Wind Speed Modeling and Energy Production Simulation with Weibull Sampling Zuwei Yu, Member IEEE, Akiner Tuzuner Abstract – This paper describes the Maximum Likely Estimation (MLE) method and the Method of Moments (MOM) for wind speed modeling. The Weibull wind speed distribution models are fitted using the two methods and the wind data from a tall tower in the Midwestern United States, with seasonal wind speed variations also considered in the modeling. It turned out that both methods provide very similar results with comparable accuracy. The Monte Carlo simulation is used for obtaining expected wind energy production using a Weibull sampling technique. Keywords – Wind speed, Weibull distribution, MLE, MOM, energy production. I.
INTRODUCTION
Wind power expansion has been accelerating worldwide due to technology progress, cost reduction and concerns over global warming partly caused by power plants fired with coal, natural gas (NG) and oil. At the same time, there has been growing interest in wind power production modeling and simulation studies. These studies include wind resource quantification, wind speed modeling and prediction, wind power production, system reliability assessment, etc. Literature abounds in the above mentioned areas, with both analytical and empirical tools and models developed [e.g., 1-9]. This paper addresses two major issues in wind power system analysis: wind speed distribution (WSD) modeling and energy production simulation. 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 (WS) distribution and direction. Hence, accurate wind speed distribution modeling is the first step to achieve accurate wind energy production simulation. It has been well recognized that wind speed distributions are Weibull [6-7] regardless of the geographical locations. In this paper, wind speed distribution is first modeled using the Maximum Likely Estimation to fit Weibull distributions with actual wind speed data. Instead of fitting one grand distribution model for the whole year, this paper emphasizes the importance ________________________________________________________ Zuwei Yu is with the Energy Center at Purdue University, the United States. His email is
[email protected]. Akiner Tuzuner is
[email protected].
©2008 IEEE.
with
Energy
Center
and
his
email
is
of fitting various Weibull wind speed distribution functions to account for seasonal or even monthly wind speed variations. We call this method the stratified fitting, which is particularly important because seasonal wind speed varies largely for most regions of the world and for the Midwestern United States, the location of interest in this paper. In the Midwestern United States, the wind speed in summer time is the lowest among the four seasons while the peak power demand occurs in summer. Apparently, the annual grand wind speed distribution surely exaggerates wind power production in the summer and would provide false reliability assessment accordingly. In the Midwestern state of Indiana, wind speed may average around 5.0 m/s – 5.7 m/s in the summer, as shown in Fig 1. This range of wind speed may convert to wind power availability in the range of 15 – 22% in the summer, depending on wind turbine characteristics. In the same state, wind speed in April may be in the range of 7 9.6 m/s, resulting in wind power availability roughly around 35-50%. The variations in seasonal wind speed have no doubt a great impact on power production, system reliability and system capacity planning. The Method of Moments (MOM) is also used to estimate the Weibull distribution. One complication of using the MOM is that one needs to use the inverse of the MOM function involving Gamma functions. However, an approximate inverse function can easily be obtained using software like Mathematica that can evaluate those expressions with Gamma functions. It turns out that the MOM provides very similar estimates to the ones obtained by the MLE method. Our conclusion is that both methods can be used to fit the Weibull wind speed distributions to obtain accurate results even though the MLE method is more familiar to many researchers. The remainder of the paper is arranged as follows. Section II describes the negative correlation between the average monthly wind speed and the monthly average energy in MWHs (Mega Watt Hours). Section III presents the MLE modeling of the wind speed and the resulting Weibull distribution. The section also describes the Weibull distribution parameter estimation using the MOM method. Section IV describes an analytical equation for wind power production. A small numerical study is used to illustrate the model, coupled with the Monte Carlo simulation method. Section V concludes the paper with a summary.
II.
