Review ArticleA Critical Review on Wind Turbine Power CurveModelling Techniques and Their Applications in Wind BasedEnergy Systems

Vaishali Sohoni, S. C. Gupta, and R. K. Nema

Department of Electrical Engineering, Maulana Azad National Institute of Technology, Bhopal 462051, India

Correspondence should be addressed to Vaishali Sohoni; vaishalisohonibpl@gmail.com

Received 27 March 2016; Revised 8 June 2016; Accepted 12 June 2016

Academic Editor: Kamal Aly

Copyright © 2016 Vaishali Sohoni et al.This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Power curve of a wind turbine depicts the relationship between output power and hub height wind speed and is an importantcharacteristic of the turbine. Power curve aids in energy assessment, warranty formulations, and performance monitoring of theturbines. With the growth of wind industry, turbines are being installed in diverse climatic conditions, onshore and offshore,and in complex terrains causing significant departure of these curves from the warranted values. Accurate models of powercurves can play an important role in improving the performance of wind energy based systems. This paper presents a detailedreview of different approaches for modelling of the wind turbine power curve. The methodology of modelling depends uponthe purpose of modelling, availability of data, and the desired accuracy. The objectives of modelling, various issues involvedtherein, and the standard procedure for power performance measurement with its limitations have therefore been discussed here.Modelling methods described here use data frommanufacturers’ specifications and actual data from the wind farms. Classificationof modelling methods, various modelling techniques available in the literature, model evaluation criteria, and application of softcomputing methods for modelling are then reviewed in detail. The drawbacks of the existing methods and future scope of researchare also identified.

1. Introduction

Wind energy has emerged as a promising alternative sourcefor overcoming the energy crisis in the world. Wind powerbased energy is one of the most rapidly growing areas amongthe renewable energy sources and will continue to do sobecause of the growing concern about sustainability andemission reduction requirements. The uncertain nature ofwind and high penetration of wind energy in power systemsare a big challenge to the reliability and stability of thesesystems. To make wind energy a reliable source, accuratemodels for predicting the power output and performancemonitoring of wind turbines are needed. The theoreticalpower captured (𝑃) by a wind turbine is given by [1]

𝑃 =1

2𝜌𝐴𝑤𝐶𝑃(𝜆, 𝛽) 𝜐

3

. (1)

The power production of a wind turbine (WT) thusdepends upon many parameters such as wind speed, wind

direction, air density (a function of temperature, pressure,and humidity) and turbine parameters [2]. Much complexityis involved in considering the effects of all the influencingparameters properly. It is therefore difficult to evaluate theoutput power using the theoretical equation given above.Power curve of a wind turbine, which gives the output powerof turbine at a specific wind speed, provides a convenientway to model the performance of wind turbines. A typicalpower curve for a pitch regulated wind turbine is shown inFigure 1. In the first region when the wind speed is less thana threshold minimum, known as the cut-in speed, the poweroutput is zero. In the second region between the cut-in andthe rated speed, there is a rapid growth of power produced. Inthe third region, a constant output (rated) is produced untilthe cut-off speed is attained. Beyond this speed (region 4) theturbine is taken out of operation to protect its componentsfrom high winds; hence it produces zero power in thisregion.

Hindawi Publishing CorporationJournal of EnergyVolume 2016, Article ID 8519785, 18 pageshttp://dx.doi.org/10.1155/2016/8519785

2 Journal of Energy

Region1

Region 2

Region3

Region4

Wind speed

Pow

erou

tput

Prated

0 �cut in �rated �cut off

Figure 1: Typical power curve of a pitch regulated wind turbine.

The power curve of a WT indicates its performance.Accurate models of power curves are important tools forforecasting of power and online monitoring of the turbines.A number of methods have been proposed in various worksto model the wind turbine power curve. These methodswhich use data frommanufacturers’ specifications and actualdata from the wind farms have been utilized by manyresearchers in various wind power applications [3, 4]. Theliterature reviewed reveals that appropriate selection of powercurve models can help in improved performance of windenergy based systems. This paper presents current statusof research and future directions of different wind turbinepower curve modelling approaches. The need of modelling,modelling methodology, classification of models, and meth-ods of evaluation have been discussed. The impact of variousparameters on these curves, the standard procedure forpower performance measurement of wind turbines, and theneed for developing site specific curves is also discussed.Various models proposed and used in various studies havebeen compared critically and finally inferences are drawn.

2. Need of Power Curve Modelling

The power curve reflects the power response of a WT tovariouswind speeds. Accuratemodels of the curves are usefulin a number of wind power applications. The objectives ofmodelling the wind turbine power curve have been discussedhere.

2.1. Wind Power Assessment and Forecasting. TheWT powercurve can be used for wind power assessment.Wind resourceassessment of a region in terms of wind speed, wind powerdensity, and wind energy potential is done to identify areassuitable for wind power development [3]. In this process,estimation of energy is done by using the available wind dataand wind turbine power curve. Predicting the output powerof the turbine at a candidate site is also required in sizing andcost optimization studies during the design stage of a windenergy based system. The accuracy in power prediction isimportant as an overestimation can result in poor reliabilityand an underestimation can lead to oversizing of the windenergy conversion system.Wind turbine operators who tradeenergy directly to the electricity market also need to forecastthe power output of their turbines accurately, so that they willbe able to deliver the traded amount of power [2].

Power curves are supplied by the manufacturers in atabular or graphical form. However, a generic equationwhich represents this curve accurately is required in var-ious problems of wind power systems. Derivation of anappropriate function to describe the actual shape of thecurve is a very important task. However, the manufacturer’scurves are created under standard conditions therefore theymay not represent the realistic conditions of the site underconsideration. The turbine performance at the wind farmsis also not ideal due to wear and tear and aging of turbines.Another method tomodel the power curves is to derive themusing the actual data of wind speed and powermeasured fromthe turbines [4]. The data of wind turbines collected by theSCADA (supervisory control and data acquisition) systemcan be utilized for this purpose.This method can incorporatethe actual conditions at the wind farms, thus providing betteraccuracy in power prediction.

2.2. Capacity Factor Estimation. The capacity factor of a WTis defined as the ratio of the average power output to therated output power of the generator and is an indicator ofits efficiency [5]. It is used to estimate the average energyproduction of aWT required for the sizing and cost optimiza-tion studies, optimum turbine-site matching, and rankingof potential sites [5, 6]. The wind turbine power curvemodels are used to estimate the capacity factor of a WT. Acomparative analysis of four power curvemodellingmethodsin estimation of capacity factor of wind turbine generator ispresented in [7].

2.3. Selection of Turbines. The power curve can be used tomake generic comparison between models and can aid in thechoice of turbine from the available options. The selection ofthe turbine characteristics whichmatch with the wind regimeof the site helps in optimizing the efficiency of wind energysystem [8].

2.4. Online Monitoring of Power Curves. Power curves can beused for monitoring the performance of turbines. For this,a benchmark curve which represents the performance of anormally operating turbine is required. This reference curvecan be extracted from measured power output and windspeed data of wind turbines. The actual curve of the turbineto bemonitored can be comparedwith this benchmark curve.The deviations of the actual values from the expected outputcan indicate underperformance or faults [1]. The wind poweroutput of a turbine can be affected by underperformance orvarious faults/anomalies of the turbine such as blade faultsand yaw and pitch system faults [4, 9]. Different types of faultsaffect the turbine system differently and will cause the powercurve to depart from the expected value in a different way.Tools which can characterize and quantify these departurescan aid in early identification of faults. Statistical analysis ofthe outlier data can give indications of the specific reason ofanomaly.Wind turbine conditionmonitoring by use of powercurve copula modelling is suggested in [10, 11] and is a topicof further research. Early recognition of the emerging faultsand timely repair andmaintenance of the equipment can helpin improving the performance of wind turbines.

Journal of Energy 3

3. Modelling Issues

A number of aspects require consideration while modellingthe power curves of wind turbines. The selection of modeland methodology adopted depends upon the purpose ofmodelling, available data, impact of various parameters onthese curves, and other related issues.

The following important issues should be taken intoconsideration while modelling of the power curve.

