# Wind Farm Power Forecasting

Authors: Nabiha Haouas; Pierre R. Bertrand
Journal: Mathematical Problems in Engineering
 (2013)

---

## Abstract

Forecasting annual wind power
production is useful for the energy industry. Until recently,
attention has only been paid to the mean annual wind power energy
and statistical uncertainties on this forecasting. Recently,
Bensoussan et al. (2012) have pointed that the annual wind
power produced by one wind turbine is a Gaussian random variable
under a reasonable set of assumptions. Moreover, they can derive
both mean and quantiles of annual wind power produced by one wind
turbine. The novelty of this work is the obtainment of similar
results for estimating the annual wind farm power production.
Eventually, we study the relationship between the power production
for each turbine of the farm in order to avoid interaction between
them.


---

## Body

## 1. Introduction

The energy industry is one of the most important types of modern industries. In recent years, wind power has become increasingly popular as a renewable energy source that can both develop the economy and protect the environment. Thereby, the development of suitable mathematical models becomes necessary. A detailed review on the existing tools used in wind power prediction is provided by [1], which proposes a perspective of future developments. A more recent update can be found in [2] where they used two alternative numerical prediction models: an empirical one and a computational one, in order to forecast the power output of two Greek wind farms before their installation. Different models for monitoring and forecasting the turbine output are considered such as those in the studies by the authors of [3–7] or [8] and recently in [9]. But, to our knowledge, the performance of a wind power farm has not been adequately studied. In this work, we suggest a statistical analysis based on central limit theorem as in [9]. Firstly, by using the wind speed, as input variables, we can forecast the annual energy production and its quantiles. Secondly, we study the relationship between the power outputs for each turbine in the farm to avoid the effect of interaction between them.The rest of this paper is organized as follows. In Section2, we present the dataset. In Section 3, we first recall the state on the art on the wind farm power forecasting, then we give the theoretical results for the forecasting of the annual wind power production, and we apply these results to real datasets. Moreover, we study the relationship between the power outputs for each turbine in the farm. Finally, discussions are available in Section 4.


## 2. Data Presentation

In this case study, we have processed ten-minute wind speed and ten-minute wind power production corresponding to a wind farm with four turbines. The duration of observation is 29 months leading to large series. The wind farm is located in a flat area close to the sea.The wind farm power production depends on wind speed. Thus we begin with data representation. The wind speed series is intermittent; that is, it presents very irregular variations, as shown in Figure1. This intermittency induces forecasting difficulties. A different approach has been considered, but also with a different horizon of time; see, for example, [10, 11] or [12]. Let us point out that the previous methods take only into account the absolute value of wind speed, leading to 1D time series. Other modern methods include the 2D or 3D behavior of wind; see, for example, [13] or [14].Figure 1
Mean wind speed in the farm.By a simple calculation, the average wind speed over the period is equal to 7.25 m/s, which is enough to guarantee a good profitability of the project but does not allow a detailed forecast.Regarding the energy output, we can note the large amount generated by this wind farm which reaches 2870 kW on average every ten minutes. This perfectly fits the building of a wind farm on a flat area close to a windy sea. We can also plot the ten-minute wind speed versus the ten-minute power; see Figure3. We then get a cloud of points around the nominal power law. Let us recall that the nominal power law is provided by the manufacturer and indicates the power P produced for a given wind speed v, which corresponds to the map v↦P(v) (see Figure 2).Figure 2
Average wind farm power every ten minutes.Figure 3
Wind power versus wind speed for turbine 1.Moreover, the turbine is cut for wind speed outside the interval(Vcut.in,Vcut.off). The large dispersion in Figure 3 around the nominal power law is due to outliers and error of measurement. We see that the power (i)
is null if the wind speed is less than the starting speed (Vcut.in=3.5 m/s) and beyond the cut out speed Vcut.off=25 m/s,(ii)
is proportional to the wind speed rise betweenVcut.in and the rated speed (about 13 m/s),(iii)
is constant between the rated speed and the cut out speed. Let us remark that the energy output is not the same for each of the four turbines. Indeed, there is a turbine that produces on average less than the others although the four turbines have the same power law, being of the same type. A possible explanation is the wake effect. In this frame, modeling by 2D wind series could enhance wind power forecasting (see Table 1).Table 1
Mean wind power for ten-minute by each turbine.
Turbine
T
1
T
2
T
3
T
4
Mean wind power (kW)
756
712
684
717


