PDFs. Assuming turbulent fluctuations to be zero mean Gaussian distributed, which is usually a good approximation, we have in particular the following relationships (Papoulis, p. 147)
Introducing equations (D.6) and (D.7) into equation (D.4), the variance of cup anemometer recordings may thus be approximated as
Combining equations (10) and (D.8), we can finally relate the standard deviation of cup anemometer recordings to the friction velocity for a flat and homogeneous terrain in a first order approximation as
This is usually a good approximation, as changes in the ensemble variance usually result from changes in the ensemble mean wind speed. Because of the required time span requested by the atmospheric boundary layer to adjust to the new equilibrium conditions, the diverted change in the ensemble variance will occur with some delay compared with the observed change in the ensemble mean wind speed.
These are conventionally selected as zero second derivatives.
The small surface roughness driving waves are waves with wavelengths slightly larger (~0.1 m) than those of capillary waves. These waves display characteristics that are a ‘mixture’ of the characteristics of capillary waves and gravity-dominated waves, respectively.
Assuming equation (28) to apply not only on mean wind speed level but also under non-stationary conditions (i.e. assuming equation (28) to adapt instantaneously to any changes in the wind speed V(t)) may potentially be used in an attempt to resolve non-stationary properties of (mechanically driven) σ(t). This will, however, not be pursued any further here—instead, we will, in analogy with traditional de-trending, assume σ(t) stationary within individual averaging time spans T as outlined in Section 2.1.
In any case, the approximate solution may be used as an initial guess in an iterative numerical approach.