Some Mathematical Methods in Wind Power ModelingMatt Rosenzweig

Abstract

In this paper, we discuss some mathematical methods used to estimate the wind power resources of agiven location. We introduce the two-parameter Weibull distribution as a model for hourly average windspeeds, prove an existence an existence and uniqueness theorem for the maximum likelihood estimates ofthis distribution, and provide an algorithm for their computation. We derive a probability distributionfor the power output of a wind turbine with given cut-in, cut-off, and rated wind speeds and compute themoments of this distribution, in addition to deriving an expression for the capacity factor as a functonof these inputs. Lastly, we consider the problem of optimizing the choice of wind turbine for a givenlocation with known wind speed distribution.

Contents

1 Introduction 2

2 The Weibull Distribution 22.1 Definition and Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.1 Likelihood Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.2 Maximum Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.3 Computation of MLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Goodness of Fit 9

4 Single-Site Power Distribution 94.1 Ideal Wind Turbine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.2 Power Distribution of Ideal Wind Turbine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.3 Turbine-Site Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5 Conclusion 13

A Newton-Raphson Method 14

B Example Graphs 15

1

1 Introduction

Before one decides to a build a wind farm at a given location, he or she needs to know what the distributionof wind speeds is for the location in order to be able to estimate the amount of electricity the wind farm willproduce. In particular, if the distribution of the wind speed is such that only a small portion of the righttail is above the cut-in speed for the model of wind turbine, then it would be physically and economicallynonsensical to build a windfarm at the location. Analogously, if the distribution of the wind speed is such thatonly a small portion of the left tail is below the cut-off speed, then it would also be nonsensical, in additionto potentially dangerous, to build a windfarm at the location. Given that different geographic locations canhave completely different wind speed profiles, the distribution of wind speed is an inherently local question.

This paper is intended as an introduction to the use of the two-parameter Weibull distribution to modelsingle-site hourly average wind speeds (i.e. wind speeds at a given wind farm). Our goal is to write downan expression for the probability distribution of the power produced by a wind turbine at a fixed location,so that the modeling problem reduces to collecting data to estimate the two parameters of the Weibulldistribution and running some algorithms and statistical tests with standard software (e.g. Matlab). Thisreasoning a priori assumes that the Weibuill distribution is an appropriate choice, an assumption which hassupport in the literature ([7], [5], [8]) and will not be challenged in this paper. Our focus will be abstractand mathematical rather than concrete and empirical.

In section 2, we define the Weibull probability distribution, compute its moments, and prove some otherbasic properties. We then discuss the maximum likelihood estimate (MLE) technique for estimating theunknown parameters of the Weibull distribution. We prove an existence and uniqueness theorem for theMLE under mild hypotheses and give an algorithm based on the Newton-Raphson root-finding method forcomputing the MLE. The mathematical details of the Newton-Raphson method can be found in appendix A.For analysis of the performance of the MLE technique in comparison to the least-squares method and methodof moments in terms of minimizing mean-squared error (MSE), we refer the reader to [5]. We then use actualwind speed data from Bahrain to compute the parameters and a give graphical sense of the goodness of fitof the Weibull distribution with estimated parameters for the given data. In section 3, we briefly discussthe general problem of testing the goodness of fit of a null hypothesis distribution and provide referencesfor the reader interested in applying such tests when the parameters of the null hypothesis distribution areestimated. In section 4, we derive a probability distribution for the eletric power output of a wind turbinewith given cut-in, cut-off, and rated wind speeds. We also compute the moments of this distribution andderive an expression for the capacity factor of the wind turbine. Lastly, we consider the problem of matchinga wind turbines rated wind speed to the wind speed distribution of a given site.

There are two, not necessarily disjoint, intended audiences for this paper: quantitatively literate per-sons with a general interest in applied mathematics and a desire to make informed decisions concerningrenewable energy policy and practitioners who need mathematical modeling to estimate the wind resourcesas a particular site. The reader is assumed to have a knowledge of real analysis at the level of [15], basicmeasure-theoretic probability, elementary statistics to understand all proofs. The general reasoning of thispaper can be followed without attention to all the mathematical details, but to ignore them is contrary tothe intent this paper: to give a rigorous brief exposition of some mathematical methods in wind speed andpower modeling. The reader is only assumed to have knowledge of basic physics.

For questions about notation or any arguments in the proofs presented, the author can be reached bye-mail at matthew.rosenzweig@college.harvard.edu.

