The Rainfall Wind Turbine or Turbulence project (RW-Turb; https://hmco.enpc.fr/portfolio-archive/rw-turb/, last access: 26 November 2024), supported by the Agence Nationale de la Recherche (ANR, French National Research Agency in English), is designed to understand the long- and short-term effects of rainfall on wind power production, with simultaneous high-resolution measurements on an operational wind farm. Interested readers are directed to for an overview of the campaign. To briefly summarize this, the RW-Turb measurement campaign (at Pay d'Othe, 110 km south-east of Paris, France) consists of a meteorological mast (Fig. b on the right side) on a wind farm (jointly operated by Boralex, https://www.boralex.com/our-projects-and-sites/, last access: 26 November 2024), and JP Énergie Environnement, https://pays-othe-89.parc-eolien-jpee.fr/, last access: 26 November 2024). Figure a shows the location of the project; the nine wind turbines of the Pays d'Othe wind farm (aligned south-east and within a 4 km radius) are marked as black vertical crosses and the meteorological mast as a star (in the middle). Data from four Vestas V-90 (marked 1, 2, 8, and 9 in Fig. a) instruments are available: two are closer to the meteorological mast and two are farther from it (≈3.5 km from the mast). The five turbines of the Molinons wind farm in the north are also visible within a 5 km radius (grey vertical crosses). It should also be noted that a small groove is located just south of the mast at roughly 160 m; a larger one is to the east at roughly 100 m. Nearby the mast (i.e. within the 1 km radius), there is a small slope in the north–south direction. The meteorological mast consists of two sets of optical disdrometers (OTT Parsivel², 30 s, not used in current study), 3D sonic anemometers (ThiesCLIMA, 100 Hz), and a mini meteorological station (1 Hz) at heights of roughly 45 and 80 m (managed by the Hydrology Meteorology and Complexity laboratory of the École des Ponts ParisTech – HM&Co, ENPC).

Technical and working information on the turbine can be found in . The turbines have a rated power of 2.0 MW, which is pitch-regulated with variable speed. The hub height of the turbines is 80 m, which is closer to the vertical height of the upper set of devices on the mast (≈78 m). The turbines have a cut-in wind speed of 4 m s⁻¹ and a rated wind speed of 12 m s⁻¹. This can be seen in the power curves in Fig.  (last row), where the turbine register power at the cut-in speed maintains the rated power of 2000 kW after the rated wind speed. The cut-out speed of the turbine is 25 m s⁻¹ (the extreme x-axis point of the power curves): this is the speed at which a turbine stops registering power. Generally, the turbines register positive values of wind power. However, when the power retrieved from wind is less than what is required to work a turbine, it registers negative power. This can be seen in the power curves as clusters around 0. Along with the wind power, the turbine also provides information on the local velocity (at a sampling measurement rate of 15 s), which is used for internal regulation; this is used to plot the power curves in Fig. . The wind power data used for the studies come from the four turbines of Boralex – 1 and 2 are located closest to the mast (as can be seen in Fig. b), and 8 and 9 are located at the farthest end (at a sampling frequency of 15 s).

The temporal evolution and power curves (power vs. velocity, with the expected curve provided by the manufacturer in red) for the turbines are shown in Fig.  for 3 months (from 1 January to 1 April 2021). There are instances where the turbine failed to produce any power and had to consume energy for its basic operation. This results in negative data values, and for realistic analysis they were considered to be zero. This is why there is clustering of points at zero in the power curve (Fig. e–h).

In addition to the empirical power provided by turbines, the theoretical power available for extraction can be obtained by 1Pa=12ρAv3Cp, where ρ is the air density at the wind turbine height (hhub), A the swept area of the turbine rotor, v the wind velocity (m s⁻¹) approximated at the turbine height, and Cp the power coefficient or Betz coefficient (for the Vestas V-90 instruments examined here, hhub=80 m, A=6362 m², the rated power is 2 MW, and Cp is 0.593). Here, Pa is estimated at the same sampling rate as that of Pt (15 s) despite the 3D sonic anemometer and mini meteorological station registering data at finer sampling rates.

