Y Tian

B Cotté

A Chaigne

Wind turbine noise modeling: prediction of amplitude modulation and influence of atmospheric conditions

Keywords: no, 43, 28, Py, 43, 28, Ra

Aeroacoustic noise from a wind turbine is mainly caused by the interaction between the wind turbine blade and the air ow. For a modern wind turbine, trailing edge noise is often the dominant noise source. In this paper, a detailed study of trailing edge noise is carried out using Amiet's frequency domain analytical source model. Model results are compared with experimental data. Features of wind turbine noise, such as amplitude modulation, ground directivity, inuence of blade twist and pitch are studied. In the last part, the inuence of realistic wind proles is investigated. Wind shear is seen to increase the sound power level and the amplitude modulation.

Introduction

A modern wind turbine converts wind energy into electrical power with satisfying eciency. The number of wind farms, given the fact that it is a clean and renewable energy, grows very fast in many countries to reduce the reliance on traditional energy sources. Meanwhile, noise from a wind turbine, which is generated aerodynamically is the main concern for the acceptance of wind farms by the neighborhood. The annoyance is enhanced by the amplitude modulation associated with wind turbine noise, that is caused by the rotation of the blade [START_REF] Lenchine | Amplitude modulation in wind turbine noise[END_REF].

The main aeroacoustic sources of a wind turbine are the trailing edge noise, the turbulent inow noise, stall noise and tip noise. Trailing edge noise is caused by the interaction of the boundary layer turbulence with the airfoil trailing edge, and is usually the dominant source for a large wind turbine [START_REF] Oerlemans | Prediction of wind turbine noise and validation against experiment[END_REF]. The strength of this noise source is inuenced by atmospheric conditions, that can be categorized into stable, unstable and neutral. Usually an unstable atmosphere appears in daytime and is characterized by a weak wind shear, while a stable atmosphere is typically present at nighttime with stronger wind shear eects.

(c) European Acoustics Association

In this paper, we focus on wind turbine trailing edge noise, and we adapt Amiet's analytical model to a rotating blade. This paper aims at validating the model against large size wind turbine measurements, at understanding the cause of amplitude modulation, and at investigating the inuence of wind prole, blade twist and pitch on the radiated noise. It is organized as follows : in Section 2, Amiet's model is presented and applied to a full size wind turbine. Then, in Section 3, the model is validated by comparing predicted and experimental power spectra, and the eect of twist and pitch are investigated. Finally, calculations with dierent wind proles from onsite measurements are carried in Section 4 out to study the inuence of atmospheric stability on wind turbine noise.

2. Amiet's analytical model for trailing edge noise 2.1. Presentation of Amiet's analytical model Amiet's model was rst developed for turbulent inow noise [START_REF] Amiet | Acoustic Radiation from an Airfoil in a Turbulent Stream[END_REF], and extended later to trailing edge noise [START_REF] Amiet | Noise Due to Turbulent Flow Past a Trailing Edge[END_REF]. Figure 1 shows the geometry at the starting point of the model. An incoming ow with a uniform velocity U encounters a at plate at the leading edge. Turbulence grows inside the boundary layer while convected to the trailing edge, and is scattered at the trailing edge (shown in red in the gure). The plate has a span L and a chord c, and a receiver is located at (x R , y R , z R ), where the origin of the coordinate is set at the middle of the trailing edge. Amiet showed that the far eld power spectrum density S pp for large aspect ratio can be written as [START_REF] Amiet | Noise Due to Turbulent Flow Past a Trailing Edge[END_REF] :

S pp (x R , y R , z R , ω) = ωcz R 4πc 0 S 2 0 2 L 2 L ω U c , ky R S 0 2 Φ pp (ω)l y ω, ky R S 0 , (1)
with ω the angular frequency, c 0 the sound speed, S 0 a modied distance between the source and the observer, Φ pp the span-wise wall pressure spectra, l y the span-wise correlation length, k = kc/2 the normalized acoustic wavenumber, and L a transfer function that connects the airfoil surface pressure uctuation to the acoustic pressure in the far eld. Moreau and Roger [START_REF] Moreau | Back-scattering correction and further extensions of Amiet's trailing-edge noise model[END_REF] have shown that the large aspect ratio approximation is valid for L/c > 1.

We validated the model for a xed airfoil in [START_REF] Tian | Wind turbine noise modeling based on Amiet's theory[END_REF]. The main diculty with the model is to obtain the wall pressure spectra Φ pp . Here we use a scaling law model proposed by Rozenberg et al. [START_REF] Rozenberg | Wall-pressure spectral model including the adverse pressure gradient eects[END_REF], that considers an adverse pressure gradient ow condition. The span-wise correlation length is estimated using Corcos model [START_REF] Corcos | Resolution of pressure in turbulence[END_REF].

Model adaption to a rotating blade

Amiet's model was originally developed for a xed plate. A simplied method to account for the blade rotating motion is to consider a series of discrete angular positions. Blandeau showed that this approximation is valid within a given range of frequencies [START_REF] Blandeau | Validation of Amiet's model for propeller trailing-edge noise[END_REF].

Trailing edge noise varies along the blade due to the variations of turbulence parameters, wind speed and blade geometry, thus it is necessary to divide the long blade into short segments in order to describe these variations. However, another limitation on the segment length is that it has to be larger than the spanwise wall pressure correlation length l y , so that two neighboring segments can be assumed uncorrelated.

Once the number and size of the blade segments has been chosen, the noise is calculated according to the following steps : 1. launch XFOIL to calculate the boundary layer parameters required by the wall pressure model; 2. use Amiet's model to calculate the far eld pressure spectrum S pp for each segment at a certain blade position; 3. apply Doppler correction and calculate the total noise spectrum by logarithmic summation; 4. move to the next blade position and repeat the step 1 and 3.

3. Application on a wind turbine

Conguration studied

The model wind turbine is a 2.3 MW Siemens SWT 2.3-93. The tower height (ground to hub) is 80 m, the blade length is 45 m and it has 3 blades that have controllable pitch angle. The chord length is 3.5 m at the root of a blade, and 0.8 m at the tip. A linear variation is assumed as shown in Figure 3. These data in addition to the measurements are found in [START_REF] Leloudas | Optimization of wind turbines with respect to noise[END_REF]. Three cases mentioned in this reference are summarized in Table I. In this paper, only the results from case 2 and case 3 are presented. A constant wind prole (no shear) is considered in this section.

Pitch angle is applied by rotating the blade around the blade axis as a whole. It changes the angle of attack (AoA) by the same amount for all the segments. Here, by default, a positive pitch decreases AoA while a negative pitch increases AoA. The blade twist is chosen so that the AoA is 7 o by default. An illustration of a twisted blade is shown in Figure 2.

Before the calculation, it is needed to decide how to divide the blades. The largest wind speed provided in [START_REF] Leloudas | Optimization of wind turbines with respect to noise[END_REF] length is less than 0.5 m according to Corcos model. In this study, we propose to divide the blade into 17 segments, whose length ranges from 5 m to 2 m from the root to the tip. The aspect ratio for each segment is shown in Figure 3, and remains greater than unity.