3.1. Difference in Models. The power curves vary with dif-ferent manufacturers and models. Therefore the model usedto describe them should also be different [12]. Also, thereis difference between pitch regulated and stall regulatedturbines. Pitch regulated turbines maintain constant outputfrom the rated to cut-off speed, whereas the stall regulatedturbines have a decreased power output above the rated windspeeds (stall region).

3.2. Cut-In and Cut-Off Behaviour. The turbine behaviournear cut-in and cut-off wind speeds can be difficult to model[8]. These limits are different for different turbine models.When the power curve is derived using the measured data,some nonzero and negative values of power outputs belowcut-in speed can be obtained. The cut-off hysteresis whichoccurs during the period between shut down and restart ofturbine affects the productivity of turbine [13]. Hysteresiseffects can bemore significant with certain wind patterns andterrains such as unsteady and gusty winds requiring frequentstarting and shutting down, resulting in considerable loss ofenergy production. Power curve correction which takes intoaccount this behaviour of turbine at cut-off can reduce thepower prediction errors.

3.3. Single versus Group of Turbines. The manufacturers’curves are suitable for predicting the power output of asingle turbine of a specific type. In a big wind farm anumber of turbines are spread over a wide area. Wind energyproduction involves uncertainties due to stochastic nature ofthe wind and variation of the power curve [14]. The speedand direction of wind encountered by the turbines of a windfarm may not be the same due to variation of wind. Hence,in a wind farm, power produced by turbines with identicalspecifications can also differ, even if the wind speed is thesame. The shadowing effect of turbines causes this differenceas the turbines which operate in wake of other turbinesmay get reduced wind speeds [15]. This difference can alsohappen due to factors such as wear and tear, aging, anddirt or ice deposition on blades. With the growth of windenergy projects it has become essential to develop methodsto monitor the performance of not only a single turbine butalso the wind farm as a whole.Therefore building appropriatemodels to obtain the relationship between wind speed andoutput power when a group of turbines are deployed on awind farm is required.

3.4. Influencing Factors. A number of factors can cause thepower curve to deviate from the theoretical value [1, 2]. The

important influencing factors are given here and need dueattention during modelling.

(i) Wind Conditions at the Site. Wind is highly stochastic innature. The wind speed and direction change continuously.Thewind at a particular site is affected byweather phenomenaand topology of the site. The turbulence of wind at a givenlocation affects the power production [16]. Obstacles liketrees, buildings, and other high structures influence the wind.

(ii) Air Density.The pressure, temperature, and humidity ofsite affect the air density [17], hence affecting the powerproduced. Effect of varying air density has been consideredfor developing site specific curves [18]. It is shown in [2]that temperature has the highest influence on air density andconsidering its effect along with the wind direction resultedin improved performance of models.

(iii) Extrapolation of Wind Speed.The wind speed changeswith height.Thiswind shear effect is affected by the roughnessof terrain. The power curve uses the wind speed measured atthe hub height of turbine, but this height varies with differentmodels and manufacturers, and it is not always possible tomeasure the wind speed at this height. A number of methodshave been used in the literature to express the variation ofwind speed with height [12]. Also the wind speed measuredat the masts is different from the speed at the turbine locationand sometimes when the wind speed values at the particularsite are not available the wind speed measurements from anearby location are used to determine the wind profile ofthis site. The accuracy of conversion of the measured windspeed to wind speed at hub height and at the turbine locationdepends on factors such as the vertical wind profile at the site,position of masts relative to the turbine, and themethod usedfor extrapolation.

(iv) Turbine Condition.The power curve is affected by thecondition of turbine and associated equipment. Aging andwear and tear of turbine, anomalies and faults, blade condi-tion, yaw and pitch misalignments, controller settings, andso forth cause the power curve to depart from actual values[1, 11].

3.5. IEC 61400-12-1 Standard [19]. IEC 61400-12-1, the com-monly adopted international standard for power perfor-mance measurement, is found to be of relevance. The pro-cedure for measuring the power performance characteristicsof single wind turbines is specified in this standard. It isthe most accepted standard for power curve measurementof single wind turbines. The standard describes the measure-ment methodology for the measured power curve which isdetermined by simultaneousmeasurement of wind speed andpower output at the test site. A previous site calibration isrequired for certain terrain conditions. The annual energyproduction is calculated by applying the measured powercurve to reference wind speed frequency distributions sup-plemented by sources of uncertainty and their effects. Thestandard prescribes derivation of power curve using the hubheight wind speed measured with a cup anemometer in the

4 Journal of Energy

suitable measurement sector, but if the wind speed has alarge variation over the rotor swept area then there can be asignificant difference between the hub height wind speed andwind speed averaged over the whole rotor swept area. Themeasurement methods and accuracy of measuring instru-ments can cause variance in measurements and can lead tolarge prediction errors. The impact of other measurementoptions such as consideration of rotor equivalent wind speedin which speed is measured at heights over the full rotorplane with the use of remote sensing technology (LIDAR andSONAR) and nacelle based anemometry is a topic of furtherresearch [20].

The IEC standard uses ten-minute averaged data groupedinto wind speed intervals of 0.5m/s (method of bins). This10-minute averaging of data introduces systematic averagingerrors and short wind fluctuations are killed off. Wind at aspecific site can be affected by a number of factors such astopology of the site and obstacles and weather phenomena.Although the IEC power curve considers the wind conditionof the current site it may not always be appropriate to applyto the wind conditions of other sites. Research efforts aretherefore required to develop site specific power curves.These curves can incorporate the wind conditions of theparticular site, thus giving better results [18, 19].

Appropriate selection of modelling method is an impor-tant requirement for during planning and operation stage ofwind based system and helps in improving the performanceof the system. The methods which consider only wind speedas input may not take into account the variance caused byvarious influencing parameters. Methods which consider theinfluence of these parameters on the power curve can resultin more accurate models. Wind at a specific site can beaffected by a number of factors such as topology of the siteand obstacles and weather phenomena. It is shown in [18]that the use of developed site power curves, which usedthe knowledge of site and turbine parameters for modelling,resulted in more accurate energy assessment than the turbinepower curve.The issues discussed above if addressed properlycan result in efficient models of power curves.

4. Wind Speed Modelling

Wind power generated is highly correlated with the windspeed distribution across the region where the wind farmis situated and depends upon the type of WT deployed inthe wind farm. The accuracy in prediction of wind energycan be achieved by modelling the wind speed and powersimultaneously.The wind speed at a site varies randomly andits variation in a certain region over a period of time canbe represented by different probability distribution functions(PDF). Selection of appropriate PDF to describe the actualwind speed distribution of the site is crucial for accuracy inpower prediction.

Themost commonly used and accepted distribution is thetwo-parameter Weibull distribution [5, 21]. It is a versatilePDF, is simple to use, and is found to be accurate for most ofthe wind regimes encountered in nature. However, Weibulldistribution is not suitable for certain wind regimes, forexample, those having high frequencies of null winds, and

0 5 10 15 20 25 30 350

5

10

15

20

25

30

Wind speed (m/s)

Out

put p

ower

(MW

)

Figure 2: Power curve using actual data for a group of wind turbinesat a wind farm (NREL) [66].

for short time horizons [1, 22]. The Weibull PDF is givenby

𝑓 (V) =𝑘

𝑐(V𝑐)

𝑘−1

exp(−V𝑐)

𝑘

. (2)

Another widely used distribution is Rayleigh PDF [18]in which the shape parameter of (2) is taken as 𝑘 = 2.It is a simple PDF and can describe the wind regime withsufficient accuracy when little detail is available about thewind characteristics of a site.Thewind speed distribution hasalso been described in the literature using several other PDFswhich include lognormal, beta, and gamma distributions[23]. A detailed review of different PDFs for wind speedmodelling and techniques for estimation of their parametersis given in [22]. Appropriate PDF and parameter estimationtechnique should be selected for modelling the wind speedfor a particular site.

5. Power Curve Models Classification

The power curve modelling methods can be classifiedinto discrete, deterministic/probabilistic, parametric/nonpara-metric, and stochasticmethods or they can be classified on thebasis of data used for modelling.