The value of air density is often approximated as 1.255 kg m⁻³ (standard value at sea level 15 °C). However, it is known to show fluctuations and is reported to have an effect on power generation at varying levels . For the purpose of this analysis, air density was considered a varying quantity and estimated using the current official formula of the International Committee for Weights and Measures (CIPM), referred to as the CIPM-2007 equation, which accounts for humidity : 2ρ(T,P,Hr)=PMaZ(T,P,Hr)RT(K)×1-xv(T,P,Hr)1-MvMa, where T (°C), P (Pa), and Hr (0≤Hr≤1) are the temperature, pressure, and humidity from the meteorological station at hhub. The other derived parameters are the following:

T(K) the air temperature (K; from T);

Spectral analysis is widely used for characterizing scaling properties; here, the second-order statistics of rain in the frequency domain were examined for power law scaling as follows : 3E(k)≈k-β, where k corresponds to the wave number and β is the spectral exponent.

However, to fully characterize the complexity of the process across its intensities and spatio-temporal variation, information on higher- and lower-order statistics is required. For this, we use UM, which rely on the assumption of the field being generated by an underlying cascade process with conserved statistical properties at each scale while inheriting the scale-invariant properties of Navier–Stokes equations . In this framework, the probability of a field exceeding a particular threshold across all the scales is captured using the scale-invariant notion of singularity (γ), and for a multifractal field this probability scales according to the resolution (λ: the ratio of L, the outer scale, to l, the observational scale), with a corresponding fractal co-dimension as the scaling exponent c(γ) : 4p(ϵλ≥λγ)≈λ-c(γ). This relation implies that the statistical moments q of the field also scale with resolution and the moment scaling function K(q) as 5〈ϵλq〉≈λK(q). Both functions are related by Legendre transform . For a conservative field in the UM framework, Kc(q) can be fully determined with only two parameters, the multifractality index α and the mean intermittency co-dimension C1. 6Kc(q)=C1α-1(qα-q)α≠1C1qln⁡qα=1 C1 measures the clustering of average intensity across scales (C1∈[0,1] for one-dimensional fields); when C1=0, the field is homogeneous with little variability. α measures how this clustering changes with respect to intensity levels (α∈[0,2]): the higher the value of α, the higher the variability, with α=0 being a mono-fractal field where intermittency of the extreme is the same as that of the mean. If the UM parameters are known, the co-dimension function of the conservative multifractal field cc(γ) can also be obtained using Lengendre transform : 7cc(γ)=C1γC1α′+1αα′α≠1,C1exp⁡γC1-1α=1, where 1α+1α′=1.

For a non-conservative field ϕλ, i.e. a field whose average (〈ϕλ〉) changes with scales, a non-conservative parameter H (not to be confused with the Hurst exponent: though for values >0 both quantify long-range correlations, the latter does not have a simple general expression for the multifractal process; see the Appendix A for more details) is used in the expression of scaling : 8ϕλ=dϵλλ-H, where =d denotes equality in the distribution (X=dY⇔∀x:Pr(X>x)=Pr(Y>x)) and ϵ is a conservative field characterized by C1 and α for a conservative field H=0. For a non-conservative field with positive H, fractional differentiation is required to retrieve a coarser field. Similarly, from a non-conservative field with a negative value of H, the conservative field is retrieved through fractional integration. H is related to the spectral slope β (Eq. ) via the relationship 9β=1+2H-Kc(2). The scaling behaviour of conservative multifractal fields can be examined using a trace moment (TM), where a log–log plot of upscaled fields against resolution λ is taken for each moment q (Eq. ). The quality of the scaling is given by the estimate r2 of the linear regression; the value for q=1.5 is used as a reference. A double trace moment (DTM) is a more robust version of a TM tailored for UM fields, where the moment scaling function K(q,η) of the field ϵλ(η) (the initial field raised to power η at maximum resolution and re-normalized) is expressed as a function of the multifractal index α . 10〈(ϵλ(η))q〉≈λK(q,η)=ληαK(q) From the above equation, a value of α can be obtained as the slope of the linear part, where K(q,η) is represented for a given q as a function of η in the log–log plot. Both the TM and DTM techniques give reliable estimates as long as H<0.5 for the studied field.

