This paper presents a new method to accurately char- acterize and predict the annual variation of wind conditions. Estimation of the distribution of wind conditions is necessary (i) to quantify the available energy (power density) at a site, and (ii) to design optimal wind farm configurations. We develop a smooth multivariate wind distribution model that captures the coupled variation of wind speed, wind direction, and air density. The wind distribution model developed in this paper also avoids the limiting assumption of unimodality of the distribution. This method, which we call the Multivariate and Multimodal Wind distribution (MMWD) model, is an evolution from existing wind distribution modeling techniques. Multivariate kernel density estimation, a standard non-parametric approach to estimate the probability density function of random variables, is adopted for this purpose. The MMWD technique is successfully applied to model (i) the distribution of wind speed (univariate); (ii) the distribution of wind speed and wind direction (bivariate); and (iii) the distribution of wind speed, wind direction, and air density (multivariate). The latter is a novel contribution of this paper, while the former offers opportunities for validation. Ten-year recorded wind data, obtained from the North Dakota Agricultural Weather Network (NDAWN), is used in this paper. We found the coupled distribution to be multimodal. A strong correlation among the wind condition parameters was also observed.

1. A Multivariate and Multimodal Wind Distribution
Model
Jie Zhang*, Souma Chowdhury*, Achille Messac# and Luciano Castillo*
* Rensselaer Polytechnic Institute, Department of Mechanical, Aerospace, and Nuclear Engineering
# Syracuse University, Department of Mechanical and Aerospace Engineering
ASME 2011 5th International Conference on Energy Sustainability
August 7-10, 2011
Washington, DC

2. Impact of Wind Variations
 Careful determination of energy variability at a site serves two
important objectives:
1. Accurate assessment of the potential of a wind farm site, and
2. Effective design of the wind farm layout, and selection of the
appropriate turbine types for the site.
 For a given farm layout, the direction of wind has a strong influence on
the wakes created & subsequently on the overall flow pattern.
 The annual variation in air density was estimated to be 30%.
 This paper develops an accurate multivariate probability distribution of
wind speed, wind direction and air density.
2

3. Motivation
3
 Wind Power Density (WPD)

max max
WPD U f U d dU
     

min
3
U
0
1
( , )
2
 Existing wind distribution modeling approaches can be broadly
classified into:
 univariate and unimodal distributions of wind speed
 bivariate and unimodal distributions of wind speed and wind
direction
 These wind distribution models make limiting assumptions regarding
the correlativity and the modality of the distribution of wind - such
assumptions can lead to approximations that deviate significantly from
the actual scenario.

4. Outline
 Research objectives
 Multivariate and Multimodal Wind Distribution
(MMWD) Model
 Wind condition data
 Case study: application of the MMWD model
 Concluding remarks
4

5. Research Objectives
 Develop and explore a new method to represent the
multivariate (and likely multimodal) distribution of wind
conditions.
 Multivariate and Multimodal Wind Distribution
(MMWD) model.
5

6. Literature Review
 Univariate: Weibull distribution, Rayleigh distribution, Gamma distribution,
Lognormal distribution, Beta distribution, inverse Gaussian distribution,
singly truncated normal Weibull mixture distribution, and maximum
entropy probability density function.
 Lackner and Elkinton characterized the wind speed data by direction sector
and fitted a Weibull distribution for each direction sector.
 Vega and Letchford used Weibull distribution to estimate the wind speed
probability, and modeled the shape parameter and the scale parameter as
functions of wind direction.
 Carta et al. presented a joint probability density function of wind speed and
wind direction for wind energy analysis.
 Erdem and Shi compared three differing bivariate joint distributions
(angular-linear, Farlie-Gumbel-Morgenstern, and anisotropic lognormal
approaches) to represent wind speed and wind direction data.
6

7. Multivariate and Multimodal Wind Distribution
(MMWD) Model
 This model is uniquely capable of representing
multimodal wind data.
 This model can capture the joint variations of wind
speed, wind direction and air density.
7

8. MMWD Model
8
• Kernel Density Estimation (KDE) method is adopted to
develop the MMWD model
• Kernel Density Estimation
• Multivariate Kernel Density Estimation
• Optimal Bandwidth Matrix Selection

9. Wind Condition Data
 The onshore wind data is obtained from the North Dakota Agricultural
Weather Network (NDAWN).
 The offshore wind data is obtained from the National Data Buoy Center
(NDBC).
 The data are recorded between the year 2000 and 2009.
Details of the two stations
9 (a) Onshore (Baker) (b) Offshore (44013)

10. Application of the MMWD Model
 Based on the onshore and the offshore wind data, the
MMWD technique has been applied to investigate three
different cases:
 Case I: distribution of wind speed (univariate).
 Case II: joint distribution of wind speed and wind direction
(bivariate).
 Case III: distribution of wind speed, wind direction, and air
density (multivariate).
10

11. MMWD Case I: Univariate Distribution
11
 For onshore data, the probability distribution curves of MMWD,
Lognormal and Gamma match the histogram.
 For offshore data, the probability distribution curves of MMWD and
Lognormal match the histogram.

12. Quantile-Quantile (Q-Q) Plot
 Q-Q plot is a graphical method for comparing two probability distributions
12
by plotting their quantiles against each other.
 The MMWD distribution follows the theoretical distribution ( line y = x)
more closely than other distributions
45o

13. Coefficient of Determination
13
 The coefficient of determination is a measure of the agreement between an
estimated distribution and the recorded data.

14. MMWD Case II: Bivariate Distribution
14
 The estimated probability distribution of the onshore wind data is highly
multimodal in nature;
 The estimated offshore wind distribution can be treated practically as
unimodal, which is actually common for offshore.

15. Wind Rose
 For the onshore site, we observe that winds from the Northwest and the South
dominate over the whole year. Minimal wind is observed from the Northeast direction.
 For the offshore site, we observe that winds from the West and the Southwest dominate
over the whole year. Minimal wind is observed from the North direction.
15

16. MMWD Case III: Multivariate Distribution
 We model the multivariate probability distribution of wind speed, wind
direction, and air density.
 A strong correlation among the three wind condition parameters is
evident.
16

17. Year-to-Year Variations (Onshore Site)
17
Wind distributions estimated using the Multivariate and
Multimodal model for a site at Baker, ND
Zhang et al., 2011

18. Year-to-Year Variations (Offshore Site)
Zhang et al., 2011 18
Estimated Wind
Distribution
(preceding years’ data)
Predicted Long Term
Variation of Wind
(succeeding years)
May not be the right
way to account for
Deterministic assumption
wind variations

19. Conclusion
 This paper developed a Multivariate and Multimodal Wind
Distribution (MMWD) model.
 A strong correlation was observed among wind speed,
wind direction, and air density.
 Such a nonparametric stochastic modeling approach can be
helpful to represent the variation in other natural energy
resources as well.
 Specifically, the MMWD model could be helpful for
evaluating the wind resource potential for farm siting.
19

20. Acknowledgement
• I would like to acknowledge my research adviser
Prof. Achille Messac, and my co-adviser Prof.
Luciano Castillo for their immense help and
support in this research.
• I would also like to thank my friend and colleague
Souma Chowdhury for his valuable contributions to
this paper.
• I would also like to thank NSF for supporting this
research.
20

21. Questions
and
Comments
21
Thank you