An innovative correction method of wind speed for efficiency evaluation of wind turbines


ACTA IMEKO 
ISSN: 2221-870X 
June 2021, Volume 10, Number 2, 46 - 53 

 

ACTA IMEKO | www.imeko.org June 2021 | Volume 10 | Number 2 | 46 

An innovative correction method of wind speed for efficiency 
evaluation of wind turbines 

Alessio Carullo1, Alessandro Ciocia2, Gabriele Malgaroli2, Filippo Spertino2 

1 Dipartimento di Elettronica e Telecomunicazioni, Politecnico di Torino, corso Duca degli Abruzzi 24, 10129 Turin, Italy 
2 Dipartimento Energia, Politecnico di Torino, corso Duca degli Abruzzi 24, 10129 Turin, Italy 

 

 

Section: RESEARCH PAPER  

Keywords: Renewable energy sources; wind energy systems; wind speed; power measurement; uncertainty 

Citation: Alessio Carullo, Alessandro Ciocia, Gabriele Malgaroli, Filippo Spertino, An innovative correction method of wind speed for efficiency evaluation of 
wind turbines, Acta IMEKO, vol. 10, no. 2, article 8, June 2021, identifier: IMEKO-ACTA-10 (2021)-02-08 

Section Editor: Ciro Spataro, University of Palermo, Italy 

Received January 15, 2021; In final form April 28, 2021; Published June 2021 

Copyright: This is an open-access article distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, 
distribution, and reproduction in any medium, provided the original author and source are credited. 

Corresponding author: Alessio Carullo, e-mail: alessio.carullo@polito.it    

 

1. INTRODUCTION 

The increasing energy demand and the requirements of 
minimal environmental impact have pushed towards to a huge 
increase of Renewable Energy Sources (RES). A drawback of 
these sources is their intermittency, which can be mitigated by 
means of the integration of storage units, e.g. electrochemical 
batteries [1]-[3]. Among the RES, Wind Turbines (WTs) 
represent a reliable and clean source of electricity with low 
marginal costs [3]. New wind power plants have been installed in 
Europe in 2020, with a cumulative rated power of about 7 GW, 
and an increase of about 10 GW is expected in 2021, thus 
reaching a cumulative capacity of about 250 GW [4]. In this 
framework, offshore applications will represent about 20 % of 
new installations in the period 2020-2023, especially in the 
Netherlands, Ireland, Norway and France. 

WTs can work at fixed speed or variable speed, and the latter 
are able to adjust the rotor speed, thus following the maximum 
aerodynamic power of the wind [5]. On the other hand, their 

control requires the wind speed to be measured through an 
anemometer, thus increasing the overall cost and the size of the 
system. The anemometer is usually located on the back of the 
turbine, where a wind speed that is lower than the wind speed 
entering in the rotor is measured. For this reason, the use of the 
measurement of this anemometer leads WTs to exhibit 
experimental performance that seems better than their nameplate 
specification, since the manufacturer states the power curve with 
reference to the wind speed at the entrance of the rotor. In 
addition, manufacturer-stated performance refers to ideal 
conditions of minimum turbulence, flat terrain and absence of 
wakes due to obstacles [6]. A reliable estimation of WT 
performance then requires the measured wind speed to be 
corrected and, for this reason, two different correction methods 
have been defined in technical specifications and International 
Standards. The first method does not take into account the 
effects of the wakes of other turbines and obstacles, while the 
second method filters the considered direction of the wind in 
order to remove these wake effects. Wake is a sort of loading 
effect, as occurs in electric circuits. Wakes are long trails of wind 

ABSTRACT 
The performance of horizontal axis Wind Turbines (WTs) is strongly affected by the wind speed entering in their rotor. Generally, this 
quantity is not available, because the wind speed is measured on the nacelle behind the turbine rotor, providing a lower value. Therefore, 
two correction methods are usually employed, requiring two input quantities: the wind speed on the back of the turbine nacelle and 
the wind speed measured by a meteorological mast close to the turbines under analysis. However, the presence of this station in wind 
farms is rare and the number of WTs in the wind farm is high. This paper proposes an innovative correction, named “Statistical Method” 
(SM), that evaluates the efficiency of WTs by estimating the wind speed entering in the WTs rotor. This method relies on the 
manufacturer power curve and the data measured by the WT anemometer only, thus having the possibility to be employed also in wind 
farms without a meteorological station. The effectiveness of such a method is discussed by comparing the results obtained through the 
standard methods implemented on two turbines (rated power = 1.5 MW and 2.5 MW) of a wind power plant (nominal power = 80 MW) 
in Southern Italy. 

mailto:alessio.carullo@polito.it


 

ACTA IMEKO | www.imeko.org June 2021 | Volume 10 | Number 2 | 47 

in turbulent regime and with lower speed with respect to the 
entrance of turbines rotor. This effect needs to be minimized 
because if the wind flow entering in the WT rotor is affected by 
the wake of another turbine, its speed is lower than in the wake-
free conditions, i.e. energy production is reduced. A common 
requirement of these methods is the availability of two quantities: 
the wind speed vWT measured by the anemometer and the wind 
speed vstat detected by a meteorological mast, which has to be 
close to the turbine under investigation. Unfortunately, a 
meteorological mast is not always present in wind power plants, 
thus preventing the implementation of these correction methods.  