5.1. Discrete Models. In this method as described in IEC61400-12 all the wind speeds are discretized into 0.5m/sbins [19]. The power output for each bin is then modelled.This is a simple method as it does not require mathematicalfunctions for describing the curve. Also it takes into accountthe nonlinear wind speed-power output relation. However alarge number of data are required in this method to developa reliable model.

5.2. Deterministic and Probabilistic Models. A deterministicpower curve model assumes a fixed relation between theoutput power and wind speed. But when a fleet of windturbines are deployed on a wind farm, turbines of the sametypemay produce different amount of power even if the windspeed is the same (Figure 2). A probabilistic power curvemodel incorporates these power variations to characterize therelationship between wind speed and actual output powers.Most of themodels available in the literature are of determin-istic nature and are constructed by using the manufacturers’

Journal of Energy 5

power curve data. A probabilistic model proposed in [14]characterizes the dynamics of output power by a normaldistribution with varying mean and constant standarddeviation. The method given in the paper accommodatesthe uncertainty of output power. The probabilistic nature ofwind power output can also be modelled by deriving curvesusing actual data of power output and wind speed of turbinesdeployed in a wind farm. This method requires a largenumber of historical data but results in accurate models[4, 24].

5.3. Parametric and Nonparametric Models. A parametricmodel defines the relationship between input and outputby a set of mathematical equations with a finite numberof parameters. In a nonparametric model, no assumptionis made about the functional form of the phenomenonunder observation. Parametric models of WT power curvecan be built by utilizing a set of mathematical expressionshaving a fixed number of parameters, which are usuallycollected together to form a single parameter vector 𝜃 =(𝜃1, 𝜃2, 𝜃3, . . . , 𝜃

𝑛). Nonparametric models are used when it

is difficult to define the underlying theory upon which theparametric model can be constructed [24].

5.4. Models Based on Presumed Shape, Curve Fitting, andActual Data. The models of power curves can be classifiedaccording to the data being used for modelling. Models ofpower curve based on presumed shape of curve utilize onlythe cut-in, cut-off, and rated speeds and the rated powerof the selected turbine for calculating the parameters ofexpressions used in the model [12, 25, 26]. These ratings areavailable from the specifications of the turbines. When themanufacturer’s power curve data is available, models can bedeveloped by fitting one or more appropriate expressionsto the actual curve. The parameters of the expression beingfitted to the actual curve are generally calculated by usingthe least squares method [4].Themodels derived from actualdata of wind farm need the actual wind speed and poweroutput data from an operational wind farm. If the effect ofthe influencing parameters is also included in the model,then the data of the included parameters is also required.This data can be obtained from the wind farm’s SCADA sys-tem.

5.5. Stochastic Models. The stochastic method consists ofcharacterizing the power performance of wind turbine byevaluating dynamic response against the fluctuating windspeed inputs [27, 28].The dynamic power output is separatedinto a deterministic stochastic part in this model. In [29] theMarkov chain theory is used to describe the power outputof WT. The resulting model is independent of turbulenceintensity; however, the effect of other influencing parametersis not taken into account in this method.

6. Power Curve Modelling Approaches

Various approaches have been used in the literature formodelling of WT power curve. These methods, their merits,limitations, and application areas are discussed here.

6.1. Parametric Models. The power delivered by a WT can beexpressed as

𝑃 =

{{{{

{{{{

{

0 V < V𝑐, V > V

𝑓

𝑞 (V) V𝑐< V < V

𝑟

𝑃𝑟

V𝑟≤ V ≤ V

𝑓.

(3)

The relationship between power output and wind speedof a WT between cut-in and rated speed is nonlinear (region2 of Figure 1). The relation 𝑞(V) can be approximated byvarious functions using polynomial and other than polyno-mial expressions. The pitch regulated turbines maintain aconstant power output in region 3 of Figure 1, whereas thestall regulated turbines have decreased power output in thisregion; thus the power in this region for a stall regulatedturbine should not be modelled as a constant. The governingequations for different approximations of power curve aregiven in Table 1.

6.1.1. Polynomial Function Approximation. The nonlinearwind speed-power relationship, 𝑞(V), can be approximatedby various polynomial expressions. Different models usinglinear, quadratic, cubic, and higher powers of speed or theircombinations have been used in the literature.

(i) The most simplified model based on a linear curve,which describes region 2 of power curve by a straightline, is used in many applications [12, 39–42].

(ii) A quadratic model represents the nonlinear portionof the curve by an equation of degree 2. 𝑞(V) hasbeen approximated by a quadratic equation in [25] todescribe the relation between output power and windspeed of a WT. A binomial expression discussed in[30] has been adopted by many researchers [43, 44]to determine output power of wind turbines. Regions2 and 3 of Figure 1 for a stall regulated WT can bedescribed by using two different binomial expressionsas in [45].

(iii) Model based on cubic law approximates region 2 ofpower curve by cubic law. A model which describesthe nonlinear power-wind speed relationship by acubic law is discussed in [46]. This model is usedfor power output calculations in [26, 31, 47]. A cubicexpression for region 2 and a linear expression forregion 3 are chosen to describe the power curve in[48].

The models given above use the WT specifications ofrated power and cut-in, cut-off, and rated wind speed onlyto determine the equations for power curve.

(i) A methodology based on Weibull’s parameter isproposed in [6]. This model based on Weibull shapeparameter is used by many researchers [49, 50] forcalculating the power output of WT.

(ii) A linearized segmented model discussed in [24, 51]carries out a piecewise linear approximation of the

6 Journal of Energy

Table 1: Expressions of parametric models.

Model Expressions of 𝑃 and 𝑞 Parameters

Linear [12] 𝑞 (V) = 𝑃𝑟

(V − V𝑐)

(V𝑟− V𝑐)

—

Quadratic [25] 𝑞 (V) = 𝑃𝑟(V − V𝑐

V𝑟− V𝑐

)

2

—

Binomial [30] 𝑞 (V) = (𝑎 + 𝑏V + 𝑐V2) 𝑃𝑟

𝑎 =1

(V𝑐− V𝑟)2[V𝑐(V𝑐+ V𝑟) − 4V

𝑐V𝑟

(V𝑐− V𝑟)2

2V𝑟

]

𝑏 =1

(V𝑐− V𝑟)2[4 (V𝑐+ V𝑟)(V𝑐− V𝑟)3

2V𝑟

− 3 (V𝑐+ V𝑟)]

𝑐 =1

(V𝑐− V𝑟)2[2 − 4

(V𝑛+ V𝑟)3

2V𝑟

]

Cubic [31] 𝑞 (V) = 𝑎V3 − 𝑏𝑃𝑟

𝑎 =𝑃𝑟

(V𝑟

3 − V𝑐

3)

𝑏 =V3𝑟

(V𝑟

3 − V𝑐

3)

Weibull based [6] 𝑞 (V) = 𝑎 + 𝑏V𝑘𝑎 =

𝑃𝑟V𝑐

𝑘

(V𝑐

𝑘 − V𝑟

𝑘)

𝑏 =𝑃𝑟

(V𝑐

𝑘 − V𝑟

𝑘)

Double exponential [32] 𝑃 = exp (−𝜏1exp (−𝜐𝜏

2)) 𝜏

1and 𝜏

2to be estimated

4PL [1, 33] 𝑃 = 𝑎(1 + 𝑎𝑚𝑒𝜐/𝜏

1 + 𝑎𝑚𝑒𝜐/𝜏)

𝜃 = (𝑎,𝑚, 𝑛, 𝜏)

(1) From manufacturers’ curve [33]𝑎 = 𝑃

𝑟

𝑛 = 𝑒2𝑠V𝑖𝑝/(𝑃𝑟−𝑃𝑖𝑝)

𝑚 = 𝑛(

2𝑃𝑖𝑝

𝑃𝑟

− 1)

𝜏 =

(𝑃𝑟− 𝑃𝑖𝑝)

2𝑠

(2) Parameters obtained by evolutionary techniques forSCADA data in [1]

4PL [34, 35] 𝑃 = 𝑓 (V, 𝜃) = 𝐷 + (𝐴 − 𝐷)1 + (V/𝐶)𝐵

𝜃 = (𝐴, 𝐵, 𝐶,𝐷)

𝐴 = minimum asymptote𝐵 = Hill slope𝐶 = inflection point (point of curve where the curvaturechanges direction)𝐷 = maximum asymptote(parameters obtained by evolutionary techniques)

5PL [24, 35, 36] 𝑃 = 𝑓 (V, 𝜃) = 𝐷 + (𝐴 − 𝐷)

(1 + (V/𝐶)𝐵)𝐺

𝜃 = (𝐴, 𝐵, 𝐶,𝐷, 𝐺)

𝐴 = minimum asymptote𝐵 = Hill slope𝐶 = inflection point (point of curve where the curvaturechanges direction)𝐷 = maximum asymptote𝐺 = asymmetry factor(parameters obtained by evolutionary techniques)

curve by using the equation of a straight line. Theresulting curve follows the actual curve more accu-rately.

