A Framework to Determine the Probability DensityFunction for the Output Power of Wind Farms

Sairaj V. Dhople and Alejandro D. Domínguez-GarcíaDepartment of Electrical and Computer Engineering

University of Illinois at Urbana-ChampaignUrbana, IL, 61801, USA

E-mail: {sdhople2, aledan}@ILLINOIS.EDU

Abstract—This paper proposes a numerical framework topropagate wind-speed uncertainty through to the power outputof a wind farm while factoring in the availability of the windturbines in the farm. In particular, given a probability den-sity function (pdf) that describes wind speed, and a statisticalavailability model for the wind turbines, we propose a methodto determine the wind-farm power output pdf. The proposedframework offers several advantages over conventional methods,e.g., it is agnostic to the wind-speed distribution and easilyincorporates wind farm availability. Case studies compare thewind farm power distributions computed using the proposedframework with field data for an actual wind farm. In addition,we demonstrate how the framework allows the computation ofcommon wind generation indices such as the expected availablewind energy, expected generated wind energy, and capacityfactor.

Index Terms—Availability, wind farms, random variable trans-formations.

I. INTRODUCTION

Quantifying the impact of wind-speed uncertainty on wind-farm power output is critical in determining if the installedgenerating capacity can accommodate the system load at anagreeable reliability level [1], [2]. Time-domain simulations,e.g., sequential Monte Carlo methods [3], can be utilized insuch studies. However, while these methods preserve chrono-logical aspects, they are not only computationally expensivebut also unsuitable when historical wind-speed time-series datais unavailable [3], [4]. To address these concerns, probabilisticmethods have been widely applied in generation adequacy aswell as bulk power system reliability studies [5]–[8].

This work proposes an improved numerical framework topropagate wind-speed uncertainty to the wind-farm poweroutput—a critical step in any probabilistic method for system-level reliability assessment. Inputs to the framework are thewind-speed pdf and a statistical availability model for the windturbines in the wind farm. These inputs are propagated througha mathematical model that describes the power output of thewind farm to obtain a pdf for the power produced by the windfarm. This is accomplished following well known random-variable transformation methods.

The proposed method has several advantages. First, thewind-turbine power curve is modeled by a continuous functionup to the furling (cut-out) wind speed. In related works (see,e.g., [4], [9], [10]), the turbine output is generally assumed

to be zero up to a cut-in wind speed and a constant be-tween rated and furling speeds. On applying random-variabletransformations to this model, two impulses are obtained inthe power distribution—at zero and rated power. However,this is perhaps not a realistic model for real-world conditions(as is illustrated in the case studies). Second, wind-turbineavailability is incorporated in a unified way into the analysis.Additionally, the proposed method is agnostic to the distribu-tion adopted to model the wind speed. Finally, the frameworkeasily allows computation of common wind generation indicessuch as the expected available wind energy, expected generatedwind energy, and capacity factor [11], [12].

The remainder of this paper is organized as follows. InSection II we provide a brief overview of the proposedframework, formulate the wind turbine (and farm) power-wind speed characteristic, and describe the method based onrandom-variable transformation to obtain the pdf of the wind-farm power output. Then, in Section III, we demonstrate thevalidity of the proposed framework to infer wind farm powerstatistics by comparing results with field data from an actualwind farm. Finally, concluding remarks and directions forfuture work are discussed in Section IV.

II. METHODOLOGY

We begin this section with a brief overview of the proposedframework. Then, we describe the piecewise continuous func-tions utilized to model the power output of the wind turbinesand the wind farm. Next, the statistical availability model isfactored into the framework. Finally, we demonstrate how theuncertainty in wind speed can be propagated to the poweroutput of the wind farm. This section also includes pseudocodesuitable for computer implementation to implement the pro-posed method.

A. Overview of Proposed Framework

The building blocks of the proposed framework are illus-trated in the block diagram shown in Fig. 1. The first inputto the framework is wind speed, modeled as a continuousrandom variable V , with pdf fV (v), which is assumed to beknown. The framework also requires a statistical availabilitymodel for the wind farm, i.e., for a wind farm comprised of nturbines, the probabilities πm, m = 0, 1, . . . , n, where πm isthe probability that m turbines are operational. These inputs

Figure 1. Block diagram that illustrates proposed framework.

are propagated through a mathematical model which capturesthe relation between power and wind-speed (referred as p− vcharacteristic subsequently) of the wind farm to obtain the pdfof the wind-farm power—denoted by fP (p), where P is therandom variable that describes the power.

B. Wind Turbine p− v Characteristic

Following standard terminology (see e.g., [13]), denote thecut-in, rated, and furling wind speed of a wind turbine by vc,vr, and vf , respectively. Further, denote the rated output powerof the turbine by Pr. Let us consider the following piecewisecontinuous function to model the power output of the windturbine p, as a function of wind speed v:

p(v) =

{p1(v) for 0 ≤ v ≤ vlim,p2(v) for vlim < v ≤ vf ,

(1)

where

p1(v) = Pr − Pr[1 + exp

(v − vmid

c

)]−1, (2)

p2(v) = Pr − Prα (v − vlim)q, (3)

and where vmid, c, α, and q are parameters that determine theshape of the functions [14]. We provide a brief note below onhow the different parameters can be determined.

