Fractional Weibull Wind Speed Modeling For Wind Power Production Estimation Zuwei Yu, Sr. Member IEEE, and Akiner Tuzuner, Student Member IEEE
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I. INTRODUCTION IND power expansion has been accelerating worldwide due to technological progress, cost reduction and concerns over global warming partly caused by greater concentration of CO2 in the atmosphere. Power plants using coal, natural gas and oil have been contributing a large portion of the concentration. As a result, there has been growing interest in wind power production modeling and simulation studies. These studies include wind resource quantification, wind speed modeling, wind power production, and system reliability assessment [1]-[16]. This paper addresses two major issues in wind power system analysis: wind speed distribution (WSD) modeling and energy production calculation. Both issues deserve further research due to technical and practical complications. Wind energy production (e.g., power, air compression and hydrogen production) depends on wind speed distribution. Hence, accurate wind speed distribution modeling is the first step to achieve accurate wind energy production estimation. It has been generally recognized that wind speed distributions are Weibull [6]-[7], [10] regardless of the geographical locations. It has also been claimed that the shape
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Keywords – fitting error, fractional Weibull, Mean-Variance, MLE, uniqueness, wind speed, wind power.
factor of the Weibull wind speed distribution is in the range of 1.2 to 2.3 [11]. However, after a thorough study of the wind data both in our area and other areas including the wind speed plots in the literature, we found that wind speed measurements may not necessarily be Weibull in general. In particular, some data show disproportionately high frequencies of close-to-zero wind speeds, and even a dip-down of frequencies for wind speeds slightly greater than these values, which doesn’t conform to the unimodular (“single-peaked”) shape of the Weibull distribution. Nonetheless, the greater upper range of the wind speed can be suitably fitted with the Weibull distribution, which leads to our conclusion that the Fractional Weibull Distribution (FWD) is in general more suitable for wind speed modeling. Let us first analyze the wind speed distribution in West Lafayette, Indiana. The wind speed probability distribution is plotted in Fig. 1 (more precisely, the ensemble frequency distribution) [12]. The data used for this plot is summer wind speed from June 20 to September 10, 1997–2006. Our purpose is to fit seasonal wind speed distribution models to catch the large variations in seasonal wind speed (the average wind speed in the summer is about 60% of that in the spring [9]). This is a stratified method that can yield more accurate results in wind power production estimation. One can see from Fig. 1 that the probability of the zero wind speed is the largest, with a value of about 0.0934, and has a dip-down after zero, which is not Weibull because the probability of the Weibull distribution is zero at the origin. The lighter dotted line from the origin is a modified distribution to be discussed later.
Frequency (or probability)
1 Abstract – This paper describes the method of the Fractional Weibull Distribution (FWD) for modeling wind speed distributions. The Maximum Likelihood Estimation (MLE) method is used for estimating the parameters of the FWD, fitted to the wind data from a tall tower. Seasonal wind speed variations are considered in the modeling. Compared to the standard Weibull distribution estimates, the FWD estimates yield greater accuracy in wind power estimation. The discrete distributions of the probabilities of the FWDs are used for obtaining expected wind energy production, which shows a considerable reduction in errors by about tenfold. A simple Mean-Variance analysis of power production is performed for a wind farm that can have a mix of three different turbine models. The results indicate that the standard deviation of power production can be considerably reduced by choosing an appropriate mix of turbines.
Wind speed (m/s)
Fig. 1. Sample wind speed distribution (10 m above ground) Zuwei Yu is with Choren USA LLC and is also associated with the College of Engineering at Purdue University, the United States. His email is
[email protected]. Akiner Tuzuner is a Ph.D. candidate in the School of Industrial Engineering, Purdue University and his email is
[email protected].
978-1-4244-4241-6/09/$25.00 ©2009 IEEE
As shown in [7] and [10], high occurrences of zero wind speed are witnessed almost everywhere, with one typical example shown in Fig. 2. In this case, the frequency of zero wind speed is about 6% on average even though the measurements were taken at 80m above the ground. Fig. 2 confirms again that the overall distribution is not truly Weibull.
