### Details of the wind farm studied

Wind farm located at Soda site in the Thar desert region of western Rajasthan, India is selected for this study. It is in Jaisalmer district where May and June are hottest and January is the coldest month. Rainfall is very low and monsoon winds that bring rains in India bypass this region. Wind farm has twenty 1.25-MW capacity Suzlon-S66 turbines as shown in Figures 1 and 2. The total capacity of wind farm is 25 MW and turbines are having hub height of 65 m, cut-in speed *v*
_{
c
} of 3 m/s, rated speed *v*
_{
r
} of 14 m/s, and cut-off speed *v*
_{
f
} of 22 m/s (http://www.suzlon.com/pdf/s66%20product%20brochure.pdf. Accessed 09 September 2014). Wind and meteorological data measurement mast of 65-m height at Soda wind farm is shown in Figure 3. Its specific position in the wind farm is marked in Figure 2.

### Wind data modeling and analysis

Mean wind speed and standard deviation of grouped data are defined by Jangamshetti and Rau (1999), Manwell et al. (2009), and Bird (2003) as:

$$ \overline{v}=\frac{{\displaystyle {\sum}_{i=1}^n\left({f}_m\left({v}_i\right)\times {v}_i\right)}}{{\displaystyle {\sum}_{i=1}^n{f}_m\left({v}_i\right)}} $$

(1)

$$ \sigma =\sqrt{\frac{{\displaystyle {\sum}_{i=1}^n{f}_m\left({v}_i\right)\cdot {\left({v}_i-\overline{v}\right)}^2}}{{\displaystyle {\sum}_{i=1}^n{f}_m\left({v}_i\right)}}} $$

(2)

where \( \overline{v} \) is the mean wind speed in meter per second, *σ* is the standard deviation of wind speed in meter per second, *v*
_{
i
} is the wind speed in meter per second at *i*th bin midpoint, *f*
_{
m
}(*v*
_{
i
}) is the measured frequency of wind speed for *i*th bin, and *n* is the number of wind speed bins.

Weibull probability density function and its cumulative distribution function, used for describing the wind speed frequency distribution of a site, are defined by Masters (2004) as:

$$ f(v)=\frac{k}{c}\ {\left(\frac{v}{c}\right)}^{k-1} \exp \left(-{\left(\frac{v}{c}\right)}^k\right) $$

(3)

$$ F(v)=1- \exp \left(-{\left(\frac{v}{c}\right)}^k\right) $$

(4)

where *f*(*v*) is the Weibull wind speed probability density function at hub height, *F*(*v*) is the Weibull cumulative distribution function, *v* is the wind speed in meter per second, *k* is the shape parameter at hub height, and *c* is the scale parameter at hub height.

Power available in the wind (*P*
_{
w
}(*v*)) is expressed as *P*
_{
w
}(*v*) = 0.5*ρAv*
^{3}, where *ρ* is the air density in kilogram per cubic meter, *A* is the rotor swept area in square meter, and *v* is the wind speed in meter per second. Wind power density (WPD) of a site that is based on Weibull distribution is defined by Jowder (2009), Huang and Wan (2012), and Chang et al. (2003) as:

$$ \mathrm{W}\mathrm{P}\mathrm{D}={\displaystyle {\int}_0^{\infty }{P}_w(v)}f(v)dv=0.5\rho {c}^3\varGamma \left(1+3/k\right) $$

(5)

where *Γ* is a gamma function.