To overcome this limitation, the present work proposes an 
alternative method that relies on the manufacturer power curve 
and the wind speed detected by the turbine anemometer, thus 
not requiring the measurements provided by a meteorological 
mast. 

In Section 2, a review of the two standard methods is 
presented and the new correction method is described. Section 3 
defines the concept of yearly average efficiency (which takes into 
account the energy generated by a WT), and describes the 
parameters availability and capacity factor. In Section 4, a case 
study is presented related to a wind power plant in Southern Italy. 
Section 5 reports the obtained results for two different WTs that 
are located in two different areas of the considered power plant. 
Finally, Section 6 summarizes the main outcomes of this work. 

2. CORRECTION METHODS 

An expansion of the stream tube occurs before and after the 
passage in the three-blade rotor. Therefore, the cross section of 
the wind flow increases, while its kinetic energy (thus, its speed) 
decreases. The presented methods aim to correct the wind speed 
detected by the anemometer behind the turbine rotor, calculating 
the corresponding speed at its entrance. 

Before the application of one of the proposed correction 
methods, a preliminary normalization to the reference air density 
ρref = 1.225 kg/m3 is performed, since manufacturer 
specifications refer to this condition. In particular, for WTs with 
active power control, experimental results are corrected 
according to the following expression [6]: 

𝑣cor = 𝑣𝑒𝑥𝑝 ∙ (
𝜌air
𝜌ref

)

1
3⁄

  , (1) 

where vcor is the corrected wind speed, vexp is the measured wind 
speed and ρair is the air density during the measurement. 

2.1. Method #1 – Straight Line Method (SLM) 

The first method requires vWT and vstat as input quantities, and 
consists of the following steps: 

1) Step A - Selection of the wind-speed direction. 
The wind direction β is properly selected in order to 
consider valid the assumption vstat ≈ ventr, where ventr is 
the wind speed that enters in the turbine rotor. In 
particular, experimental results are filtered in order to 
analyse the wind contributions flowing from the station 
to the WT. Assuming to simplify the problem as a 2D 
system without the vertical coordinate, a straight line is 
traced between the anemometric station and the WT 
under test and its orientation βWT with respect to the 
North direction is calculated. However, if the set of 
experimental data is limited, a low number of 
experimental points is available. In this case, it is 
generally convenient to extend the analysis to wind 

speeds with orientations β = βWT ± Δβ, where 2 Δβ is 
the top angle of a triangle whose base is the rotor 
diameter D of the WT (D = 2 ∙ rd, by assumption, where 
rd is the length of a blade, neglecting the hub radius) and 
the third vertex of the triangle is the meteorological 
mast. 

2) Step B - Selection of data with ventr > vWT. 
As described in the step A, the wind speeds of interest 
flow from the anemometric station to the WT. 
Therefore, ventr has to be larger than vWT because the 
kinetic energy of the wind decreases when it flows 
through the meteorological mast. 

3) Step C - Removal of experimental data with turbulence 
larger than 10 %. 
In each time interval, the turbulence is the ratio between 
the standard deviation of the wind speed and its average 
value in the interval. The power curve provided by the 
manufacturer is measured in conditions of minimum 
turbulence, which is generally lower than 10 % [7]. 

4) Step D - Linear regression of experimental data. 
In this step, a linear equation that describes vstat as a 
function of vWT is identified in order to estimate ventr by 
the line of regression of vstat on vWT, where the 
measurement of vWT is corrected thanks to the 
measurement of vstat. The goodness-of-fit of the linear 
regression to the experimental data is measured through 
the parameter R2, which ranges from 0 (no suitable 
model) to 1 (best model). 

During the design of a wind power plant, the position of the 
turbines has to be optimized in order to minimize their mutual 
wakes and maximize their energy production. However, due to 
different constraints, such as terrains and land morphology, these 
effects cannot be always minimized. Therefore, the first method 
needs to be modified in order to remove possible errors due to 
mutual wakes effect. 