(iii) The power curve of wind turbine has also been mod-elled by other than polynomial functions. The powercurve that is modelled in [52] uses a exponentialequation whereas a double exponential equation isused in [32].

(iv) A polynomial regression parametric model is devel-oped from real data as a benchmark method in [53].Three nonparametric methods are also proposed inthis study.

6.1.2. Approximations Considering Inflection Point on theCurve. All the polynomial expressions given above do nottake into account the inflection point on the power curves.

Journal of Energy 7

In most real power curves there is an inflection point onthe curve at which its curvature changes sign. Models whichconsider this inflection point can describe the actual shapeof the curve more accurately than the above models. Anew formula for power curve interpolation which considersinflection point on the curve is proposed in [54]. A doubleexponential model is proposed in [32] to fit the data in twoinflection zones using a single equation. Functions basedon four- and five-parameter logistic approximations alsoconsider this inflection point on the curve and are promisingapproaches for modelling of power curve.

(i) The shape of awind power curve can be approximatedby a four-parameter logistics (4PL) function [1, 34]. Aprocedure to obtain the parameters of the 4PL func-tionmodelmodelled frommanufactures power curvedata has been proposed in [33]. The four parametersof the function are obtained directly from the powercurve instead of using an optimization process in thiswork. The paper also proposes manufacturer’s powercurve approximation using a three-parameter model.The power curve is derived from the SCADA data ofwind farms using a 4PL approximation in [1, 4, 24]. Itis shown in [1] that this model can be used for onlinemonitoring of power curves. Another form of four PLexpression is used for extracting power curve fromactual wind speed and power curve data of a windfarm in [35]. The method is applied for wind energyestimation of the selected wind farm site. Literaturereviewed reveals that the four-PLmodel produces lesserrors in representing the power curve than themeth-ods based on polynomial approximation. However a4PL curve is symmetric about the inflection pointwhereas the power curves are asymmetric. Modelswhich can incorporate this asymmetry can thereforeproduce even better results. Cubic splines can be usedfor asymmetric data. A cubic spline is the smoothestcurve that passes through the exact data points. Asthe power curves are quite smooth their asymmetrycan be approximated by a cubic spline interpolationtechnique [55–57], but the disadvantage with a splinefit is that it does not represent the random variationof data.

(ii) A five-parameter logistic (5PL) approximationincludes a fifth parameter (𝐺 in Table 1) to control thedegree of asymmetry [58].Thismethod canmodel theasymmetry effectively and can be used for modellingthe WT power curve [36]. A 5PL model howeverhas the possibility of becoming ill-conditioned; thusevaluation of parameter vector becomes difficult. A5PL model derived from the SCADA data of windfarm is applied for energy estimation of the farm in[35] and it is shown that it produces less error in theestimated energy compared to the 4PL model.

6.1.3. Model Based on Curve Fitting of Manufacturer’s Curve.The power curve models obtained by means of curve fittingof the manufacturer’s curve are used in several applications.The characteristic equation of wind generator is fitted with

three binomial expressions for 𝑞(V) in [59] to get the accuracyin fitting. A ninth-order polynomial for power curve fittinghas been used in [60] and it has been found that it givesaccurate correlation with the real data, producing exclusivelypositive values for the generated power between cut-in to cut-off range. High-order polynomials may produce better fittingresults for a particular set of data; however they may notrepresent the variance of data and should be used carefully.Thepower curvemodels by curve fitting of themanufacturer’scurve are analyzed in [55–57].

6.2. Parameter Estimation. The parametric models of WTpower curve express the shape of the curve by a set of mathe-matical equations. Determination of the coefficients of theseequations requires fitting the data to the selected model. Thetechniques and algorithms used in various works for param-eter estimation of power curve models are discussed here.

6.2.1. Techniques for Parameter Estimation

(i) Least SquaresMethod.The least squaresmethodminimizesthe summed square of residuals to obtain the parametersof the model and is the most commonly used and acceptedmethod [4, 24].

(ii) Maximum Likelihood Method (MLM). Another approachused in the literature is to determine the parameters ofpower curve model by maximum likelihood method. In thismethod the parameters of a statistical model are estimated bymaximizing the likelihood function. It was found in [1] thatthis method did not perform well in comparison to the leastsquares method.

6.2.2. Algorithms for Parameter Estimation. The parametersof the parametric models especially those using the 4PL and5PL approximations are difficult to evaluate. The parameterestimation becomes more difficult when the models arederived from the actual data of wind turbines. Developingaccurate models of wind turbines and optimization for hugedata sets are a very complicated process. Modern nontradi-tional solution techniques for parameter estimation enhancethe accuracy, reduce the computational time, and are easyto implement. Various evolutionary techniques have beenapplied for determining the parameter vector 𝜃 of logisticfunction based power curve models [4, 24].

6.3. Data Preprocessing. Thepower curve derived from actualwind speed and power output data of wind turbines usesSCADA data from the wind turbines. This data is proneto errors due to measurement, sensor, and communicationssystem errors. The data is also affected by nonproduction ofturbines when it is shut down by the control system for somereason other than anomalous operation. SCADA system canhave null entries or erroneous data which can result ininaccurate models. Hence it is necessary to remove thesemisleading entries before using this data for further analyses.The most common method is to remove the data manually.These outliers can be identified by visual inspection [11] ofwind speed power output plot and can be removed before

8 Journal of Energy

proceeding for development of model. However the methodcan lead to inaccurate results as the data from SCADAsystem is voluminous and it is difficult to differentiatebetween correct and erroneous data.These outliers have beenremoved by different statistical methods in various worksbefore development of models. In [4] the analysis of residualstogether with control charts is used to filter potential outliers.The outliers can be detected by classical least mean square(LMS) method which minimizes the sum of the squares overall the measurements and if a measurement is found to befar away from the correct value it prevails in the resultingfitting; however, in this method a single outlier point candestroy the fitting. In [32] least median of squares method isused for data preprocessing in which instead of the sum asin LMS method sum of medians is minimized to identify theoutliers. It is shown that this method is very robust; howeverit requires iterative solution. The wind data preprocessingis done in four steps in [63] which include validity check,data scaling, missing data processing, and lag removal. In[64] a probabilisticmethod developed around a copula-basedjoint probability model for power curve outlier rejection isproposed. A data mining approach to process the raw datahas been proposed in [65]. Appropriate method for datapreprocessing is important requirement for development ofan efficient model.

6.4. Evaluation of Model. After developing the model fromthe data, it is important to determine whether this modelappropriately represents the behaviour of the actual data forthe power curve. The evaluation of the developed modelsin various applications is done on the basis of a number ofperformance metrics. The root mean square error (RMSE),goodness of fit statistics used by awide number of researchers,estimates of the standard deviation of random component inthe data, and a value closer to zero indicate a better fit. The𝑅-squared statistic is the square of the correlation betweenthe actual and the predicted values that measures how closelythe fit explains variation in the data. An 𝑅-squared valuecloser to one indicates a good fit [38]. Other criteria used inthe literature include the mean absolute error (MAE), meanabsolute percentage error (MAPE), the sum of squares error(SSE), SD (standard deviation), and chi-square 𝑅 [2, 4, 8, 11,17]. The most commonly used criteria are given in Table 2.

