Power curve modelling techniques

Power curve modelling techniques are used to model the relationship between wind speed and wind power production. This is a form of simple regression, since only one predictor is used. Wind speed conversion to wind power through Wind Turbine Power Curve (WTPC) modelling is a key pillar of any wind power prediction model (Marciukaitis et al., 2017). The easiest way to do this is to use theoretical (manufacturer’s) wind power curve. However, in most cases this leads to additional errors due to differences in theoretical and real-life wind power measurement data. Many different mathematical modeling techniques for WTPC are available. Literature classifies these techniques into parametric techniques and non-parametric techniques (Lydia et al., 2014).

Figure 2.12: WTPC modelling techniques (Lydia et al., 2014).

Each turbine has a different power curve depending on model type and environmental factors like orography, site turbulence and complexity of terrain. Therefore, accurately modelling the power curve for power output prediction is essential (Marciukaitis et al., 2017). Figure 2.11 shows an overview of techniques, these are not all techniques that are available in the literature. We highlight some

Page | 21 techniques that show promising results according to the literature. Parametric techniques are mostly used in the physical approach, while non-parametric techniques are often used in statistical approaches. First, we focus on the parametric modeling techniques, after this we discuss the non- parametric techniques.

2.5.1 Parametric techniques

Parametric techniques are based on solving mathematical expressions. These techniques are often used in the statistical approach to estimate the power curve. Some techniques are only able to calculate a part of the power curve, which is shown in Equation 2.3. The actual power output, P(v), can be expressed as given below (Carrillo et al., 2013):

𝑃(𝑣) = { 0, 𝑣 < 𝑣𝑐𝑖, 𝑣 > 𝑣𝑐𝑜 𝑞(𝑣), 𝑣𝑐𝑖 ≤ 𝑣 ≤ 𝑣𝑟 𝑃𝑟, 𝑣𝑟≤ 𝑣 ≤ 𝑣𝑐𝑜 (2.3) where: 𝑣 = wind speed in m/s

𝑣𝑐𝑖 = cut-in wind speed in m/s

𝑣𝑐𝑜 = cut-out wind speed in m/s

𝑣𝑟 = rated wind speed

𝑃𝑟 = rated power

Here, q(v) is the variable region between the cut-in speed and the rated speed at which rated power is reached. This distinction has to be made, since some techniques focus on approximating this part of the power curve instead of the entire curve. The most typical mathematical equations for representing q(v) are the polynomial power curve, exponential power curve and approximate cubic power curve (Carrillo et al., 2013). All of the equations listed in this subsection, except for the approximate cubic power curve, are used for curve fitting, which means the parameters have no physical meaning.

Approximate cubic power curve

The cubic power curve is estimated by assuming the power coefficient (Cp) is equal to the maximum

value of the effective power coefficient (Cp,max) of the turbine type. The term effective means that

electrical and mechanical losses are included in this coefficient. The resulting equation is:

𝑞(𝑣) =1₂𝜌𝐴𝐶𝑝,𝑚𝑎𝑥𝑣3 (2.4)

This equation is similar to Equation 2.2. To be able to calculate the resulting power output, the air density, area of the swept rotor and the maximum power coefficient have to be known. Of course, the entire power curve can also be calculated using this equation, whether or not this impacts the results negatively is not certain. The approximate cubic power curve showed the best results according to Carrillo et al. (2013) and Lydia et al. (2014). However, Thapar et al. (2011) argue that models based on Equation 2.2 are cumbersome and are not suitable for accurately calculating hourly energy production. Polynomial power curve

Polynomial functions can be used to approximate both the non-linear part of the power curve as well as the entire curve. Which part of the curve is estimated depends on the degree of the polynomial, the polynomial function is expressed as follows:

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𝑃(𝑣) = 𝑎0+ 𝑎1𝑣 + 𝑎2𝑣2+ 𝑎3𝑣3+ ⋯ + 𝑎𝑛𝑣𝑛 (2.5)

Here, n is the order of the polynomial and 𝑎𝑛 are the parameters of the polynomial function to be estimated. Among the polynomial functions, the quadratic (n=2) power curve showed the worst results when estimating 𝑞(𝑣) (Carrillo et al., 2013). The ninth-order polynomial showed the most promising results when estimating the entire curve 𝑃(𝑣) (Lydia et al., 2014).