## 3. Wind Farm Power Forecasting

### 3.1. State of the Art

The traditional method is based on modeling ten-minute wind speed probability density function (pdf) and then calculating the average ten-minute wind power production as𝒫=(1-p0)∫Vcut.inVcut.off‍P(v)f(v)dv, where p0 denotes the probability of zero wind; Vcut.in, Vcut.off denote the wind speed for cutting in and cutting off the turbine; v↦P(v) denotes the nominal power law; f(v) denotes the wind speed probability density function (pdf). Most often the ten-minute wind speed is assumed to follow a Weibull distribution, or a hybrid Weibull law; see, for example, [15, 16], or [7]. This model gives a good estimation of both mean ten-minute wind speed and mean ten-minute wind power production and after that provides us with a good forecasting of the mean annual wind power production. However, this traditional method discards the time structure of wind speed and wind power and thus it does not allow to forecast the variance, nor the quantiles of annual wind power production.Recently, the authors of [9] have proposed a new method which takes into account the dynamical structure of the annual wind power production. This method provides forecasting not only of the mean annual wind power production, but also of its variance and the quantiles of annual wind power production. The techniques rely on the central limit theorem (CLT) which asserts that the pdf of annual wind power production is almost Gaussian under natural assumptions. In the next section, we slightly adapt this method to the case of a wind farm.


### 3.2.  Forecasting Wind Farm Power

The annual wind power production of the farm is defined by(1)𝒫annualF=∑t=1TPtF,
where T=52,560 denotes the number of ten-minute periods during one year. The farm production is the sum of the individual production of the J turbines, that is, PtF=∑j=1JPt(j). The most simple but quite large model is to assume that each series Pt is wide sense stationary. Wide sense stationary means that on the one hand both mean 𝒫(j) and variance 𝒱(j) are not depending on time and that on the other hand the dynamical structure read on the covariance does not depend on the time t. As a corollary, the farm production series PtF is also wide sense stationary with mean 𝒫F=𝔼(PtF)=∑j=1J𝒫(j) and variance as follows:
(2)𝒱F=Var(PtF)=∑i,j=1Jcov(Pt(i),Pt(j))
and correlation coefficient ρPF(k)=cov(PtF,Pt+kF)/Var(PtF) which does not depend on the time t. Moreover, the family of random variables PtF is weakly dependent, and it admits a finite second-order moment (i.e., 𝔼(𝒫tF)<∞ for each t≥1). In addition, the annual wind production 𝒫annualF is also random with mean 𝔼(𝒫annualF)=T×𝒫F and variance as follows:
(3)var(𝒫annualF)=var(∑t=1TPtF)=T×𝒱F×(ΓTF)2,
where ΓTF={1+2∑k=1T[ρpF(k)×(1-k/T)]}1/2 with ρPF(k)=cov(PtF,Pt+kF)/Var(PtF) which was introduced to variance analysis for characterizing wind energy conversion as in [17].Let us stress that the variance of annual production of the farm depends both on the variance of ten-minute wind power𝒱Fand its correlogram ρpF(k), which corresponds to the time structure of the series. We will also need the two following assumptions. (A1)
The second-order moment ofPt is finite, that is, size 𝒱=𝔼((Pt)2)<∞;(A2)
The family of random variablesPt is weakly dependent. After having made precise assumptions and notation, by using the same tricks as in Proposition  3.1 in [9], we can deduce the following CLT.Theorem 1 (CLT for wind farm annual production).
(i)
If the family of r.v.Pt is wide sense stationary, and assumptions (A1) and (A2) are fulfilled, then one has
(4)𝒫annualF=T×𝒫F+T1/2×(𝒱F)1/2·ΓTF×ɛT,
where
(5)𝒫F=∑j=1J𝒫(j),𝒱F=Var(PtF)=∑i,j=1Jcov(Pt(i),Pt(j)),ΓTF={1+2∑k=1T[ρpF(k)×(1-kT)]}1/2
and ɛT is a zero mean r.v. with variance 1 which converges towards a standard Gaussian law; that is, ɛT→𝒩(0,1) when T→∞.
(ii)
Moreover, one can deduce the quantiles of annual production as follows:(6)Q0.05=T×𝒫F-1.65×T1/2×(𝒱F)1/2×ΓTF,Q0.95=T×𝒫F+1.65×T1/2×(𝒱F)1/2×ΓTF.Remark 2.
Theorem1 is warrant under the assumptions (A1) and (A2), which correspond to the stationarity of wind series and consequently of power series Pt. However, this result can be enhanced by taking into account the wind direction as in [13, 14] or by replacing the stationarity assumption by seasonality.To sum up, the quantiles and the mean annual wind power production depend on three parameters, that is𝒫F the mean of ten-minute wind power of the farm, 𝒱F the variance of ten-minute wind power, and ΓTF which depends on the correlation structure of the wind power time series. Before starting the estimation procedures, we check the correlation coefficient ρX(k) as described previously, which proves that wind speed and the average wind farm power of four turbines are strongly correlated at a time scale smaller than ten hours and become uncorrelated at scale 48 hours (confirmed by Ljung-Box test; see e.g., [18]) as shown in the following Figure 4.Figure 4
Correlogram of mean wind speed and mean wind farm power, duration 19 months.