The parameter vmid is the wind speed at which the poweroutput of the turbine is half the rated value, i.e., p1(vmid) =Pr/2. Hence it can be determined by inspecting wind-turbinedata sheets. The parameter c can be tuned to ensure that thecharacteristic is within some predetermined percentage of therated power, Pr at the rated wind speed, vr. For instance, ifwe require p1(vr) = β · Pr (typically β ≥ 0.99 to ensurep1(vr) ≈ Pr), it is easy to show

c = (vr − vmid)[log

(β

1− β

)]−1. (4)

Table IWIND-TURBINE PARAMETERS

Symbol Parameter Value

Pr Rated power 2 MWvc Cut-in wind speed 4 m/svr Rated wind speed 12 m/svlim Limiting wind speed 24 m/svf Furling wind speed 25 m/sc Shape parameter that ensures p1(vr) = β · Pr 0.7617 m/s

vmid Wind speed such that p1(vmid) = Pr/2 8.5 m/sq Order of drop off for v > vlim 2α Parameter that ensures p2(vf ) = 0 1 (m/s)−2

Figure 2. Sample p− v characteristic for a representative turbine.

Finally, the parameter α is given by

α = (vf − vlim)−q, (5)

which ensures that p2(vf ) = 0. The function p2(v) is formu-lated to model a q-order drop-off in power output for windspeeds greater than a limiting wind speed, vlim. The limitingwind speed can be chosen to be arbitrarily close to the furlingspeed. As will be shown in the case studies in Section III,a quadratic drop off (q = 2) models the output of an actualoperating wind farm better than conventional models in whichvlim = vf so that p2(v) = 0 ∀v ≥ vf . Figure 2 illustratesthe p − v characteristic for the Vestas V90-2.0 MW turbine[15]. Relevant turbine specifications and model parameters—extracted following the approach outlined above—are listed inTable I.

C. Wind Farm p− v Characteristic

Let us consider a wind farm comprising n identical windturbines. Recall that the p − v characteristic of each turbinecan be modeled by (1)-(3). If the turbines are staggeredappropriately, interference effects can be minimized [13], and

the p− v characteristic of the farm is given by1

p(v) =

{p1(v) for 0 ≤ v ≤ vlim,p2(v) for vlim < v ≤ vf ,

(6)

where

p1(v) = P ξr − P ξr[1 + exp

(v − vmid

c

)]−1, (7)

p2(v) = P ξr − P ξr α (v − vlim)q. (8)

The rated power of the farm, denoted by P ξr , is defined as

P ξr = ξPr, (9)

where ξ is the expected number of operational turbines overthe period of study, and Pr is the rated power output of asingle turbine. The expected number of turbines is given by

ξ = π0 · 0 + π1 · 1 + . . .+ πn · n, (10)

where πm is the probability that m turbines are operational.The probabilities πi, i = 0, 1, . . . , n, can be obtained from aMarkov availability model [11], [12]. Alternately, they can beobtained from field data. We demonstrate this approach in thecase studies.

D. Propagating Wind-Speed Uncertainty to the Wind-FarmPower Output

Suppose the wind speed is described by a random variableV , with pdf fV (v), which is assumed to be known. Applyingrandom-variable transformations to (6), we get

fP (p) =fV (v1)

|p′1(v1)|+fV (v2)

|p′2(v2)|, (11)

where v1 and v2 can be obtained by inverting the p1(v), p2(v)characteristics in (7)-(8) as follows

v1 = vmid + c · log(

p

P ξr − p

), (12)

v2 = vlim +

(1

α

P ξr − pP ξr

)1/q

. (13)

The expressions for p′1(v) and p′2(v) in (11) can be obtainedby differentiating (7)-(8) which results in

p′1(v) =P ξrc

exp(v − vmid

c

)[1 + exp

(v − vmid

c

)]−2,

(14)p′2(v) = −2αP ξr (v − vlim). (15)

The equation in (11) follows from well-known random-variable transformation methods [16]. In the context of propa-gating wind-speed uncertainty, these methods require invertingthe p − v characteristic for each value of p—which is easygiven the form of p(v) in (7)-(8).

1We abuse notation and denote the p−v characteristic of the wind farm byp(v). However, as expressed in (6)-(8), the rated power is different from thatin (1)-(3). Subsequently, we will only be dealing with the p−v characteristicof the wind farm as described in (6)-(8).

Wind speed pdf fV (v), can be obtained by fitting fielddata with standard distributions. For instance the Weibulldistribution

fV (v) =b

abvb−1exp

(−va

)b, (16)

where b is called the shape parameter and a is called the scaleparameter, has been widely adopted to model wind-speedstatistics [13], [17]. While we utilize the Weibull distributionin the case studies that follow, note that the method proposedabove is agnostic to the wind-speed distribution.