1
In this paper, the FWDs are fitted using the MLE method with actual wind speed data. We emphasize the importance of fitting various FWDs to account for seasonal wind speed variations. In the Midwestern U.S., the average wind speed in the summer is at its lowest when the peak power demand occurs. Hence, the annual grand wind speed distribution model surely exaggerates wind power production in the summer and would provide false reliability assessments accordingly.
power production (i.e. from the cut-in speed to the cut-out speed). A. The Type 1 FWD For type 1, the criterion of threshold filtering is used. We start by dropping out the data at a very low wind speed such as 1.0 MPH or 2.0, that is, by dropping the data that are above the lighter line segment as shown in Fig. 1. A FWD is then fitted using the remaining data, resulting in FWD1. The data at low wind speed (no greater than the cut-in wind speed) but above the frequency described by the FWD1 line is again filtered out. The process can be repeated a few times leading to quick convergence (FWDm = FWDm-1). In the end, the sum (δ1) of the frequency differences between the sample distribution and the final FWD is the fraction subtracted from the Weibull distribution, resulting in equation (1) below:
f 1 (v , λ , k ) =
Fig. 2. Wind speed distribution in Dodge City, Kansas (see [10])
Wind speed in Indiana may average around 5.0 – 5.7 m/s in the summer at 100 m [9], resulting in a wind power availability around 15 – 22%.. However, wind speed in April may be in the range of 7 – 9.6 m/s, resulting in wind power availability up to 50%. While a Weibull sampling method can be used in the Monte Carlo simulation for finding the expected power production [16], we use the discrete probability method for the same purpose in this paper due to its simplicity and tractability. The results show that the FWPD can reduce the standard errors in power production estimation by about 7% (or tenfold) as compared to the WPD fitting. Section II describes the criteria for choosing the correct FWDs. Section III presents the MLE method for fitting the FWDs. Section IV presents the results of the estimates for wind power production and estimation error. Section V presents a Mean-Variance analysis of the power production of a wind farm having different models of turbines. Section VI concludes the paper with a summary, followed by an appendix in section VII that presents a proof of the uniqueness of the FWD parameter estimates. II. CRITERIA FOR CHOOSING FWDS Wind speed is generally chaotic [6]. However, the ensemble distributions may be categorized as two types: 1) the one in Fig. 1 where the frequencies after zero speed decline dramatically before reversing the trend –however, there is no zero frequency before the trend is reversed; 2) the one shown in Fig. 2, where the frequency of low wind speed (greater than zero) is zero or near zero. We will analyze the correct FWDs for each type next. The idea is to discard some wind data that do not conform to the Weibull distribution at low wind speed, and at the same time, to make sure the wind speed is fitted with high accuracy for the speed range suitable for wind
k (1 − δ 1) v k −1 −( v / λ ) k ( ) e
λ
(1)
λ
where v is wind speed, k>0 is the shape parameter and λ>0 is the scale parameter of the Weibull distribution. Compared to the standard Weibull distribution, equation (1) has a scaling factor (1- δ1) that is smaller than 1.0, which is the reason why equation (1) is called the fractional Weibull distribution, or the fractional Weibull probability density (FWPD). The corresponding cumulative distribution is given in equation (2) below. k (2) F1 (v, λ , k ) = (1 − δ 1)[1 − e − ( v / λ ) ] The FWD will better fit the actual wind speed distribution in the wind production speed zone. Without using our FWD fitting strategy, the large frequency components around zero wind speed would force an excessively left-tilted Weibull fit (i.e. introduce excess positive-skewness), as shown later in the paper. B. The Type 2 FWD In this type of the FWD, the “near zero-frequency-cutoff” criterion is used, that is, the wind speed data in the frequency bins to the left of the near-zero frequency bin are discarded. To fit the remaining data, we may just use a fractional Weibull distribution shifted by a value a that is around the near-zerofrequency bin (Fig. 2). The equation is then: f 2 (v , λ , k ) =