2 The Weibull Distribution

There appears to be a consensus in the literature that the Weibull distribution is a good probabilistic model ofwind speed at one location. In this section, we will introduce the two-parameter Weibull distribution, denotedWeib(, k), and prove some of its basic properties. We will then survey a few techniques for estimating theparameters , k. We will not focus on algorithms for the numerical computation of these estimates, as thesecan be easily done with software such as R or Matlab. We will defer questions of goodness of fit to a latersection of this paper.

2

2.1 Definition and Basic Properties

Let (,F ,P) be a probability space. A real-valued random variable X : R is said to have a (two-parameter) Weibull distribution if it has a probability density function (pdf)

f(x;, k) =

{k (

x )k1e(

x )k

x 00 x < 0

where k > 0 is a dimensionless shape parameter and > 0 is the scale parameter. As an exercise, thereader can check by means of integration that this is indeed a pdf. For our modeling purposes below, x willcorrespond to the wind velocity v. We will use the notation X Weib(, k) to denote that X has Weibulldistribution with parameters , k.

Proposition 2.1. The Weibull distribution with paramters , k > 0 has cummulative distribution function(cdf) given by

F (x;, k) =

{0 x < 0

1 e( x )k x 0

Proof. By the fundamental theorem of calculus,

P(X x) = x

f(y;, x)dy =

x0

k

(y

)k1e(

y )k

= x

0

d

dy

[e(

y )k]dy =

[e(

x )k 1

]= 1 e( x )k

Figure 1: Plots of the Weibull pdf and cdf, respectively, for various shape paramters k

Proposition 2.2. Let X Weib(, k). Then X has mth moment given by

E[Xm] = m(1 +m

k)

In particular, X has mean, variance, and maximum respectively given by

1. E[X] = (1 + 1k ),

3

2. Var(X) = 2((1 + 2k ) 2(1 + 1k )

),

where () denotes the gamma function.Proof. Recall Eulers integral for the gamma function, (z) =

0ettz1dt for Re z > 0. Hence, for

m Z1,

E[Xm] =Rxmf(x;, k)dx =

0

xmk

(x

)k1e(

x )k

dx

We make the change of variable y = (x )k to obtain

=

0

mymk eydy = (1 +

m

k)

Setting m = 1 gives the expression for E[X]. For Var(X), by definition

Var(X) = E[X2] E2[X] = 2(

(1 +2

k) 2(1 + 1

k)

)

The following proposition tells us that the wind speed follows a Weibull distribution regardless of thechoice of units.

Proposition 2.3. Let > 0 and X Weib(, k). Then X =: Y Weib(, k).Proof. Using Proposition 2.1, we see that

FY (x) = P(X x) = P(X x

) = 1 e( x )k ,

which is the cdf of a Weib(, k) random variable. Since the distribution function uniquely characterizes thelaw of the random variable, the conclusion follows immediately.

The following result on the minimum order statistic of Weibull random variates will be useful in giving arough answer to questions, such as given finitely many wind farms, what is the probability that all of themare producing more than P watts. In reality, wind speeds as dispersed locations are correlated, so that iswhat makes the following a rough answer.

Proposition 2.4. Let X1, , Xn be independent random variables with Xi Weib(i, ki), for 1 i n.Then

P(min(X1, , Xn) > x) = exp(

ni=1

(x

i)ki

)

If k1 = = kn = k, then min(X1, , Xn) Weib(min, k), where

min :=

(ni=1

ki

) 1kProof. Clearly, min(X1, , Xn) > x Xi > x i = 1, , n. Since the Xi are independent,

P(X1 > x, , Xn > x) =ni=1

P(Xi > x) =ni=1

(1

(1 e( xi )ki

))=

ni=1

e( xi )

ki

= exp

(

ni=1

(x

i)ki

)

4

This completes the proof of the first assertion. Now suppose that ki = k, 1 i n. Then we may write theabove as

= exp

(

ni=1

xk

ki

)= exp

( x

k(ni=1

ki

)1)

= exp( xk

kmin), min :=

(ni=1

ki

) 1kWe therefore have that

P(min(X1, , Xn) x) = 1 P(min(X1, , Xn) > x) = 1 exp(( xmin

)k),

which shows that min(X1, , Xn) Weib(min, k) by the uniqueness of distribution functions and Propo-sition 2.1.

2.2 Parameter Estimation

Having established some basic properties of the Weibull distribution and found a probability distribution forthe power output at a single site, we now turn to the problem of estimating the parameters , k.