Since multifractal processes are generated by cascade processes, the average values can become too concentrated over a certain area, leading to spurious estimates of moments above a particular value of q (at qD, q is above K(q)≈+∞). This effect is called divergence of moments. The convex natures of the functions K(q) and c(γ) are also limited by the sample size of the data or the maximum value of the scale-invariant threshold or singularity (γs) and the corresponding moment (qs). The details of their computation can be found in , , and . For reliable statistical estimates of the moment scaling function and hence the UM parameters, the moment orders used should not exceed qs (moment order corresponding to the maximum singularity) or qD (moment order where divergence happens).

An in-depth discussion of the methodological choices we have made, along with an overview of other multifractal formalisms and their strengths and weaknesses, is provided in Appendix A.

For the Vestas V-90 instruments, the rated power is 2 MW; this means that the maximum power the turbine can produce is 2000 kW. However, if we calculate the available power as per Eq. (), there are many instances where it can go beyond the rated value (see Fig. c). When analysing the variability of a field using statistical methods, the presence of instrumental limits in the data (here an upper limit) can introduce biases. As briefly mentioned in the Introduction section, an instrumental lower threshold in the data can increase α and decrease C1 . In addition to this, there is also bias from the presence of zeroes in the data (underestimation of α, deterioration of scaling), which replaced negative values of the turbine power. Figure  shows the real and theoretical turbine power state curves along with the bias this poses in statistical analysis. Along with the power produced, the turbine data provide wind velocity at the hub (from a basic sensor installed on the hub), which is used for its internal monitoring. For research purposes, the wind velocity from the 3D anemometer at the mast is more desirable as it offers a more reliable measurement (on almost the same horizontal plane as the turbine hub). Figure a shows the empirical state curve of the turbine with this velocity, and Fig. b shows the same curve with the velocity from the anemometer. There is considerably more scatter with the latter. It should be stated that the turbines are not in the exact location of the mast (Turbines 8 and 9 are ≈ 3.5 km away), and hence approximation of the wind velocity from the mast (for computing the theoretical power in Eq. ) comes with some biases.

From Fig. c and d, it can be clearly seen that the rated power imposes an upper threshold on the turbine power (Pt), while the power available (Pa) is the actual underlying field. For this 1-week-long series of Pt, 21.7 % of the data were at an upper threshold and 2.9 % were either zero or negative (taken as zeroes in the analysis); this percentage was found to change according to the data selected. The effect of these limits on UM analysis is shown in Fig. a, where the data in Fig.  are treated as an ensemble of 32 min. UM analysis was performed on direct fields as the values of H were within acceptable limits (H<0.3). A unique scaling regime from 15 s to 32 min was considered. The presence of the rated power clips the values of the field and results in a reduced value of α for Pt (Fig. a: α=1.36, C1=0.00715) from that of Pa (Fig. b: α=1.93, C1=0.01753). Imposition of a similar threshold (Pa<=2000=2000) on Pa was found to artificially reduce the estimates (α from 1.93 to 1.39, C1 from 0.10753 to 0.0076) in Fig. c, bringing them closer to that of biased turbine power, Pt (Fig. a). Even closer values of α were obtained when a lower threshold was also imposed (replacing Pa values with zeroes at positions where Pt was negative), giving α a value of 1.35 and C1 a value of 0.0077 (Fig. d). The results are compiled in Table . The scaling quality remained similar for all the cases mentioned here, with the r2 value (of the TM curve at q=1.5) remaining around 0.99 (second column of Fig. ).

It should be noted that the effect of a threshold could be different according to the size of the sample and the scaling regimes studied. In the same spirit as σe in , the effect of rated power as an upper threshold on Pt is explored here in the theoretical framework of UM. The effect on different scaling regimes as well as the additional complexity from the known effect of zeroes , though identified here, are not considered to avoid complexity.

Let us take the upper threshold (rated power in this case) at the largest possible scale ratio as 11T=ΛγT, where γT is the singularity corresponding to threshold T and Λ is the maximum resolution (length of the time series). For multifractal fields, the probabilities of exceeding scale-independent thresholds, λγ, scale with resolution λ (see Eq. ). At the upper threshold T, this yields 12Pr(ϵλ≥T)≈λ-c(γT). If we set the upper threshold, i.e. set all the values of the field greater than T equal to T (represented here by this expression: {ϵλ≥T}=T), the probability of having values greater than T, Pr(ϵλ>T), becomes 0, reducing the above relation into Pr(ϵλ=T)≈λ-c(γT). This leaves the value of c(γ) equal to +∞ for singularities above γT (for γ>γT, c(γ)=+∞). Here, c(γT) is the limiting non-zero value above which c(γ) becomes +∞.