Sound power level calculations and comparison to measurements

The third octave band spectrum of the sound power level L W is calculated from the sound pressure level at a receiver 100 m downwind of the wind turbine; the spectra are averaged over a complete rotation. Measured and calculated spectra are compared in Figures 4 and5 for cases 1 and 3, respectively. The results agree well for frequencies below 400-500Hz, and increases with increasing constant wind speed. We also notice that the pitch angle changes the shape as well as the peak frequency of the spectra. For U = 6m/s, results considering pitch t better over the whole frequency range. For U = 9.5m/s, the pitch-included spectrum has a shape closer to the measured one. However, there are discrepancies for both cases, which can be attributed to other noise sources that are not taken into account in the calculations, and to the atmospheric conditions during the measurements that could dier from the constant wind prole considered here.

Figures 4 and5 show that the pitch angle has an inuence on the shape of the spectrum. The reason is that the pitch angle changes the AoA at leading edge, thus changing the surface pressure uctuations inside the boundary layer. In Amiet's model, the radiated noise spectrum is determined by the surface wall pressure spectrum, so the change of wall pressure inside the boundary layer modies the radiated noise. Figure 6 shows the inuence of the pitch angle for U = 9.5 m/s case. Along with increasing pitch angle (thus decreasing AoA), the peak frequency moves to the right, and the maximum L W increased, except for a pitch of 5 o .

Amplitude modulation and directivity

Amplitude modulation is caused by the rotation of the blades. It has a frequency of 1/3 of the blade rotating frequency. In Figure 7, the relative variations of sound pressure level are shown for two observer locations. The variations can reach up to 8 dB for a crosswind observer, while it is almost constant for a downwind direction. Figure 8 shows the directiv- ity of amplitude modulation strength for cases 1 and 2. We can see that the strongest amplitude modulation appears in the direction a little upwind from the rotor plane, while the overall directivity pattern does not vary a lot with dierent wind speeds.

Amiet's model assumes that the surface pressure uctuations along the blade radiate in a dipole-like fashion. The directivity of trailing edge noise is thus determined by the orientation of the plate, and depends on frequency as seen in Figure 9. Considering blade twist, the orientation of the dipole will change along the blade, which explains that the horizontal directivity plot obtained in Figure 10 loses the typical dipole pattern that was shown in [START_REF] Tian | Wind turbine noise modeling based on Amiet's theory[END_REF]. The minimum overall sound pressure level (SPL) moved to the direction slightly upwind from the rotor plane where the maximum amplitude modulation was found.

4. Atmospheric condition impact on the trailing edge noise

Wind proles obtained by SIRTA data

In reality, wind turbines are always working in an atmosphere where the wind is turbulent and nonuniform. Our goal is to know how the realistic wind prole impacts the generated noise. The SIRTA (Site Instrumental de Recherche par Télédétection Atmosphérique) is an atmospheric observatory located 20 km south of Paris [START_REF] Haeelin | SIRTA, a ground-based atmospheric observatory for cloud and aerosol research[END_REF]. An anemometer provides 10 min averaged wind speed at 10 m height, as well as the averaged wind direction. A Doppler lidar measures the wind speed prole between 40 m and 200 m height. In addition, a sonic anemometer measures the wind uctuations which allow us to estimate the friction velocity u * , and temperature scale T * . Monin-Obukhov similarity theory (MOST) is used to obtain the wind prole in the vertical direction. Atmospheric conditions can be classied by the Obukhov length scale L * , which is related to u * and T * :

L * = T u 2 * /(κgT * ), (2)
where T is the mean temperature potential, κ is the von Kármán constant, and g is the gravity. A physical interpretation of L * can be given as the height at which the buoyant production of turbulence kinetic energy is equal to the wind shear production. To obtain realistic wind proles, measurements from SIRTA are analyzed for a chosen day (a period of 24 h) when the wind was mostly from the west where the terrain is open and at. Vertical wind proles obtained by MOST for 5 chosen durations within 24 h are shown in Figure 11. Corresponding friction velocities, temperature scales and Obukhov length scales obtained from the measurements as well as by MOST are listed in Table II. There are limitations for MOST, as we can see that for some periods MOST proles do not t the measurements. The following results will only focus on very unstable case A and very stable case D .

Eects of wind shear on sound power spectra

The calculated sound power level spectra for cases A and D are shown in Figures 12 and13, and compared to the results for a constant wind speed prole equal to the wind speed at hub height (80 m). For case A, the wind prole has really weak shear, representing an unstable atmosphere, and only slight dierences at high frequencies are observed. For case D, the measured wind prole has strong shear, which indicates The directivity of amplitude modulation strength for cases A and D are plotted in Figure 15. For τ = 270 o direction, where the blades rotate downwards, the amplitude modulation is stronger when the strong wind shear of case D is present.

Conclusions and perspectives

In this study, Amiet's analytical model for trailing edge noise is applied for wind turbine noise prediction. The trailing edge noise spectrum is determined from the surface pressure uctuation. Since the twist angle and the pitch angle change the AoA at the leading edge, surface pressure at the trailing edge is modied. Twist and pitch angle also have an inuence on the ground directivity. When the blade twist is considered, the ground directivity loses the typical dipole shape. Wind shear increases the trailing edge noise, especially in the wind direction. The increase is most noticeable at frequencies higher than the peak frequency. From the results, wind shear increases the amplitude modulation in the crosswind direction where the blades rotate downwards (τ = 270 o ). With blade twist, the minimum directivity is not exactly in the crosswind direction where amplitude modulation is the most signicant, meaning that in the direction where the minimum noise is perceived (in case D τ = 100 o and 260 o ), the amplitude modulation strength is only half the maximum value.

In the future, we will study the eect of atmospheric variability on wind turbine noise, and the inuence of the orientation of the rotor plane with respect to the wind direction. Also, other noise sources such as turbulent inow noise will be investigated.

Figure 1 .

 1 Figure 1. Schematics for Amiet's trailing edge noise model.

Figure 2 .

 2 Figure 2. Geometry of a blade of 45 m length with twist (top) and without twist (bottom).

Figure 3 .

 3 Figure 3. Chord length (blue) and aspect ratio (green) of each segment. The vertical blue lines indicate the 17 segments.

Figure 4 .

 4 Figure 4. Sound power level spectra for case 1.

Figure 5 .

 5 Figure 5. Sound power level spectra for case 3.

Figure 6 .

 6 Figure 6. Sound power level spectra for dierent pitch angle settings with a constant wind speed of 9.5 m/s.

Figure 7 .

 7 Figure 7. Amplitude modulation in wind direction (top) and cross-wind direction (bottom). The observer is 100 m away from the wind turbine in the two directions. The results are subtracted from the mean sound pressure level.

Figure 8 .Figure 9 .

 89 Figure 8. Directivity of amplitude modulation strength in dB, at a distance of 100 m away from the wind turbine.

Figure 10 .

 10 Figure 10. Directivity plot of overall SPL in dB(A). The black line indicates the rotor plane when looking from above. The wind is coming from the left.