2.2.1 Likelihood Function

Let (X1, , Xn) : Rn be a random variable with probability density function f(x; ) for a k-tuple ofparameters Rk. Recall that for a sample X() = x R, the likelihood function of , denoted by`x() = `(), is defined by

` : R, `() = f(x)

Let denote the topological closure of in R. If satisfies `() = sup `(), we say that is amaximum likelihood estimate (MLE) of . If : R, 7 , where `X()() = sup `(), thenwe say that is a maximum likelihood estimator (MLE) of .

We define the log-likelihood function to be log `(). We first note that this definition makes sense sincewe may asume that `() > 0 for all . The following lemma often simplifies computations of MLEs since thelogarithm converts products to sums.

Lemma 2.5. is an MLE if and only if log `() = sup `(log ).

Proof. This is immediate from the fact that log x is a strictly increasing function on (0,).

2.2.2 Maximum Likelihood

It is not true that every probability density function has a nice closed-form expression for the MLE . Unfor-tunately for us, this is also the case for the Weibull distribution. However, with the use of numerical softwaresuch as Matlab, we can use an iterative scheme (or trial and error, if we just want a rough approximation)

to approximate .Let X1, , Xn be independent identically distibuted (i.i.d.) Weib(, k) samples. Since the joint density

function of independent random variables factors, we have that

`(, k) =

ni=1

f(xi;, k) =

ni=1

(k

)(xi

)k1e(xi )

k

= (k

)n exp

( 1k

ni=1

xki

) ni=1 x

k1i

n(k1)

Taking the natural logarithm of both sides, we obtain that the log-likelihood function is

log `(, k) = n log(k) n log() 1k

(ni=1

xki

)+

ni=1

(k 1) log(xi) n(k 1) log()

= n log(k) nk log() 1k

(ni=1

xki

)+

ni=1

(k 1) log(xi)

5

Taking the partial derivatives with respect to , k, we obtain

[log `(, k)] =

nk n log() + k

k+1

(ni=1

xki

)and

k[log `(, k)] =

n

k+

log

k

(ni=1

xki

) 1k

(ni=1

log(xi)xki

)+

ni=1

log(xi)

Proposition 2.6. The MLE (, k) exists and is unique if x1, , xn satisfy min(x1, , xn) < max(x1, , xn)and xi > 0 i = 1, , n.Proof. Note that the log-likelihood function is in C1,1(0,). We set the two preceding equations to 0. Wehave

0 = nk

+k

k+1

(ni=1

xki

) =

(ni=1

xkin

) 1k

It is clear that, for k fixed, is the unique root of gk() = nk + kk+1 (ni=1 x

ki ). I claim that gk() 0 for

all . Indeed, since k+1 = o(k) as 0, gk() < 0 for all sufficiently small. If gk() > 0 for some < implies by the intermediate value theorem that there exists , 0 < < < such that gk() = 0,which is a contradiction. Noting that = o(k+1) as and therefore gk() < 0 for all sufficientlylarge, the same argument shows that gk() 0 for . We conclude that is the unique global maximumof gk.

Similarly, we have

0 =n

k n log() + log

k

(ni=1

xki

) 1k

(ni=1

log(xi)xki

)+

ni=1

log(xi)

=n

k n log() + n log()

kk 1

k

(ni=1

log(xi)xki

)+

ni=1

log(xi)

=n

k n

ni=1 x

ki log(xi)ni=1 x

ki

+

ni=1

log(xi)

ni=1 x

ki log(xi)ni=1 x

ki

1k

=

ni=1 log(xi)

n

I claim that this equation has a unique solution. For existence, observe that

h(k) = log `(, k) = n log(k) nkk log

(ni=1

xkin

) n

ni=1 x

kin

i=1 xki

+

ni=1

(k 1) log(xi)

= n log(k) n log(

ni=1

xkin

) n+

ni=1

(k 1) log(xi)

It is evident that h(k) as k 0. We now consider h(k) for large values of k, in particular k 2. Sincethe function x 7 xk is strictly convex, for k 2, by Jensens inequality,

ni=1

xkin>

(ni=1

xin

)k n log

[ni=1

xkin

]< n log

( ni=1

xin

)k = nk log [ ni=1

xin

]

