Evaluation of suitability of wind speed probability distribution models: a case study from Tamil Nadu, India

Abstract

The optimal design and performance monitoring of wind farms depend on the precise assessment of spatial and temporal distribution of wind speed. The aim of this research is to investigate the appropriateness of nine popular probability distribution models (exponential, gamma, generalised extreme value, inverse Gaussian, Kumaraswamy, log-logistic, lognormal, Nakagami, and Weibull) for the assessment of wind speed distribution (WSD) at 10 sites situated at topographically distinct locations in Tamil Nadu, India, based on 39 years of data. The results suggest that a single distribution cannot produce best fit for all the stations. On an individual level, the generalised extreme value distribution provided the most suitable fit for majority of the stations, followed by the Kumaraswamy distribution. The Kumaraswamy distribution has performed well even if the WSD of the station is negatively skewed. Hence, based on the ranking and performance consistency, the Kumaraswamy distribution can be preferred irrespective of the topographical heterogeneity of the stations.

Appendix

1. Exponential distribution

The PDF of the exponential distribution is determined using the following function (Masseran et al. 2012):

$$ f\left(\nu, \lambda \right)=\left\{\begin{array}{c}\lambda {e}^{-\lambda \nu}\kern0.75em \nu \ge 0\ \\ {}0\kern3em \nu <0\end{array}\right. $$

(4)

where λ is known as the scale parameter and λ can be calculated from the inverse of the mean wind speed:

$$ \lambda =\frac{1}{\overline{\nu}} $$

(5)

2. Gamma distribution

The PDF of the gamma distribution is determined using the following function (Masseran et al. 2012):

$$ f\left(v,\alpha, \beta \right)=\frac{v^{\alpha -1}}{\beta^{\alpha}\Gamma \left(\alpha \right)}\exp \left(-\frac{v}{\beta}\right) $$

(6)

where α is the shape parameter, and β is the scale parameter, and г is the Gamma function. The shape and scale parameters are calculated by following equations simultaneously:

$$ \alpha \beta =\overline{v} $$

(7)

$$ nln\left(\beta \right)+ n\psi \left(\alpha \right)={\sum}_{i=1}^n\ln \left({v}_i\right) $$

(8)

where n is the number of samples and ψ is the di gamma function.

3. Generalised extreme value distribution

The PDF of the generalised extreme value distribution is defined as (Shukla et al. 2010):

$$ f\left(v,\xi, \mu, \sigma \right)=\frac{1}{\sigma}\mathit{\exp}\left[-{\left(1+\xi \left(\frac{v-\mu }{\sigma}\right)\right)}^{-1/2}\right]{\left(1+\xi \left(\frac{v-\mu }{\sigma}\right)\right)}^{-1-\left(\frac{1}{\xi}\right)} $$

(9)

For $1+\xi \left(\frac{v-\mu }{\sigma}\right)>0$ and ξ ≠ 0.

4. Inverse Gaussian distribution

The PDF of the inverse Gaussian distribution is given by (Masseran et al. 2012):

$$ f\left(v,\lambda, \mu \right)=\sqrt{\frac{\lambda }{2\pi {v}^3}\mathit{\exp}\left\{-\frac{\lambda }{2{\mu}^2v}{\left(v-\mu \right)}^2\right\}} $$

(10)

where λ and μ are the shape and scale parameter, respectively. The parameters of the inverse Gaussian distribution is determined using the following expressions:

$$ \mu =\overline{v} $$

(11)

$$ \lambda =\frac{n}{\sum_{i=1}^n\frac{1}{v_i}-\frac{1}{\overline{v}}} $$

(12)

5. Log-Logistic distribution

The PDF of the log-logistic distribution is determined using the following functions (Meeker and Escobar 1998):

$$ f\left(v,\mu, \sigma \right)=\frac{\mathit{\exp}\left(\frac{\ln (v)-\mu }{\sigma}\right)}{\sigma v{\left(1+\mathit{\exp}\left(\frac{\ln (v)-\mu }{\sigma}\right)\right)}^2} $$

(13)

where μ and σ are log location and log scale parameters, respectively. The parameters of the log-logistic can be determined using the following equations simultaneously:

$$ {\sum}_{i=1}^n\frac{\exp \left(\frac{v_i-\mu }{\sigma}\right)}{1+\exp \left(\frac{v_i-\mu }{\sigma}\right)}=\frac{n}{2} $$

(14)

$$ n\sigma - n\mu +{\sum}_{i=1}^n{v}_i-2{\sum}_{i=1}^n\frac{\left({v}_i-\mu \right)\exp \left(\frac{v_i-\mu }{\sigma}\right)}{1+\exp \left(\frac{v_i-\mu }{\sigma}\right)}=0 $$

(15)

6. Lognormal distribution

The PDF of the lognormal distribution, with shape and scale parameters σ and μ, is obtained using the following function (Masseran et al. 2012):

$$ f\left(v,\mu, \sigma \right)=\frac{1}{v\sigma \sqrt{2\pi }}\mathit{\exp}\left\{\frac{-1}{2}{\left(\frac{\ln (v)-\mu }{\sigma}\right)}^2\right\} $$

(16)

7. Nakagami distribution

The PDF of the Nakagami distribution is defined by the following functions (Lopez-Martinez et al. 2013):

$$ f\left(v,m,\varOmega \right)=\frac{2{m}^m}{\varGamma (m){\varOmega}^m}{v}^{2m-1}\exp \left(-\frac{m}{\varOmega }{v}^2\right) $$

(17)

where Γis the gamma function and m and Ω indicate the shape and scale parameters calculated using the following expressions:

$$ \varOmega =\overline{v^2} $$

(18)

$$ m=\frac{\left(\overline{v^2}\right)}{{\left({v}^2-\overline{v^2}\right)}^2},m\ge 1/2 $$

(19)

8. Weibull distribution

The PDF of the Weibull distribution is given by (Manwell et al. 2002):

$$ f\left(\nu, k,c\right)=\frac{k}{c}{\left(\nu /c\right)}^{k-1}\exp \left[-{\left(\frac{\nu }{c}\ \right)}^k\right] $$

(20)

where ν, k, and c are the wind speed, shape, and scale parameter, respectively.

The shape and scale parameters are determined as follows:

$$ k={\left[\frac{\sum_{i=1}^n{v_i}^k\ln \left({v}_i\right)}{\sum_{i=1}^n{v_i}^k}-\frac{\sum_{i=1}^N\ln \left({v}_i\right)}{N}\right]}^{-1} $$

(21)

$$ c={\left[\frac{\sum_{i=1}^n{v_i}^k}{n}\right]}^{1/k} $$

(22)

where v_i is the wind speed at time step i (m/s) and n is the number of non-zero WD points.