Root mean square error (RMSE) is based on the variation between measured and estimated values. RMSE of wind speed probability is defined by Rocha et al. (2012) and Bird (2003) as:

$$ \mathrm{RMSE} = \sqrt{\left[\frac{1}{n}{\displaystyle {\sum}_{i=1}^n{\left({f}_m\left({v}_i\right)-{f}_c\left({v}_i\right)\right)}^2}\right]} $$

(6)

where *f*
_{
m
}(*v*
_{
i
}) is the measured wind speed frequency for *i*th bin, *f*
_{
c
}(*v*
_{
i
}) is the estimated Weibull wind speed probability, *v*
_{
i
} is the wind speed at *i*th bin midpoint, and *n* is the number of observations/bins. The percentage error between measured and estimated value is calculated using expression:

$$ \mathrm{Error}\ \%=\frac{\mathrm{measured}\ \mathrm{value}-\mathrm{estimated}\ \mathrm{value}}{\mathrm{measured}\ \mathrm{value}}\times 100. $$

(7)

### Estimation of Weibull scale and shape parameters

Graphical method (GM) (Johnson 1978) uses Weibull cumulative distribution function and least square approximation for calculating the scale and shape parameters. Using Equation 4 and on taking twice the logarithm of each side, it becomes a form of straight line equation written as *y* = *ax* + *b* where *y* = ln[−ln(1 − *F*(*v*))], *a* = *k*, *x* = ln(*v*), and *b* = − *k* ln(*c*). For *n* pairs of (*x*, *y*) where all summations are from 1 to *n*, the values of *a* and *b* are expressed as:

$$ a=\frac{{\displaystyle \sum xy-\frac{{\displaystyle \sum x{\displaystyle \sum y}}}{n}}}{{\displaystyle \sum {x}^2}-\frac{{\left({\displaystyle \sum x}\right)}^2}{n}} $$

(8)

$$ b=\overline{y}-a\overline{x}=\frac{1}{n}{\displaystyle \sum y-\frac{a}{n}}{\displaystyle \sum x}. $$

(9)

Shape and scale parameters are then expressed as *k* = *a* and *c* = exp(−*b*/*k*).

Empirical method (EM) uses shape and scale parameter defined by Jangamshetti and Rau (1999) and Rocha et al. (2012) as:

$$ k={\left(\sigma /\overline{v}\right)}^{-1.086} $$

(10)

$$ c=\frac{\overline{v}}{\varGamma \left(1+\frac{1}{k}\right)}. $$

(11)

Modified maximum likelihood (MML) method uses frequency distribution of wind speed. Shape parameter is calculated by using numerical iterations and then scale parameter is obtained by solving equation explicitly. Value of shape parameter is around 2 for majority of sites and is a good initial estimate for iterative process. Shape and scale parameters are defined by Rocha et al. (2012) as:

$$ k={\left[\frac{{\displaystyle {\sum}_{i=1}^n{v}_i^k \ln \left({v}_i\right)}f\left({v}_i\right)}{{\displaystyle {\sum}_{i=1}^n{v}_i^kf\left({v}_i\right)}}-\frac{{\displaystyle {\sum}_{i=1}^n \ln \left({v}_i\right)}f\left({v}_i\right)}{f\left(v\ge 0\right)}\right]}^{- 1} $$

(12)

$$ c={\left(\frac{1}{f\left(v\ge 0\right)}{\displaystyle {\sum}_{i=1}^n{v_i}^kf\left({v}_i\right)}\right)}^{\frac{1}{k}} $$

(13)

where *v*
_{
i
} is the wind speed at *i*th bin midpoint, *n* is the number of bins, *f*(*v*
_{
i
}) is the frequency of wind speed occurrence in bin *i*, and *f*(*v* ≥ 0) is the probability of wind speed ≥ 0.