Since the function x 7 log(x) is strictly convex and by our hypothesis that min(x1, , xn) < max(x1, , xn),we have by another application of Jensens inequality that

n log[

ni=1

xkin

]< nk log

[ni=1

xin

]< nk

ni=1

log(xi)

n= k

ni=1

log(xi),

6

which implies that

k

(ni=1

log(xi)

) n log

(ni=1

xkin

)< k

(ni=1

log(xi) n log(

ni=1

xin

))

C

< 0

Hence, C < 0. Since log(k) = o(k) as k , it follows that h(k) as k . Hence, there existsk1, k2 with 0 < k1 < k2 < , such that supk[k1,k2] h(k) = supk(0,) h(k). By Weierstrass extreme valuetheorem, there exists k [k1, k2] such that

h(k) = supk[k1,k2]

h(k) = supk(0,)

h(k),

which implies that k is a global maximum of h(k). By adjusting k1 and k2, if necessary, we may assume that

h(k) < h(k), k (0,) \ (k1, k2)Since (0,) is an open subset of R, we have by Fermats lemma for critical points that

0 = h(k) =`(, k)

k(k) =

n

k n

ni=1 x

ki log(xi)ni=1 x

ki

+

ni=1

log(xi)

For uniqueness, we note that since k is a local maximum, there exists an > 0 such that h(k) > 0 fork (k , k) and h(k) < 0 for k (k, k + ). Suppose that k is another global maximum of h. If k > k,then h(k) > 0 for some k k + , which which implies by the intermediate value theorem that h(k) = 0for some k (k, k), and in particular, this k is a local minimum. To arrive at a contradiction, we computeh(k):

h(k) = nk2 n

ni=1 x

ki (log xi)

2ni=1 x

ki

+ n

(ni=1 x

ki log(xi)

) (nj=1 x

ki log(xi)

)(n

i=1 xki

)2= n

k2 n

i

Then

g(k) = nk2 n

ni=1 x

ki (log xi)

2ni=1 x

ki

+ n

(ni=1 x

ki log(xi)

) (nj=1 x

ki log(xi)

)(n

i=1 xki

)2= n

k2 n

i

Wind Speed (ms ) Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec0 21 31 36 29 20 24 43 42 51 78 41 741 138 93 96 126 109 93 170 212 204 245 129 2012 277 171 216 245 305 183 244 381 341 389 349 3133 339 305 318 347 388 306 407 440 417 530 407 2924 331 294 337 381 432 309 376 375 368 383 320 2745 315 279 328 316 291 282 322 250 285 230 236 2556 256 243 260 249 230 283 262 203 192 132 200 2407 203 197 229 174 193 250 190 159 151 91 169 2018 166 157 172 122 108 218 152 101 98 72 119 1639 95 104 122 69 77 124 55 49 41 51 94 13210 58 97 64 37 39 43 11 14 8 25 48 5911 22 41 28 30 22 34 0 3 4 5 33 2412 8 15 14 16 14 8 0 3 0 1 10 313 3 10 7 14 4 3 0 0 0 0 4 114 0 3 3 4 0 0 0 0 0 0 1 015 0 0 1 0 0 0 0 0 0 0 0 016 0 0 1 1 0 0 0 0 0 0 0 0

Table 1: Frequency (number of hours) of wind speed classes according to month from 2003-2005

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

5.4017 5.9035 5.7283 5.3186 5.1571 5.8659 4.8827 4.4430 4.4203 4.0345 5.0466 5.1341

k 2.0042 1.9946 2.0209 1.9239 2.0171 2.1825 2.0318 1.8459 1.8616 1.6810 1.7876 1.6707

Table 2: Maximum Likelihood Estimates for the Weibull parameters , k for the monthly distributions ofhourly average wind speed

3 Goodness of Fit

The order of this paper suggests that we a priori assume that the Weibull distribution is a good modelfor, say, monthly average wind speeds. In statistical hypothesis testing, we need a null hypothesis againstwhich to evaluate the observed samples. In our case, the null hypothesis is that the samples come from aWeib(, k) distribution. This choice of null hypothesis is far from arbitrary, though, as a number of empiricalstudies have been carried out to support this choice. We refer the reader to. Nevertheless, it a crucialpart of statistical modeling to test whether the null hypothesis makes sense given the data: given a samplex1, , xn, what is the probability of observing samples at least as extreme if the null hypothesis is true?

