Potential of wind energy in Cameroon based on Weibull, normal, and lognormal distribution

Abstract

Careful analysis of long-term wind data in a broad area is essential to estimate the wind energy potential of a region. For this purpose, knowledge of wind speed distribution is an essential task. This paper proposes a comprehensive statistical evaluation of monthly, annual, and interannual variabilities of mean wind speeds and wind power densities of 2745 different sites over an area covering the whole of Cameroon. We used wind speed data obtained from ERA-5 for the period 2000–2017. In addition to the popular Weibull probability density function (WPDF), other continuous distributions such as the Normal PDF (NPDF) and the Lognormal PDF (LPDF) were used to investigate their suitability for the description of the wind regimes. We also established the analytical expression of the Normal power density. Three statistical analysis tools were used to determine the goodness of fit of the curves and the parameters of the PDFs were determined by the maximum likelihood method. We found that the theoretical power (TP) and power densities based on WPDF and NPDF are closely related. We observed the highest wind speeds and power densities in a few areas in the Lake Chad Basin and the Gulf of Guinea (GoG) which did not exceed class 2. It is seen that most of the areas studied show relatively low wind speeds and power densities that belong mostly to class 1 during all seasons. These results are useful for sizing wind turbines specific for different locations in Cameroon and making good decisions in terms of budgetary optimization of investments in the energy sector.

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Acknowledgements

The authors sincerely express theirs thanks to the reviewers for their constructive reviews.

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Correspondence to Christian Kenfack-Sadem.

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Appendix

Appendix

WPD

$$\begin{gathered} {\text{WPD}} = \frac{1}{2}\rho \mathop \int \limits_{0}^{\infty } v^{3} f\left( v \right)\;{\text{d}}v \hfill \\ {\text{WPD}} = \frac{1}{2}\rho \mathop \int \limits_{0}^{\infty } v^{3} \left( {\frac{k}{c}} \right)\left( {\frac{v}{c}} \right)^{{k - 1}} \exp \left[ { - \left( {\frac{v}{c}} \right)^{k} } \right]\;{\text{d}}v \hfill \\ {\text{WPD}} = \frac{1}{2}\rho ~\overline{{v^{3} }} \hfill \\ \end{gathered}$$

We know that

$$\begin{gathered} \overline{{v^{n} }} = \mathop \int \limits_{0}^{\infty } v^{n} f\left( v \right)\;{\text{d}}v = c^{n} \Gamma \left( {1 + \frac{n}{k}} \right) \hfill \\ {\text{WPD}} = \frac{1}{2}\rho ~c^{3} \Gamma \left( {1 + \frac{3}{k}} \right) \hfill \\ \end{gathered}$$

LPD

\({\text{LPD}} = \frac{1}{2}\rho \mathop \int \nolimits_{0}^{\infty } v^{3} f\left( v \right)\;{\text{d}}v\) with

$$\begin{gathered} {\text{LPD}} = \frac{1}{2}\rho \mathop \int \limits_{0}^{\infty } v^{3} \frac{1}{{\sigma v\sqrt {2\pi } }}\exp \left[ { - \frac{1}{2}\left( {\frac{{\ln \left( v \right) - \mu }}{\sigma }} \right)^{2} } \right]\;{\text{d}}v \hfill \\ {\text{LPD}} = \frac{1}{2}\frac{1}{{\sigma \sqrt {2\pi } }}\rho \mathop \int \limits_{0}^{\infty } v^{2} \exp \left[ { - \frac{1}{2}\left( {\frac{{\ln \left( v \right) - \mu }}{\sigma }} \right)^{2} } \right]\;{\text{d}}v \hfill \\ \end{gathered}$$

Let’s \(I = \mathop \int \nolimits_{0}^{\infty } v^{2} \exp \left[ { - \frac{1}{2}\left( {\frac{{\ln \left( v \right) - \mu }}{\sigma }} \right)^{2} } \right]\;{\text{d}}v\) then \({\text{LPD}} = \frac{1}{2}\frac{1}{{\sigma \sqrt {2\pi } }}\rho I\).

Let’s \(s = \frac{{\ln \left( v \right) - \mu }}{{\sigma \sqrt 2 }}\) then \(v = \exp \left( {\mu + \sigma s\sqrt 2 } \right)\) and \({\text{d}}v = \sigma \sqrt 2 \exp \left( {\mu + \sigma s\sqrt 2 } \right){\text{d}}v\).