Energy pattern factor (EPF) is expressed as mean of the sum of cubes of all individual wind speed considered in a sample, divided by the cube of mean wind speed of sample (Centre for Wind Energy Technology 2011):

$$ \mathrm{E}\mathrm{P}\mathrm{F}=\frac{1}{{\left(\overline{v}\right)}^3}\times \left({\displaystyle {\sum}_{i=1}^n{v}_i^3}/n\right) $$

(14)

where *v*
_{
i
} is the wind speed in meter per second for *i*th observation, *n* is the number of wind speed samples, and \( \overline{v} \) is the monthly mean wind speed. The monthly wind power density (WPD) is given by:

$$ \mathrm{W}\mathrm{P}\mathrm{D}=0.5\rho \left({\displaystyle {\sum}_{i=1}^n{v}_i^3}/n\right) $$

(15)

where *ρ* is the monthly mean air density at hub height in kilogram per cubic meter. By substituting Equation 15 in Equation 14, EPF is expressed as:

$$ \mathrm{E}\mathrm{P}\mathrm{F}=\frac{1}{{\left(\overline{v}\right)}^3}\times \left(\frac{\mathrm{WPD}}{0.5\times \rho}\right). $$

(16)

Shape parameter is calculated from EPF parameter using an expression defined by Rocha et al. (2012) as:

$$ k=1+\frac{3.69}{{\left(\mathrm{E}\mathrm{P}\mathrm{F}\right)}^2}. $$

(17)

Scale parameter is then calculated by using the expression given in Equation 11.

### Polynomial model of power curve for pitch-regulated wind turbines

Relation between wind turbine electric power output (*P*
_{
e
}(*v*)) and wind speed (*v*) for pitch regulated wind turbines are defined by Albadi (2010) as:

$$ {P}_e(v)={P}_r\times \left\{\begin{array}{c}\hfill 0,\kern6.25em \left(v<{v}_c\ or\ v>{v}_f\right)\kern0.5em \hfill \\ {}\hfill {P}_{cinr}(v),\kern7.1em \left({v}_c\le v\le {v}_r\right)\kern0.5em \hfill \\ {}\hfill 1,\kern9.5em \left({v}_r\le v\le {v}_f\right)\ \hfill \end{array}\right. $$

(18)

where *P*
_{
r
} is the rated electrical power, and *P*
_{
cinr
}(*v*) is the turbine output power as a fraction of rated power between (including) cut-in wind speed *v*
_{
c
} and rated wind speed *v*
_{
r
}. *v*
_{
f
} is cut-out wind speed.

There are many generic power curve models reported in the literature for representing the non-linear region between cut-in and rated wind speed of Figure 4. These models are not accurate as they do not fit the manufacturer’s power curve data and only provide an approximate model of power curve that has errors. The approach used in this paper is to use a polynomial of eighth degree to model manufacturer wind turbine power curve data between cut-in and rated wind speed region.

A function is called polynomial of *n*th degree when it is expressed in the form as

$$ P(v)={a}_0+{a}_1v+{a}_2{v}^2+{a}_3{v}^3+\dots +{a}_n{v}^n $$

(19)

where *a*
_{0}, *a*
_{1}, *a*
_{2}, …, *a*
_{
n
} are the constant coefficients of polynomial function. The procedure of calculating coefficients of *n*th-degree polynomial by combined use of linear least square and matrix factorization methods through MATLAB are explained below.

### Linear least square method

Consider given *m* sets of data (*x*
_{
i
}, *y*
_{
i
}) where *i* = 1,.., *m* and the polynomial model that is fitted to data is of *n*th degree expressed as:

$$ P(x)={a}_0+{a}_1x+{a}_2{x}^2+{a}_3{x}^3+\dots +{a}_n{x}^n $$

(20)

where *a*
_{0}, *a*
_{1}, *a*
_{2}, …, *a*
_{
n
} are the coefficients that are to be found out. The *m* sets of data and polynomial *P*(*x*) are expressed in matrix form as *y* = *Xα* where:

$$ \boldsymbol{y}=\left[\begin{array}{c}\hfill {y}_1\hfill \\ {}\hfill {y}_2\hfill \\ {}\hfill \vdots \hfill \\ {}\hfill {y}_m\hfill \end{array}\right], $$