An often used tests for such hypotheses is the Kolmogorov-Smirnov test. The mathematics beyond thistest involves more advanced results from the the theory of probability and stochastic processes and thus willnot be discussed. We refer the reader to Section 6.5.2 in [16]. However, the reader will note that in our case,the paramters are not known; we estimate them with the MLE, provided that it exists. The Kolmogorov-Smirnov test as normally formulated is invalid and requires modification. We refer the reader interested inthe details of the Kolmogorov-Smirnoff test for the Weibull distribution with estimated paramters to [3].

4 Single-Site Power Distribution

In this section, we introduce the aerodynamics (at a very elementary and simplified level) of wind turbines

4.1 Ideal Wind Turbine

The presentation here is a brief skethc of Chapter 4 [6]. We refer the reader further interested in theaerodynamics of wind turbines to this text and also to [4].

A wind turbine functions by converting the kinetic energy of the wind. It is well know that the power

9

(W) of wind with mass flow m (kgs ) and velocity v (ms ) is given by

P =1

2m |v|2

A wind turbine which converted all the kinetic energy of the wind into mechanical energy would reduce thespeed of the wind to 0 and therefore have power

P =1

2m |v|2 = 1

2A |v|3 ,

where is the air density ( kgm2 ) and A is the area (m2) swept by the rotator blades. Suppose we have a

horizontal-axis wind turbine which has infinitely many blades with no spacing between. If we assume thatthe air travels in a cylinder, called the stream tube and that changes in velocity are continuous, and a fewother ideal technical hypotheses, it can be shown that there is a theoretical limit to the amount of kineticenergy the Betz turbine can extract from the wind. This quantity C = 1627 .59 is known as Betz limit. Wecall C the power coefficient of the Betz turbine. Thus, the amount of energy the Betz turbine extracts is

1

2CA |v|3

Modern wind turbines come reasonable close to the Betz limit with power coefficients upwards of CP .5([4]). For the remainder of this paper, we will assume that electric power produced by a wind turbine isgiven by

1

2CP A |v|3 ,

where CP is the power coefficient of the turbine, (0, 1) is some efficiency constant, A is the rotor area,and , v are as above.

4.2 Power Distribution of Ideal Wind Turbine

Suppose V Weib(, k) (we reserve the lower case v for specific values of V ; i.e. V () = v for ). LetP := 12AV

3, where and A are as above and are constants. In reality, is stochastic, but we are assumingthe variability of is negligible.

Proposition 4.1. P has cdf

FP (x) = 1 exp( 1k

(2x

A

) k3

)

and pdf

fP (x) =kx

k31

3k

(2

A

) k3

exp

( 1

(2x

A

) k3

)

Proof. By Proposition 2.1, we have that

FP (x) = P(P x) = P(V

(2x

A

) 13

)= F

((2x

A

) 13

)= 1 exp

( 1

(2x

A

) k3

)

Differentiating both sides with repect to x, we obtain that P has probability density function fp given by

fP (x) =k

3k2

A

(2x

A

) k31

exp

( 1

(2x

A

) k3

)=kx

k31

3k

(2

A

) k3

exp

( 1

(2x

A

) k3

)

10

A wind turbine, though, does not operate at all wind speeds V . The implied wind speed of wind turbinewith power coefficient CP and efficiency coefficient producing power P is given by

Vturbine =

0 Vturbine < vcut-inV vcut-in V vratedvrated vrated < V < vcut-off0 V vcut-off

where vcut-in < vrated < vcut-off are specified by the manufacturer. For example, the GE 1.5 MW SLEwind turbine has cut-in wind speed vcut-in = 3.5

ms , rated wind speed vrated = 14

ms , and cut-out wind speed

vcut-off = 25ms (see [1] for the technical details of the 1.5 SLE model). Analogously,

Pturbine =

0 Vturbine < vcut-in12ACP V

3 vcut-in V vrated12ACP v

3rated vrated < V < vcut-off

0 V vcut-offIt is evident Vturbine and Pturbine are discontinuous random variable, but we can still compute their distributionfunctions and moments. They will fail to have a probability density function, though, their laws are notabsolutely continuous with respect to the Lebesgue measure: the law of Vturbine assigns nonzero probabilityto the singleton {0}, which is a set of Lebesgue measure zero.Proposition 4.2. Pturbine has cdf

FPturbine(x) =

0 < x < 01 + e(

vcut-off )

k e( vcut-in )k 0 x < 12CP Av3cut-in1 e 1k ( 2xCP A )

k3

+ e(vcut-off

)k 1

2CP Av3cut-in x < 12CP Av3rated

1 12CP Av3rated x

Proof. By Theorem 1 pg. 192 and Theorem 7 pg. 196 in [17],

E [Pmturbine] =

Pmturbined(FPturbine(x))