Figure 11 .

 11 Figure 11. Wind proles calculated by MOST (black line) compared to SIRTA data (red line). Red marker stands for the 10 min averaged wind speed at 10 m height. The two horizontal dash lines correspond to the lowest and highest altitude that a blade can reach.

Figure 12 .Figure 13 .

 1213 Figure 12. Sound power level of case A. Solid: wind prole with shear; dash: constant wind prole

Figure 14 .Figure 15 .

 1415 Figure 14. Normalized overall SPL (top) and normalized amplitude modulation strength (bottom) with respect to ground azimuthal angle τ .

Table I .

 I Parameters for the 3 test cases

			Wind speed		Rotor speed		Pitch
				(m/s)			(rpm)			( o )
	case 1			6				13			3
	case 2			8				14			-2
	case 3			9.5				17			5
	0	5	10	15	20	25	30	35	40	45	-2

Table II .

 II Atmospheric conditions for 5 chosen periods.

		U hub (m/s) U hub (m/s)	u *	T *	L *
		Lidar	MOST	m/s	K	m
	case A	7.2	6.7	0.47 -0.11 -141
	case B	8.2	7.6	0.51 -0.05 -330
	case C	7.3	4.0	0.06 0.05 5.7
	case D	6.8	6.5	0.17 0.07	32
	case E	4.6	3.4	0.25 -0.05 -84

Acknowledgement

The authors would like to thank Dr. Jean-Charles Dupont for providing us the onsite atmoshperic measurements and for constructive discussions on the data analysis.


Y Tian
email: tian@ensta.frb.cotté

Modelling of ground and atmospheric effects on wind turbine noise

Amiet's analytical model for trailing edge noise is used to predict the noise radiated by a Siemens SWT 2.3-93 wind turbine. Good agreement with experiment is found for sound power level (SWL) spectrum for frequencies higher than 1kHz. The immission level is then calculated with an image source model and compared with point source calculation. Ground reflection and atmospheric absorption are considered for the propagation model. The effect of ground reflection is seen to modify the sound pressure level spectrum and the amplitude modulation strength. The point source approximation yields accurate results for the overall sound pressure level, but exaggerates interference dips in the spectrum and thus overestimates the strength of amplitude modulation.

Introduction

A modern wind turbine converts wind energy into electrical power with satisfying efficiency. However, the noise emission from a wind turbine has been a great concern for its acceptance by the neighbourhood. Among all the wind turbine noise mechanisms, aerodynamic noise, mainly, turbulent inflow noise and trailing edge noise are believed to be the most important noise sources. In addition, due to the rotation of the blade, wind turbine noise has a feather called amplitude modulation. It leads to the fluctuation of total noise immission which can be quite disturbing even when the noise level is low. It is of interests to know if this amplitude modulation varies over certain distance, and if its strength is the same in difference directions with respect to the wind.

In this paper, we focus on trailing edge noise since the total noise is usually dominated by trailing edge noise for a modern wind turbine [START_REF] Oerlemans | Prediction of wind turbine noise and validation against wexperiment[END_REF]]. Amiet's analytical model for a fixed plate is adopted for wind turbine blades to predict noise emission level. Then immission level is calculated using an analytical propagation model that includes ground reflection and atmospheric absorption. At the end, the validity of the commonly used point source assumption is examined.

Amiet's model for trailing edge noise and its application to wind turbine

Introduction to Amiet's analytical model

Amiet's model [START_REF] Amiet | Acoustic radiation from an airfoil in a turbulent stream[END_REF] was first developed for turbulent inflow noise. Since the two mechanisms, turbulent inflow noise and trailing edge noise are both caused by turbulence scattering, the original model can be extended to trailing edge noise [START_REF] Amiet | Noise due to turbulent flow past a trailing edge[END_REF]. Figure 1 shows the geometry of the model setup. Flow with a uniform velocity U encounters a flat plate at the leading edge, turbulence grows inside the boundary layer while being convected downstream, and then scattered at the trailing edge (as shown in red in figure 1). The plate has a span L and a chord c, and the receiver is located at the point (x , y , z ) as indicated. The origin of the coordinate is set at the middle of the trailing edge. The flow direction is set as x, the y direction is along the trailing edge, and z is for the vertical direction.

Amiet showed that the far field power spectrum density S for large aspect ratio, that is L/c > 3 [START_REF] Roger | Back-scattering correction and further extensions of Amiet's trailing-edge noise model. Part 1: theory[END_REF] can be written as [START_REF] Amiet | Noise due to turbulent flow past a trailing edge[END_REF]:

( , , , ) = 4! " " # $ # % 2 'ℒ ) * , + , - " $' # Φ // ( )0 1 , + , - " $ (1)
where source coordinates are expressed as (x, y, z), is angular frequency, " is sound speed, " is a modified distance between the source and the observer, Φ // is the span-wise wall pressure spectra, 0 1 is span-wise correlation length, estimated by Corcos model [START_REF] Corcos | Resolution of pressure in turbulence[END_REF], + , = + /2 is the normalized acoustic wavenumber, and ℒ is a transfer function that connects the airfoil surface pressure fluctuation to the acoustic pressure at a far field point. A more detailed derivation can be found in [START_REF] Roger | Back-scattering correction and further extensions of Amiet's trailing-edge noise model. Part 1: theory[END_REF]Moreau 2005, Rozenberg et al.,2007].

Application of Amiet's model to a wind turbine

The model wind turbine used in this study is a 2.3MW Siemens SWT 2.3-93. The tower height (ground to hub) is 80m, it has 3 B45 blades of length 45m that have controllable pitch angle. The chord length is 3.5m at the root of a blade, and 0.8m at the tip. A linear variation of chord is assumed when dividing the blade into small segment. These data in addition to the measurements are found in [START_REF] Leloudas | Optimization of wind turbines with respect to noise, Master thesis DTU Heutschi K[END_REF]. To apply Amiet's model on a wind turbine, each blade is first divided into several segments, and each segment is treated as a fixed plate. Doppler effect due to the rotation is taken into account [START_REF] Schlinker | Helicopter Rotor Trailing Edge Noise[END_REF]. The overall sound pressure level (SPL) is then obtained by logarithmic summation over all the segments. Wall pressure spectra is predicted with a scaling formula that considers an adverse gradient flow condition [START_REF] Rozenberg | Wall-pressure spectral model including the adverse pressure gradient effects[END_REF]. Boundary layer parameters required by the wall pressure spectra model are calculated with XFOIL for an airfoil NACA63 3 415, which has a really similar shape as B45 airfoil [START_REF] Creech | Simulations of an offshore wind farm using large eddy simulation and a torque-controlled actuator disc model[END_REF].

3.Noise propagation over a grass ground

Wind turbine noise propagation in the atmosphere is influenced by many factors, such as refraction by vertical sound speed gradients, scattering by turbulence, topography, etc. paper, we consider only the ground effect and the atmospheric absorptio atmosphere is considered homogeneous, meaning there is no temperature variation, and the atmosphere is at rest.