SEASONAL WIND VARIATIONS
Wind speed is generally chaotic as described in the literature [e.g., 6]. However, average monthly wind speed has some statistical features in many geographical areas. As show in Fig. 1, the wind speed in Indiana has the lowest monthly averages in the summer. This fact, however, has an unpleasant implication: wind speed is negatively correlated with electricity demand. Taken the average wind speed of the site 9004 shown in Fig.1 as an example, the site is in the territory of an Indiana utility, with a monthly electricity demand tabulated in TABLE 1, together with the wind speed data. Based on the data in the table, an analysis of the correlation between energy and wind speed produces the following results: 1) If data for all 12 month is used, the correlation coefficient is – 0.709; 2) When only data from March to October is used, the correlation coefficient is – 0.8463. The correlation results show that the wind speed distribution in the area does not favor summer peak demand. However, it is much better for winter power production and most favorable for spring power production. Notice that the correlation coefficient is a standard statistical terminology that can be found in many probability and statistics books. Readers are referred to [10] and similar books for further reading. The negative correlation between wind speed and energy demand in Indiana has some other climatological root. In a separate study, it has been shown that wind speed is negatively correlated with the relative humidity during the summer in West Lafayette, Indiana [11], with a correlation coefficient of – 0.3443. It is well known that humidity is second only to temperature as one of the primary drivers for determining electricity demand in summer. Therefore, it is very natural for demand to be negatively correlated with wind speed in the area under study.
TABLE 1. MONTHLY ELECTRICITY VS. WIND SPEED Month
Electricity demand (MWH) 3142760 2873476 2834315 2563276 2956268 3463478 3514687 3441295 2756636 2838033 2829422
Jan Feb March April May June July Aug Oct Nov Dec
Further implications of the results may include: 1) Wind power will compete with baseload power plants including coal, nuclear and even NGCC plants. 2) More peaking power plants will be needed for meeting summer peak demand as more wind power capacity is added. 3) Longer time can be allocated to the regular maintenance of thermal power plants during spring (and perhaps fall as well) because wind turbines can produce more power to replace thermal units in spring. Therefore, better maintenance of the thermal units may be achieved so that they can be more reliable for summer power production to meet peak demand. III.
WIND SPEED MODELING
It has been recognized that wind speed distribution closely resembles the Weibull distribution [6-7]. A Weibull probability density function can be expressed by the following equation:
f ( ws, λ , k ) =
k ws k −1 − ( ws / λ )k ( ) e , ∀ ws ≥ 0
λ λ
(1)
Where ws is wind speed, k>0 is the shape parameter and λ>0 is the scale parameter. The Weibull cumulative probability distribution function is:
F ( ws, λ , k ) = 1 − e − ( ws / λ )
Fig. 1. Average monthly WSDs in five Indiana locations [9] (Legend: 9001 = Haubstadt, 9002 = Carthage, 9003 = Goodland, 9004 = Geetinsville, 9005= LaGrange).
Wind speed at site 9004 ( m/s) 7.1 7.1 7.4 9.7 7.7 5.7 5.1 6.1 7.3 7.6 8.5
k
(2)
Figures of Weibull distributions can be found in [10] and other resources. It would be good for us to take a look at the actual wind speed distribution. The wind speed distribution of the site 9004 (for July-August) described previously in the paper is plotted in Fig. 2. More precisely, it is a histogram or frequency of occurrences. It can be seen that the top plot of Fig. 2 closely resembles the Weibull probability density function except for a large spike when the wind speed is less than or equal to 1.0 MPH (mile per hour) or 0.4444 m/s. It is a reality that wind speed near zero has greater occurrences than the Weibull distribution. The
of samples used in the estimate. Take the natural logarithm of (3), we have
0.062 0.052
k ln f ML = n ln( ) + (k − 1)
Probability
0.042
λ
n
n
i
i =1
(4)
i k
i =1
0.032
Take partial derivatives of (4), we have
0.022
∑
∑
n n ws ws ∂ ln f ML nλ −k λ − wsi = ( 2 ) + (k − 1) ( 2 ) + k ( i ) k −1 2i ∂λ k λ ws λ λ λ i =1 i =1 i
0.012 0.002 1
4
7
10 13 16 19 22 25 28 31 34 37 40 43 46 49
-0.008 Wind speed (M PH)
After simplifying and setting it equal to zero, we have −n
1
λ
0.9 0.8 0.7 Probability
∑ ln( wsλ ) − ∑ ( wsλ )
k −1
−
λ
∑ wsλ (1) + λk ∑ ( wsλ ) n
n
i =1
i
=0
k
i =1
i
That is,
0.6 0.5
^k
1 n
λ =
0.4 0.3 0.2
0 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 Wind speed (MPH)
Fig. 2. Wind speed distribution for July-August at site 9004 (Top: probability density; Bottom: Cumulative probability distribution. Tower height = 100 meters. Legend: PD = probability density, CPD = cumulative probability distribution).
