# Reformulation of parameters of the logistic function applied to power curves of wind turbines

### Significance Statement

The power curve of a wind turbine indicates the relationship between the wind speed and the electric power supplied. Therefore, it is widely used when analyzing or studying a wind turbine or a wind farm. The power curve of a wind turbine is the one provided in the manufacturer’s paperwork. This paper will focus on the modeling of this curve although measurements in the wind turbine in a wind farm will eventually reveal conditions that can slightly differ from those stated by the manufacturer. There are circumstances that can affect the operating conditions, and they include, turbulence, gusty winds, wind shear, wake effects, icing and component fatigue in the wind turbine.

Daniel Villanueva and Andrés Feijóo from de Vigo university in Spain proposed a method that would lead to the reformulation of parameters of the logistic function applied to power curves of wind turbines. They presented an alternative procedure to obtain parameters of the 4-parameter logistic function in order to improve the model. Their work is now published in peer-reviewed journal, Electric Power Systems Research.

The models used were based on the 4-parameter logistic function model, where the relationship between wind speed and generated power is a continuous curve. The parameters of the model obtained by using optimization techniques have no technical meaning. In order to know the influence of the wind turbine’s features and behaviors, another model is needed. The reason behind using parameters with some technical meaning is in order to assess the power curve from a theoretical point of view and obtain the weight of each parameter in the power curve and in the power output.

The method adopted in this paper consists of obtaining the parameters of the model from the features of a wind turbine power curve. Each feature investigated imposes a constraint that must be satisfied by the model, and this contributes to the configuration of the end result. Therefore, a deterministic process is proposed to obtain the parameters of the 4-parameter logistic model, which are obtained directly from the features of the power curve.

Three wind turbine power curve models based on a reformulation of the parameters of the logistic curve function were presented. The first one 4P-DP is similar to the 4-parameter logistic function but takes into account a deterministic procedure to obtain its parameters making the model meaningful since it provides information on the wind turbine behavior.

The second model (4P-DS), obtained by the means of simplification, some issues were improved, for instance, approximations to the power curve lower values of wind speed and ease of obtaining probability density function (PDF) expressions. The models consist of a continuous function that simplified the implementation of the curve in a computer program compared to the piecewise models.

The third one (3P-DP) simplifies even more the former expression and provides an easy way to model the power curve with just three parameters.

One result that can be obtained from these models is the expression of the PDF of the output power. This will provide a cumulative behavior of the power and can be used as input data to solve more complex problems such as probabilistic load flow, where the probabilistic behavior of the power is taken into consideration.

In order to check the proposed models, several wind turbines were taken into account. This was in a bid to make detailed comparison of wind turbines of the same rated power from difference manufacturers, and to check models for a wide range of rated powers.

Daniel Villanueva joined the Departamento de Enxeñería Eléctrica of the Universidade de Vigo, Spain, in 2009, obtaining a PhD built on the analysis of the correlation among neighboring wind speeds and its joint impact in the electrical networks. As a consequence, he has co-developed several mathematical/technical tools, as the following: analysis of the wind speed behaviour, simulation of correlated non-Normal series of data, different wind power curve models, expressions for the wind power Probability Density Function, analysis of Probabilistic Load Flow with wind power, simulation of wind speed data for Economic Dispatch assessment, etc.