Modelling of ground reflection

Like a light beam, a sound wave will be reflected immision level is then the sum of a direct wave and a ref write the complex pressure amplitude at a receiver as * = 7 exp then it can be shown that the relative sound pressure level Δ% 100;<10 =1

where > ? and > # are the distance of figure 2, + is acoustic wavenumber, calculated by the simplified model proposed by [START_REF] Chessell | Propagation of noise along a finite impedance boundary[END_REF] .

Noise propagation over a grass ground

Wind turbine noise propagation in the atmosphere is influenced by many factors, such as refraction by vertical sound speed gradients, scattering by turbulence, topography, etc. paper, we consider only the ground effect and the atmospheric absorptio atmosphere is considered homogeneous, meaning there is no temperature variation, and the Modelling of ground reflection ound wave will be reflected when it encounters a then the sum of a direct wave and a reflected wave, as shown in figure the complex pressure amplitude at a receiver as [Chessel, 1977;Salomons, 2001]

7 B+> ? > ? C D exp B+> # > # E
the relative sound pressure level Δ% has a form =1 C D > ? > # exp B+> # F B+> ? = # the distance of the direct ray and reflected ray respectively, as shown in number, and D is the spherical-wave reflectio calculated by the simplified model proposed by [START_REF] Chessell | Propagation of noise along a finite impedance boundary[END_REF] for the sound propagation over t is related to ground impedance Z H , the latter can be model [Salomons,2011]. Q depends on ground . For a rigid ground, there is no energy lost, @ is infinite; while for a typical

200+JK • M • N O# , which is used in this paper.
with respect to frequency over a typical grassland is shown in figure 4 for a source m respectively, with various source height 100m , the first interference dips for the 3 sources occur at R 1000m , they occur at almost the same frequency. This is the angle between the reflected ray and the ground is almost the same much larger than the source height. Wind turbine noise propagation in the atmosphere is influenced by many factors, such as refraction by vertical sound speed gradients, scattering by turbulence, topography, etc. In this paper, we consider only the ground effect and the atmospheric absorption. For simplicity, atmosphere is considered homogeneous, meaning there is no temperature variation, and the when it encounters a ground. The total lected wave, as shown in figure 2. If we [Chessel, 1977;Salomons, 2001]:

(2) of [Salomons]:

(3) direct ray and reflected ray respectively, as shown in wave reflection coefficient calculated by the simplified model proposed by [START_REF] Chessell | Propagation of noise along a finite impedance boundary[END_REF] for the sound propagation over , the latter can be modelled by depends on ground resistivity, which is infinite; while for a typical , which is used in this paper. An example for Δ% for a source -receiver various source height H Q , and a fixed terference dips for the 3 sources occur at , they occur at almost the same frequency. This is the angle between the reflected ray and the ground is almost the same when the

Modelling of atmospheric absorption

Atmospheric absorption is due to the dissipative process during the wave propagation. Energy loss leads to a decrease of the amplitude of the wave. The strength of absorption depends on frequency, temperature and the humidity of the atmosphere. The absorption coefficient V can be estimated by the International Standard ISO 9613-1:1993 with the information of temperature and air humidity. Atmospheric absorption is more pronounced at high frequency range, and for longer source-receiver distance R. An example of absorption coefficient V as a function of frequency is shown in figure 5, with temperature of 20C X and air humidity of 80%.

4.Results and discussions

Emission level prediction

The sound Power level % ` is obtained by % `= % a + 100;<10(4!> # ) assuming a free field propagation, where the sound pressure level % a is the output from Amiet's model. The observer is located 100m in the downwind direction. Wind speed is 8m/s, and the blade rotating speed is 1.47rad/s .The results are averaged over one complete rotation and compared with measurements in Figure 6. It shows that at frequencies less than 1kHz, the trailing edge noise emitted from airfoil suction side is greater than the pressure side, while at frequency higher then 1kHz , it is dominated by pressure side. This phenomenon agrees with fixed plate measurements of [Brooks. et al,1989]. For the total trailing edge noise, the results are in good agreement at frequency higher than 1kHz. However, the model underestimates the noise level at low frequencies. This can be attributed to different factors including: 1, there are other noise mechanisms who are important at low frequency, namely, turbulent inflow noise and noise due to flow separation; 2, a constant wind profile cannot represent the real wind conditions during the measurements.

Directivity of total SPL is shown in figure 7, with the wind coming from left to right. Rotor plane is represented by the vertical bold line. A clear dipole shape is seen. The lowest SPL is not exactly at crosswind direction, which is due to the effect of blade twist, the trailing edge being off the rotor plane. Amplitude modulation, that is, the fluctuation of total noise level during one blade rotation is shown in figure 8 for downwind and crosswind observer. The results are normalized by the mean overall SPL during one rotation. We can see that the amplitude modulation is much pronounced in crosswind direction, and almost constant in downwind direction. Figure 9 shows the directivity of amplitude modulation, with wind coming from left to right. The maximum is observed at direction a little upwind from the rotor plane, where we observed the lowest overall SPL in figure7.

Immission level prediction

The sound propagation results are shown in this section. The receiver is at 1.5m above ground for all distances, the air temperature is assumed at 20 X C and humidity at 80%. In figure 10,Aweighted 1/3 octave bands are plotted for a receiver distance of 100, 200, 500, and 1000m. The first interference dip is shifted from around 80Hz for R = 100m to 300Hz for R = 1000m. This is due to the fact that when the observer is further, the length difference between the direct wave path and reflected wave path is smaller, thus the destructive interference occurs at a smaller wavelength, meaning higher wave frequency. The effect of atmospheric absorption is more pronounced at higher frequency and at a longer distance. Figure 11 shows the overall SPL with respect to source -receiver distance R for free field, and with both ground reflection (G.R) and atmospheric absorption (A.A). At distance R = 100m, the A.A can be neglected, and ground reflection along adds 2dB. But with atmospheric absorption, at 1000m distance, the total SPL is up to 5dB less than that of free field, for a ground with σ = 500 kPa • s • m O# . Figure 11 shows that a more rigid ground (higher @ value) leads to a lower overall SPL at > = 1000N, which seems at first counterintuitive. To explain this phenomenon, we plot the the third octave band spectrum for the same @ values at > = 1000N in Figure 12, and the relative sound pressure level for a point source at 80-meter height in Figure 13. We can see clearly that for a more rigid ground, the first interference dip occurs at higher frequency, which tends to reduce the total SPL more significantly when summing up all the frequency bands. Figures 14 and 15 show the effect of ground and atmospheric absorption on the directivities of SPL and amplitude modulation strength. Observer is 1000m away from wind turbine. In agreement with the results of figure 11, at 1000m distance in the downwind direction, the overall SPL with ground reflection and atmospheric absorption is around 5dB lower than that of free field. On the other hand, for amplitude modulation, the ground reflection and atmospheric absorption increase its strength for most directions of τ. The amplitude modulation strength for τ = 0 X , 45 X , 90 X , and 105 X as a function of sourcereceiver distance is shown in figure 16. It is seen that when sound is propagating along downwind direction (0° and 45°), the strength of amplitude modulation increase with increasing distance but remains lower than 1dB; while in the crosswind direction (90° and 105°), the strength tends to decrease with increasing distance.