k (1 − δ 2) v − a k −1 −[( v− a ) / λ ]k ( ) e
λ
λ
(3)
where the shift parameter a > 0 is used for a more accurate fit of the distribution of greater wind speed data. The cumulative distribution function of the Type 2 FWD is the following: k
F2 (v, λ , k ) = (1 − δ 2)[1 − e −[(v − a ) / λ ] ]
(4)
III. WIND SPEED MODELING A. Maximum Likelihood Estimation (MLE) Method The MLE procedure employed is to determine the values of the Weibull parameters, λ and k. We will use the Type 1 FWD
2
to illustrate the method. The first order optimality conditions below are used for the purpose: ∂ ln L(λ , k ,V ) = 0 and ∂λ
∂ ln L(λ , k ,V ) = 0, ∂k
where ln is the natural logarithm, V is the vector of sample wind speed data, and L( ), as defined in equation (5) below. In order to use the MLE method we need to construct the Maximum Likelihood (ML) function as shown below: n k k n v L(λ , k ,V ) = (1 − δ 1)n ( )n ∏ ( i )k −1 ∏ e−( vi / λ )
λ
i =1
λ
(5)
i =1
where n is the number of actual samples (measurements) used in the estimation. Notice that wind speed sample to be used in the above is the truncated wind speed data set, not the original wind speed measurements. Taking the natural logarithm of (5), we obtain ln f1ML = n ln(1 − δ 1) + n ln( ) + (k − 1)
λ
∑ ln( λ ) − ∑ ( λ ) vi
n
n
i =1
vi
WPD Actual Sample PD
0.06
k
(6)
i =1
Taking the derivative of (6) with respect to λ, we have
∑
0.07
0.05 Probability
k
the actual sample PD. More precisely, it is a histogram or the frequency of occurrences falling in bins of size 1 MPH (miles per hour). One can see that Fig. 3 closely resembles the Weibull probability density function (except for the disproportionately high frequencies at the wind speed less than or equal to 1.0 MPH etc.). Several sets of results are obtained using equations (7) and (8) (but only the critical ones are listed for brevity): WPD - the Weibull probability density fitted to all wind speed data, FWPD1 - the fractional Weibull PD fitted to wind speed data discarded at 1.0 MPH or less, FWPD2 - the one with wind data at or lower than 3 MPH partially filtered out using FWPD1, and finally, FWPD5 – the one fitted to wind speed data at or lower than 7.0 MPH partially filtered out using the FWPD in the previous iteration. The results are summarized in Table I, with errors plotted in Fig. 4.
∑
n n ∂ ln f1ML nλ − k λ −vi v v = ( 2 ) + ( k − 1) ( 2 ) + k ( i ) k −1 i2 λ ∂λ k λ i =1 vi λ i =1 λ
0.04 0.03 0.02 0.01 0
After simplifying and setting it equal to zero according to the first order conditions for optimality, we obtain −n
λ
−
k −1
λ
∑v n
λ
i =1
i
(1) +
k
λ
∑λ
1 λ = n
∑v n
i
i =1
k
,
^
or
λ =(
1 n
∑v ) n
i =1
^
^
,
(7)
where the carets are for estimates of the parameters. Using the same procedure with respect to k, we obtain 1 ^
=(
k
∑v n
i =1
i
^
k
ln vi ) /(
∑ v ) − 1n ∑ ln v n
n
^
i =1
k i
i =1
i
(8)
Equation (8) is nonlinear and can be solved using the NewtonRaphson method to obtain the k caret. Equation (8) can be rewritten as f k1 =
1 1 + k n
∑ ln v − (∑ v n
i =1
n
i
i =1
i
k
ln vi ) /(
∑ v ), ∀ k > 0 , n
i =1
k i
7
9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47
TABLE I. PARAMETER ESTIMATES (S is fitting type)
k 1/ k
i
5
Wind spee d MPH
i =1
Solving for λ in the above, we obtain ^k
3
Fig. 3. Wind speed distribution for July-August at site 9004.
