Analytical Model for the Wind Farm Capacity Factor Based on Local Wind Characteristics

Abstract

An important parameter to quantitatively assess the performance of a wind farm is its capacity factor. This study presents an analytical model to determine the capacity factor of a wind farm using the Weibull probability distribution function according to the local wind speed associated with the characteristics of the selected wind turbine. The model is based on the analytical integral of the wind energy characteristics calculated in relation to the wind speed probability distribution. The model allows determining the best rated speed for the wind turbine considering the local wind characteristic. The study has also introduced an analytical method for the wind turbine selection in order to maximize the plant’s capacity factor. In addition, the model allows determining the turbine operating time in each of the regions of the power curve. Knowing the distribution of operating time is useful for formulating failure monitoring strategies and operation and maintenance (O&M) planning. The proposed model has been validated by experimental results obtained in a wind farm in Southern Brazil.

Appendices

Appendix

To compute the capacity factor, we calculated the integral of the product between the power associated with the wind speed $v\left( {P\left( v \right)} \right)$ and its probability of occurrence $f\left( {v,c,k} \right)$, dividing the result by the rated power of the wind turbine.

$$ {\text{Fc}} = \mathop \int \nolimits_{0}^{\infty } f\left( {v,c,k} \right) \cdot P\left( v \right){\text{d}}v $$

This integral can be divided into four parts:

$$ {\text{Fc}} = \mathop \int \nolimits_{0}^{{V_{{\text{cut - in}}} }} 0 \cdot {\text{d}}v + \mathop \int \nolimits_{{V_{{\text{cut - in}}} }}^{{V{\text{nom}}}} f\left( {v,c,k} \right) \cdot P\left( v \right){\text{d}}v + \mathop \int \nolimits_{{V{\text{nom}}}}^{{V_{{\text{cut - out}}} }} f\left( {v,c,k} \right) \cdot P_{{{\text{nom}}}} \cdot {\text{d}}v + \mathop \int \nolimits_{{V_{{\text{cut - out}}} }}^{\infty } 0 \cdot {\text{d}}v $$

with $f$ being the Weilbull's probability density function, the integral that determines Fc becomes:

$$ {\text{Fc}} = \mathop \int \nolimits_{{V_{{\text{cut - in}}} }}^{{V{\text{nom}}}} f\left( {v,c,k} \right) \cdot P\left( v \right){\text{d}}v + \mathop \int \nolimits_{{V{\text{nom}}}}^{{V_{{\text{cut - out}}} }} f\left( {v,c,k} \right) \cdot P_{{{\text{nom}}}} \cdot {\text{d}}v $$

$$ {\text{Fc}} = P_{{{\text{II}}}} + P_{{{\text{III}}}} $$

designate

$$ P_{{{\text{II}}}} = \mathop \int \nolimits_{{V_{{\text{cut - in}}} }}^{{V{\text{nom}}}} f\left( {v,c,k} \right) \cdot P\left( v \right){\text{d}}v $$

$$ P_{{{\text{III}}}} = \mathop \int \nolimits_{{V{\text{nom}}}}^{{V_{{\text{cut - out}}} }} f\left( {v,c,k} \right) \cdot P_{{{\text{nom}}}} \cdot {\text{d}}v $$

Therefore,

$$ P_{{{\text{II}}}} = \mathop \int \nolimits_{{V_{{\text{cut - in}}} }}^{{V_{{{\text{nom}}}} }} \left[ {\frac{k}{c} \cdot \left( \frac{v}{c} \right)^{k - 1} \cdot e^{{\left( { - \frac{v}{c}} \right)^{k} }} } \right]\left[ {\left( {\frac{v}{{V_{{{\text{nom}}}} }}} \right)^{3} \cdot {\text{Pot}}_{{{\text{nom}}}} } \right]{\text{d}}v $$

$$ P_{{{\text{III}}}} = \mathop \int \nolimits_{{V_{{{\text{nom}}}} }}^{{V_{{\text{cut - out}}} }} \left[ {\frac{k}{c} \cdot \left( \frac{v}{c} \right)^{k - 1} \cdot e^{{\left( { - \frac{v}{c}} \right)^{k} }} } \right]\left[ {{\text{Pot}}_{{{\text{nom}}}} } \right]{\text{d}}v $$