For \(v \to 0~,~s \to ~ - \infty\); For \(v \to + \infty ~,~s \to ~ + \infty\)

$$I = \sigma \sqrt 2 \exp \left( {3\mu + \frac{9}{2}\sigma ^{2} } \right)\mathop \int \limits_{{ - \infty }}^{{ + \infty }} \exp \left[ { - \left( {s - \frac{{3\sigma \sqrt 2 }}{2}} \right)^{2} } \right]\;{\text{d}}s$$

For \(J = \mathop \int \nolimits_{{ - \infty }}^{{ + \infty }} \exp \left[ { - \left( {s - \frac{{3\sigma \sqrt 2 }}{2}} \right)^{2} } \right]\;{\text{d}}s\) then \(I = \sigma \sqrt 2 \exp \left( {3\mu + \frac{9}{2}\sigma ^{2} } \right)J\).

Let’s \(u = s - \frac{{3\sigma \sqrt 2 }}{2}\) then \({\text{d}}u = {\text{d}}s\). For \(s \to - \infty ~,~u \to ~ - \infty\); For \(s \to + \infty ~,~u \to ~ + \infty\)

$$\begin{gathered} J = \mathop \int \limits_{{ - \infty }}^{{ + \infty }} \exp \left( { - u^{2} } \right)\;{\text{d}}u = \sqrt \pi \hfill \\ I = \sigma \sqrt 2 \exp \left( {3\mu + \frac{9}{2}\sigma ^{2} } \right)\sqrt \pi \hfill \\ \end{gathered}$$

By replacing I by its value in equation \({\text{LPD}} = \frac{1}{2}\frac{1}{{\sigma \sqrt {2\pi } }}\rho I\)

$${\text{LPD}} = \frac{\rho }{2}\exp \left( {3\mu + \frac{9}{2}\sigma ^{2} } \right)$$

We have also demonstrated that \(\mathop \int \nolimits_{0}^{\infty } x^{{k - 1}} \exp \left[ { - \frac{1}{2}\left( {\frac{{\ln x - \mu }}{\sigma }} \right)^{2} } \right]\;{\text{d}}x~ = \sqrt {2\pi \sigma ^{2} } \exp \left[ {k\mu + \frac{1}{2}k^{2} \mu ^{2} } \right]\)

Then \({\text{LPD}} = \frac{\rho }{2}\exp \left( {3\mu + \frac{9}{2}\sigma ^{2} } \right)\).

NPD

\({\text{NPD}} = \frac{1}{2}\rho \mathop \int \nolimits_{0}^{\infty } v^{3} f\left( v \right)\;{\text{d}}v\) with \(f\left( v \right) = \frac{1}{{\sigma \sqrt {2\pi } }}\exp \left[ { - \frac{{\left( {v - \mu } \right)^{2} }}{{2\sigma ^{2} }}} \right]\)

$${\text{NPD}} = \frac{1}{2}\rho \frac{1}{{\sigma \sqrt {2\pi } }}\mathop \int \limits_{0}^{\infty } v^{3} \exp \left[ { - \frac{{\left( {v - \mu } \right)^{2} }}{{2\sigma ^{2} }}} \right]\;{\text{d}}v$$

\({\text{NPD}} = \frac{1}{2}\rho \frac{1}{{\sigma \sqrt {2\pi } }}\overline{{v^{3} }}\) with \(\overline{{v^{3} }} = \mathop \int \nolimits_{0}^{\infty } v^{3} \exp \left[ { - \frac{{\left( {v - \mu } \right)^{2} }}{{2\sigma ^{2} }}} \right]\;{\text{d}}v\).

Let’s \(v = x + \mu\) then \({\text{d}}v = {\text{d}}x\). For \(v \to 0~,~x \to ~ - \mu\); For \(v \to + \infty ~,~x \to ~ + \infty\)

$$\begin{gathered} \overline{{v^{3} }} = \mathop \int \limits_{{ - \mu }}^{{ + \infty }} \left( {x + \mu } \right)^{3} \exp \left[ { - \frac{{x^{2} }}{{2\sigma ^{2} }}} \right]\;{\text{d}}x \hfill \\ \overline{{v^{3} }} = \mathop \int \limits_{{ - \mu }}^{0} \left( {x + \mu } \right)^{3} \exp \left[ { - \frac{{x^{2} }}{{2\sigma ^{2} }}} \right]\;{\text{d}}x + \mathop \int \limits_{0}^{{ + \infty }} \left( {x + \mu } \right)^{3} \exp \left[ { - \frac{{x^{2} }}{{2\sigma ^{2} }}} \right]\;{\text{d}}x \hfill \\ \end{gathered}$$

Let’s \(a = \frac{1}{{2\sigma ^{2} }}\), \(J = \mathop \int \nolimits_{{ - \mu }}^{0} \left( {x + \mu } \right)^{3} \exp \left( { - ax^{2} } \right)\;{\text{d}}x\) and \(I = \mathop \int \nolimits_{0}^{{ + \infty }} \left( {x + \mu } \right)^{3} \exp \left( { - ax^{2} } \right)\;{\text{d}}x\).