lower the tower, the higher are the occurrences of zero wind speed. This is why the wind tower for power production should not be low. A) The MLE modeling of the Weibull distribution The MLE procedure is to determine the values of the parameters of λ and k of the Weibull distribution, the first order optimality conditions below are used for the purpose:
λ
i =1
k
∑ ws ) n
, or λ = ( 1 ^
n
i =1
^
^
k 1/ k
i
,
(5)
1 ^
=(
k
∑ ws n
^
k
i
i =1
ln wsi ) /(
∑ ws ) − n1 ∑ ln ws n
n
^
k i
i =1
(6)
i
i =1
Equation (6) is nonlinear and can be solved using the Newton-Raphson method to obtain k caret. Results from the MLE modeling: i) Results based on the July-August data Equation (6) can be rewritten as f^ = k
1 ^
k
+
1 n
∑ ln ws − (∑ ws n
i =1
n
i
i =1
^
k
i
ln wsi ) /(
∑ ws ), ∀ k > 0 n
i =1
(7)
^
^
k i
Our aim is to find a k caret such that f ^ = 0. It would be k
^
solutions. We first plot the function of f ^ for 0< k 20, and k
the conclusion below will not change. It can be seen that f ^ has only one root in the k caret range of [0.2, 20]. It k
has much greater positive values for k caret in (0, 0.2), but these are not plotted due to poor visual effect. As a matter of fact, the derivative of f ^ remains negative for k caret k
>0, which shows that f ^ is monotone decreasing in the k
^
data range. Given that f ^ is positive for k =0.2 and that
0.07 Sample
k
Weibull
0.06
its derivative is always negative ( f ^ is monotone k
0.05 Probability
decreasing), it can be concluded that f ^ has just one root k
^
in the k range of [0.2 ,20]. A more vigorous proof of the uniqueness of k caret estimate can be found in [12]. Using a modified Newton-Raphson method, kˆ ≅ 2.0893 is estimated for equation (6). Consequently, λˆ ≅ 13.2671 is estimated using equation (5). The sample wind speed PD is plotted against the fitted Weibull PD in Fig. 4 from which we can see that the overall fitting is good except when the wind speed is less than one MPH, as shown in Fig. 5. The maximum error is 0.0204285, and the sum of all errors is 0.020427. In other words, errors at other wind speed values nearly cancel each other. This is an interesting observation because it tells that the fitted Weibull has about 98% accuracy, and the main error is attributed to the one at the wind speed less than one MPH literally. (However, errors at higher wind speed would cause greater errors in power production simulation due to the fact that power production is a function of the cubic wind speed before the cut-off speed).
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 39 41 43 45 47 49 Wind speed (MPH)
Fig. 4. Comparison of sample and fitted Weibull PDs. 0.025 0.02
Sample - Fitted Weibull
0.015 0.01 0.005 0 1
4
7
10 13 16 19 22
25 28 31 34 37 40
43 46 49
-0.005 -0.01 -0.015
20 k
Value of f sub(k) and k
18
Wind speed (MPH)
f unction
16
Fig. 5. Error (sample PD-fitted Weibull PD).
14 12 10
0.062
8 6
0.052
4 2
0.042
1
16
31
46
61 76
91 106 121 136 151 166 181 196
Counts of k increm ents (0.1 each for k =0.2 to 20)
Fig. 3. Plot of equation (7).
Probability
0
0.032 0.022 0.012
ii) Results based on the April-May data The sample probability density function based on the April-May data set is plotted in Fig. 6. We can see that the PD for the April-May data is shifted to a higher MPH compared with the PD for the July-August data. The mean value is now 18.7 MPH as compared to the mean of 11.8 MPH for the July-August data, even though the maximum wind speeds of the two data sets are not too different. Using the same procedure as in i), kˆ ≅ 2.8063 and λˆ ≅ 20.9032 . The same argument can be made to conclude that the values estimated are unique, and the fitted Weibull distribution using the MLE method is unique too. The deviations between the fitted Weibull wind speed PD and the sample wind speed PD is plotted in Fig. 7. It can be seen that the errors at lower wind
0.002 1
4
7
10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55
-0.008 MPH
Fig. 6. Probability density (PD) of wind speed April-May
speed values are not as pronounced as the July-August case, while the error at higher wind speed values of 20-23 MPH is greater. The sum of the errors is – 0.00177, much smaller than the 0.020427. In other words, this fit of Weibull PD has about 99.8% accuracy. However, when one takes a close look at the error plots, one may find similar error patterns – The fitted Weibull PDs are smaller than the samples at wind speed values close to zero, a few MPHs above the mean value, and greater than the samples at other wind speed values. This pattern may be further explored to improve fitting accuracy.