=

vratedvcut-in

(1

2CP Av

3

)m(k

)(v

)k1e(

v )k

dv +

vcut-offvrated

(1

2CP Av

3rated

)m(k

)(v

)k1e(

v )k

dv

We make the change of variable y = ( v )k to obtain

E [Pmturbine] =(

1

2CP A

)m [3m

( vrated )k(vcut-in )

k

y3mk eydy +

( vcut-off )k(vrated )

k

v3mratedeydy

]

=

(1

2CP A

)m [3m

((1 +

3m

k, (vrated

)k) (1 + 3mk, (vcut-in

)k)

)+ v3mrated

(e(

vrated )

k e( vcut-off )k)]

In particular, Pturbine has mean

E[Pturbine] =1

2CP A

[3((1 +

3

k, (vrated

)k) (1 + 3k, (vcut-in

)k)

)+ v3rated

(e(

vrated )

k e( vcut-off )k)]

We can define the capacity factor of a wind turbine to be the ratio

CF =E[Pturbine]

12CpAv

3rated

Using our preceding work, we can obtain a closed-form expression for CF:

CF =1

v3rated

[3((1 +

3

k, (vrated

)k) (1 + 3k, (vcut-in

)k)

)+ v3rated

(e(

vrated )

k e( vcut-off )k)]

=

(

vrated

)3((1 +

3

k, (vrated

)k) (1 + 3k, (vcut-in

)k)

)+ e(

vrated )

k e( vcut-off )k

4.3 Turbine-Site Matching

Suppose we have chosen a location where the wind speed V for a given month has a Weib(, k) distribution.Furthermore, suppose a manufacturer can manufacture wind turbines with fixedcut-in and cut-off speedsvcut-in and vcut-off , respectively, but can adjust the rated wind speed vr [vcut-in, vcut-off). Suppose also thatthe power coefficient CP , the efficiency coefficient and the rotor area A are independent of vr. What shouldthe rated wind speed be if we want to maximize the capacity factor CF? What should the rated wind speedbe if we want to maximize average power of the wind turbine E[Pturbine]? Are these two wind speeds equal?

These questions can be answered by use of the calculus. Define a function CF : [vcut-in, vcut-off) R by

CF(v) :=

(

v

)3((1 +

3

k, (v

)k) (1 + 3

k, (vcut-in

)k)

)+ e(

v )k e( vcut-off )k

Then

CF(v) = 33

v4

((1 +

3

k, (v

)k) (1 + 3

k, (vcut-in

)k)

)+

(

v

)3((v

)k

3k e(

v )k k

(v

)k1

) k

(v

)k1e(

v )k

= 33

v4

((1 +

3

k, (v

)k) (1 + 3

k, (vcut-in

)k)

)

12

We see that the last expression has a root precisely at v = vcut-in and since CF(v) < 0 for v > vcut-in, it

follows that the capacity factor is maximized for vrated = vcut-in.We now compute the value vrated which maximizes E[Pturbine]. Define a function Pavg : [vcut-in, vcut-off)

R by

Pavg(v) :=1

2CP A

[3((1 +

3

k, (v

)k) (1 + 3

k, (vcut-in

)k)

)+ v3

(e(

v )k e( vcut-off )k

)]

Differentiating with respect to v,

P avg(v) =1

2CP A

[3(

v

)k

3k e(

v )k k

(v

)k1 + 3v2

(e(

v )k e( vcut-off )k

) v3 k

(v

)k1e(

v )k

]= 3v2

(e(

v )k e( vcut-off )k

)Setting the RHS equal to 0, we see that Pavg has no critical points in the domain [vcut-in, vcut-off). Rather,since Pavg > 0 on [vcut-in, vcut-off), we only conclude that supv[vcutin,vcut-off ) Pavg(v) =

12CP Av

3cut-off .

These results indicate that by having lowly rated wind turbines, we can maximize the capacity factor atthe cost of minimizing expected power output. Conversely, we can increase expected power output at thecost of decreasing the capacity factor. It is not a priori clear which approach is better, since we have not, norwill we, discussed the economics of wind power. One obvious downside to using lowly rated wind turbines isincreased land usage, which would also raise costs if the wind farm operator is leasing the land. For example,in a study of wind power potential in Bahrain, the author of [8] noted that using a large number of low-ratedturbines would not necessarily be ideal given that the kindgom of Bahrain is only 665km2.