Exponential power curve

Exponential functions are used in literature to estimate the power curve. A lot of adaptations of these kinds of functions are used. Recently, Marciukaitis et al. (2017) used the following function to estimate the entire curve:

𝑃(𝑣) = 𝑃𝑟(1 + ( 𝛽 𝑣) 𝛼 ) −𝑦 , 𝛼, 𝛽, 𝑦 > 0 (2.6)

Here, 𝛽, 𝛼, and 𝑘 are positive parameters which have to be estimated. A lot of different other exponential functions have been used in literature, this function yielded the best results after cross- validation according to Marciukaitis et al. (2017). They claimed that this model outperforms the polynomial and approximate cubic power curve functions.

Logistic power curve

The shape of the power curve can be approximated by using a logistic expression with varying parameters. Lydia et al. (2013) experimented with four and five parameter logistic expressions successfully. The four parameter logistic function is expressed as follows:

𝑃(𝑣) = 𝛼 (1+𝑚𝑒 −𝑣_𝜏 1+𝑛𝑒− 𝑣 𝜏 ) (2.7)

Parameters 𝛼, 𝑚, 𝑛, and 𝜏 have specific ranges giving the function favorable results. The five parameter logistic function is expressed as follows:

𝑃(𝑣) = 𝑑 + ( 𝑎−𝑑 (1+(𝑣 𝑓) 𝑏 ) 𝑔) , 𝑓, 𝑔 > 0 (2.8)

The five parameter logistic function showed the best results of the parametric functions (Lydia et al., 2013). However, this method was not compared to the exponential, polynomial or approximate cubic power curve. It did outperform some non-parametric techniques like neural networks, fuzzy logic and data mining algorithms.

2.5.2 Non-parametric techniques

Several non-parametric techniques are used to find the relationship between the input wind speed data and output power. We highlight the techniques that are most widely used in the literature and show the most promising results. Most of these techniques are used in the statistical approach. These techniques are far more complex than their parametric counterparts, therefore we only give a short description of each.

Artificial neural networks

An Artificial Neural Network (ANN) is an information-processing model simulating the biological nervous system (Lydia et al., 2014). It has the capacity to derive meaning from complicated and imprecise data and extracts patterns and trends that are too complex to be identified by humans. Lydia

Page | 23 et al. (2014) mentioned three ANNs that were widely used, Generalized Mapping Regressor (GMR), feed forward Multi-Layer Perceptron (MLP) and a General Regression Neural Network (GRNN).

Fuzzy methods

Fuzzy logic is a multi-valued logic which deals with approximate reasoning. Lydia et al. (2014), who made a comprehensive review on WTPC modeling techniques, distinguishes three types of fuzzy methods, fuzzy cluster center method, fuzzy c-means clustering and subtractive clustering. Fuzzy cluster center method clusters data using a clustering algorithm, the accuracy of the model increases with the number of clusters. The performance of the fuzzy cluster center method is the best out of the fuzzy methods.

Data mining algorithms

Data mining is the process of analyzing data present in huge databases and extracting valuable information and patterns. For most wind farms, huge volumes of data are available which presents opportunities for the application of data mining algorithms (Lydia et al., 2014). Non-parametric models of a WTPC have been obtained using five data mining algorithms, random forest, Multi-Layer Perceptron (MLP), M5P tree, boosting algorithm and k-Nearest Neighbor (k-NN). The last algorithm mentioned yielded the best results.

2.5.3 Summary of WTPC modeling techniques

Models based on the basic concept of power available in the wind, like the approximate cubic power model do not give accurate results. Models based on the historic wind speed-power data of a wind turbine using curve-fitting techniques perform better. These models include the polynomial, exponential and logistic power curve models. Out of the polynomial functions, the ninth-degree polynomial had the most accurate results. The non-parametric models give accurate results as well; however, these are not desirable because they are complex to implement due to their underlying algorithms. Lydia et al. (2013) used four optimization algorithms for parameter estimation with logistic parametric models. These algorithms included a genetic algorithm (GA), evolutionary programming (EP), particle swarm optimization (PSO) and differential evolution (DE). The five parameter logistic function got the best results when using the DE algorithm. According to Lydia et al. (2013), the five parameter logistic function using the DE algorithm outperforms the non-parametric techniques. It is not clear whether or not the logistic power curve techniques outperform the polynomial or exponential models.