2.2. Method #2 – No Wakes Method (NWM) 

This method is similar to and consists of the same steps as the 
SLM. However, since NWM aims to avoid the mutual wakes that 
affect the measurements, step A is modified. Indeed, NWM does 
not focus the correction on the direction joining the 
meteorological mast and the WT; on the contrary, it investigates 
all the directions in which the anemometric station and the WT 
are not affected by the wakes of other turbines. The procedure 
used to determine the wind directions disengaged from any 
obstacles is based on the document [6]. In particular, for each 
obstacle in the neighbourhood of the WT, such as other 
operating WTs or a meteorological station, the wind direction 

angles α that must be excluded from the analysis are calculated 
according to this expression: 

𝛼 = 1.3 ∙ arctan (
2.5 ∙ 𝐷

𝐿
+ 0.15) + 10 ,  (2) 

where D is the rotor diameter and L is the mutual distance 
between the obstacle and the WT under test. 

After the selection of the proper wind direction, it is possible 
to verify the validity of the results thanks to a more sophisticated 
analytical model, which is named the “Jensen Model” or “Park 
Model” [8]. It permits to estimate the wind speed v* perturbed by 
the wake of a turbine using the following expression: 



 

ACTA IMEKO | www.imeko.org June 2021 | Volume 10 | Number 2 | 48 

𝑣∗ = 𝑣0 ∙

[
 
 
 

1 −
1 − √1 − 𝐶𝑇

(1 +
𝑘 ∙ 𝑥
𝑟𝑑

)
2
 

]
 
 
 

 , (3) 

where v0 is the wind speed not affected by wakes, CT is the thrust 
coefficient of a WT and depends on the wind intensity, rd is the 
radius of the turbine rotor, and x is the downwind distance. The 
parameter k is the decay constant of the wake that is estimated 
according to the following equation: 

𝑘 =
0.5

ln (
ℎ
𝑧0

)
 , 

(4) 

where h is the hub height of the WT and z0 is the roughness of 
the ground. According to Jensen model, the wake increases 
linearly with x and its diffusion radius rx can be estimated as: 

𝑟𝑥 = 𝑟𝑑 + 𝑘 ∙ 𝑥 .  (5) 

It should be noted that the model considers the perturbation 
of the flow profile along the direction of the wind, while its 
perpendicular component is assumed constant (1-D model). 
Finally, this model assumes that k is a constant parameter that 

depends only on h and z0. 

2.3. Method #3 – Statistical Method (SM) 

The alternative method does not require experimental data 
provided by a meteorological station [9]: the input quantities are 
the wind speed measured by the WT anemometer and the power 
curve provided by the manufacturer. The assumption behind this 
methodology is that the power curve of the WT manufacturer is 
the locus of the points where the generator operates with the best 
performance. Therefore, the analytic relation between vWT and 
the wind speed provided by the WT manufacturer (for the same 
output electric power Pk) is derived. Figure 1 reports the scheme 

of the aforementioned methodology. More in detail, the 
methodology consists of the following steps: 

1) Step A - Removal of experimental data with turbulence 
larger than 10 % [7]; 

2) Step B - Selection of the experimental set Sk. 
One of the available working point Pk = P(vk) is selected 
on the power curve provided by the WT manufacturer. 
Then, a set Sk of experimental data is identified such 
that the electric output power lies in the neighbourhood 
of Pk, i.e. in the interval between Pk∙(1-ε) and Pk∙(1+ε). 
In this work, the value of ε was set to 0.01, based on the 
consideration that output powers within a ± 1 % 
interval are not distinguishable due to the typical 
uncertainty of the equipment that is used to measure 
this quantity. The set Sk is described as: 

𝑆𝑘 = {[𝑣WT,𝑖, 𝑃(𝑣WT,𝑖)]: 

𝑃(𝑣WT,𝑖) ∈ [𝑃𝑘 ∙ (1 − 𝜀) ÷ 𝑃𝑘 ∙ (1 + 𝜀)] } . 
(6) 

3) Step C - Calculation of the Empiric Cumulative 
Distribution Function (ECDF) of the wind speed. 
The ECDF of the wind speed corresponding to the 
selected value of vk is calculated, as shown in Figure 2 
(blue dots), which refers to the value vk = 12 m/s. The 
same figure also highlights how the calculated ECDF is 
well approximated by the CDF F(vWT) (red line) 
corresponding to the Probability Density Function 
(PDF) f(vWT) of the known factorial function Γ [10]: 

𝑓(𝑣WT) =
𝑣WT

𝑎−1

𝑏𝑎 ∙ 𝛤(𝑎)
∙ 𝑒

−
𝑣WT

𝑏      (𝑣WT ≥ 0) , (7) 

where the parameter a is estimated as the square ratio 
between the mean value and the standard deviation of 
Sk, while the parameter b is derived as the ratio between 
the mean value of Sk and a; 

4) Step D - Estimation of the wind-speed fifth percentile. 