Selection of appropriate model analyzed on the basis ofsuitable criteria is a very important task for improving theperformance of wind power plants. Many models used in theliterature have not been evaluated for goodness of fit withactual curve data or their suitability for specific applications.Thepolynomialmodels of [54, 57] are compared by observingthe visual fit. Models in [2, 11] are compared by MAE, RMSE,MAPE, and SD performance metrics; however, suitability ofthese models for wind power applications is not evaluated.Also as these models will ultimately be used in wind energyapplications it is not appropriate to judge their suitability onthe basis of goodness of fit parameters alone, but it should alsobe examined how successfully they can be employed for theparticular applications.

Table 2: Model evaluation indices.

ExpressionAbsolute error (AE)[1] AE =

𝑦𝑚 (𝑖) − 𝑦𝑎(𝑖)

Relative error (RE) [1] RE =𝑦𝑚− 𝑦𝑎

𝑦𝑎

× 100

Mean absolute error(MAE) [24] MAE =

1

𝑁

𝑁

∑

𝑖=1

𝑦𝑚 (𝑖) − 𝑦𝑎 (𝑖)

Mean absolutepercentage error(MAPE) [37]

MAPE = 1𝑁

𝑁

∑

𝑖=1

𝑦𝑚(𝑖) − 𝑦

𝑎(𝑖)

𝑦𝑎(𝑖)

Root mean squareerror (RMSE) [24] RMSE = [

1

𝑁

𝑁

∑

𝑖=1

(𝑦𝑚(𝑖) − 𝑦

𝑎(𝑖))2

]

1/2

𝑅-squared [38] 𝑅2 = 1 −∑𝑁

𝑖=1(𝑦𝑎(𝑖) − 𝑦

𝑚(𝑖))2

∑𝑁

𝑖=1(𝑦𝑎(𝑖) − 𝑦

𝑎𝑚(𝑖))2

𝑁= number of data, 𝑦𝑚(𝑖) = 𝑖th modelled value, 𝑦

𝑎(𝑖) = 𝑖th actual value, and

𝑦𝑎𝑚(𝑖) = mean of actual value.

6.5. NonparametricModels. Various nonparametricmethodscan be used for modelling of WT power curves. The SCADAdata collected from the wind farms is voluminous and usuallycontains errors. It may be difficult to obtain the relationbetween input and output using a functional form. Nonpara-metric methods can be suitable for deriving power curvesfrom this data. After preprocessing of data, model extractioncan be done using different methods. Nonparametric modelscan also incorporate the effect of parameters other than windspeed on the power curves more easily than the parametricmodels. The models can be trained by taking these otherparameters as inputs to the models [17, 18]. Developmentsin soft computing techniques offer promising approaches forpower curve modelling. Various advanced algorithms canbe utilized to generate accurate nonparametric power curvesfor various applications. Table 3 summarises details of somenonparametric models from the literature.

6.5.1. Neural Networks. Artificial neural networks (ANN)inspired by biological nervous system emulate the naturalintelligence of human brain [67] and can learn the nonlinearrelationship between input and output data sets by use of acti-vation function within the hidden neurons. Neural networksare used to estimate power generation of turbines at a windfarm in [17]. A separatemultilayer perception (MLP) networkfor each turbine uses ten-minute averages of wind speedand direction from two meteorological towers as inputs andpower generated by the turbine as the output. A comparativeanalysis of regression and ANNmodels for WT power curveestimation is done in [61] and it is shown that neural networkmodels perform better than the regression models. HoweverANN has a black box approach and it is difficult to developan insight about meaning associated with each neuron andweight [68]. ANN based multistage modelling has been usedin [62] to model the wind turbine power curve. The windspeed and air density are used as inputs in the first stage andthe normalized power output obtained from this stage, windspeed data turbulence intensity, is used to train the secondANN stage. It is claimed that this method has produced

Journal of Energy 9

Table 3: Details of some nonparametric models from various studies.

Ref.

Data set Model

Interval Data Number ofvalues Model ParametersStructure/transferfunction (TF)/trainingmethod

[4] 10min SCADA data100 WTs

Total 4347Training 3476Testing 871

𝑘-NN 𝑘 = 100 Euclidian distancemetric

[17] 10min

12 WTs(wind speed anddirection from

twometeorological

towers)

Training1500 patterns for

each WTANN

Number of hidden layers= 1

Number of hidden layerneurons = 8

(i) Separate MLPnetwork for each WT(ii) Training-patternmode(iii) TF-hyperbolic (alllayers)

[8] — Measured data100 kWWT —Fuzzy

clusteringNumber of cluster

centers = 8

(i) CFL(ii) Subtractiveclustering

[24] 10min

Data set 1generated withmethod in [9]

Total 1008Training 50%Testing 50%

ANN Number of hidden layerneurons = 5

(i) Feed forward backpropagation(ii)Training:Levenberg-Marquardt(iii) TF: hidden layertransig(iv) TF: output layerpurlin

Data set 2Total 4388

Training 50%Testing 50%

Data sets 3, 4,and 5

Total 2208Training 50%Testing 50%

Data sets1–5 as above As above

Fuzzyclustering

Number of clustercenters = 8 Fuzzy 𝐶-means

[2] 10min

SCADA (three2MWWTs)(model type 1:wind speedModel type 2:wind speed and

direction,temperature)

32796Training 60%Validation 40%

Fuzzyclustering

Number of clustercenters

Type 1 = 3Type 2 = 6

CCFL

MLPNN

Number of hidden layers= 2

(i) Training-gradientdescent(ii) TF: hiddenlayer-sigmoid(iii) TF: outputlayer-linear

𝑘-NN Type 1 𝑘 = 150Type 2 𝑘 = 3 —

ANFIS

(i) FIS structure Sugenotype(ii) Training-hybridlearning(iii) MembershipfunctionsInput space-generalizednormalOutput space-linear(iv) Number of MFs = 3

better results compared to the parametric, nonparametric,and discrete models.

6.5.2. Clustering Methods. Clustering is grouping of similardata into classes or clusters. A wind farm having many windturbine generators has variable power outputs due to varia-tion of wind speed. Efficient power curve can be found by

applying clustering methods. Power curve characterizationby cluster centre, fuzzy C-means, and subtractive clusteringmethods is done in [69]. Fuzzy clustering applies the conceptof fuzzy sets to cluster analysis and belongingness of eachpoint of data set to a group is given by amembership function.The method has the advantage of adapting noisy data. FuzzyC-means clustering uses fuzzy partitioning to partition acollection of vectors into 𝑐 fuzzy groups and finds a cluster

10 Journal of Energy

centre 𝑐𝑖in each group. The performance of FCM depends

upon the initial cluster centres.The method proposed in [70]to decide the number of clusters and their initial values forinitializing iterative optimization based clustering algorithmsis used in [8] to set up cluster centre fuzzy logic (CCFL)modelof power curve. This cluster estimation method is used as abasis for identifying fuzzy models and the effect of number ofcluster centres and cluster neighbourhood distance on RMSE iscalculated. On comparison of CCFL model with polynomialmodel fitted by least square method it is concluded thatthe RMSE obtained with the fuzzy logic method is muchlower than that obtained with the least square polynomialmodel.

6.5.3. Data Mining. Data mining refers to extracting ormining knowledge from large amounts of data [71]. Devel-opments in data mining offer promising approaches formodelling power curves of wind turbines. Selection of appro-priate data mining method and algorithm is important toget accurate, stable, and robust power curve. Data drivennonparametric models using multilayer perception (MLP),random forest, M5P, boosting tree, and 𝑘-nearest neighbour(𝑘-NN) algorithms are developed in [1] along with the 4PLparametric models. Performance of these models for onlinemonitoring of the power curves was analyzed and the leastsquare 4PL parametric and 𝑘-NNmodels were found to be ofhigh fidelity to be used as reference curves for monitoring ofpower curves. Control chart approach is used for detectingthe outliers and indicating the abnormal conditions of theturbine. However in another study [2] the performance of 𝑘-NN was found to be poor. Different data mining algorithms,namely, MLP, REP tree, M5P tree, bagging tree, and 𝑘-nearest neighbour algorithms, are used to build models forpower prediction and online monitoring in [4]. Principlecomponent analysis and 𝑘-NN algorithm are used for datareduction and filtering of outliers is done by residual andcontrol chart approach. In [2] four data mining techniques,namely, bagging, M5P algorithm, REP tree algorithm, andM5 rules, were used for constructing the nonparametricpower curve models. Model trees are type of decision treeswith linear regression functions at the leaves and are appliedfor modelling the power curves in a few applications [2,24]. Much detail of these models is not given in thesestudies. More information on model trees can be found in[72, 73].