v ( i )k = 0
n
1
(9)
where our aim is to find the kˆ such that f′k =0. It is desirable that there be only one such kˆ to avoid multiple sets of parameter estimates. As shown in [13] and the appendix of the paper, the root of equation (9) is truly unique. B. Results Based on the July-August Data The summer wind speed distribution from site 9004 (Geetinsville, Indiana [9]) is plotted in Fig. 3 and labeled as
S WPD FWPD1 FWPD5
k caret 2.0893 2.2376 2.34886
λ caret 13.2671 13.55195 14.0266
1-δ1 100% 97.84% 95.25%
We now explain why the FWPDs can improve the estimation accuracy. In Fig. 3. one can see that the fitted WPD peaks at the wind speed of 10 MPH while the actual ensemble PD at 12, leaving large fitting errors at higher wind speed. The large errors from the WPD are caused by the large number of wind speed observations at low wind speed levels, which introduces significant positive skewness in the fitted distribution. Indeed, if some of the original wind data at low speed levels (say 6 MPH and less) are discarded, the fitted FWPD would tilt a bit to the right, peaking at 11 or even 12 MPH, and be more evenly spread out over the higher wind speed. (i.e. k caret and especially λ caret would be greater). Consequently, errors at higher wind speed will be reduced on average. Although in this case fitting errors at low wind speed will be greater, this is not a concern because the accuracy of the fit for low wind speed is unimportant due to the cut-in effect. More explicitly, since power production is zero for wind speed below the cut-in speed, probabilistic calculations such as the expected value or the variance of power production only require estimates of the cumulative
3
0.018 WPD 0.015
FWPD1
Actual PD-Fitted PDs
0.012
FWPD5
0.009 0.006 0.003 0 -0.003
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38
-0.006 -0.009 -0.012 -0.015 Wind speed MPH
Fig. 4. Fitting errors for the three probability distributions.
Probability
probability of wind speed being greater than the cut-in speed, which is incorporated by δ 1 . The FWPD5 is plotted against the actual distribution in Fig. 5. We can see that the peak probability distribution of FWPD5 is around 11.0-12.0 MPH which nearly coincides with the actual peak probability distribution of the actual wind speed. As a result, the fitted distribution errors for FWPD5 around the peak probability are reduced considerably compared to the WPD. The errors from FWPD5 for wind speed greater than 23.0 MPH are increased a little. However, since the errors are reduced a lot more around the wind speed at about 12 MPH where frequencies are highest, the power production estimation error may be reduced significantly. 0.06
FWPD5
0.05
Ensemble PD
0.04 0.03 0.02 0.01 0 1
3
5
7
9
11
13
15
17
19
21
23
25
27
29
31
33
35
37
Wind speed MPH
Fig. 5. Actual ensemble PD vs FWPD5 for the July-August data set.
Surely we need a quantitative measure for verifying the accuracy of the models fitted. The sum of the expected error of power production may be used: Vsatur
Er ( s ) =
∫
Cv 3 [ actual sample PD − f (v)]dv
Vcut −in
(10)
IV. WIND ENERGY PRODUCTION & ERROR ANALYSIS According to the Wind Association, Wind turbine power is approximated by a cubic function between the cut-in and the rated wind speed (Vsatur) values [14]. In general, wind power production can be expressed as:
⎧ 0, ⎪ P = ⎨Cw(v )3 , ⎪P , ⎩ max
v ∈ [0,Vcutin )
∫
[actual sample PD − f (v )]dv, ∀s
Vsatur
where s is the type of fitting as described in Table I, Vcutin, Vsatur and Vcutout are respectively the cut-in, saturation (or rated) and cut-out wind speeds for power production. Vcutin is around 3.5 - 4.0 m/s (8.0 - 9.0 MPH) for most wind power generators. Equation (10) is based on (11) and can be approximated by discrete values as in (12).
(11)
where Pmax is the rated power output. Notice that equation (11) is an ideal power curve, and in practice there can be deviations from it. The power curve for the Vestas 3.0 MW can be found in [15], and a similar power curve with 1.5 MW in rated capacity is plotted in Fig. 6, together with the actual wind speed sample distribution. The expected electricity production from such a wind power generator can be obtained either by the Monte Carlo simulation as described in [3,16] or by finding the sum of the products of the discrete wind speed probabilities and the corresponding discrete power output. Closed-form equations for expected power are yet to be developed due to the complication of the cubic function in (11). We will only discuss the discrete probability method in wind power estimation due to space limitations. A. The Sum of Discrete Products The wind power production equation using the discrete probabilities can be expressed as:
Vcutout
+ P max
or v > Vcutout
v ∈ [Vcutin ,Vsatur ) v ∈ [Vsatur ,Vcutout ]
E ( P, s ) =
∑ Pr(s, l)P(l), ∀ s
L max
(12)
l =1
where Pr(s,l) is the discrete probability of model s, P(l) is the corresponding expected power output, and l is the bin index for discrete wind speed distribution. The use of equation (12) is illustrated in Fig. 6. Given the wind power curve in Fig. 6, E(P,s) is easily calculated using Excel, resulting in a value of 0.24082 MW for s = “actual sample PD”. This number is the true expected power production, and will be used for benchmarking errors of the FWPDs next.