To gain a little more understanding on the increase of amplitude modulation strength when considering ground reflection and atmospheric absorption, a spectrum of amplitude modulation strength is plotted in Figure 17 for > = 1000N, in crosswind direction. It shows large frequency variations when ground reflection and atmospheric absorption are taken into account, while the free field spectrum is quite smooth. If we focus on the third octave band 2000vw, by looking at the spectra of the blade positions that produce maximum and minimum SPL in Figure 18, the Free field With G.R and A.A cause of the 10 dB amplitude modulation peak is seen to be the ground interference dip at this frequency. It is necessary to notice that for different frequencies, the blade positions for maximum and minimum SPL are not necessary the same, thus the overall amplitude modulation strength is not simply the logarithm summation of the spectrum in Figure 17.

Accuracy of point source assumption for wind turbine noise

Wind turbine noise is an extended noise source, given the rotor diameter can be as large as 100 meters. But if the receiver is far enough, wind turbine noise may be modelled by a point source, and the calculation will be greatly simplified. The narrowband spectra of SPL for extended source and point source calculations are compared in Figure 19 for various sourcereceiver distance in downwind direction. From the figure we can see clearly that at R = 100m, the 2 spectra have similar level, but for point source, there are many interference dips. On the other hand the extended source spectrum is quite smooth, as already noticed by [Heutschi, 2014]. This is because the geometrical positions of the source and receiver are unique for a point source, while for the extended source, the distance between receiver and each segment are different. Thus the interference dips for each source (segment) -receiver distance occur at different frequencies using the extended source, as previously shown in figure 4(a). As a result, the overall SPL is smoothed out by the compensations. At a larger distance, for example, R = 1000m , the 2 spectra almost overlap for frequency less than 1kHz , and the first interference dips appear at the same frequency. This is because when the receiver is far, the distance differences between source (segment) -receiver are small, so even for an extended source, the first interference dip occurs at almost the same frequency, see figure 4(b). However at higher frequencies, there are still some interference dips for point source spectrum that are not observed for extended source. Similar results are observed for other directions. The overall SPL for extended source and point source with respect to distance in the downwind and crosswind direction are shown in Figure 20. From where we can see that there are no significant discrepancies, we can say that point source is a good simplification for overall immission level prediction. However, in Figure 21, it shows that the point source calculations over estimate the amplitude modulation strength in all the tested ground azimuthal directions. Close to crosswind direction, for τ 90 X and 105 X , the increases are significant.

Conclusions

Amiet's analytical model for trailing edge noise is able to predict noise emission from a wind turbine. The immission level is reduced at large distance mainly due to atmospheric absorption. Ground reflection modifies the shape of SPL spectrum. The amplitude modulation strength is increased when considering a grassland with atmospheric absorption. The point source assumption for wind turbine noise is good for predicting the overall immission level, but it overestimates the amplitude modulation strength, and cannot account for the frequency dependence of ground effects. A more accurate propagation model based on parabolic equation method [Cotte, et al., 2007 ] will be used in the future to take into account the effects of atmospheric turbulence and wind.

Figure 1 .

 1 Figure 1. Schematic for Amiet's trailing edge noise model

Figure. 2

 2 Figure.2 Geometry of a sound wave reflected by a ground.

Figure. 3

 3 Figure. 3 Definition of ground azimuthal angle P.

Figure 4 .

 4 Figure 4. Relative sound pressure level RS for a different source height T U , and a receiver height of 1.5m.

Figure 5 .

 5 Figure 5. Atmospheric absorption coefficient Z for a temperature of [\] ^ and an air humidity of _\%.

Figure 6 .Figure 7 .

 67 Figure 6. Third octave band spectrum of g h for measurements and for Amiet's trailing edge noise model

Figure 8 .

 8 Figure 8. Amplitude modulation during one complete blade rotation. Upper: observer in downwind direction; lower: observer in crosswind direction.

Figure

  Figure 9. Directivity of amplitude modulation strength.

Figure

  Figure 10. 1/3 octave band spectra for the different propagation distances. Solid lines: with only ground reflection; dash lines: with ground reflection and atmospheric absorption.

Figure

  Figure 11. Overall SPL with respect to propagation distance in downwind direction, for different fluid resistivity. Both ground reflection and atmospheric absorption are considered.

Figure 12 .

 12 Figure 12. SPL spectra for different fluid resistivity n, at o = p\\\q.

Figure 13 .

 13 Figure 13. Relative sound pressure level for different fluid resistivity n. Source height: _\q, observer height: p. rq, o = p\\\q.

Figure 14 .

 14 Figure 14. Directivity of overall SPL for free field and for a grassland with atmospheric absorption. Observer is 1000m away from wind turbine. t = [\\kPa • s • m -2 .

Figure 15 .

 15 Figure 15. Directivity of amplitude modulation strength for free field and for a grassland with atmospheric absorption. Observer is 1000m away from wind turbine.

Figure 17 .

 17 Figure 17. Spectrum of amplitude modulation strength. o p\\\q crosswind direction.

Figure 16 .Figure 18 .

 1618 Figure16. Amplitude modulation strength with respect to source -receiver distance for P \ ^, xr ^, y\ ^ z{| p\r ^

Figure 19 .

 19 Figure 19. Narrow band SPL for point source and extended source at different source -receiver distance in downwind direction. Solid lines: extended source; dash lines: point source.

  Overall SPL (immission level) with respect to distance in (a) downwind direction, (b) crosswind direction.

Figure 21 .

 21 Figure 21. Amplitude modulation strength for extend source and point source calculation at different ground azimuthal angles.

Geometry of a sound wave reflected by a ground. Figure. 3 Definition of ground azimuthal angle

	for the sound propagation over
	a finite impedance ground. It is related to ground impedance
	Delany & Bazley empirical model [Salomons described by @. For a rigid ground, there is no energy lost, grassland, @ has a value of 200 with respect to frequency over a typical grassland is shown in figure 4 distance > 100m and 1000m receiver height of 1.5m. For R different frequencies, while for R because the angle between the reflected ray and the ground is almost the same source -observer distance R is much larger than the source height
	3


Benjamin Cotté
email: benjamin.cotte@ensta.fr

Prediction of variability in wind turbine noise calculations

We propose in this work a method to predict the variability in wind turbine noise calculations due to wind speed and direction fluctuations. First, wind lidar data measurements during a 24hour period are analyzed, and four periods with different atmospheric stability conditions are selected. Then, a wind turbine noise model based on Amiet's theory for trailing edge noise is presented and used to predict the sound pressure level at a fixed receiver during the 24-hour period. Finally, a Monte Carlo sampling method is described that allows us to accurately predict the statistics of sound pressure level during each selected period. The variability is seen to be much more pronounced during the day than during the night, and statistical quantities are shown to depend on the period duration considered.