6.5.4. Adaptive Network-Based Fuzzy Inference System(ANFIS) Model. Adaptive network-based fuzzy inferencesystem (ANFIS) is a fuzzy inference system implementedin the framework of adaptive networks and thus integratesthe best features of fuzzy systems and neural networks [68].A fuzzy inference system using fuzzy if-then rules is basedon human knowledge and reasoning processes. In ANFIS,tuning of nonlinear signal relations can be done by con-structing a set of fuzzy rules with appropriate membershipfunction parameters tuned in a training phase [67].Application of ANFIS for wind turbine power curvemonitoring is proposed in [2]. This method of modellingis compared with earlier best performing methods, namely,

ANN, CCFL, and 𝑘-NN methods, found in the liter-ature. Effect of including direction of wind and ambient tem-perature on the prediction error is evaluated. Data for pitchregulated turbines is used for the modelling.

6.5.5. Joint Probability of Wind Speed and Power. Anotherapproach of modelling is to consider the joint probabilitydistribution of power and wind speed instead of consideringthe implied function of the two variables. Considering jointprobability of the two variables instead of their individualprobabilities can incorporate measures of uncertainty intoperformance estimates [11]. SCADA wind speed and powermeasurement data from wind turbines are used to estimatebivariate probability distribution functions and constructpower curve using copula modelling technique in [10]. Theapplication of empirical copulas is proposed to approximatethe complex form of dependency between active power andwind speed. Usefulness of copula analysis for turbine condi-tion monitoring and early recognition of faults is suggestedin this paper. It is shown that different fault modes producedifferent signatures in 𝑅, 𝑅2, and chi2 statistics and can beused to identify the type of turbine fault.

6.5.6. Wavelet Support Vector Machine. Wavelet analysis is atechnique to analyze the nature of signals and is a promisingtool for nonstationary signals. In support vector machinesthe data is mapped into higher-dimensional feature spacevia nonlinear mapping. The power curve of WT is used forwindpower prediction in [37]. Anovelwavelet support vectormachine based model for wind speed prediction is proposedin this paper and its performance is found to be effective forshort time prediction.

A number of works in literature also include comparativeanalyses of various parametric and nonparametric methods[2, 8, 57, 74]. Seven different functions have been comparedin [75] for modelling the power curves of six differentturbines. In [53] four models are developed from availableoperational power output data. The polynomial regressionbased parametricmodel is used as a benchmarkmodel.Threenonparametric methods in addition to the above methods,namely, locally weighted polynomial regression, cubic splineregression, and a penalized spline regression model, areproposed in this study.

7. Selection of Modelling Method

Anumber ofmodels andmodellingmethodologies have beenproposed in various works formodelling ofWT power curve.The choice of appropriate model and methodology adoptedfor a specific application is important and is a difficult task.The model selection for a particular application is done onthe basis of availability of data, complexity of model, desiredaccuracy, and type of turbine and its power curve. On thebasis of reviewed literature the following points are identifiedfor selection of modelling methodology.

(i) Wind power curve models required for initial windresource assessment need handy methods for estima-tion of energy. Wind power output calculation and

Journal of Energy 11

energy estimation which are done during designingof wind based systems need a power curve modelwith fair degree of accuracy. When only specificationvalues (cut-in, cut-off, and rated speeds and therated power) for a wind turbine are available, thepolynomial models based on presumed shape canbe used. These models can also be used as a handytool for calculation of wind turbine output duringdesign stage of wind farms because of the simplicity ofcalculations. When the manufacturers’ curve data isavailable it is preferable to fit a polynomial function tothe data as it results in better accuracy. These modelsare thus suitable for modelling of single turbinesfor predicting power for small systems where fairlyaccurate accuracy is desired.

(ii) For power prediction and selection of turbines fordesigning of large wind based systems a very goodaccuracy is required as oversizing can result in lossof revenue and undersizing can hamper the reliabilityof the system. Accurate prediction is also required forwind farm operators for energy trading. The 4PL and5PL functions may therefore be used for these appli-cations to develop models from the manufacturers’curve data.

(iii) When the SCADA data from a nearby wind farm isavailable and it is desired to assess the power outputof a prospectivewind farmwith good accuracy havinga group of turbines or when the models is to be usedfor online monitoring of curves, it is appropriate toextract model from SCADA data of the wind farmwith appropriate extrapolations.These models can bederived using the 4PL and 5PL parametric methodsor one of the nonparametric methods such as ANNor ANFIS. The models can also incorporate the effectof other influencing parameters in themodels. Choiceof parameters depends upon terrain, wind conditions,obstacles, correlation of parameters, and so forth.Thecopula method of defining the power curve using ajoint probability distribution function of wind speedand power can be used for condition monitoringapplications.

8. Discussions and Prospects

Various methods of modelling ofWT power curve have beenreviewed. Several methods of modelling have been proposedand used in various studies (Figure 3). A summary of note-worthy contributions is given in Table 4. The salient featuresof these models are summarised in Table 5. The inferencesdrawn, the deficiencies, and the suggestions proposed aregiven below.

(i) Most of the parametric models used in the literatureuse polynomial approximations tomodel wind powercurve models. These models are mostly used forpredicting power output of turbines for sizing andcost optimization applications. These models do notconsider the inflection point on the power curveaccurately and can result in large prediction errors.

However they are simple to use and can be utilized forpredicting power during initial resource assessmentand designing of small systems if very good accuracyis not desired. 4PL and 5PL parametric models canfollow the actual shape of the power curve moreaccurately.These novel methods can result in reducedpower prediction errors and can be applied for powerand energy assessment during design of large systemsand forecasting of power for energy trading wheregood accuracy is a crucial requirement.

(ii) The deterministic methods which use manufacturer’sdata for modelling are suitable for single turbines andnot appropriate formodelling a group of turbines.Theprobabilistic methods which consider the variation ofboth power andwind speed are suitable formodellingpower curve for a fleet of turbines.

(iii) Power curve models extracted from actual data ofwind farms can incorporate actual conditions at a siteand are suitable for modelling a group of turbines.These curves can be derived from the available databy parametric and nonparametric methods.

(iv) The parametric models derived from actual windfarm data used in the literature include linear seg-mented and 4PL and 5PL models. Nonparametricmethods used are neural networks, clustering meth-ods, data mining, ANFIS, and copula models. Datamining techniques can offer good results as the dataavailable from the wind farms is voluminous andfrequent updating of data is easier. ANN and ANFISmodels perform well for power prediction and onlinemonitoring applications. These curves derived fromactual data can help in minimizing power predictionerrors.

(v) The 4PL and data mining technique based modelsextracted from actual data of wind farms have beenanalyzed in the literature for their application inonline monitoring. It is indicated that the monitoringof the power curves can be used to detect anomaliesand statistical analysis of the outlier data can giveindications of the specific reason of anomaly. Appli-cation of 5PL model for online monitoring of curve isnot yet researched.

(vi) Thewind power output of a turbine can be affected byvarious faults/anomalies or underperformance of theturbine, such as blade faults and yaw and pitch systemfaults. Different types of faults affect the turbine sys-tem differently and cause the power curve to departfrom the expected value in a different way. Toolswhich can characterize and quantify these departurescan aid in early identification of faults. Likely linkbetween copula statistics and WT faults/anomaliesis indicated in the literature should be investigated.Further research should also focus on considering thejoint probability distribution of these variables.

(vii) Applications of advanced algorithms for developingimproved parametric and nonparametric methodsneed to be explored.

12 Journal of Energy

Table4:Summaryof

noteworthycontrib

utions.