5 Conclusion

In the preceding sections, we have given a blueprint for an analyst to estimate the wind resources of a fixedlocation. With a collection of, say, hourly average wind speeds for a year, he or she can use the maximumlikelihood method to determine Weibull parameters for the monthly wind speed distributions. He or she canthen use the results of section 4 to compute the expected power output of a given model of wind turbine,with specifed cut-in, cut-off, and rated speeds, and can also compute the capacity factor. We emphasize thatthe end result of this analysis is an estimate, and a rough one at that. To make computations tractable, wehave introduced a number of simplifications. In reality, the wind is not always normal to the rotor plane,nor is the power for wind speeds in between the cut-in and cut-off speeds a simple cubic function of the windspeed; it depends on the specific wind turbines power curve (see [1] for the power curves of GE 1.5 MWturbine models). Furthermore, our model ignores variations in wind speeds over large areas, over which asufficiently large wind farm would be spread. Perhaps most importantly, the Weibull distribution is not aperfect description of wind speed distributions, as one can tell by the graphs in Appendix B. However, even ifthe absolute estimates given by the model are to be interpreted with caution, the model nevertheless remainsvery useful for comparing the wind resources of one site to another.

Having discussed the limitations of the model mathematically described in this paper, it is natural toask in what ways it can be improved and generalized. In terms of improvement, the obvious answer is tomore precisely account for the aerodynamics of wind turbines. But the remaining limitation in this directionis that we still ignore the fact that we are still only really considering one wind turbine at one location.Even within the same wind farm, the power output of any individual wind turbine is a random variable,not perfectly correlated with the power of other turbines: the wind does not strike each turbine at the sameangle, nor blow at the same speed for turbines far apart from one another, and each turbine has a failure ratethat is not independent of the failure rate of other wind turbines. We encounter similar modeling difficultieswhen we consider multiple dispersed wind farms. Modeling dispersed wind farms is particularly importantbecause a question with which we should be concerned is how valid a criticism is it that the wind doesntblow all the time and when it does blow, it doesnt blow at a uniform, predictable speed? Does this criticismlose its merits under the model of a wind farms dispersed over multiple high wind speed locations, where thewind blows at peak speed at different times, providing electricity over an integrated system. Of course, sucha grid system does not exist in the status quo; but it is not inconceivable that a federal initiative, analogousto that of the interstate highway system, could effect such a grid in the not too distant future.

How is one to tackle such a difficult modeling problem? One might naturally start by considering amultivariate extension of the Weibull distribution, but there is not a unique multivariate distribution with

13

prescribed Weibull marginal distributions (see Section 4, Chapter 47 in [9] for several classes of multivariateWeibull distributions). Moreover, when one does start to look at the various multivariate extensions of theWeibull distribution, he or she sees that they can be quite ugly, as the reader can see in [11]. In [2], Carlin andHaslett used the multivariate normal distribution as an approximation to a joint distribution of correlatedWeibull random variables, but adopting their approach seems like a step in the wrong direction. Computingpower has become much more efficient and cheaper since their paper was written, and thus using numericaland Monte Carlo methods to handle seemingly intractable computations seems the better approach. Itsuffices to say that multi-site wind power modeling and estimation remains a difficult open problem in needof further research.

A Newton-Raphson Method

Theorem A.1. Let [a, b] R such that the function f : [a, b] R is twice differentiable, has a unique zero [a, b], and satisfies |f (x)| > 0, |f (x)| M x [a, b]. Choose x1 (, b) and inductively define

xn+1 := xn f(xn)f (xn)

, n 1

Then limn xn = and moreover,

0 xn+1 1A

(A(x1 ))2n

, A :=M

2

Proof. Since [a, b] is connected, either f (x) > x [a, b] or f (x) < x [a, b]. Without loss ofgenerality, assume the former. We first show that xn . Since is the unique zero of f in [a, b] and f isstrictly increasing on [a, b], we have that f(a) < 0 and f(b) > 0. Hence, it follows by induction that

xn xn+1 = f(xn)f (xn)

> 0 xn+1 < xn n 1

Since f(xn) > 0 n 1, we must have that xn for all n. By the monotone limit theorem, xn . Iclaim that = . Indeed, since f, f are continuous, we have that

= limnxn+1 = limn

[xn f(xn)

f (xn)

]= limnxn

limn f(xn)limn f (xn)