4
Probability & 10MW
B. Error Analysis The expected power from models WPD, FWPD1 and FWPD5 are calculated by (12), and the errors in expected power production from the three models are shown in Table II. We can see that FWPD5 has reduced the absolute power production estimation error by about 6.8% (i.e., 7.5-0.7%), or about tenfold as compared to the WPD. Apparently, the WPD underestimates wind power production by a better fit of the low speed data (the wind data that does not produce power). On the other hand, the FWPD5 overestimates power production by a little due to the fact that its PD errors are greater on the right tail of the distribution (i.e. for wind speed greater than 23 MPH). 0.14
Pow er curve (10MW)
0.12
Actual PD
0.1 0.08 0.06 0.04 0.02
45
41
37
33
29
25
21
17
9
13
5
1
0
Wind speed MPH
Fig. 6. Discrete distributions of the wind speed and the power curve. TABLE II. ERRORS IN EXPECTED POWER PRODUCTION (Vcutin = 9, Vsatur=28, and Vcutout = 55 MPH)
WPD FWPD1 FWPD5
Expected Power (MW) 0.2227 0.2267 0.24243
Error
% error
(-) 0.01812 (-) 0.01412 (+) 0.00161
7.5% 5.86% 0.7%
Errors from the FWPD with a shift “a” have not been analyzed due to a lack of the wind speed data of Type 2 and space limitations. This remains a future research topic. V. WIND ENERGY PRODUCTION MEAN-VARIANCE ANALYSIS Rapid variability of wind speed causes fluctuations in a wind turbine’s power output. Such fluctuations are undesirable due to technological limitations and costs of storing electrical power in order to provide a steady supply to the transmission grid. Therefore, a turbine’s expected power production, although a more meaningful measure than its rated (or, nominal) power, can not be taken as the sole measure of its performance. One also needs to take into account the variability of the power output via an appropriate measure of risk. Statistical variance (or, standard deviation, which is more intuitive to interpret as it has the same units as the expected value) is a commonly used measure of risk. Then, if strictly one turbine model shall be installed on a wind farm, the rule of thumb is as follows: select the one with the greatest expected power to standard deviation ratio. Of course, this is an over simplified guideline based on revenue maximization. A complete analysis requires consideration of other factors, including costs (investment and OM) and other technical and logistical matters. Nevertheless, restricting the turbine choice to strictly one model overlooks the variance reduction that can
be achieved by installing several different turbine models, or in economics terms, by diversifying the turbine portfolio. A more detailed engineering economic analysis is out of the scope and space limitations of this paper. We thus focus on the following analysis. We consider a wind farm where the wind speed comes from the empirical distribution for the April-May data from the aforementioned Indiana Site 9004. We fit the data to a fractional Weibull probability distribution using the method described in Section III. The parameter estimates of the fit are δ 1 = 6.386% (wind speeds below 7.50 MPH, which is just below the Vcutin of the generators to be used in the coming analysis, are excluded from the fitting), λˆ = 21.8788, and kˆ =3.30213. We consider the three models of GE wind turbines to be installed on the farm, whose specifications, obtained from brochures on the GE web site [17], are given in Table III. The chosen turbines have significantly different rated powers as well as different saturation and cutout speeds. We consider the problem of turbine selection for a desired expected total wind farm power of Pfarm. Denoting the number of each turbine by ni, the feasible combinations of turbines are given 3 by P farm = ∑i=1 ni E[Pi ] , where E[Pi] is the expected power of turbine type i. Using the fitted Wiebull distribution and the power formula in equation (11), the expected value and standard deviation (in parenthesis) for the three turbines are calculated by numerical integration with Mathematica as E[P1]= 0.591 MW (0.483), E[P2] = 0.985 MW (0.805), and E[P3] = 1.074 MW (0.965). Then, for different values of Pfarm, we can first determine all feasible turbine combinations (n1, n2, n3), and, for each combination we can calculate the standard deviation of the farm’s power output, σfarm , using the formula in equation (13) below.