For \(I = \mathop \int \nolimits_{0}^{{ + \infty }} \left( {x^{3} + 3\mu x^{2} + 3\mu ^{2} x + \mu ^{3} } \right)\exp \left( { - ax^{2} } \right)\;{\text{d}}x\)

$$\begin{aligned} I & = \mathop \int \limits_{0}^{{ + \infty }} x^{3} \exp \left( { - ax^{2} } \right)\;{\text{d}}x + 3\mu \mathop \int \limits_{0}^{{ + \infty }} x^{2} \exp \left( { - ax^{2} } \right)\;{\text{d}}x \\ \quad + 3\mu ^{2} \mathop \int \limits_{0}^{{ + \infty }} x\exp \left( { - ax^{2} } \right)\;{\text{d}}x + \mu ^{3} \mathop \int \limits_{0}^{{ + \infty }} \exp \left( { - ax^{2} } \right)\;{\text{d}}x \\ \end{aligned}$$

We know that for \(I_{m} \left( a \right) = \mathop \int \nolimits_{0}^{{ + \infty }} x^{m} \exp \left( { - ax^{2} } \right)\;{\text{d}}x\), the various derivatives are given by de recurrence relation \(I_{{m + 2}} \left( a \right) = \frac{{m + 1}}{{2a}}I_{m} \left( a \right)\) with \(I_{0} \left( a \right) = \frac{1}{2}\sqrt {\frac{\pi }{a}}\) and \(I_{0} \left( a \right) = \frac{1}{{2a}}\).

Then \(I = 2\sigma ^{4} + \frac{{3\mu \sigma ^{3} }}{2}\sqrt {2\pi } + 3\mu ^{2} \sigma ^{2} + \frac{{\mu ^{3} \sigma }}{2}\sqrt {2\pi }\).

For \(J = \mathop \int \nolimits_{{ - \mu }}^{0} \left( {x^{3} + 3\mu x^{2} + 3\mu ^{2} x + \mu ^{3} } \right)\exp \left( { - ax^{2} } \right)\;{\text{d}}x\)

$$\begin{aligned} J & = \mathop \int \limits_{{ - \mu }}^{0} x^{3} \exp \left( { - ax^{2} } \right)\;{\text{d}}x + 3\mu \mathop \int \limits_{{ - \mu }}^{0} x^{2} \exp \left( { - ax^{2} } \right)\;{\text{d}}x \\ \quad + 3\mu ^{2} \mathop \int \limits_{{ - \mu }}^{0} x\exp \left( { - ax^{2} } \right)\;{\text{d}}x + \mu ^{3} \mathop \int \limits_{{ - \mu }}^{0} \exp \left( { - ax^{2} } \right)\;{\text{d}}x \\ \end{aligned}$$
$$\begin{aligned} J & = \mathop \int \limits_{{ - \mu }}^{0} x^{3} \exp \left( { - ax^{2} } \right)\;{\text{d}}x + 3\mu \mathop \int \limits_{{ - \mu }}^{0} x^{2} \exp \left( { - ax^{2} } \right)\;{\text{d}}x \\ \quad + 3\mu ^{2} \mathop \int \limits_{{ - \mu }}^{0} x\exp \left( { - ax^{2} } \right)\;{\text{d}}x + \mu ^{3} \mathop \int \limits_{{ - \mu }}^{0} \exp \left( { - ax^{2} } \right)\;{\text{d}}x \\ \end{aligned}$$

Let’s \(q^{2} = a = \frac{1}{{2\sigma ^{2} }}\), \(u = - \mu\)

$$\begin{aligned} J & = - \mathop \int \limits_{0}^{u} x^{3} \exp \left( { - q^{2} x^{2} } \right)\;{\text{d}}x + 3u\mathop \int \limits_{0}^{u} x^{2} \exp \left( { - q^{2} x^{2} } \right)\;{\text{d}}x \\ \quad - 3u^{2} \mathop \int \limits_{0}^{u} x\exp \left( { - q^{2} x^{2} } \right)\;{\text{d}}x + u^{3} \mathop \int \limits_{0}^{u} \exp \left( { - q^{2} x^{2} } \right)\;{\text{d}}x \\ \end{aligned}$$