Author

Mod

elsData

Evaluatio

nFeatures

ofapplications

(sug

geste

d/analysed/used)

Diafetal.[12]

Linear

Ratin

gsof

WT

(V𝑐,V𝑓,V𝑟,𝑃𝑟,etc.)

—(i)

Allarep

olyn

omialm

odels

(ii)D

ono

tfollowcurvatureo

fpow

ercurve

(iii)Ac

curacy

ispo

or(iv

)Mostly

appliedforp

ower

predictio

nforsizingof

windbasedsyste

ms

Diafetal.[25]

Quadratic

—Giorsetto

andUtsu

rogi[30]

Bino

mial

—Ch

edid

etal.[46

]Cu

bic

—

Powell[6]

Weibu

llbased

Ratin

gsof

WTand

𝑘—

(i)Re

quire

sshape

parameter

ofsite

(ii)A

ccuracydepend

sonwindregime

(iii)Ap

pliedforc

apacity

factor

evaluatio

n

Aietal.[59]

Manufacturer’s

curvefi

tting

(3bino

mialexpressions)

Manufacturer’s

curve

—(i)

Needs

manyexpressio

nsto

representthe

shapeo

fcurve

accurately

(ii)A

ppliedforsizingof

hybrid

windPV

syste

mDiafetal.[55]

Manufacturer’s

curveinterpo

latio

nby

cubic

splin

esManufacturer’s

curve

—

Katsigiannise

tal.[60]

Manufacturer’s

curvefi

tting

9th-orderp

olyn

omial

Manufacturer’s

curve

—(i)

Accurateforthe

particular

dataset

(ii)H

igh-orderp

olyn

omialsmay

notrepresent

thev

arianceo

fdata

(iii)Ap

pliedforsizingof

hybrid

wind,PV

,and

biod

ieselbased

syste

m

GottschalkandDun

n[58]

Linear,W

eibu

llbased,

cubics

plines,

manufacturer’s

curvefi

tting

V 𝑐,V𝑓,V𝑟,𝑃𝑟,and

manufacturer’s

curve

Visualcomparis

on,

%errorinenergy

prod

uctio

n

(i)Po

lyno

mialm

odels

(ii)F

ittingaccuracy

with

good

nessof

fitindicatorsisno

tevaluated

NandKishoreand

Fernandez[26]

Linear

segm

ented

Manufacturer’s

curve

—(i)

Bette

raccuracy

(ii)N

eeds

manyexpressio

ns

Liu[54]

New

nonlinearformula

Ratin

gsof

WT

Visual

comparis

on

(i)Con

sidersinfl

ectio

npo

into

ncurve

(ii)F

ollowsshape

ofcurvem

orea

ccurately

than

polyno

mialm

odels

(iii)Simplem

etho

d(iv

)App

liedfore

cono

micload

dispatch,reliabilityanalysis,

andmultip

leturbines

with

andwith

outcorrelation

Villanu

evaa

ndFeijó

o[33]

3PLand4P

Lcurve

Manufacturer’s

curve

MAE,

MAPE

,and

RMSE

(i)Parametersa

rederiv

eddirectly,

noneed

ofiterativ

eprocedu

re

Kusia

ketal.[1]

4PLcurve

Parameter

estim

ation:

Techniqu

e:leastsqu

ares

andMLM

Algorith

m:E

SAc

tualdataof

wind

farm

AE,

RE(i)

Leastsqu

areb

etterthanMLM

metho

d(ii)𝑘

-NNou

tperform

sother

nonp

aram

etric

mod

els(iii)MLP

bestaccuracy

(iv)R

esidualand

controlchartapproach

foro

nlinem

onito

ringispresented

(v)4

PLleastsqu

ares

and𝑘-N

Nmod

elsp

ropo

sedford

etectin

ganom

alies

Datam

ining:𝑘-N

N,M

LP,

rand

omforest,

M5P

tree,andbo

ostin

gtre

e

Kusia

ketal.[4]

4PLcurve

Parameter

estim

ation:

Techniqu

e:leastsqu

ares

Algorith

m:E

SDatam

ining:𝑘-N

N,M

LP,R

EPtre

e,M5P

tree,

andbaggingtre

e

Manufacturer’s

curve

Actualdataof

wind

farm

MAE,

AE,

RE,and

meanrelativ

eerror

(i)𝑘-N

Nou

tperform

sother

nonp

aram

etric

mod

els(ii)O

utlierfi

lterin

gby

resid

ualapp

roachandcontrolchart

(iii)Ap

plicationforp

ower

predictio

nandon

linem

onito

ringissuggested

Lietal.[17]

ANN(M

LP)

Actualdataof

wind

farm

MSE

(i)Be

tterthanthetraditio

nalm

odel

(ii)A

ppliedforp

ower

predictio

n

Lietal.[61]

Regressio

nANN(M

LP)

Actualdataof

wind

farm

RMSE

(i)Com

paris

onof

both

mod

els(ii)N

Nmod

elismorea

ccurate

(iii)Eff

ecto

fwinddirectionisconsidered

Pelletie

retal.[62]

ANN(m

ultistage

2-lay

erMLP

s)Ac

tualdataof

wind

farm

Visual

meanerror,MAE

(i)Com

paredwith

parametric

,non

parametric

,and

discretemod

els(ii)S

ixinpu

tsvaria

bles

considered

Üstü

ntaş

andŞahin[8]

Leastsqu

ares

fittin

g(2nd

-order

polyno

mial)

CCFL

Actualdataof

wind

farm

RMSE

(i)CC

FLmod

elismorea

ccuratethantheleastsquaresfi

tting

basedmod

el

Journal of Energy 13

Table4:Con

tinued.

Author

Mod

elsData

Evaluatio

nFeatures

ofapplications

(sug

geste

d/analysed/used)

Lydiae

tal.[24]

Parametric

mod

el:lin

earsegmented,4P

Land

5PL

Parameter

estim

ation:

Techniqu

e:leastsqu

ares

Algorith

ms:GA,E

P,PS

O,and

DE

Actualdataof

wind

farm

RMSE

,MAE

(i)5P

logisticmod

elwith

parameterse

stim

ated

usingDEisthem

ostaccuratea

mon

gparametric

mod

els

Non

parametric

mod

el:ANN,cluste

ring,

datamining(m

odeltre

es)

Actualdataof

wind

farm

RMSE

,MAE

(i)ANNmod

elisthem

ostaccuratea

mon

gthen

onparametric

mod

els

Lydiae

tal.[36]

5PL

Actualdataof

wind

farm

RMSE

,MAE

(i)Usedforw

indresource

assessment

Soho

nietal.[35]

4PLand5P

LAc

tualdataof

wind

farm

%errorinenergy

estim

ation

(i)Ap

pliedforw

indenergy

estim

ation

Schlechtingenetal.[2]

CCFL

ANN𝑘-N

NANFIS

Actualdataof

wind

farm

MAE,

RMSE

MAPE

,SD

(i)ANNandNNperfo

rmbest

(ii)𝑘

-NNworstperfo

rmance

(iii)Eff

ecto

fwinddirection,

temperature

considerationprod

uces

lesserrors

(iv)A

pplicationin

onlin

emon

itorin

g

Stephenetal.[10]

Cop

ulam

odel

Actualdataof

wind

farm

—(i)

Jointp

robabilitydistr

ibutionof

windspeedandpo

wer

isconsidered

(ii)A

pplications

forc

onditio

nmon

itorin

gares

uggeste

d

Gill

etal.[11]

Cop

ulam

odel

Actualdataof

wind

farm

𝑅,𝑅2,and

chi-squ

are

(i)Jointp

robabilitydistr

ibutionof

windspeedandpo

wer

isconsidered

(ii)A

pplications

forc

onditio

nmon

itorin

gares

uggeste

d

Zeng

andQiao[37]

Wavele

tSVM

forw

indspeedpredictio

n

Actualdataof

wind

farm

and

manufacturer’s

power

curve

MAE,

MAPE

,and

SD(i)

Outperfo

rmsthe

persistence

mod

elforsho

rt-te

rmpredictio

n(ii)A

ppliedforsho

rttim

epow

erpredictio

n

14 Journal of Energy

Table 5: Comparison of modelling methods.