σ
(
⎛ ∞ 3 farm = ⎜ ∫0 ∑i =1 ni ⎝
Pi (ws )
)f 2
FWD
(ws )
⎞ 2 dws − P farm ⎟ ⎠
1
2
, (13)
where fFWD is the fitted fractional Weibull density function. TABLE III. GE WIND TURBINE SPECIFICATIONS Turbine, i GE 1.5 xle GE 2.5 xl GE 3.6 sl
Vcutin (MPH) 7.829 7.829 7.829
Vsatur (MPH) 27.962 27.962 31.317
Vcutout (MPH) 44.739 55.923 60.397
Pmax (MW) 1.5 2.5 3.6
As we do not have an explicit analytical form for equation (13), we obtain numerical results for a set of discrete set of points. We vary Pfarm by discrete values in the range 50-150 MW and calculate the standard deviation for a discrete subset of feasible (n1, n2, n3). Considerable computation effort is required for a good coverage of the entire feasible set. The use of a fitted FWD, as opposed to the empirical distribution itself, eases this process significantly as we can exploit the numerical integration capabilities of specialized software such as Mathematica (in contrast to retrieving values from a histogram table). The results are depicted in the Mean-Variance plot in Fig. 7. The shaded region on the plot corresponds to all feasible combinations of turbines. The thick line on the left boundary of the region corresponds to the efficient frontier; that is, those
5
combinations that yield the lowest standard deviation for a given total expected power. Fig. 7 illustrates how different combinations of turbines with the same expected total power can have different variations in the power output, where this variation increases linearly as the total power increases. Fig. 8 depicts the efficient turbine combinations (n1,n2) for Turbines 1 and 2, as well as the ratio of their fractions in the total power (we have n3=0 and f3=0 for all efficient points). Turbine 3 (GE-3.6 MW) is never part of an efficient combination.
Fig. 7. Mean vs. Standard Deviation of Total Power for Different Combinations of GE Wind Turbines.
The thicker line in Fig. 8 (with the left-hand-side axis) depicts the efficient combinations of the total numbers of Turbine 1 (GE-1.5MW) and Turbine 2 (GE-2.5MW). The number of each model of turbine grows exactly proportional to each other as Pfarm increases. The thinner flat line in Fig. 8 (with the right-hand-side axis) is the ratio of the turbines’ total power, i.e. (n1 E[P1] )/(n2 E[P2]), and it verifies this fact as the efficient ratio is invariant with respect to expected total power. This efficient power ratio is calculated as 0.525; that is, roughly half of the total power comes from each turbine. We would like to note that the proportionality of the results is not incidental but owes to the linearity of Pfarm and σfarm to the vector (n1, n2, n3,), for which we omit the proof due to space limitations.
indicated in Fig. 8, the standard deviation can be reduced considerably. To be exact, for any value of Pfarm the efficient point has 10.1% less standard deviation than the point with the largest standard deviation, and this difference is 8.2% of Pfarm. For example, for a wind farm of 150 MW total expected power, the reduction in standard deviation can be as much as 12.35 MW. The results above may be more or less significant under other wind speed distributions and turbine options, which is a calling subject for further research. Nevertheless, our result is intended to point out the fact that a portion of the variability in wind farm power can be eliminated by installing a combination of a variety of types of turbines. This is essentially because turbine “diversification” allows better harvesting of wind energy at different wind speeds. VI. SUMMARY The paper describes the MLE method for estimating the parameters of the fractional Weibull wind speed distributions. The new method of FWPD fitting can reduce wind power production estimation errors by about 6.8% as compared to the method with a classical Weibull estimate using the MLE. The paper also proves that the parameters estimated are unique, as shown in the Appendix. A simple Mean-Variance analysis of total wind farm power is performed by considering three different options of turbine models to be installed. The results indicate that the choice of an efficient mix of turbines can considerably reduce the standard deviation of total power production. VII. APPENDIX – PROOF OF UNIQUENESS The proof of uniqueness in fitting the FWPD parameters is similar to our earlier proof [13], with some changes in details including the deletion of the requirement that there exist at least one distinct wind speed in the data set. We put a smaller version of the proof here for the completeness of this paper. Eq. (9) is the key to the proof, where n is the number of wind speed measurements. If there is zero wind speed, we can just add a very small positive value so that f k is a continuous function for any wind speed greater than zero. For simplicity in the proof below, we abandon the caret sign in the expressions. Proposition: f k is strictly monotone decreasing, has a unique root in k∈(0,+∞), and converges to a negative value as k → +∞ . Proof: As shown in [13], it is easy to show that
∑ (ln(v ) − ln(v n
lim f k → ∞, and lim f k = 1 k →+∞ k →0 +
n
i =1
i
max
)) < 0
where vmax is the maximum wind speed. Based on the above result, and since fk is continuous over k∈(0,+∞), we conclude that fk passes the k-axis at least once (i.e. equation (9) has at least one root). Fig. 8. Efficient Combinations of Turbine 1 (n1) and Turbine 2 (n2) From Calculus we know that if the derivative of a continuous function is continuous and negative, the function While for all feasible combinations the standard deviation itself is strictly monotone decreasing. We will use this to prove is generally high relative to the expected power, Fig. 7 that equation (9) has one and only one root over k ∈ (0,+∞) . indicates that choosing a good mix of turbine fractions, as Let’s first present the derivative of (9) below.