### 3.3.  Results

In order to carry out the overestimation of the mean wind farm power production, first, we take again the nominal power law proposed in [9] based on the use of a numerical sample of the ten-minute wind power derived from the sample of wind speed. Secondly, we use the traditional method which is modeled by a Weibull distribution law described in Section 3.1. We compare the overestimation rate for each turbine and for all the farm as illustrated in Table 2.Table 2
The overestimation rate.
E
1
E
2
E
3
E
4
Farm
Power law
−4.4
−3.8
−0.7
+2.1
−1.8
Weibull law
−1.8
−0.2
−46.3
+4.7
−10.43We should note that the overestimation of the mean power farm production based on a Weibull model is 10.43% and with the numerical wind power series it becomes 1.8%. The sources of this overestimation are out of the scope of this article. However, starting from the estimation of the autocorrelation coefficient, defined in Section3.2, we proceed to calculate the mean and the quantiles of the annual power production for the whole farm. It is based on a sample of ten-minute wind power as a postproduction approach. Then, we use the new method, that is, the preproduction approach which has already been used in the computed precedents.


#### 3.3.1. The Postproduction Approach

For the measurementsPtF of the existing turbines, we calculate the quantity ΓTF provided by the entire farm and ΓTEj for each turbine, that is, j=1,…,4, and we get the values stored in Table 3.Table 3
Estimated values ofΓ for existing turbines.
Γ
T
F
Γ
T
E
1
Γ
T
E
2
Γ
T
E
3
Γ
T
E
4
9.159
10.068
10.015
9.891
10.008Then, we can deduce the mean and the quantiles of the annual power production (see Table4).Table 4
The annual mean power production and its quantiles (GWH).
T
1
T
2
T
3
T
4
Farm
E
(
𝒫
annual
)
6.62
6.24
6.0
6.29
25.14
Q
0.05
6.18
5.82
5.58
5.86
23.47
Q
0.95
7.06
6.67
6.41
6.71
26.82
Spread (%)
13.4
13.6
13.9
13.5
13.3