Introduction

Wind turbine noise depends on numerous factors such as wind turbine characteristics, receiver position, atmospheric conditions, ground type and terrain features. The accurate modelling of all these factors is a difficult task. Furthermore, the accuracy of the models may be limited by uncertainties, that can be classified as either epistemic or aleatory. Epistemic (or reducible) uncertainty comes from incomplete information or imprecise measurement of a quantity (e.g. source-receiver geometry). On the other hand, aleatory (or irreducible) uncertainty, also called variability, is inherently random (e.g. atmospheric turbulence).

In this paper, we propose a method to predict the variability in wind turbine noise calculations due to atmospheric conditions. We consider a trailing edge noise model based on Amiet's theory, applied to a full-scale 2.3 MW wind turbine. Both wind speed and direction are considered random, and the variability of wind turbine noise is calculated using the Monte Carlo sampling method. To do so, high frequency wind lidar data measured at the SIRTA, an atmospheric observatory located South of Paris, are analyzed in order to obtain realistic probability density functions for different atmospheric conditions (stable and unstable atmospheres). Convergence and validation tests are performed to ensure the reliability of the method. Finally, predictions of the statistics of the sound pressure level for various periods are detailed.

Meteorological data measured at SIRTA

SIRTA (Site Instrumental de Recherche par Télédétection Atmosphérique) is an atmospheric observatory located 20 km south of Paris [START_REF] Haeffelin | SIRTA, a ground-based atmospheric observatory for cloud and aerosol research[END_REF]. In this work, we used data from two SIRTA instruments :

• a wind lidar that provides the wind speed and direction between 40-meter and 200-meter height with a time resolution of 4 seconds ; • a sonic anemometer at 10-meters height that provides estimates of the friction velocity u*, temperature scale T* and Obukhov length L* every 10 minutes.

The parameters u*, T* and L* are useful to determine the type of stratification occurring in the atmospheric boundary layer [START_REF] Stull | An Introduction to Boundary Layer Meteorology[END_REF]. The atmosphere is typically stable (L* > 0) during a clear night, with temperature increasing with increasing altitude, which discourages vertical air motion. On the contrary, the atmosphere is typically unstable (L* < 0) during daytime, with significant vertical air motion.

We analyzed 24 hours of data between 8:30am the 19th and 8:30am the 20th of October 2011. This day was selected because no rain occurred, the cloud cover was low and the wind was mostly blowing from the West where the terrain is open and flat. In Figure 1, the wind speed U at 80-meter height and the angle τ are plotted for the whole 24-hour period, where τ is the difference between the wind direction at 80-meter height and its mean over the whole period.

The mean wind speed is 6.7 m/s, and the mean wind direction is 296°, where 0° corresponds to the North ; by definition the mean of τ is 0° over the whole period.

We observe in Figure 1 than during the night -between approximately 6:54 pm and 8:17 am at this time of the year -the fluctuations of wind speed and direction are much weaker than during the day, which can be linked to the stability of the atmospheric boundary layer. Four periods of duration 40 to 60 minutes, noted by letters A to D in Figure 1, will be analyzed in more details in the following. During each period, the friction velocity u*, temperature scale T* and angle τ vary by a small amount. The corresponding mean meteorological quantities for these four selected periods are given in Table 1. In a recent paper [START_REF] Tian | Wind turbine noise modeling : prediction of amplitude modulation and influence of atmospheric conditions[END_REF], we showed that the proposed by [START_REF] Amiet | Noise due to turbulent flow past a trailing edge[END_REF] can be applied to a full model is to estimate the spectrum of wall pressure fluctuations results were obtained considering the empirical model proposed by Rozenberg that takes into account the effect of an mechanisms such as turbulence inflow noise and separation consider only trailing edge noise in this work.

This model is applied to a Siemens SWT 2.3 height is 80 meter, and it has three previous work [START_REF] Tian | Wind turbine noise modeling : prediction of amplitude modulation and influence of atmospheric conditions[END_REF], where w wind speed is constant with altitude. atmospheres, but not necessarily in stable atmospheres characterized by a stronger wind shear. The rotational speed is supposed to increase linearly from 6 speed of 4 m/s to 16 rpm at the rated wind speed of 12

The calculations are performed for a spectrum of 58 octave band spectrum between 50

The receiver is fixed at a distance R with respect to the wind direction as shown in Figure to the wind direction. For the sake of simplicity, the receiver is placed on a ground that is supposed rigid, such that the sound pressure level (SPL) is 6 SPL. Note that it would be possible to consider a more realistic ground effect, a companion paper (Tian and Cotté

The third octave band spectrum of sound power level (SWL) at τ=0° (downwind) is plotted in Figure the spectrum is around 800 Hz for all wind speeds. The overall SWL is also plotted to wind speed in Figure 3(b). Finally, the directivity of SPL amplitude modulation is plotted in Figure is the difference between the maximum and minimum SPL during one blade rotation. When the SPL is maximum, for downwind and upwind directions, the AM is close to 0, while AM reaches a maximum of 6 to 8 dB(A) for crosswind directions, where t directivities are very similar for the three wind speeds considered. , 2014), we showed that the trailing edge noise analytical model proposed by [START_REF] Amiet | Noise due to turbulent flow past a trailing edge[END_REF] can be applied to a full-size wind turbine. The main difficulty of this model is to estimate the spectrum of wall pressure fluctuations at the trailing edge sidering the empirical model proposed by Rozenberg effect of an adverse pressure gradient. Although other noise such as turbulence inflow noise and separation noise may be important, we iling edge noise in this work.

mens SWT 2.3-93 wind turbine of nominal power 2.3 and it has three B45 blades of length 45 meters. In contrast with our , 2014), where wind shear were considered, we assume here that the wind speed is constant with altitude. This is usually a good approximation for unstable atmospheres, but not necessarily in stable atmospheres characterized by a stronger wind s supposed to increase linearly from 6 rpm at the cut rpm at the rated wind speed of 12 m/s, with a pitch angle equal to zero. The calculations are performed for a spectrum of 58 frequencies in order to estimate the third e band spectrum between 50 Hz and 2500 Hz.

a distance R = 200 meters from the wind turbine, and makes an angle wind direction as shown in Figure 2 ; the rotor plane is always perpendicular direction. For the sake of simplicity, the receiver is placed on a ground that is supposed rigid, such that the sound pressure level (SPL) is 6 dB greater than the free field SPL. Note that it would be possible to consider a more realistic ground effect, a and Cotté, 2015).

spectrum of sound power level (SWL) in dB(A) estimated from the SPL =0° (downwind) is plotted in Figure 3(a) for a wind speed of 4, 8 and 12

Hz for all wind speeds. The overall SWL is also plotted Finally, the directivity of SPL normalized by its maximum amplitude modulation is plotted in Figure 4 for different wind speeds. Amplitud is the difference between the maximum and minimum SPL during one blade rotation. When the SPL is maximum, for downwind and upwind directions, the AM is close to 0, while AM reaches dB(A) for crosswind directions, where the SPL is minimum. directivities are very similar for the three wind speeds considered.

between the wind direction and the direction of the receiver from the wind turbine.