6
∑ ∑
⎧⎛ n ⎞ k v i ln v i ⎟ ⎪⎜ ⎪⎝ i =1 ⎠ +⎨ n k ⎪ ⎛ − ⎜ v i ln v i ⎪ i = 1 ⎝ ⎩
df k 1 =− 2 dk k
( )
2
( )2 ⎞⎟⎛⎜ ⎠⎝
∑v n
i =1
k i
⎫ ⎪ ⎪ ⎛ ⎬ /⎜ ⎞⎪ ⎝ ⎟ ⎪ ⎠⎭
∑v n
i =1
k i
⎞ ⎟ ⎠
2
(14)
The expression in (14) is apparently continuous for nonzero wind speed, and its first term, -1/k2, is always negative for finite k. Let us then look at the other term. The denominator of the other term in (14) is always positive; so we just need to investigate the sign of its numerator given below. ⎛ ⎜ ⎝
∑v n
i =1
2
ln (vi )⎟ − ⎜ ⎞
k i
⎛
⎠
⎝
∑v n
i =1
k i
ln (vi )
2
⎞⎛ ⎟⎜ ⎠⎝
∑v n
i =1
k i
⎞ ⎟ ⎠
(15)
The expression in (15) can be opened into the following terms: ⎛ ⎜ ⎝
∑ n
i =1
⎞
2
vi ln (v i )⎟ = k
⎠
∑v n
i =1
2k i
2
n
( )
(v i v j ) k ln (v i ) ln v j
(16)
i =1 j ≠ i
⎛ n k ⎜ ∑ vi ⎜ ⎝ i =1
)
( )
∑
∑∑ i =1 j ≠ i
+
n
∑ ∑ ( vi v j )k ln(vi )2
(17)
i =1 j ≠ i
(vi v j ) ln (v i )(ln (v j ) − ln (v i )) .
(18)
k
(vi v j ) k ln (vi )(ln (v j ) − ln (vi )) and (v j v i ) k ln (v j )(ln (vi ) − ln (v j )),
whose sum is 2
(19)
− ⎣⎡ln(vi ) − ln(v j ) ⎦⎤ (vi v j )k ≤ 0 .
The inequality in (19) holds because the squared term is always non-negative, the sample data v i ≥ 0, ∀i , and the parameter estimate kˆ > 0 . Also, the above inequality holds strictly for at
least one pair of data where v i ≠ v j (since, otherwise, the data
sample is degenerate). vi , v j > 0 and df k / dk is continuous, therefore, the following strict inequality holds:
∑∑ (v v ) [ln(v ) − ln(v )] /(∑ v ) n
i =1 j ≠ i
k
i
j
[3]
[5]
2001, pp. 611-618.