kˆ that solves G (kˆ) = mˆ 2 mˆ 12 and a unique ˆ λ=
0.015
(
1 kˆ
)
. A
complete proof can be found in [12].
0.01
Sample - Weibull
mˆ 1 Γ 1+
Results from the MOM Modeling: We use a modified Newton method to find an approximate solution for kˆ . To save space, we only report
0.005
0 1
4
7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55
-0.005
-0.01
-0.015
^
^
the estimated λ and k from the MOM-based estimation. The final results are kˆ ≅ 2.1298 , λˆ ≅ 13.3259 for the July-August data set; kˆ ≅ 2.8331 and λˆ ≅ 20.9872 for the April-May data set. These results are very close to the ones obtained by the use of the MLE fitting method.
Widn speed (MPH)
Fig. 7. Error (sample PD-fitted Weibull PD based on April-May data).
B) The Method of Moments (MOM) Modeling of the Weibull distribution From known results in probability theory, the first two (un-centered) moments of the above Weibull distribution are given as (see, for example, Wikipedia) m1 = E[ ws ] = λ Γ(1 + 1k ) , and
m2 = E[ ws 2 ] = λ2 Γ (1 +
2 k
)
Rearranging the above, we can write m2 = Γ(1 + m12
The
2 k
) Γ(1 + 1k )2 , and λ = m1 Γ(1 + 1k )
estimation
procedure is as follows. Let G ( k ) = Γ(1+2/k)/Γ(1+1/k)2 . Then, using the Method of Moments, k can be estimated as kˆ that solves ˆ 1 and mˆ 2 are, respectively, the G (kˆ) = mˆ 2 mˆ 12 , where m unbiased sample mean and the sample 2nd moment. kˆ can then be used to estimate λˆ = mˆ 1 Γ(1 + 1ˆ ) . From results in k
statistics, we know that mˆ 2 mˆ 1 2 ≥ 1 (where equality holds if and only all if observations are the same) and that it can be arbitrarily big. So, it is left to show that G (kˆ) = mˆ 2 mˆ 12 has a unique solution over kˆ ∈ (0, ∞) . To this end, we
state that G (k ) is continuous, (strictly) monotone decreasing over k ∈ (0, ∞) , and its limiting values are
→ +∞ →0 ⎯→1 . Then, using the G(k ) ⎯k⎯ ⎯→ +∞ and G(k ) ⎯k⎯ Intermediate Value Theorem, we conclude that for a ˆ 1 and mˆ 2 there exists a unique given sample with m +
IV. WIND ENERGY PRODUCTION MODELING According to the Wind Association, Wind turbine power in the range between the cut-in and cut-out wind speed is approximated by [13]:
P =0.5Aρ CpNbNg (ws )3
(8)
where: P = power in watts A = rotor swept area exposed to the wind (m2) ρ = air density, ws = wind speed (m/s) Cp = Coefficient of performance (.59 {Betz limit} is the maximum theoretically possible, .35 for a good design), Ng = generator efficiency (80% or possibly more for a grid-connected induction generator), and Nb = gearbox/bearings efficiency (could be as high as 95% for good product). Equation (8) can be simplified to P =Cw(ws)3 where Cw is a combined coefficient. In general, power production from a wind generator can be approximated as:
⎧0, ∀ ws ∈ [0, Vcutin), or, ws > Vcutout ⎪ P (t ) = ⎨ Cw( ws )3 , ∀ ws ∈ [Vcutin, Vsatur ] ⎪ P max, ∀ ws ∈ (Vsatur , Vcutout ] ⎩
(9)
Where Pmax is the maximum power output, Vcutin is the cut-in wind speed at which power production occurs, Vsatur is the saturation wind speed when power output reaches Pmax, and Vcutout is the wind speed at which wind power production stops. The power curve of the Vesta 3.0 MW is illustrated in Fig. 10 [14]. Notice that (9) is an ideal power curve, and there can be power variations in the wind speed range [Vsatur, Vcutout].