This effect of the upper threshold (c(γ)→+∞ for γ>γT) is similar to the effect of the sampling dimension (Ds) in the UM framework. The maximum observable singularity can be defined by taking the probability at the corresponding threshold as in Eq. (): 13Pr(ϵλ≥λγS)≈1NsλD, where Ns is the number of samples and λD is the number of values per sample. Ns=λDs (Ds being the sampling dimension, which quantifies the number of independent samples with resolution λ; for one sample, Ds=0). Using the notions of Ds and D, with γ corresponding to the sampling resolution, γs can be estimated from c(γs), c(γs)=D+Ds .

The moment scaling function K(q) and co-dimension function c(γ) were discussed earlier in terms of UM parameters in Eqs. () and (). Both are equivalent functions, and for multifractals they are related by a simple Legendre transform : 14K(q)=max⁡γ[qγ-c(γ)],c(γ)=max⁡q[qγ-K(q)]. Hence, for every singularity γ, there is a corresponding order of moment q associated with it and vice versa: q=c′(γq) and γ=K′(qγ).

When γ>γs, c(γ) = +∞; by Legendre transform K(q) becomes linear from q>qs=c′(γs): 15γs=α′C1D+DsC11α′-C1α-1,qs=D+DsC11α. In the case of the sampling dimension, c(γ) varies as follows: 16c(γ)=+∞forγ>γsD+Dsforγ=γsc(γ)forγ<γs. Similarly, in the presence of the upper threshold here ({ϵλ≥T}=T), c(γ) reaches +∞ at an earlier limiting value c(γT), where γT<γs (Fig. a). 17cT(γ)=+∞forγ>γTc(γT)forγ=γTc(γ)forγ<γT Here, γT is defined from the threshold as initially stated in Eq. (), and cT(γ) is estimated as above. From this, the corresponding limit moment qT can be obtained as in Eq. (). 18qT=c(γT)C11α To summarize, in standard data analysis with sampling limitations, c(γ) is bounded by a maximum value c(γs), above which it becomes infinite. K(q), which is connected to c(γ) through Legendre transform (Eq. ), becomes linear beyond this q (q≥qs) value (K(q)=(q-qs)γs+K(qs)). Similarly, in this specific case, when an upper threshold is imposed ({ϵλ≥T}=T), K(q) becomes linear at an earlier value of q (at qT<qs) as defined by γT (Fig. b). 19KT(q)=γT(q-qT)+K(qT)forq>qTK(qT)=qTγT-c(γT)forq=qTK(q)forq<qT Using the DTM technique, for a given q, K(q,η)=K(qη)-qK(η), which for UM fields is ηαK(q). When no thresholds are applied, K(q,η) varies as 20K(q,η)=(q-1)(D+Ds)forη≥η+(q)ηαK(q)forη<η-(q), where η+(q) corresponds to the maximum values of η above which K(q,η) becomes a plateau due to sampling limitations (Eq. ). To elaborate, K(q,η) consists of two parts K(qη) and K(η), and η+(q) corresponds to the value of η above which both are linear (which is qs). The transition to the plateau starts at the lower value η-(q) (which is qs/q), above which only K(qη) is linear. In the presence of an upper threshold ({ϵλ≥T}=T), the DTM curve will be (Fig. c) 21KT(q,η)=(q-1)c(γT)forη≥η+(q)ηαK(q)forη<η-(q), where η+(q) = qT and η-(q) = qT/q.

It is important to note here that the value of K(q,η) does not reach the upper plateau abruptly at qT or qs. Rather, it flattens gradually, starting from a value of η = qs/q or qT/q (as per the value of qη in K(q,η)). The presence of the upper threshold shifts this starting point and decreases the range of possible values for estimation of α (slope of the DTM curve). Hence, the presence of a plateau will result in biased (reduced) estimates.