Defining pu wind values and pu power:

$$ V_{{{\text{cout}}}} = \frac{{V_{{\text{cut - out}}} }}{{V_{{{\text{nom}}}} }} $$

$$ V_{{{\text{cin}}}} = \frac{{V_{{\text{cut - in}}} }}{{V_{{{\text{nom}}}} }} $$

$$ V = \frac{v}{{V_{{{\text{nom}}}} }} $$

$$ C = \frac{c}{{V_{{{\text{nom}}}} }} $$

$$ \overline{{P_{{{\text{II}}}} }} = \mathop \int \nolimits_{{V_{{{\text{cin}}}} }}^{1} \left[ {\frac{k}{C} \cdot \left( \frac{V}{C} \right)^{k - 1} \cdot e^{{ - \left( \frac{V}{C} \right)^{k} }} } \right]\left[ {V^{3} } \right]{\text{d}}V $$

$$ \overline{{P_{{{\text{II}}}} }} = \mathop \int \nolimits_{{V_{{{\text{cin}}}} }}^{1} \left[ {k \cdot C^{2} \left( \frac{V}{C} \right)^{k + 2} \cdot e^{{ - \left( \frac{V}{C} \right)^{k} }} } \right]{\text{d}}V $$

$$ \overline{{P_{{{\text{III}}}} }} = \mathop \int \nolimits_{1}^{{V_{{{\text{cout}}}} }} \left[ {\frac{k}{C} \cdot \left( \frac{V}{C} \right)^{k - 1} \cdot e^{{ - \left( \frac{V}{C} \right)^{k} }} } \right]{\text{d}}V $$

Integrating regions II and III separately, we have:

REGION II

$$ \overline{{P_{{{\text{II}}}} }} = k \cdot C^{2} \mathop \int \nolimits_{{V_{{{\text{cin}}}} }}^{1} \left[ {\left( \frac{V}{C} \right)^{k + 2} \cdot e^{{ - \left( \frac{V}{C} \right)^{k} }} } \right]{\text{d}}V $$

Changing the variables:

$$ \frac{V}{C} = x $$

$$ \overline{{P_{{{\text{II}}}} }} = k \cdot C^{3} \mathop \int \nolimits_{{V_{{\frac{{{\text{cin}}}}{C}}} }}^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 C}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$C$}}}} \left[ {\left( x \right)^{k + 2} \cdot e^{{ - \left( x \right)^{k} }} } \right]{\text{d}}x $$

the primitive, just for the integral, is

$$ - \frac{1}{{k^{2} }}\left[ {k \cdot x^{3} \cdot e^{{ - x^{k} }} + 3\Gamma \left( {\frac{3}{k},x^{k} } \right)} \right] $$

Since the gamma function is incomplete

$$ \Gamma \left( {a,z} \right) = \mathop \int \nolimits_{z}^{\infty } t^{a - 1} \cdot e^{ - t} \cdot {\text{d}}t $$

Therefore,

$$ \overline{{P_{{{\text{II}}}} }} = \frac{{C^{3} }}{k}\left\{ {\begin{array}{*{20}c} {\left[ {\begin{array}{*{20}c} {k \cdot \left( {V_{{{\text{cin}}}} } \right)^{3} \cdot e^{{ - \left( {V_{{{\text{cin}}}} } \right)^{k} }} + } \\ {3\Gamma \left( {\frac{3}{k},\left( {V_{{{\text{cin}}}} } \right)^{k} } \right)} \\ \end{array} } \right]} \\ { - \left[ {\begin{array}{*{20}c} {k \cdot \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 C}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$C$}}} \right)^{3} \cdot e^{{ - \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 C}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$C$}}} \right)^{k} }} + } \\ {3\Gamma \left( {\frac{3}{k},\left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 C}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$C$}}} \right)^{k} } \right)} \\ \end{array} } \right]} \\ \end{array} } \right\} $$

REGION III

In region III, the integral is simpler, as it is only the integral of the Weibull function.

$$ \overline{{P_{{{\text{III}}}} }} = \mathop \int \nolimits_{1}^{{V_{{{\text{cout}}}} }} \left[ {\frac{k}{C} \cdot \left( \frac{V}{C} \right)^{k - 1} \cdot e^{{ - \left( \frac{V}{C} \right)^{k} }} } \right]{\text{d}}V $$

Therefore,

$$ \overline{{P_{{{\text{III}}}} }} = \left[ {1 - e^{{ - \left( {{\raise0.7ex\hbox{$V$} \!\mathord{\left/ {\vphantom {V C}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$C$}}} \right)^{k} }} } \right]_{V = 1}^{{V = V_{{{\text{cout}}}} }} = e^{{ - \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 C}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$C$}}} \right)^{k} }} - e^{{ - \left( {{\raise0.7ex\hbox{${V_{{{\text{cout}}}} }$} \!\mathord{\left/ {\vphantom {{V_{{{\text{cout}}}} } C}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$C$}}} \right)^{k} }} $$