#### 3.3.2. The Preproduction Approach

First, we calculatePtE1, PtE2, PtE3, and PtE4 based on the wind speed measurements provided by each turbine through power law. Thereafter, we follow the same procedure as mentioned previously. We get Table 5.Table 5
Estimated values ofΓ obtained by the preproduction approach.
Γ
T
F
Γ
T
E
1
Γ
T
E
2
Γ
T
E
3
Γ
T
E
4
10.095
10.085
9.995
9.904
10.026Then, from these values, we obtained the mean and the quantiles by using Theorem1 for each turbine and for the farm, for the annual wind power production (see Table 6).Table 6
Forecasting power production for one year and its quantiles (GWH).
T
1
T
2
T
3
T
4
Farm
𝔼
(
𝒫
annual
)
6.33
6.0
5.95
6.42
24.70
Q
0.05
5.89
5.58
5.53
5.58
23
Q
0.95
6.77
6.42
6.37
6.85
26.41
Spread (%)
14
14
14.1
13.5
13.8We can also forecast the twenty-year wind farm power production (20yPP), which corresponds to the lifespan of some turbines (see Table7).Table 7
Forecasting of twenty-year power production and its quantiles (GWH).
T
1
T
2
T
3
T
4
Farm
20yPP
127
120
119
128
494
Q
0.05
125
118
117
126
486
Q
0.95
129
122
121
130
502
Spread (%)
3.1
3.1
3.1
3
3We find that the uncertainties decrease with the length of the forecast period, rising from 13.8% for one year to 3% for twenty years on total wind farm power, whereas it is decreased from 14% for one year to 3% for twenty years at each turbine.


### 3.4.  The Relationship between the Measurements of Each Turbine in the Farm

To achieve a good characterization of the relationship between the power output and the wind speed for the four turbines, we calculate the correlation coefficients of wind power production (denoted byP1, P2, P3, and P4) and the correlation coefficients of wind speed (denoted by S1, S2, S3, and S4) of the four turbines. Let us recall that the correlation coefficient for two series X and Y is calculated as ρ(X,Y)=cov(X,Y)/σXσY. The correlations for the existing turbines can be summarized in Table 8.Table 8
Correlation coefficients forP1, P2, P3, and P4.
P
1
P
2
P
3
P
4
P
1
1
0.91
0.90
0.91
P
2
0.91
1
0.92
0.90
P
3
0.90
0.92
1
0.90
P
4
0.91
0.90
0.90
1Let us point out that the coefficient correlation is always superior to 0.9 for the wind power production of the four turbines. This assessment is of importance to appreciate the quality of the location of the turbines. We should take into account the effect of interactions between neighbour turbines which can eventually present loss in wind power production. The results obtained by the data generated through the power law are very close to this one. This once again underlines the contribution of the proposed approach to study the relationship that may exist between the energy outputs of wind turbines on a farm before their installation.Similarly, regarding the correlation between the four wind speeds (denoted byS1, S2, S3, and S4), we can distinguish the strong correlation between all the turbines which is nearly equal to 1 (see Tables 9 and 10).Table 9
Correlation coefficients using numerical sample forP1, P2, P3, and P4.
P
1
P
2
P
3
P
4
P
1
1
0.96
0.94
0.96
P
2
0.96
1
0.97
0.96
P
3
0.94
0.97
1
0.97
P
4
0.96
0.96
0.97
1Table 10
Correlation coefficients forS1, S2, S3, and S4.
S
1
S
2
S
3
S
4
S
1
1
0.97
0.91
0.97
S
2
0.97
1
0.92
0.97
S
3
0.91
0.92
1
0.92
S
4
0.97
0.97
0.92
1


## 3.1. State of the Art

The traditional method is based on modeling ten-minute wind speed probability density function (pdf) and then calculating the average ten-minute wind power production as𝒫=(1-p0)∫Vcut.inVcut.off‍P(v)f(v)dv, where p0 denotes the probability of zero wind; Vcut.in, Vcut.off denote the wind speed for cutting in and cutting off the turbine; v↦P(v) denotes the nominal power law; f(v) denotes the wind speed probability density function (pdf). Most often the ten-minute wind speed is assumed to follow a Weibull distribution, or a hybrid Weibull law; see, for example, [15, 16], or [7]. This model gives a good estimation of both mean ten-minute wind speed and mean ten-minute wind power production and after that provides us with a good forecasting of the mean annual wind power production. However, this traditional method discards the time structure of wind speed and wind power and thus it does not allow to forecast the variance, nor the quantiles of annual wind power production.Recently, the authors of [9] have proposed a new method which takes into account the dynamical structure of the annual wind power production. This method provides forecasting not only of the mean annual wind power production, but also of its variance and the quantiles of annual wind power production. The techniques rely on the central limit theorem (CLT) which asserts that the pdf of annual wind power production is almost Gaussian under natural assumptions. In the next section, we slightly adapt this method to the case of a wind farm.