Starting from the PDF f(vWT), the fifth percentile 𝑣WT
5 % 

of the wind speed, i.e. the value that has the 5 % of 
probability to not be exceeded in Sk, is selected; 
Steps from B to D are repeated for each available 
working point P(vk) in the power curve provided by the 
manufacturer. 

5) Step E - Linear regression of experimental data. 
This step is similar to step D of the other methods, but 

in this case a linear equation is obtained between 𝑣WT
5 % 

and the corresponding vk. 

 

Figure 1. Scheme of the methodology for the SM. 

 

Figure 2. Example of calculated ECDF for vk = 12 m/s. 

0

0.2

0.4

0.6

0.8

1

10 11 12 13 14 15

C
D

F

wind speed (m/s)

ECDF

Gamma CDF

0

1

2

3

0 5 10 15 20

P
o

w
e
r 

o
u

tp
u

t 
(M

W
)

wind speed (m/s)

Manufacturer Curve

Experimental Data

Set Sk
Pk

0

2

4

6

8

10

12

14

2 4 6 8 10 12

v
st

a
t
(m

/s
)

vstat = 0.926 ∙ + 0.936

R2 ≈ 1

)

Steps A-B

Step E

Steps C-D

0

0.2

0.4

0.6

0.8

1

11 12 13 14 15

vWT (m/s)

ECDF F(vWT)

vk = 12 m/s



 

ACTA IMEKO | www.imeko.org June 2021 | Volume 10 | Number 2 | 49 

One should note that when the WT reaches its nominal 
power, the correspondence between the wind speed and the 
output power is not unique. The output power is indeed limited 
to the rated value and it can be obtained with several values of 
the wind speed. This represents a limit of the proposed method, 
which is not applicable in the range of high wind speeds due to 
its strong dependence on the power curve of the WT 
manufacturer. 

3. ESTIMATION OF WT EFFICIENCY 

The efficiency of a WT is the ratio between the electrical 
power it produces and the aerodynamic power of the wind at the 
entrance of the rotor. The aerodynamic power Paer of the wind 
can be calculated as [11]: 

𝑃aer =
1

2
 ∙ 𝜌air ∙

π

4
∙ 𝐷2 ∙ 𝑣entr

3  . (8) 

The efficiency can be also estimated as the ratio between 
electrical and wind energies in a certain time interval Δt. 
Indicating the measured WT output power as Pout, the efficiency 
can be obtained as [12]-[13]: 

𝜂 =
𝑃out
𝑃aer

=
𝑃out ∙ Δ𝑡

𝑃aer ∙ Δ𝑡
=

𝐸el
𝐸aer

 , (9) 

where Eel and Eaer are electrical and aerodynamic energies, 
respectively. 

With the aim of comparing the three proposed correction 
methods, the results will be expressed in terms of weighted yearly 
efficiency η*: 

𝜂∗ =
∑ (𝜂𝑘 ∙ 𝐸𝑘)year

∑ (𝐸𝑘)year
=

∑ (𝜂𝑘 ∙ 𝐸𝑘)year

𝐸y,exp
  (10) 

where ηk is the WT efficiency, Ek is the output energy in the k - 
th time interval (Δt = 10 min), and Ey,exp is the experimental yearly 
energy generated by the WT. 

Thanks to the availability of an anemometric station and to 
the accurate selection of the wind direction, the NWM is 
considered as the reference method. The other two methods will 
be then compared to the NWM by means of the efficiency 
deviation Δη*, which is defined as: 

Δ𝜂∗ = 100 % ∙
𝜂∗ − 𝜂NWM

∗

𝜂NWM
∗   (11) 

where 𝜂NWM
∗  is the average efficiency estimated with the NWM. 

Moreover, the availability factor and the capacity factor are 
calculated for the turbines under analysis. The first quantity 
provides information regarding the probability that a system is 
operational at a specific time: in particular, it is the ratio between 
the uptime of WTs and their total operation time, which includes 
non-operation periods due to failures or maintenance actions 
[14]-[15]. On the other hand, the capacity factor is, for a given 
time interval, the ratio between the real energy produced and 
injected into the grid, and the ideal energy that could be 
generated by the turbine working continuously at its rated power. 
An example of centralized power station with very high capacity 
factor (about 90 % of one year) are units for base load operation. 