The paper claims that the parameters estimated are unique, with a proof provided in a companion paper [12]. The uniqueness is important in that no parameters other than the estimated parameters exist so that the chance of a different Weibull distribution of the same data set is ruled out. VI.
Fig. 10. Power Curve of V90-3.0 [14].
The expected electricity production from such a wind power generator can be done by Monte Carlo simulation, as described in [3] and others. In order to do this, the sampling of the Weibull distributions has to be resolved first. By taking the natural logarithm of both sides of equation (2), we have wss = λ[− ln(1 − p)]1/ k
(10)
Where wss is the sample wind speed, and p is a random number drawn from the population uniformly distributed in [0,1]. That is, wss is the sample wind speed from a designated Weibull wind speed distribution. A generator wind power production can be simulated by the following steps: 1) define the wind speed distribution, 2) pick the wind generator and the parameters, 3) draw a random wind speed from (10), 4) determine the power or electricity output from (9) with some minor adjustment, 5) repeat 3) and 4) until a predetermined number of simulations or a convergence tolerance is reached. Based on the procedure, the expected electricity production from the V90-3.0 is about 0.475 MWh for a typical peak hour using the July-August wind speed data set and is about 1.16 MWh for a regular hour using the April-May wind data. That is, the expected power production from the wind generator is about 15.83% of its rated capacity in July-August time frame, and about 38.33% in April-May. V.
SUMMARY
The paper describes the MLE and the MOM methods for estimating the parameters of the Weibull wind speed distributions. Both methods result in a unique estimate for each data set. A sampling method is used for drawing samples from the distributions. A simple Monte Carlo simulation method is used for calculating the expected electricity production using the V90-3.0 wind generator. Results show that the expected power production in JulyAugust time frame for the selected area is about 16%, about 13% less than the one in April-May.
REFERENCES
[1] 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. [2] A. Estanqueiro, “Dynamic Wind Generation Model for Power System Studies,” IEEE Trans. Power Syst., Vol. 22, No. 3, August 2007, pp. 920-928. [3] 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. [4] 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. [5] 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. [6] P. Edwards, R. Hurst, “Level-crossing statistics of the horizontal wind speed in the planetary surface boundary layer,” Chaos, Vol. 11, No. 3, Sept. 2001, pp. 611-618. [7] 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] A. Breipohl, K. Shanmugan, Random Signals, Detection, Estimatin and Data Analysis, John Wiley & Sons, 1988. [11] Sika Dofonsou, “An Analysis on the Correlation between Wind Speed and Relative Humidity in the West Lafayette Area in Indiana,” an internal research done for the State Utility Forecasting Group, Purdue University, September 2007. [12] A. Tuzuner, Z. Yu, “A Theoretical Analysis on the Uniqueness of the Parameter Estimated for the Weibull Wind Speed Distribution,” submit to IEEE PES General Meeting 2008. [13] American Wind Energy Association, available at: http://www.awea.org/faq/windpower.html [14] “General Specifications – V90-3.0,” a technical manual prepared by the Vestas Wind Systems A/S, 2004. VII. BIOGRAPHY Zuwei Yu received his Ph.D. degree in energy engineering in 1995 from the University of Oklahoma with a minor in Operations Research. He is currently a senior researcher with the Energy Center at the Discovery Park, a CO-PI of the Clean Energy Project and a graduate faculty with the College of Engineering, Purdue University. His research lies in mathematical programming and economic modeling, with applications to energy, environment, risk and other systems. Akiner Tuzuner is a Ph.D. candidate in Industrial Engineering, Purdue University and a research assistant with the Energy Center. He received his M.S. in economics in 2000 and his M.S. in industrial engineering in 2002, both from Purdue University. His dissertation is on a stochastic gaming model of the crude oil market. His research interests lie in optimization, stochastic processes and game theory, particularly applied to economic and financial modeling of energy markets. Acknowledgement: Mr. Ryan Brown, Manager of Energy Division, Indiana Office of Energy & Defense provided the wind data. His support makes the study possible.