Models Data requiredfor modelling Merits Demerits Applications

Polynomialmodels(linear,quadratic,binomial, cubic,and Weibullbased)

V𝑐, V𝑓, V𝑟, and 𝑃

𝑟

of turbine

(i) Simplicity(ii) Limited data required(iii) Parameter calculation is easy

(i) Do not follow curvature ofpower curve(ii) Accuracy is poor(iii) Sometimes more than oneexpression are used todescribe the shape of curve

Suitable for power predictionand energy estimation duringinitial resource assessmentand designing of small systems

Manufacturer’scurve fitting

Manufacturer’scurve (i) Need less data

(i) Requires manufacturer’spower curve data(ii) Fairly accurate(iii) Many expressions may berequired for accuraterepresentation of curve

Suitable for power predictionand energy estimation duringinitial resource assessmentand designing of small systems

Cubic splines Manufacturer’scurve (i) Exact fit(i) Variance of data is nottaken into account Power prediction

4PL model

Manufacturer’scurve, actualdata of windfarm

(i) Consider inflection point oncurve; hence shape of curve isrepresented more accurately thanthe earlier models(ii) One expression is required

(i) Asymmetry of curve notmodelled

Online monitoring;further research on powerprediction during design andpower forecasting applicationsis required

5PL model

Manufacturer’scurveactual data ofwind farm

(i) Consider inflection point oncurve and asymmetry of curve ismodelled more accurate than theearlier and 4PL models(ii) One expression is required

(i) Parameter estimation isdifficult

Further research on powerprediction during design andpower forecasting and onlinemonitoring applications isrequired

ANN Actual data ofwind farm(i) Found to be accurate than othermethods (i) Black box approach

Wind power assessment forsizing and power forecasting,and online monitoringapplications, suitable forgroup of turbines

Clustering Actual data ofwind farm(i) More accurate than theregression method

(i) Accuracy depends on thenumber of cluster centres

Wind power assessment forsizing and power forecasting,and online monitoringapplications, suitable forgroup of turbines

𝑘-NN Actual data ofwind farm(i) Performance variable in differentstudies

(i) Accuracy depends on valueof 𝑘(ii) Less training time asinstance based scheme

Wind power assessment forsizing and power forecasting,prediction, andonline monitoringapplications, suitable forgroup of turbines

Modeltrees(REP, M5P,and baggingtree)

Actual data ofwind farm (i) Fairly accurate

(i) Much research notavailable

Applicability in powerprediction andonline monitoring to beexplored

ANFIS Actual data ofwind farm

(i) Integrates best features of fuzzysystems and neural networks(ii) Accurate method(iii) Fewer parameters required intraining therefore faster trainingcompared to NN(iv) Tunable membership functions

(i) Computational complexity

Wind power assessment forsizing and power forecastingfor energy trading Onlinemonitoring applications,suitable for group of turbines

Copula model Actual data ofwind farm

(i) Considers joint probabilitydistribution of wind speed andpower(ii) Includes measures ofuncertainty in performanceestimates

(i) Needs advanced methodfor parameter estimation ofmarginals

Applications for conditionmonitoring to be investigated

Journal of Energy 15

WT power curve

Binning (IEC) method

Based on stochastic method Dynamic model

Deterministic/probabilistic method

Deterministic

Probabilistic

Parametric/nonparametric method

Parametric

Polynomial

Linear

Quadratic

Cubic

Weibull

Cubic splines

fitting

Exponential

Inflection point

considered

Double exponential

3PL, 4PL, and 5PL

Nonparametric

ANN

Clustering

ANFIS

Datamining

Copula

SVM, wavelet

Presumed shape based on manufacturers specifications

Manufacturers power curve data

Manufacturers curve/SCADA data of wind farm

SCADA data of wind farm

Manufacturer’s curve

Figure 3: Wind turbine power curve models.

(viii) Appropriate evaluation of the developed models isa very important requirement in modelling. Theright choice of an evaluation metric is importantand depends on the data and analysis requirements.Models developed in the literature have used differ-ent performance metrics. The reviewed articles donot highlight the reason of preferring a particularcriterion for evaluation. Different statistical measurescan have different interpretations and appropriateselection of error metrics is crucial for analysis ofthe models. Moreover these models will ultimately beused in wind energy applications; therefore it is notappropriate to judge their suitability on the basis ofgoodness of fit parameters alone, but it should alsobe examined how successfully these models can beemployed for the particular applications.

(ix) Power curves of wind turbines provided by themanufacturers are used for power prediction in mostof the wind energy applications. These curves are

developed under standard test conditions. The IEC61400-12-1 is the most accepted standard for powercurvemeasurement of single wind turbines.Thewindconditions at the practical sites can be different fromthose at the test site. IEC curve may not alwaysrepresent the wind conditions of other sites. Furtherresearch should focus on development of site specificpower curves.

(x) The discrete model prescribed in IEC 61400-12 issimple, but a large amount of data is required todevelop a reliable model.

(xi) The stochastic model that is proposed in some worksis independent of turbulence intensity but does notinclude the effect of other influencing parameters.

(xii) Future works should also include the effect of variousinfluencing parameters on the power curves.

16 Journal of Energy

9. Conclusions

Accurate modelling of WT power curves is crucial forsuccessful design and operation of any wind energy conver-sion system. This paper presented an overview of differentapproaches used for modelling of wind turbine power curve.There are several methods ofmodelling which have their ownadvantages and disadvantages. Polynomial approximationbased power curve models which have been used widely aresimple to use and can be utilized for predicting power duringinitial resource assessment and designing of small systems.The four- and five-parameter logistic function based powercurve models which consider the inflection point on thesecurves are promisingmethods which can help in reduction ofpower prediction errors and improved performance in onlinemonitoring of these curves. Alsomost of themodels in earlierworks are developed from manufacturers curves of windturbines. However, the manufacturer supplied curves areturbine specific and represent their behaviour under standardtest conditions. They can be applied for power predictionof single turbines and for sites with steady winds. Improvedmodels are required which can represent the conditions atlarge wind farms with a group of turbines installed and siteshaving complex terrains. Power curves derived from actualwind speed and output power data from wind farms can takeinto account various site specific factors resulting in bettermodels. A wind farm’s SCADA data is a valuable resourcewhich can be exploited for this purpose. Nonparametricmethods of power curve modelling used in the literatureinclude neural networks, clustering, datamining, ANFIS, andcopula models. The nonparametric methods are suitable forextracting models from large data. Moreover, these modelscan incorporate the effect of parameters other than windspeed on the power curves more easily than the parametricmodels. Literature survey reveals that ANN and ANFISnonparametric methods perform well among other modelsfor power prediction and online monitoring applications.Further research should focus on development of site specificpower curves. Future models should be able to minimize theprediction errors and should be suitable for online monitor-ing of turbines. Methods which can quantify the power curvedepartures from expected values for identification of turbinefaults should be explored. As the power output of wind tur-bines is strongly dependent onwind speed of a potential windfarm site, selection of appropriate wind speed model alongwith the power curve model is an important requirementfor accurate prediction of wind farm output. Different windspeed modelling techniques have also been reviewed brieflyin this paper. It can be concluded that selection of appropriatemodel, solution technique, and proper algorithms for aparticular application is important for efficient modellingand can contribute significantly to developing reliable andefficient wind energy based power system.

Nomenclature

𝑃: Power output of wind turbine (W)V: Wind speed (m/s)𝐴𝑤: Area swept by the rotor blades (m2)

𝜌: Air density (kg/m3)𝜆: Tip speed ratio𝛽: Pitch angle (degree)𝐶𝑝: Power coefficient of wind turbine

𝑘: Shape parameter Weibull PDF𝑐: Scale parameter Weibull PDFV𝑐: Cut-in wind speed of turbine (m/s)

V𝑟: Rated wind speed of turbine (m/s)

V𝑓: Cut-off (or furling) wind speed of turbine(m/s)

𝑃𝑟: Rated power of wind turbine (W)

𝜂𝑤: The efficiency of WTG and thecorresponding converter

𝜃: Vector parameter of parametric models.

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

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