(21)

$$ {\boldsymbol{X}}_{\left(m,\ n+1\right)}=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill {x}_1\hfill & \hfill {x}_1^2\hfill \\ {}\hfill \vdots \hfill & \hfill \vdots \hfill & \hfill \vdots \hfill \\ {}\hfill 1\hfill & \hfill {x}_m\hfill & \hfill {x}_m^2\hfill \end{array}\kern0.75em \begin{array}{c}\hfill \cdots \hfill \\ {}\hfill \ddots \hfill \\ {}\hfill \dots \hfill \end{array}\kern0.75em \begin{array}{c}\hfill {x}_1^n\hfill \\ {}\hfill \vdots \hfill \\ {}\hfill {x}_m^n\hfill \end{array}\right], $$

(22)

$$ \boldsymbol{\alpha} =\left[\begin{array}{c}\hfill {a}_0\hfill \\ {}\hfill {a}_1\hfill \\ {}\hfill \vdots \hfill \\ {}\hfill {a}_n\hfill \end{array}\right]. $$

(23)

The coefficients *a*
_{0}, *a*
_{1}, *a*
_{2}, …, *a*
_{
n
}, that best fit Equation 20 are found out by solving minimization problem, where the objective function *S* is given by Press et al. (2009) as:

$$ S\left(\boldsymbol{\alpha} \right)={\displaystyle {\sum}_{i=1}^m{\left[{\boldsymbol{y}}_i-{\displaystyle {\sum}_{j=1}^{n+1}{\boldsymbol{X}}_{ij}{\boldsymbol{\alpha}}_j}\right]}^2}={\left\Vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\alpha } \right\Vert}^2. $$

(24)

Normal equations of least square problem can be expressed in matrix notation as

$$ \left({\boldsymbol{X}}^T\boldsymbol{X}\right)\boldsymbol{\alpha} ={\boldsymbol{X}}^T\boldsymbol{y} $$

(25)

where *X*
^{T} is the transpose of matrix *X*. The algebraic solution of Equation 24 is expressed (Demmel 1997) as

$$ \boldsymbol{\alpha} ={\left({\boldsymbol{X}}^T\boldsymbol{X}\right)}^{-1}{\boldsymbol{X}}^T\boldsymbol{y}\ . $$

(26)

Solution from normal equations can have round-off errors so QR decomposition of matrix *X* is done.

### QR decomposition

QR decomposition is a matrix factorization method (Embree 2010). It states that for any *m* × *n* matrix *X* with *m* ≥ *n*, there exists a unitary *m* × *m* matrix *Q* and an upper triangular *m* × *n* matrix *R* such that

$$ \boldsymbol{X}=\boldsymbol{Q}\boldsymbol{R}\ . $$

(27)

On substituting Equation 27 in Equation 26, the expression as explained by Demmel (1997) becomes:

$$ \boldsymbol{\alpha} ={\left({\boldsymbol{R}}^T{\boldsymbol{Q}}^T\boldsymbol{Q}\boldsymbol{R}\right)}^{-1}{\boldsymbol{R}}^T{\boldsymbol{Q}}^T\boldsymbol{y}={\left({\boldsymbol{R}}^T\boldsymbol{R}\right)}^{-1}{\boldsymbol{R}}^T{\boldsymbol{Q}}^T\boldsymbol{y} $$

(28)

$$ \boldsymbol{\alpha} ={\boldsymbol{R}}^{-1}{\boldsymbol{R}}^{-T}{\boldsymbol{R}}^T{\boldsymbol{Q}}^T\boldsymbol{y} $$

(29)

$$ \boldsymbol{\alpha} ={\boldsymbol{R}}^{-1}{\boldsymbol{Q}}^T\boldsymbol{y}\ . $$

(30)

On solving Equation 30, the required coefficients of polynomial Equation 20 are obtained. For computing QR decomposition of matrix *X*, the MATLAB command used is (Embree 2010):

$$ \left[\boldsymbol{Q},\boldsymbol{R}\right]=\mathrm{q}\mathrm{r}\left(\boldsymbol{X}\right). $$

(31)

This application has a *m* × *n* matrix *X* with *m* much larger than *n*. So, the QR decomposition produces a *m* × *m* matrix *Q* that will require more storage than *X* (Embree 2010). Also, columns *n* + 1, …, *m* of *Q* are surplus as they multiply against zero entries of *R*.