Underestimation of values of α due to application of the upper threshold was already observed in Fig. c using real data. To understand this further, conservative multifractal fields (H=0) were simulated using discrete cascades with values of UM parameters close to those observed for empirical powers (Pt). Discrete cascade simulation here involves division of a parent structure into sub-structures (retaining the value of the parent structure multiplied by a random factor; ), iteratively following a non-infinitesimal scale ratio while maintaining the validity of Eqs. () and ().

For ease of contrast with the simulations examining lower thresholds in , values of α=1.8 and C1=0.2 were used. UM analysis was implemented in ensembles (sample size 128 and number of samples 100) by progressively applying the upper threshold till the percentages were observed in Pt (∼ 30 %). K(q) becomes linear at earlier and earlier values of q (after qT), and an increasing percentage of values at the threshold can be seen in the third column (like in Fig. b). The DTM curve in the second column shows that both α and C1 decrease with progressive application of the thresholds (from 0 % to 30 %: α decreased from 1.8 to 1.56, while C1 decreased from 0.17 to 0.05).

While discussing this bias in the framework before, it was mentioned that the upper threshold was introduced at the maximum resolution (Eq. , ΛγT). Since in practice the lower scales in UM are obtained from averaging the outer scale (at the maximum resolution), the threshold values (and hence γT) at each stage do not correspond exactly to the originally defined one. This is the reason for an increased “transition part” (part of the curve from the straight line to the upper plateau) of the DTM curves (second column) in the simulations here (more than that in Fig. c). When the estimation of α was forced at η=1 (so that the TM and DTM estimates were the same), the bias in the values of α increased as the slope estimation moved to the transition part. For example, the already biased value of α at the 30 % threshold, 1.58 (slope at log η between -0.1 and -0.5), was further reduced to 0.95 (slope around log η=0). In this estimation, C1 moreover remained similar to previous estimates at all the thresholds.

It is interesting to note that the trend here (only for α) is not exactly the opposite of what was observed during numerical simulations with lower thresholds in the section . While the imposition of a lower threshold increased α and decreased C1, the upper threshold here reduces both UM parameters. In the specific case of turbine power, Pt (Fig. a) has a combination of the upper threshold from the rated power and the lower threshold (zeroes) from a negative power (the latter is not considered here). This, in practice, further reduces the range of available η for estimation of α by imposing a lower plateau as well (see Fig. , third column). Also, the effect of this bias could be different when fluctuations of the fields are selected to retrieve conservative fields since the simulations were performed directly on conservative multifractal fields here. Since two consecutive power values can be the same, thanks to the rated power, taking fluctuations will yield zeroes in the field and add to the zero bias. The effect of both biases could be different when the aggregate power of the wind farm is considered as well: this is not explored here either.

So far, the effect of thresholds in UM analysis has been illustrated in the framework of UM. To get a better idea of their effect on scaling, UM analysis was performed of a longer series of powers. For this, a 3-month-long continuous series was taken (from 1 January to 1 April 2021) and UM analysis was performed on Pa and Pt as ensembles of sample size 128 (32 min). Figure a and b show the curves for Pa (α: 1.93; C1: 0.01) and Pt (α: 1.11; C1: 0.0042) respectively. Considering Pa to be the underlying field, the effects of the thresholds (upper and due to zeroes) in Pt can be seen in the DTM curve (Fig. b, second row). The lower plateau corresponds to the presence of a negative power in the data (which was replaced by zeroes), and the low value of α is due to η being in the transition range (as already seen in Fig. c). Figure c and d show the UM analysis for the same data but remove the columns with thresholds. In the Pt ensemble data, the columns with thresholds, zeroes, and repetition of data were removed by using a limit of 0.01 %. For example, columns with more than 0.01 % of their values ≥1600 were removed to be on the safe side when analysing data without the influence of a threshold. For a more accurate comparison, the same columns were removed from Pa as well; the results are shown in Fig. c (α: 1.76; C1: 0.0095) and Fig. d. It can be seen that the lower plateau disappeared for Pt and that the values of the UM parameters (α: 1.5749 from 1.11; C1: 0.00554 from 0.0042) improved; this also increased the value of β (1.8 from 1.6) and consequently that of H (0.41 from 0.3). It should be noted that the values of the UM parameters get closer to those of Pa but are not the exact values. This suggests that there are slight differences in the properties of both fields, even though they appear comparable when biases are removed.