4. CASE STUDY 

The three methods previously described were applied to two 
WTs of a wind farm in Southern Italy (with 43 WTs, global 
nominal power of 80 MW and altitude between 1100 m and 

1200 m) using data collected during a measurement campaign in 
2017. The turbines of the wind farm have hub height of 80 m 
and a three-bladed rotor. Their wind speed range is the following: 
cut-in speed vc-in = 3.5 m/s, cut-out speed vc-out = 25 m/s. 
However, the wind farm is divided in two parts, with two 
different models of turbines. In the first area, WTs have a 
nominal power of 2.5 MW, a rotor diameter of 80 m and rated 
wind speed of 15 m/s. In this part, a meteorological mast (height 
of about 80 m) is present, measuring the quantities of interest. In 
particular, it is equipped with: 

1) a First Class cup anemometer, which acquires the 
horizontal component of the wind speed according to 
the requirements provided in [6]; 

2) First Class sensors that detect the wind direction 
according to [7]; 

3) pressure, humidity and temperature sensors, which 
measure the environmental quantities that are used to 
estimate the air density at the height of meteorological 
mast and turbine. 

The anemometer provides a resolution of 0.05 m/s and its stated 
uncertainty is ± 1 % of the measured value in the range 
(0.3 ÷ 50) m/s with a minimum uncertainty of ± 0.2 m/s. The 
environmental quantities are measured with uncertainties of 
± 2 °C for the temperature, ± 5 %RH for the relative humidity 
and ± 1 kPa for the pressure. 

In the second area of the wind farm, WTs have a rated power 
of 1.5 MW, a rotor diameter of 77 m and a rated wind speed of 
12 m/s. However, in this area, there is no meteorological station 
installed; hence, the only data available are provided by the 
anemometer located on the back of the turbines. 

Regarding the WTs of the entire plant, they are equipped with 
an ultrasonic anemometer that measures the absolute value and 
the direction of the wind speed, providing a resolution of 
0.01 m/s and an uncertainty of ± 2 % of the measured value in 
the range (0 ÷ 60) m/s (minimum uncertainty ± 0.25 m/s). The 
electrical output power of the WTs is measured with a relative 
standard uncertainty of 1 %. 

5. RESULTS 

5.1. Results of one turbine with the meteorological station 

In this subsection, the results for a WT in the first area of the 
wind farm (which is the one equipped with a meteorological 
mast) are presented. The electrical power measurements Pout 
obtained at the output of the WT (average values within 10 min 
time intervals) are shown in Figure 3 (blue points) with respect 
to the measured wind speed, which is corrected according to 
equation (1). In the same figure, which refers to results that are 
collected during a time interval of approximately one year, the 
manufacturer power curve (red line) is also reported. One should 
note that a high number of observations are on the left of the 
manufacturer power curve, since no correction methods are 
applied to these experimental results. This behaviour is not 
realistic, since the experimental performance of the WT cannot 
be higher than the manufacturer's specifications. Furthermore, 
the cut-in and cut-out wind speeds are about 3 m/s and 24 m/s, 
respectively, which are lower than the corresponding nominal 
values (vc-in = 3.5 m/s, vc-out = 25 m/s). 

According to the correction methods described in Section II, 
experimental results that show turbulence larger than 10 % are 
removed. Furthermore, also results showing null output power 
for wind speed in the range (vc-in ÷ vc-out) are removed, since they 
refer to failure conditions of the investigated plant. Before 



 

ACTA IMEKO | www.imeko.org June 2021 | Volume 10 | Number 2 | 50 

applying the described correction methods, a preliminary 
uncertainty estimation is performed, taking into account the 
instrumental uncertainty of wattmeter and anemometer of the 
WT and the contribution related to the repeatability of the 
measured output power. As a first step, the method of bins [16] 
is applied: power measurements are grouped according to the 
corresponding wind speed measured by the WT anemometer. 
Since its uncertainty has a maximum value of 0.5 m/s for wind 
speed value of 25 m/s, experimental results are grouped in 
uniform wind-speed bins with a width of ± 0.5 m/s. 

Then, the mean output power is estimated for each identified 
group and the standard deviation of the mean is considered as 
the estimation of the measurement repeatability. This 
contribution is combined to the instrumental standard 
uncertainty (1 % of the measured value), thus obtaining the 
combined standard uncertainty u(P). The obtained results are 
summarized in Figure 4, where the red bars refer to the 
manufacturer power curve, while the grey bars represent the 
experimental means of each group centred around integer values 
of wind speed. The error bars superimposed to each grey bar are 
the intervals Pmean,i ± u(Pi). Even though the anomalous data 
points are removed, the uncorrected experimental results are not 
fully consistent with the manufacturer specifications yet: for 
wind-speeds up to 6 m/s and at 21 m/s, the electrical output 
power is higher than the manufacturer specifications, and cut-in 
and cut-out wind speeds remain the same previously estimated. 