## 3.2.  Forecasting Wind Farm Power

The annual wind power production of the farm is defined by(1)𝒫annualF=∑t=1TPtF,
where T=52,560 denotes the number of ten-minute periods during one year. The farm production is the sum of the individual production of the J turbines, that is, PtF=∑j=1JPt(j). The most simple but quite large model is to assume that each series Pt is wide sense stationary. Wide sense stationary means that on the one hand both mean 𝒫(j) and variance 𝒱(j) are not depending on time and that on the other hand the dynamical structure read on the covariance does not depend on the time t. As a corollary, the farm production series PtF is also wide sense stationary with mean 𝒫F=𝔼(PtF)=∑j=1J𝒫(j) and variance as follows:
(2)𝒱F=Var(PtF)=∑i,j=1Jcov(Pt(i),Pt(j))
and correlation coefficient ρPF(k)=cov(PtF,Pt+kF)/Var(PtF) which does not depend on the time t. Moreover, the family of random variables PtF is weakly dependent, and it admits a finite second-order moment (i.e., 𝔼(𝒫tF)<∞ for each t≥1). In addition, the annual wind production 𝒫annualF is also random with mean 𝔼(𝒫annualF)=T×𝒫F and variance as follows:
(3)var(𝒫annualF)=var(∑t=1TPtF)=T×𝒱F×(ΓTF)2,
where ΓTF={1+2∑k=1T[ρpF(k)×(1-k/T)]}1/2 with ρPF(k)=cov(PtF,Pt+kF)/Var(PtF) which was introduced to variance analysis for characterizing wind energy conversion as in [17].Let us stress that the variance of annual production of the farm depends both on the variance of ten-minute wind power𝒱Fand its correlogram ρpF(k), which corresponds to the time structure of the series. We will also need the two following assumptions. (A1)
The second-order moment ofPt is finite, that is, size 𝒱=𝔼((Pt)2)<∞;(A2)
The family of random variablesPt is weakly dependent. After having made precise assumptions and notation, by using the same tricks as in Proposition  3.1 in [9], we can deduce the following CLT.Theorem 1 (CLT for wind farm annual production).
(i)
If the family of r.v.Pt is wide sense stationary, and assumptions (A1) and (A2) are fulfilled, then one has
(4)𝒫annualF=T×𝒫F+T1/2×(𝒱F)1/2·ΓTF×ɛT,
where
(5)𝒫F=∑j=1J𝒫(j),𝒱F=Var(PtF)=∑i,j=1Jcov(Pt(i),Pt(j)),ΓTF={1+2∑k=1T[ρpF(k)×(1-kT)]}1/2
and ɛT is a zero mean r.v. with variance 1 which converges towards a standard Gaussian law; that is, ɛT→𝒩(0,1) when T→∞.
(ii)
Moreover, one can deduce the quantiles of annual production as follows:(6)Q0.05=T×𝒫F-1.65×T1/2×(𝒱F)1/2×ΓTF,Q0.95=T×𝒫F+1.65×T1/2×(𝒱F)1/2×ΓTF.Remark 2.
Theorem1 is warrant under the assumptions (A1) and (A2), which correspond to the stationarity of wind series and consequently of power series Pt. However, this result can be enhanced by taking into account the wind direction as in [13, 14] or by replacing the stationarity assumption by seasonality.To sum up, the quantiles and the mean annual wind power production depend on three parameters, that is𝒫F the mean of ten-minute wind power of the farm, 𝒱F the variance of ten-minute wind power, and ΓTF which depends on the correlation structure of the wind power time series. Before starting the estimation procedures, we check the correlation coefficient ρX(k) as described previously, which proves that wind speed and the average wind farm power of four turbines are strongly correlated at a time scale smaller than ten hours and become uncorrelated at scale 48 hours (confirmed by Ljung-Box test; see e.g., [18]) as shown in the following Figure 4.Figure 4
Correlogram of mean wind speed and mean wind farm power, duration 19 months.