In (18), two terms are involved for each (i,j) pair, i ≠ j :
df k 1 =− 2 − dk k
[2]
J. Smith, M. Milligan, E. DeMeo, B. Parsons, “Utility Wind Integration and Operating Impact State of Art,” IEEE Trans. Power Syst., Vol. 22, No. 3, August 2007, pp. 900-908. A. Estanqueiro, “Dynamic Wind Generation Model for Power System Studies,” IEEE Trans. Power Syst., Vol. 22, No. 3, August 2007, pp. 920-928. R. Billinton, W. Wangdee, “Reliability-Based Transmission Reinforcement Planning Associated with Large-Scale Wind Farms,” IEEE Trans. Power Syst., Vol. 22, No. 1, February 2007. P. Giorsetto, K. Utsurogi, “Development of a New Procedure for Reliability Modeling of Wind Turbine Generators,” IEEE Trans. Power App.& Syst., Vol. PAS-102, No.1, 1983, pp. 134-143. W. Li, R. Billinton, “A Minimum Cost Assessment Method for Composite Generation and Transmission Planning,” IEEE Trans. Power Syst., Vol. 8, No.2, My 1993, pp. 628-636. P. Edwards, R. Hurst, “Level-crossing statistics of the horizontal wind speed in the planetary surface boundary layer,” Chaos, Vol. 11, No. 3, Sept.
C. Archer, M. Jacobson, “Evaluation of global wind power,” GEOPHYSICAL RESEARCH, Vol. 110, 2005, D12110. [8] R. Thresher, M. Robinson, P. Veers, “To Capture the Wind,” IEEE Power & Energy, Vol. 5, No. 6, Nov/Dec 2007, pp. 34-46. [9] “Indiana Energy Group Tall Towers Wind Study Final Project Report,” prepared by the Global Energy Concepts, Oct 2005. [10] C. Archer, M. Jacobson, “Spatial and temporal distributions of U.S. winds and wind power at 80 m derived from measurements,”
[7]
Taking the difference of (16) and (17), the expression in (15) reduces to n
IX. REFERENCES [1]
[6]
⎞⎛ n ⎞ vi 2 ⎟⎟⎜⎜ ∑ vi k ⎟⎟ = ⎠⎝ i =1 ⎠ n 2k vi ln vi 2 i =1
ln (
VIII. ACKNOWLEDGEMENT The authors would like to thank Mr. R. Brown, Indiana Office of Energy & Defense for providing the wind data. His support makes the study possible.
[4]
ln (v i ) +
∑∑
exists a unique solution kˆ ∈ (0,+∞) of the MLE equation fk(k)=0 which yields, by equation (7) a unique estimate for λˆ .
2
i
j
n
i =1
k i
2
< 0.
This implies that the derivative in (14) is strictly negative for all k∈(0,+∞), and thus, the function in equation (9) is monotone decreasing in k∈(0,+∞). This proves that fk crosses the k-axis once and only once. Hence, we conclude that there
GEOPHYSICAL RESEARCH, VOL. 108, NO. D9, 2003.
[11] C. G. Justus, C.G., W.R. Hargraves, and A. Yalcin, “Nationwide Assessment of Potential Output from Wind-Powered Generators,” J. Appl. Meteoro. 15, 1976, pp. 673–678. [12] S. Dofonsou, “An Analysis on the Correlation between Wind Speed and Humidity,” an internal research at Purdue University, 2007. [13] A. Tuzuner, Z. Yu, “A Theoretical Analysis on Parameter Estimation for the Weibull Wind Speed Distribution,” in Proc. IEEE PES General Meeting 2008, July 20-24, 2008, Pittsburgh. [14] American Wind Energy Association, available at: http://www.awea.org/faq/windpower.html [15] “General Specifications – V90-3.0,” a technical manual prepared by the Vestas Wind Systems A/S, 2004. [16] Z. Yu, A. Tuzuner, “Wind Speed Modeling and Energy Production Simulation with Weibull Sampling,” PES General Meeting 2008. [17] GE Wind Turbine Brochures (1.5 MW, 2.5 MW, and 3.6 MW), available at http://www.gepower.com/prod_serv/products/wind_turbines /en/2xmw/index.htm.
X. BIOGRAPHIES Zuwei Yu received his Ph.D. degree in Energy Engineering in 1995 with a minor in Operations Research. He is currently a Sr. Technology & Development Officer with Choren USA LLC. He is also a graduate faculty with the College of Engineering, Purdue University. His research lies in mathematical programming in project optimization and economic modeling, with applications to gasification, energy, environment, risk and other systems. Akiner Tuzuner is a Ph.D. candidate in Industrial Engineering at Purdue University. He received his M.S. in Economics in 2000 and M.S. in Industrial Engineering in 2002, both from Purdue. His dissertation is on a stochastic gaming model of the crude oil market. His research interests lie in optimization, stochastic processes and game theory.
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