Implementing the SLM, the wind speed direction considered 
in the correction is β= (231 ± 13)° and the linear regression 
(R2 = 0.969) results in the following equation: 

𝑣𝑠𝑡𝑎𝑡 = 0.971 ∙ 𝑣𝑊𝑇 + 0.758 (12) 

The results after the correction with equation (12) are 
reported in Figure 5. For wind speeds lower than 21 m/s, the 
corrected output power is lower than the manufacturer curve. 
Regarding cut-in and cut-out wind speeds, the SLM correction 
leads to an estimation of vc-in that is comparable to the rated 
specification (≈ 3.5 m/s), while vc-out remains lower (≈ 24 m/s). 
Moreover, for wind speeds higher than 13 m/s, the manufacturer 
power curve reaches a saturation power of about 2.5 MW, while 
the experimental data reach a higher saturation power. This 
behaviour is realistic, being due to the pitch regulation of the WT: 
in fact, the turbine is allowed to work with a maximum power of 
about 104 % of rated data. This performance of the WT results 
in a higher energy production; however, an earlier aging of the 
turbine due to a higher degradation of the materials may occur 
as well. 

The direction of the wind speeds considered in the SLM may 
be affected by turbulence, mainly due to the wakes of other 
turbines or obstacles. The NWM permits to solve this issue by 
identifying the wind speed directions in which the turbine and 
the meteorological mast are not affected by wakes. According to 
Jensen model, the angular section not affected by wakes 
corresponds to β ranging between -26.8° and 12.7°: thus, the 
North direction (β = 0°) is selected for the NWM. Figure 6 
presents the results of Jensen model for the wind directions 
considered in the SLM and the NWM using k = 0.075, h = 80 m, 
rd = 40 m, and z0 = 0.1 m. The blue and the red circles represent 
the WT under analysis and the meteorological mast, respectively; 
the grey circles indicate the other wind generators, and the cones 
represent the areas affected by wakes. 

With the wind direction assumed for the SLM, the WT and 
the station are affected by the wake of another turbine; on the 
other hand, with North direction, they are wake-free. The 
resulting regression equation (R2 = 0.979) for the NWM is the 
following: 

𝑣stat = 0.998 ∙ 𝑣WT + 0.550 (13) 

Figure 7 reports the results after the correction. The NWM 
correctly estimates the cut-in and cut-out wind speeds as they 
coincide with the values provided by the manufacturer 
(≈ 3.5 m/s and ≈ 25 m/s, respectively). 

The SM does not require experimental data from a 
meteorological mast, thus it can be used also in wind power 
plants without weather stations close to the WTs. However, as 
described in Section 2, this correction cannot be applied to the 

 

Figure 3. Turbine #1 - Uncorrected raw experimental data (blue dots) and 
manufacturer power curve (red line). 

 

Figure 4. Turbine #1 - Uncorrected experimental data after the preliminary 
data processing. 

 

Figure 5. Turbine #1 - SLM corrected results (grey bars) and manufacturer 
power curve (red bars). 



 

ACTA IMEKO | www.imeko.org June 2021 | Volume 10 | Number 2 | 51 

last part of the power curve, where the electrical power reaches 
a saturation value (the nominal power). Thus, the SM is applied 
to wind speeds in the range (4 ÷ 15) m/s, which is the rated wind 
speed by the manufacturer. However, the local distribution of 
experimental wind speeds for the year under study (red bars in 
Figure 8) shows that this range contains most of experimental 
data. Actually, for wind speeds of ≈ 15 m/s, the corresponding 
cumulative function (blue curve) is higher than 0.93, i.e. more 
than 93 % of wind speeds are ≤ 15 m/s. In order to achieve a 
better accuracy, regression equations are identified for two wind 
speed ranges: lower than 6 m/s, and between 6 m/s and 15 m/s. 

The resulting equations are the following (R2  1): 

𝑣stat = 0.779 ∙ 𝑣WT + 1.493; 𝑣WT <  6
m

s
  (14) 

𝑣stat = 0.923 ∙ 𝑣WT + 0.961;  6
m

s
≤ 𝑣WT <  15

m

s
  (15) 

The results of the SM, which are reported in Figure 9, 
highlight that the performance of the WT is now realistic, since 
it is lower than manufacturer power curve.  