## 3.3.  Results

In order to carry out the overestimation of the mean wind farm power production, first, we take again the nominal power law proposed in [9] based on the use of a numerical sample of the ten-minute wind power derived from the sample of wind speed. Secondly, we use the traditional method which is modeled by a Weibull distribution law described in Section 3.1. We compare the overestimation rate for each turbine and for all the farm as illustrated in Table 2.Table 2
The overestimation rate.
E
1
E
2
E
3
E
4
Farm
Power law
−4.4
−3.8
−0.7
+2.1
−1.8
Weibull law
−1.8
−0.2
−46.3
+4.7
−10.43We should note that the overestimation of the mean power farm production based on a Weibull model is 10.43% and with the numerical wind power series it becomes 1.8%. The sources of this overestimation are out of the scope of this article. However, starting from the estimation of the autocorrelation coefficient, defined in Section3.2, we proceed to calculate the mean and the quantiles of the annual power production for the whole farm. It is based on a sample of ten-minute wind power as a postproduction approach. Then, we use the new method, that is, the preproduction approach which has already been used in the computed precedents.


### 3.3.1. The Postproduction Approach

For the measurementsPtF of the existing turbines, we calculate the quantity ΓTF provided by the entire farm and ΓTEj for each turbine, that is, j=1,…,4, and we get the values stored in Table 3.Table 3
Estimated values ofΓ for existing turbines.
Γ
T
F
Γ
T
E
1
Γ
T
E
2
Γ
T
E
3
Γ
T
E
4
9.159
10.068
10.015
9.891
10.008Then, we can deduce the mean and the quantiles of the annual power production (see Table4).Table 4
The annual mean power production and its quantiles (GWH).
T
1
T
2
T
3
T
4
Farm
E
(
𝒫
annual
)
6.62
6.24
6.0
6.29
25.14
Q
0.05
6.18
5.82
5.58
5.86
23.47
Q
0.95
7.06
6.67
6.41
6.71
26.82
Spread (%)
13.4
13.6
13.9
13.5
13.3


### 3.3.2. The Preproduction Approach

First, we calculatePtE1, PtE2, PtE3, and PtE4 based on the wind speed measurements provided by each turbine through power law. Thereafter, we follow the same procedure as mentioned previously. We get Table 5.Table 5
Estimated values ofΓ obtained by the preproduction approach.
Γ
T
F
Γ
T
E
1
Γ
T
E
2
Γ
T
E
3
Γ
T
E
4
10.095
10.085
9.995
9.904
10.026Then, from these values, we obtained the mean and the quantiles by using Theorem1 for each turbine and for the farm, for the annual wind power production (see Table 6).Table 6
Forecasting power production for one year and its quantiles (GWH).
T
1
T
2
T
3
T
4
Farm
𝔼
(
𝒫
annual
)
6.33
6.0
5.95
6.42
24.70
Q
0.05
5.89
5.58
5.53
5.58
23
Q
0.95
6.77
6.42
6.37
6.85
26.41
Spread (%)
14
14
14.1
13.5
13.8We can also forecast the twenty-year wind farm power production (20yPP), which corresponds to the lifespan of some turbines (see Table7).Table 7
Forecasting of twenty-year power production and its quantiles (GWH).
T
1
T
2
T
3
T
4
Farm
20yPP
127
120
119
128
494
Q
0.05
125
118
117
126
486
Q
0.95
129
122
121
130
502
Spread (%)
3.1
3.1
3.1
3
3We find that the uncertainties decrease with the length of the forecast period, rising from 13.8% for one year to 3% for twenty years on total wind farm power, whereas it is decreased from 14% for one year to 3% for twenty years at each turbine.


## 3.3.1. The Postproduction Approach

For the measurementsPtF of the existing turbines, we calculate the quantity ΓTF provided by the entire farm and ΓTEj for each turbine, that is, j=1,…,4, and we get the values stored in Table 3.Table 3
Estimated values ofΓ for existing turbines.
Γ
T
F
Γ
T
E
1
Γ
T
E
2
Γ
T
E
3
Γ
T
E
4
9.159
10.068
10.015
9.891
10.008Then, we can deduce the mean and the quantiles of the annual power production (see Table4).Table 4
The annual mean power production and its quantiles (GWH).
T
1
T
2
T
3
T
4
Farm
E
(
𝒫
annual
)
6.62
6.24
6.0
6.29
25.14
Q
0.05
6.18
5.82
5.58
5.86
23.47
Q
0.95
7.06
6.67
6.41
6.71
26.82
Spread (%)
13.4
13.6
13.9
13.5
13.3