= f()

f (),

which implies that f()

f () = 0 f() = 0. Since is the unique zero of f in [a, b], we conclude that = .For the second assertion, we have by Taylors theorem with remainder (See Theorem 5.15 in [15]) that foreach n 0,

0 = f() = f(xn) + f(xn)( xn) + f

(n)2

( xn)2,

for some n (, xn). Since f (xn) 6= 0 and xn+1 = xn f(xn)f (xn) , we may write

0 =f(xn)

f (xn)+ ( xn) + f

(n)2f (xn)

( xn)2 = (xn xn+1) + ( xn) + f(n)

2f (xn)( xn)2

= ( xn+1) + f(n)

2f (xn)( xn)2

(xn+1 ) = f(n)

2f (xn)( xn)2

|xn+1 | M2| xn|2

By induction, we obtain that

|xn+1 | n+1

(M

2

)2n1| x1|2

n

=1

A(A | x1|)2

n

If we choose our initial guess x0 so that A |x1 | < 1, then the above gives us an upper bound for n+1which converges to 0 as n .

14

B Example Graphs

15

16

17

18

19

20

References

[1] http://geosci.uchicago.edu/~moyer/GEOS24705/Readings/GEA14954C15-MW-Broch.pdf

[2] Carlin, John and John Haslett. The Probability Distribution of Wind Power from a Dispersed Array ofWind Turbine Generators, Journal of Applied Meteorology 21 (1982), no. 3, 303-313.

[3] Chandra, M., N.D. Singpurwalla, and M.A. Stephens. Kolmogorov Statistics for Tests of Fit for theExtreme Value and Weibull Distributions, Journal of Amer. Stat. Assoc., 76 (1981), no. 375, 729-731.

[4] Gasch, Robert and Jochen Twele. Wind Power Plants: Fundamentals, Design, Construction, and Oper-ation (Second Edition). Springer, 2012.

[5] Genc, Asir, Murat Erisoglu, Ahmet Pekgor, Galip Oturanc, Arif Hepbasli. Estimation of Wind PowerPotential Using Weibull Distribution, Energy Sources 27 (2005), no. 9, 809-822.

[6] Hansen, Martin O.L. Aerodynamics of Wind Turbines (Second Edition). Earthscan, 2008.

[7] Hennessey, Joseph P. Jr. Some Aspects of Wind Power Statistics, Journal of Applied Meteorology 16(1977), no. 2, 119-128.

[8] Jowder, Fawzi A.L. Wind Power Analysis and Site Matching of Wind Turbine Generators in Kingdomof Bahrain, Applied Energy 86 (2009), 538-545.

[9] Kotz, Samuel, N. Balakrishnan, and Norman L. Johnson. Continuous Multivariate Distributions, Vol. 1:Models and Applications (Second Edition). John Wiley & Sons, 2000.

[10] Kotz, Samuel, N. Balakrishnan, and Norman L. Johnson. Continuous Univariate Distributions, Vol. 1(Second Edition). John Wiley & Sons, 1994.

[11] Lee, Cheng K. and Miin-Jye Wen. A Multivariate Weibull Distribution. http://arxiv.org/ftp/math/papers/0609/0609585.pdf.

[12] Manwell, J.F., J.G. McGowan, and A.L. Rogers. Wind Energy Explained: Theory, Design, and Appli-cation (Second Edition). John Wiley & Sons, 2009.

[13] Patel, Mukund R. Wind and Solar Power Systems: Design, Analysis, and Operation (Second Edition).Taylor & Francis, 1972.

[14] Jangamshetti, Suresh H. and V. Guruprasada Rau. Site Matching of Wind Turbine Generators: A CaseStudy, IEEE Trans. on Energy Conversion 14 (1999), No. 4, 1537-1543.

[15] Rudin, Walter. Principles of Mathematical Analysis (Third Edition). McGraw-Hill, 1964.

[16] Shao, Jun. Mathematical Statistics (Second Edition). Springer, 2003.

[17] Shiryaev, A.N. Probability (Second Edition). Springer, 1995.

21

IntroductionThe Weibull DistributionDefinition and Basic PropertiesParameter EstimationLikelihood FunctionMaximum LikelihoodComputation of MLE

An Example

Goodness of FitSingle-Site Power DistributionIdeal Wind TurbinePower Distribution of Ideal Wind TurbineTurbine-Site Matching

ConclusionNewton-Raphson MethodExample Graphs