The weighted yearly efficiencies are calculated on a yearly 
basis to evaluate the effectiveness of the corrections (Figure 10). 
For wind speeds higher than the rated value, the curves converge 
into the manufacturer data (blue curve), while at lower wind 
speeds uncorrected data (red curve) have a different shape from 
the manufacturer curve. Moreover, at low wind speeds, the 
necessity to correct raw data is evident: in fact, at a wind speed 

of ≈5 m/s, the raw weighted efficiency (η* ≈0.4) is higher than 
manufacturer value. Actually, uncorrected data are based on the 
wind speed detected on the back of the turbine rotor: this value 
is lower than the wind speed entering the turbine, leading to an 
overestimation of its efficiency. Regarding the corrections, the 
SLM (green curve) estimates the lowest efficiencies, but, as 
previously described, its results are affected by wakes. Indeed, 
the comparison with the NWM curve shows that the presence of 
wakes leads to an underestimation of the efficiency. The shape 
of the efficiency curve of the SM, which is based on the 
manufacturer data, is the closest to the shape of the manufacturer 
one. 

Table 1 reports the weighted yearly efficiencies and the 
deviations of the SLM and the SM with respect to the NWM, 
which is assumed as the reference. For wind speeds < 6 m/s, the 
two methods underestimate the efficiency, with similar 
deviations from the NWM of about -9.1 % (SLM) and -7.9 % 
(SM). In the intermediate wind speeds range (6 ÷ 15) m/s, the 
SLM underestimates the efficiency with a spread of about -3.5 %, 
while the performance of the WT is overestimated by the SM 
with a deviation of about 5.2 %.  

The turbine under analysis has very high energy performance 
in the area of the farm with the meteorological mast. Its 
availability and capacity factors are evaluated: the WT under 
study is in operation for 97.2 % of time, with an average capacity 
factor of 29.3 %. Among the turbines in this part of the wind 
farm, the performance of this WT is one of the best, being the 
average availability and capacity factors of the plant equal to 
96.5 % and 22.3 %, respectively. 

 

Figure 6. Turbine #1 - Jensen model with wind directions assumed for SLM 
(top) and NWM (bottom). 

 

Figure 7. Turbine #1 - NWM corrected results (grey bars) and manufacturer 
power curve (red bars). 

 

Figure 8. Cumulative function and PDF of wind speed distribution. 

 

Figure 9. Turbine #1 - SM corrected results (grey bars) and manufacturer 
power curve (red bars). 

0

0.1

0.2

0.3

0

0.2

0.4

0.6

0.8

1

4 10 16 22

P
D

F

C
u

m
u

la
ti

v
e
 F

u
n

c
ti

o
n

Wind speed (m/s)



 

ACTA IMEKO | www.imeko.org June 2021 | Volume 10 | Number 2 | 52 

5.2. Results of one turbine without the meteorological station 

This subsection presents the results for a WT in the second 
area of the wind farm (i.e., without the meteorological mast). The 
power curve provided by the manufacturer (red line) and 
experimental observations (blue points, corresponding to 
average Pout values within 10 min time intervals) are presented in 
Figure 11; the wind speeds are corrected according to 
equation (1). In Figure 11, a high number of data are shifted on 
the left of the manufacturer power curve. Moreover, the cut-in 
and the cut-out wind speeds (≈ 3 m/s and ≈ 20 m/s) are lower 
than the nominal values (vc-in = 3.5 m/s, vc-out = 25 m/s): thus, a 
correction of these data is required. First, failure conditions are 
excluded from the analysis by removing observations exhibiting 
turbulence higher than 10 % and null output power. As 
described in the previous subsection, a preliminary uncertainty 
estimation is performed starting from the instrumental 
uncertainty of the WT wattmeter and anemometer, and the 
repeatability of the measured output power. Figure 12 shows the 
results of this preliminary analysis: the red bars correspond to the 
manufacturer power curve, while the grey bars represent the 
experimental means of each group centered around integer 
values of wind speed. The error bars superimposed on each grey 
bar are the intervals Pmean,i ± u(Pi). The figure confirms the 
preliminary results of the other turbine under study: despite data 
corresponding to abnormal operation are excluded, the electrical 
output power remains higher than the manufacturer 
specifications. However, cut-in and cut-out wind speeds 
(≈ 4 m/s and ≈ 21 m/s) are closer to the values provided by the 
manufacturer. 