## 3.3.2. The Preproduction Approach

First, we calculatePtE1, PtE2, PtE3, and PtE4 based on the wind speed measurements provided by each turbine through power law. Thereafter, we follow the same procedure as mentioned previously. We get Table 5.Table 5
Estimated values ofΓ obtained by the preproduction approach.
Γ
T
F
Γ
T
E
1
Γ
T
E
2
Γ
T
E
3
Γ
T
E
4
10.095
10.085
9.995
9.904
10.026Then, from these values, we obtained the mean and the quantiles by using Theorem1 for each turbine and for the farm, for the annual wind power production (see Table 6).Table 6
Forecasting power production for one year and its quantiles (GWH).
T
1
T
2
T
3
T
4
Farm
𝔼
(
𝒫
annual
)
6.33
6.0
5.95
6.42
24.70
Q
0.05
5.89
5.58
5.53
5.58
23
Q
0.95
6.77
6.42
6.37
6.85
26.41
Spread (%)
14
14
14.1
13.5
13.8We can also forecast the twenty-year wind farm power production (20yPP), which corresponds to the lifespan of some turbines (see Table7).Table 7
Forecasting of twenty-year power production and its quantiles (GWH).
T
1
T
2
T
3
T
4
Farm
20yPP
127
120
119
128
494
Q
0.05
125
118
117
126
486
Q
0.95
129
122
121
130
502
Spread (%)
3.1
3.1
3.1
3
3We find that the uncertainties decrease with the length of the forecast period, rising from 13.8% for one year to 3% for twenty years on total wind farm power, whereas it is decreased from 14% for one year to 3% for twenty years at each turbine.


## 3.4.  The Relationship between the Measurements of Each Turbine in the Farm

To achieve a good characterization of the relationship between the power output and the wind speed for the four turbines, we calculate the correlation coefficients of wind power production (denoted byP1, P2, P3, and P4) and the correlation coefficients of wind speed (denoted by S1, S2, S3, and S4) of the four turbines. Let us recall that the correlation coefficient for two series X and Y is calculated as ρ(X,Y)=cov(X,Y)/σXσY. The correlations for the existing turbines can be summarized in Table 8.Table 8
Correlation coefficients forP1, P2, P3, and P4.
P
1
P
2
P
3
P
4
P
1
1
0.91
0.90
0.91
P
2
0.91
1
0.92
0.90
P
3
0.90
0.92
1
0.90
P
4
0.91
0.90
0.90
1Let us point out that the coefficient correlation is always superior to 0.9 for the wind power production of the four turbines. This assessment is of importance to appreciate the quality of the location of the turbines. We should take into account the effect of interactions between neighbour turbines which can eventually present loss in wind power production. The results obtained by the data generated through the power law are very close to this one. This once again underlines the contribution of the proposed approach to study the relationship that may exist between the energy outputs of wind turbines on a farm before their installation.Similarly, regarding the correlation between the four wind speeds (denoted byS1, S2, S3, and S4), we can distinguish the strong correlation between all the turbines which is nearly equal to 1 (see Tables 9 and 10).Table 9
Correlation coefficients using numerical sample forP1, P2, P3, and P4.
P
1
P
2
P
3
P
4
P
1
1
0.96
0.94
0.96
P
2
0.96
1
0.97
0.96
P
3
0.94
0.97
1
0.97
P
4
0.96
0.96
0.97
1Table 10
Correlation coefficients forS1, S2, S3, and S4.
S
1
S
2
S
3
S
4
S
1
1
0.97
0.91
0.97
S
2
0.97
1
0.92
0.97
S
3
0.91
0.92
1
0.92
S
4
0.97
0.97
0.92
1


## 4. Discussion

The grouping of several wind turbines on the same site reduces the investment costs. However, it is important to make an optimal configuration of the turbines locations. For this wind farm site, there will be certain directions for which other turbines affect the production of single wind turbine. A more detailed analysis of this dependence will be made in further work.



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*Source: 163565-2013-07-30.xml*