trailing edge noise analytical model size wind turbine. The main difficulty of this at the trailing edge. Satisfying sidering the empirical model proposed by [START_REF] Rozenberg | Wall-pressure spectral model including the adverse pressure gradient effects[END_REF] adverse pressure gradient. Although other noise may be important, we wind turbine of nominal power 2.3 MW. Its hub meters. In contrast with our ind shear were considered, we assume here that the This is usually a good approximation for unstable atmospheres, but not necessarily in stable atmospheres characterized by a stronger wind rpm at the cut-in wind m/s, with a pitch angle equal to zero. frequencies in order to estimate the third meters from the wind turbine, and makes an angle τ 2 ; the rotor plane is always perpendicular direction. For the sake of simplicity, the receiver is placed on a ground that is dB greater than the free field SPL. Note that it would be possible to consider a more realistic ground effect, as it is done in a estimated from the SPL 3(a) for a wind speed of 4, 8 and 12 m/s. The peak of Hz for all wind speeds. The overall SWL is also plotted with respect normalized by its maximum and of . Amplitude modulation AM is the difference between the maximum and minimum SPL during one blade rotation. When the SPL is maximum, for downwind and upwind directions, the AM is close to 0, while AM reaches he SPL is minimum. The direction of the receiver at distance R

Reference calculations using SIRTA meteorological data

The noise model is now used to calculate the overall SPL for each lidar measurement (U,τ) during the 24-hour period presented in Figure 1. First, the calculation is performed for each of the 20 459 measurement. This takes a long computation time since each calculation requires about 30'' to run on a regular PC. Second, the calculation is performed by linear interpolation between pre-calculated values. A total of 19 wind speed values between 4 and 13 m/s (step of 0.5 m/s) and 30 angular values for τ between -85° and 60° (step of 5°) are considered, which includes all the data plotted in Figure 1. The SPL values calculated by linear interpolation are in agreement with exact values within 0.1 dB, and are obtained almost immediately. Thus this interpolation technique will be used in the next section to predict SPL variability.

Figure 5 shows the SPL for all pre-calculated values. Because the angle τ is limited to the interval [-85° , 60°], the influence of the wind direction is quite weak. This τ interval is also delimited by the dashed vertical lines in Figure 4, which shows that AM will not exceed

Prediction of wind turbine noise variability 4.1 Monte-Carlo method for predicting sound pressure level variability

We consider here a stochastic (Monte Carlo) sampling technique to predict the variability of the noise radiated by wind turbines. This technique is widely used and effective for uncertainty and sensitivity analysis [START_REF] Helton | Survey of sampling-based methods for uncertainty and sensitivity analysis[END_REF][START_REF] Leroy | Congrès Français d'Acoustique[END_REF][START_REF] Wilson | Description and quantification of uncertainty in outdoor sound propagation calculations[END_REF]. Two random variables are taken into account, namely the wind speed U and the angle τ, characterized by their probability density functions (pdf) f(U) and g(τ). Thus the expected value of sound pressure level can be obtained by [START_REF] Wilson | Description and quantification of uncertainty in outdoor sound propagation calculations[END_REF]:

∫ ∫ +∞ = 0 2 0 ) ( ) ( ) , ( π τ τ τ dUd g U f U SPL SPL . ( 1
)
It is useful to introduce the cumulative distribution functions (cdf) F(U) and G(τ), that are the integral of the pdf f(U) and g(τ), to limit the integrals between 0 and 1. Since dF=f(U)dU and dG=g(τ)dτ, the integral becomes:

∫ ∫ = 1 0 1 0 )) ( ), ( ( dFdG G F U SPL SPL τ . ( 2
)
For ordinary Monte Carlo sampling, N uncorrelated random pairs of values for F and G are generated in the interval [0,1], and the integral is approximated by the average of the integrand over these N pairs:

∑ ∑ = = = ≈ N n n N n n n SPL N G F U SPL N SPL 1 1 1 )) ( ), ( ( 1 τ .
(3)

Each SPL n calculation is performed by linear interpolation using the method described in Section 3.2.

Estimation of probability density functions for wind speed and wind direction

To be able to apply the Monte Carlo sampling method to our problem, we first need to estimate the pdf of wind speed and direction from the meteorological data. For wind speed, we consider the normal and Weibull distributions, the latter being widely used in wind speed analysis [START_REF] Brano | Quality of wind speed fitting distributions for the urban area of Palermo, Italy[END_REF]. For the wind direction (angle τ), only the normal distribution is used.

The quality of the pdf fitting for the four selected periods is assessed using the Kolmogorov-Smirnov (K-S) statistical test [START_REF] Brano | Quality of wind speed fitting distributions for the urban area of Palermo, Italy[END_REF]. This test measures the distance of a random sample with a theoretical pdf by comparing the empirical cdf with the cdf of the reference distribution ; this distance is given by the K-S statistic D. Let the null hypothesis be that the samples are drawn from the reference distribution. The null hypothesis is rejected if the K-S statistic D is greater than a critical value that depends on the significance level α. The pvalue is the probability of having a test statistic D as large or larger than observed, or otherly stated the probability that the empirical cdf is as far apart from the reference cdf. If p is smaller than α taken as 1% here, the null hypothesis is rejected. The results of the K-S test are summarized in Table 2, and associated pdf for wind speed are compared to the empirical pdf in Figure 6 using 10 bins. Table 2 shows that only two wind speed distributions are rejected (p < 1%), namely the normal distribution for period C and the Weibull distribution for period D. No normal distribution is rejected for wind direction. For period A, the Weibull distribution yields a better fit for wind speed than the normal distribution,

Results and discussion

In this Section, we will present results for the four periods selected in Section 2, using the distributions obtained in Section 4.2. To assess the quality of the results, we compare the statistics of the Monte Carlo simulations with the statistics of the reference data. More specifically, we look at the mean value <SPL>, the standard deviation σ(SPL), the median µ(SPL) and the interquartile range IQR(SPL). The standard deviation and the interquartile range are two different measures of variability. The standard deviation is useful when the distribution is normal; in this case 68% of the data are in the interval <SPL>±σ(SPL), and 95% of the data are in the interval <SPL>±2σ(SPL). The interquartile range is more general: it is the range of the middle 50% of the scores in a distribution. It is calculated as the difference between the 75th percentile and the 25th percentile.

First, we look at a typical convergence curve for the Monte Carlo simulations in Figure 7(a). The median and interquartile range of SPL is obtained using different number of simulations between 500 and 10 000. It is seen that the results do not vary anymore when the number of simulations is greater than 5000. In the following, N = 10 000 simulations are used for all Monte Carlo simulations. Second, the empirical cdf obtained from reference data and from Monte Carlo simulations are plotted in Figure 7(b) for the four periods. An overall good agreement is found between both curves, but small discrepancies are visible, especially for periods A and B corresponding to daytime. These cdf curves give us a lot of information about the probability of particular events. For instance, it appears that the probability of obtaining a sound pressure level greater than 45 dB(A) is significantly different from 0 only for period B.