### QR decomposition using Gram-Schmidt orthogonalization

It is one solution to the above mentioned concern. This procedure results in a *skinny QR decomposition*, *X* = *QR*, where *Q* is *m* × *n* matrix, *R* is a *n* × *n* matrix, and *Q***Q* = *I*. Here, *Q** is the conjugate transpose matrix and *I* is *n* × *n* identity matrix (Embree 2010). This algorithm can be easily computed in MATLAB using command:

$$ \left[\boldsymbol{Q},\boldsymbol{R}\right]=\mathrm{q}\mathrm{r}\left(\boldsymbol{X},\boldsymbol{0}\right). $$

(32)

If *m* > *n*, only the first *n* columns of *Q* and the first *n* rows of *R* are computed (http://in.mathworks.com/help/matlab/ref/qr.html. Accessed 09 September 2014). If *m* ≤ *n*, then, this is same as [*Q*, *R*] = qr(*X*).

### Analytical estimation of capacity factor

Capacity factor (CF) (Masters 2004) is defined as the ratio of average output power to rated output power over a certain period of time. Monthly capacity factor (CF_{m}) is expressed as:

$$ {\mathrm{CF}}_{\mathrm{m}}=\frac{\mathrm{m}\mathrm{onthly}\ \mathrm{energy}\ \mathrm{yield}\ \mathrm{from}\ \mathrm{wind}\ \mathrm{turbine}\ \left(\mathrm{k}\mathrm{W}\mathrm{h}\right)}{\mathrm{rated}\ \mathrm{power}\ \left(\mathrm{kW}\right) \times \mathrm{total}\ \mathrm{h}\mathrm{ours}\ \mathrm{in}\ \mathrm{particular}\ \mathrm{month}} $$

(33)

and the annual capacity factor (CF_{a}) is expressed as:

$$ {\mathrm{CF}}_{\mathrm{a}} = \frac{\mathrm{a}\mathrm{nnual}\ \mathrm{energy}\ \mathrm{yield}\ \mathrm{from}\ \mathrm{wind}\ \mathrm{turbine}\left(\mathrm{k}\mathrm{W}\mathrm{h}\right)\ }{\mathrm{rated}\ \mathrm{power}\ \left(\mathrm{kW}\right) \times \mathrm{total}\ \mathrm{h}\mathrm{ours}\ \mathrm{in}\ \mathrm{a}\ \mathrm{year}}. $$

(34)

Capacity factor of a particular wind turbine at a site can be analytically estimated by using Weibull scale and shape parameters of site, wind turbine speed parameters, and coefficients of polynomial model for power curve in the expression defined by Albadi (2010) as:

$$ \mathrm{C}\mathrm{F}=-{e}^{-{\left({v}_f/c\right)}^k}+{\displaystyle {\sum}_{i=1}^n\left[{a}_i\times i\times \left({c}^i/k\right)\times \varGamma \left(i/k\right)\times \left(\gamma \left({\left({v}_r/c\right)}^k,i/k\right)-\gamma \left({\left({v}_c/c\right)}^k,i/k\right)\right)\right]} $$

(35)

where \( \varGamma (a)=\mathrm{Gamma}\ \mathrm{function}={\displaystyle {\int}_0^{\infty }{t}^{a-1}{e}^{-t}dt,} \) and \( \gamma \left(u,a\right)=\mathrm{Incomplete}\ \mathrm{gamma}\ \mathrm{function}=\left[1/\varGamma (a)\right]\times {\displaystyle {\int}_0^u{t}^{a-1}}{e}^{-t}dt. \)