This WT is in the area of the wind farm without a 
meteorological station, thus only the SM can be applied and the 
data after the correction are presented in Figure 13. As described 
in Section II, this method cannot be applied to the part of the 
power curve where the power is constant (nominal power); 
hence, wind speeds higher than rated value (12 m/s) are excluded 
from the analysis. After applying the correction, the performance 
of the WT (grey bars) is realistic, being lower than manufacturer 

curve (red bars). Two regression equations (R2  1) are 

determined for the wind speed intervals < 6 m/s, and between 
6 m/s and 12 m/s: 

𝑣stat = 0.946 ∙ 𝑣WT + 1.258; 𝑣WT <  6
m

s
  (16) 

𝑣stat = 0.893 ∙ 𝑣WT + 1.923;  6
m

s
≤ 𝑣WT <  12

m

s
  (17) 

The weighted yearly efficiencies for raw data are about 33.5 % 
(wind speeds < 6 m/s) and 42.6 % (wind speeds between 6 m/s 
and 12 m/s). After the SM correction, the weighted efficiencies 
are of about 27.1 % (< 6 m/s) and 34.3 % (6 m/s ÷ 12 m/s), 
with a decrease of about 19 % for both wind speed ranges. 

This turbine exhibits very high energy performance in the area 
of the plant without the weather station. Indeed, it has the 
highest availability of the wind farm (99.1 %), while the average 
value in the second part of the plant is 97.4 %; moreover, its 
capacity factor is 27.9 %. This value is lower than the other WT 
under analysis, despite being higher than the average quantity 
(20.1 %) in the second area of the plant. 

6. CONCLUSIONS 

This work proposes an innovative method, named “Statistical 
Method” (SM), to evaluate the average efficiency of wind 
turbines by correcting the wind speed at the entrance of the rotor 
from nacelle anemometer. In the literature, other two methods 
(Straight Line Method, SLM, and No Wakes Method, NWM) are 
defined to perform this correction, taking into account technical 

Table 1. Weighted yearly efficiencies and deviations for the correction 
methods (turbine with meteorological station). 

Wind Speed 
Range 

𝜼𝐫𝐚𝐰
∗  𝜼𝐒𝐋𝐌

∗  𝜼𝐍𝐖𝐌
∗  𝜼𝐒𝐌

∗  𝜟𝜼𝐒𝐋𝐌
∗  𝜟𝜼𝐒𝐌

∗  

< 6 m/s 39.7 30.0 % 33.0 % 30.0 % -9.1 % -7.9 % 

6 – 15 m/s 39.9 33.7 % 34.8 % 37.0 % -3.5 % 5.2 % 

 

Figure 10. Turbine #1 - Weighted yearly efficiencies for the proposed 
corrections. 

 

Figure 11. Turbine #2 - Uncorrected raw experimental data (blue dots) and 
manufacturer power curve (red line). 

 

Figure 12. Turbine #2 - Uncorrected experimental data after the preliminary 
data processing. 

0

0.1

0.2

0.3

0.4

0.5

0 5 10 15 20 25

W
e
ig

h
te

d
 y

e
a
rl

y
 e

ff
ic

ie
n

c
y
 η

*

Wind speed (m/s)

Uncorrected data

Manufacturer data

SM

NWM

SLM



 

ACTA IMEKO | www.imeko.org June 2021 | Volume 10 | Number 2 | 53 

specifications and International Standards. These correction 
methods require data measured by a meteorological mast close 
to the turbines, but the presence of this station in wind farms is 
rare. Conversely, the correction proposed in this paper evaluates 
the wind speed entering in the WTs rotor only relying on the 
manufacturer power curve and the data measured by the WT 
anemometer. Indeed, it may be applied also in wind farms that 
are not equipped with a meteorological station. In the present 
work, these three methods were applied to a one-year 
experimental campaign on a wind farm in Southern Italy. In 
particular, two turbines located in two different areas of the same 
plant were analysed: only the first turbine was close to a 
meteorological mast. The effects of the corrections were 
evaluated representing the electrical output power by means of 
the method of bins and setting the width of the bins according 
to the uncertainty of the used anemometer. An uncertainty 
estimation was also performed for the output WT power, taking 
into account the power measurement uncertainty and the 
repeatability within each wind-speed bin. The results of the 
NWM were considered as a reference. Regarding the turbine in 
the part including the mast, the SM performed similarly to the 
SLM, providing comparable absolute deviations in terms of 
weighted efficiencies with respect to the reference. In fact, the 
deviations of the SM are about ± 7 % in the total range, while 
the quantities corresponding to the SLM are ± 9 % with respect 
to the NWM. After the SM correction, the weighted yearly 
efficiency decreased between about 10 % and 20 % with respect 
to raw data before correction in the usual wind speed range. 

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Figure 13. Turbine #2 - SM corrected results (grey bars) and manufacturer 
power curve (red bars). 

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https://doi.org/10.17775/CSEEJPES.2018.00130
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https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.589.2982&rep=rep1&type=pdf
https://doi.org/10.1109/MEPCON.2017.8301377
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