The statistics for reference data and Monte Carlo simulations are given in Table 3, and also plotted as box plots in Figure 8. In the box plots, the central red line is the median, and the edges of the box are the 25th and 75th percentiles; thus the height of the box is the interquartile range. The whiskers extend to the most extreme data points not considered outliers; their length is 1.5 times the IQR. Outliers are plotted individually in red only for the reference data, since there would be too many outliers for the Monte Carlo simulations using N = 10 000. We observe in Table 3 and Figure 8 that Monte Carlo simulations predict quite well the reference data statistics. The main discrepancies are for the IQR of periods A and B where a maximum difference of 0.4 dB(A) can occur. We note also that the measures of variability, σ(SPL) and IQR(SPL), are approximately three times larger during the 2 daytime periods the during the 2 night-time periods, as was already visible in the SPL plot of Figure 1. Finally, we look at the variability at an intermediate time scale of 10 minutes inside each interval. In Figure 9, the box plots for each 10-minute interval inside periods A and D are plotted using the reference data (similar results would be obtained using Monte Carlo simulations). The scale is different for both periods, extending over 18 dB(A) for period A and only 6 dB(A) for period D. It appears quite clearly that the SPL shows as much variability in all 10-minute intervals during the daytime period A, with an IQR between 2.7 dB(A) and 3.7 dB(A). The median value varies by a small amount, remaining between 37.4 dB(A) and 38.9 dB(A). On the contrary, the median value varies significantly during night-time period D, decreasing from 40.9 dB(A) in the first 10-minute interval to 38.5 dB(A) in the last one; the variability is small during this period, with the IQR remaining below 0.7 dB(A).

Conclusion

In this paper, we describe a Monte Carlo sampling method that can be use to predict wind turbine noise variability. The wind turbine noise model we use is based on Amiet's theory for trailing edge noise. From wind lidar data measured at SIRTA, we estimate the probability density functions of wind speed and direction during four selected periods, which allows us to predict the statistics of sound pressure level during these periods. The method is shown to yield accurate results by comparison with reference calculations. The variability is seen to be much more pronounced during the daytime periods than during the night-time periods, with an interquartile range approximately three times larger during the daytime periods. We also show that statistical quantities depend on the period duration considered. During night-time period D, the median value changes by more than 2 dB(A) between the four 10-minute intervals.

In this work, we use the ordinary or "brute-force" Monte-Carlo sampling method because we are able to obtain a good convergence in a small amount of computation time. For more computationally intensive models (e.g. with more random variables), it might be useful to use more advanced sampling strategies such as importance sampling or Latin hypercube sampling [START_REF] Helton | Survey of sampling-based methods for uncertainty and sensitivity analysis[END_REF][START_REF] Wilson | Description and quantification of uncertainty in outdoor sound propagation calculations[END_REF]. In the future, it could be interesting to account for the effect of wind shear [START_REF] Tian | Wind turbine noise modeling : prediction of amplitude modulation and influence of atmospheric conditions[END_REF] and/or ground effect [START_REF] Tian | Modelling of ground and atmospheric effects on wind turbine noise, 6th Wind Turbine Noise Conference[END_REF] in the variability analysis.

Figure 2 :

 2 Figure 2: Angle τ between the

Figure 3 :

 3 Figure 3: (a) Third octave band spectrum of sound power level, and (b) overall sound power level with respect to wind speed U for τ=0° (downwind).

Figure 4 :

 4 Figure 4: Directivity of the normalized SPL (top) and of amplitude modulation AM (bottom) for a wind speed of 4, 8 and 12 m/s.

  (A) for this set of data. The SPL results for all measured data is shown in the bottom of Figure1, where it is seen that the noise level fluctuations follow clearly the wind speed fluctuations.

Figure 5 :

 5 Figure 5: SPL as a function of wind speed U and angle τ for all pre-calculated values.

Figure 6 :

 6 Figure6: Fitted wind speed probability density functions for the four selected periods estimated using 10 bins.

  be used in the next section. Both normal and Weibull distributions fit well the wind speed of period B. We choose arbitrarily to use the normal distribution in the Monte Carlo simulations of the next section.

Figure 7 :

 7 Figure 7 : (a) Convergence of Monte Carlo simulations for median and IQR of SPL of period B, and (b) empirical cumulative distribution functions for the four periods obtained from the reference data (solid lines) and from Monte Carlo simulations with N = 10 000 (dashed lines).

Figure 8 :

 8 Figure 8 : Box plots of SPL for the four selected periods using (a) reference data, and (b) Monte Carlo simulations with N =10 000. Outliers are shown only for reference data.

Figure 9 :

 9 Figure 9 : Box plots of SPL using reference data for each 10-minute interval of (a) period A, and (b) period D.

Table 1 :

 1 Mean meteorological quantities for the four selected periods.

	Periods	<U> (m/s)	<τ> (°)	<u*> (m/s)	<T*> (K)	<L*> (m)
	A : 10h15-11h05	6.4	-29	0.44	-0.16	[-98 , -79]
	B : 13h55-14h35	7.7	-10	0.46	-0.08	[-280 , -128]
	C : 19h05-20h05	7.5	+7	0.08	+0.05	[1.6 , 20]
	D : 6h15-6h55	6.9	+19	0.18	+0.07	[21 , 47]

. Wind turbine noise calculations 3.1 Wind turbine noise model based on Amiet's theory

Table 2 :

 2 Kolmogorov-Smirnov goodness of fit test with a significance level of 1% using a normal or Weibull distribution for the wind speed U and a normal distribution for angle τ.The bold values correspond to the distributions that are used for the Monte Carlo simulations.

	Period	A	B	C	D
	Time frame	10h15-11h05	13h55-14h35	19h05-20h05	6h15-6h55
	p-value of K-S test for a normal distribution of U	9.6%	12%	5.5x10 -5	1.9%
	p-value of K-S test for a Weibull distribution of U	16%	12%	6.9%	6.2x10 -7
	p-value of K-S test for a normal distribution of τ	21%	98%	16%	47%

Table 3 :

 3 Comparison of statistical quantities for SPL obtained from the reference data from Monte-Carlo simulations with N = 10 000.

	Periods	Type of calculation	<SPL> (dBA)	µ(SPL) (dBA)	σ(SPL) (dBA)	IQR(SPL) (dBA)
		Reference	37.8	38.1	2.5	3.2
	A : 10h15-11h05
		Monte Carlo	37.8	38.2	2.7	3.5
		Reference	41.7	42.0	2.7	3.0
	B : 13h55-14h35
		Monte Carlo	41.7	41.9	2.6	3.4
		Reference	41.5	41.6	1.0	1.1
	C : 19h05-20h05
		Monte Carlo	41.5	41.6	1.0	1.2
		Reference	39.7	39.7	1.0	1.3
	D : 6h15-6h55
		Monte Carlo	39.7	39.7	1.0	1.3

Acknowledgements

The authors would like to thank David Ecotière from CEREMA, Fabrice Junker from EDF and Benoit Gauvreau from Ifsttar for useful discussions regarding uncertainty prediction methods.