We know that

$$\begin{gathered} J_{3} \left( {q,u} \right) = \mathop \int \limits_{0}^{u} x^{3} \exp \left( { - q^{2} x^{2} } \right)\;{\text{d}}x = \frac{1}{{2q^{4} }}\left[ {1 - \left( {1 + q^{2} u^{2} } \right)\exp \left( { - q^{2} u^{2} } \right)} \right] \hfill \\ J_{2} \left( {q,u} \right) = \mathop \int \limits_{0}^{u} x^{2} \exp \left( { - q^{2} x^{2} } \right)\;{\text{d}}x = \frac{1}{{2q^{3} }}\left[ {\frac{{\sqrt \pi }}{2}{\text{erf}}\left( {qu} \right) - \left( {qu} \right)\exp \left( { - q^{2} u^{2} } \right)} \right] \hfill \\ J_{1} \left( {q,u} \right) = \mathop \int \limits_{0}^{u} x \cdot \exp \left( { - q^{2} x^{2} } \right)\;{\text{d}}x = \frac{1}{{2q^{2} }}\left[ {1 - \exp \left( { - q^{2} u^{2} } \right)} \right] \hfill \\ J_{0} \left( {q,u} \right) = \mathop \int \limits_{0}^{u} \exp \left( { - q^{2} x^{2} } \right)\;{\text{d}}x = \frac{{\sqrt \pi }}{{2q}}{\text{erf}}\left( {qu} \right), \hfill \\ \end{gathered}$$

where \({\text{erf}}\left( x \right) = \frac{2}{{\sqrt \pi }}\mathop \int \nolimits_{0}^{x} \exp \left( { - t^{2} } \right)\;{\text{d}}t\) is the error function.

Then, \(J = - J_{3} \left( {q,u} \right) + 3uJ_{2} \left( {q,u} \right) - 3u^{2} J_{1} \left( {q,u} \right) + u^{3} J_{0} \left( {q,u} \right)\).

By replacing q and u by its values, we have the following:

$$\begin{gathered} J_{3} \left( {\sigma ,\mu } \right) = 2\sigma ^{4} - 2\sigma ^{4} \left( {1 + \frac{{\mu ^{2} }}{{2\sigma ^{2} }}} \right)\exp \left( { - \frac{{\mu ^{2} }}{{2\sigma ^{2} }}} \right); \hfill \\ J_{2} \left( {\sigma ,\mu } \right) = \frac{{\sigma ^{3} }}{2}\sqrt {2\pi } {\text{erf}}\left( { - \frac{\mu }{{\sigma \sqrt 2 }}} \right) + \mu \sigma ^{2} ~\exp \left( { - \frac{{\mu ^{2} }}{{2\sigma ^{2} }}} \right) \hfill \\ J_{1} \left( {\sigma ,\mu } \right) = \sigma ^{2} \left[ {1 - {\text{erf}}\left( { - \frac{\mu }{{\sigma \sqrt 2 }}} \right)} \right]; \hfill \\ J_{0} \left( {\sigma ,\mu } \right) = \frac{\sigma }{2}\sqrt {2\pi } {\text{erf}}\left( { - \frac{\mu }{{\sigma \sqrt 2 }}} \right) \hfill \\ \end{gathered}$$
$$\begin{aligned} \bar{v}^{3}& = I + J = 2\sigma ^{4} \left( {1 + \frac{{\mu ^{2} }}{{2\sigma ^{2} }}} \right)\exp \left( { - \frac{{\mu ^{2} }}{{2\sigma ^{2} }}} \right) \hfill \\ &\quad +\, \frac{{3\mu \sigma ^{3} \sqrt {2\pi } }}{2}\left( {1 + \frac{{\mu ^{2} }}{{3\sigma ^{2} }}} \right)\left[ {1 - {\text{erf}}\left( { - \frac{\mu }{{\sigma \sqrt 2 }}} \right)} \right] \hfill \\ {\text{NPD}} &= \frac{1}{2}\rho \frac{1}{{\sigma \sqrt {2\pi } }}\bar{v}^{3} \hfill \\ {\text{NPD}} &= \frac{{\rho \sigma ^{3} }}{{\sqrt {2\pi } }}\left( {1 + \frac{{\mu ^{2} }}{{2\sigma ^{2} }}} \right)\exp \left( { - \frac{{\mu ^{2} }}{{2\sigma ^{2} }}} \right) \hfill \\ & \quad +\, \frac{3}{4}\rho \mu \sigma ^{2} \left( {1 + \frac{{\mu ^{2} }}{{3\sigma ^{2} }}} \right)\left[ {1 - {\text{erf}}\left( { - \frac{\mu }{{\sigma \sqrt 2 }}} \right)} \right] \hfill \\ \end{aligned}$$

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Kenfack-Sadem, C., Tagne, R., Pelap, F.B. et al. Potential of wind energy in Cameroon based on Weibull, normal, and lognormal distribution. Int J Energy Environ Eng (2021). https://doi.org/10.1007/s40095-021-00402-3

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Keywords

  • Wind speed data
  • PDFs
  • Power density (PD)
  • Goodness of fit test