We want to point out that the above method is accuratewhen the maximum wind speed at the location is less thanthe cut-out speed. As part of future work, we will augmentthe p − v characteristic to model the extenuating case whenthe maximum wind speed at the location is expected to besignificantly higher than the cut-out speed. If the power outputis assumed to be zero beyond this value, mixed distributionswill be obtained.

E. Computer Implementation

Algorithm 1 provides the pseudocode for computer imple-mentation of the method outlined in (11)-(15) to computefP (p). Since (11) has to be evaluated pointwise, p is appro-priately discretized between pmin and pmax in steps of dpto obtain the vector p̄ = [pmin : dp : pmax]. In the for loop,fP (p) is evaluated point wise for each entry of p̄, which wedenote by p̂. This involves computing v1 and v2 through (12)-(13), p′1(v1) and p′1(v1) through (14)-(15), and then applying(11).

Algorithm 1 Computation of fP (p)

define p̄ = [pmin : dp : pmax]model parameters: vc, vr, vlim, vf , α, β, c, q, PrExpected number of operational wind turbines, ξWind-speed pdf, fV (v)

for p̂ = pmin : dp : pmax docompute v1 = vmid + c · log

(p̂

P ξr−p̂

)v2 = vlim +

(1αP ξr−p̂P ξr

)1/qcompute p′1(v1) =

P ξrc exp

(v1−vmid

c

) [1 + exp

(v1−vmid

c

)]−2p′2(v2) = −2αP ξr (v2 − vlim).

compute fP (p̂) = fV (v1)

|p′1(v1)|+ fV (v2)

|p′2(v2)|end for

F. Wind Generation Indices

The proposed framework can be utilized to computecommon wind generation indices that gauge the reliabil-ity/performability of wind farms [11], [12]. We introducesome metrics of interest below and explain how they can becomputed.

1) Installed Wind Power, IWP: The installed wind poweris the sum of nominal rated power of all turbines

IWP = n× Pr [MW]. (17)

2) Installed Wind Energy, IWE : The installed wind energyis the maximum possible energy that can be extracted in oneyear from the wind farm

IWE = IWP [MW]× 8760

[hryr

]. (18)

3) Expected Available Wind Energy, EAWE: Energy ex-pected to be generated by the wind farm in one year withoutconsidering wind-turbine failure

EAWE =

∫p

p · fP (p)|ξ=n dp [MW]× 8760

[hryr

]. (19)

To compute this index, we need the expected power output ofthe wind-farm without considering wind-turbine failure. Thepdf of the power output without any failures, fP (p)|ξ=n, canbe computed following the procedure in Section II-D, withξ = n⇒ P ξr = n · Pr.

4) Expected Generated Wind Energy, EGWE: Energy ex-pected to be generated by the wind farm in one year consid-ering wind-turbine failures and repairs

EGWE =

∫p

p · fP (p) dp [MW]× 8760

[hryr

]. (20)

To compute this index, we need the expected power outputof the wind-farm while accommodating failures and repairsthrough an availability model.

5) Capacity Factor, CF: Capacity factor of the wind farmwithout considering wind-turbine failure

CF =EAWEIWE

. (21)

6) Wind Generation Availability Factor, WGAF: Capacityfactor of wind farm considering failures and repairs

WGAF =EGWE

IWE. (22)

This is the capacity factor of the the wind farm while factoringin failures and repairs of the wind turbines.

III. CASE STUDIES

In this section, we present two case studies. First, wevalidate the proposed approach by comparing the wind farmpower pdf with actual field data. In the second, we demonstratehow the generation indices in Section II-F are computed for ahypothetical wind farm.

A. Model Validation with Field Data

This case study demonstrates how the proposed method canbe applied to study the power distribution of a real wind farm.We have access to 1-second data for number of operationalturbines, wind speed, and wind-farm power output recordedat a wind farm comprising n = 75 turbines over a period oftwo months. The goal of this case study is to demonstrate thatthe method proposed in Algorithm 1 can be used to accuratelyextract the pdf of the wind-farm power output by comparingthe results with the field data.

Figure 3. Availability of turbines in the wind farm.

Figure 4. Weibull fit to distribution of wind speeds collected in the field.

Figure 5. Field data compared to proposed p− v characteristic.

The methodology comprises the following steps: i) obtaina statistical availability model of the turbines in the farm, ii)fit the pdf of recorded wind speed at the site with a Weibulldistribution, iii) formulate the p− v characteristic of the windfarm through (6)-(8), iv) propagate wind speed uncertaintythrough (11)-(15). These steps follow along Algorithm 1, and

Figure 6. Distribution for power computed from Algorithm 1 compared withfield data.

they are explained in detail subsequently.As stated above, we first determine a statistical availability

model for the wind farm. In the context of the discussionin Section II-C, this involves determining the probabilitiesπm, m = 0, 1, . . . , n, where πm is the probability that thewind farm has m operational turbines. From the available dataon number of wind turbines, πm can be computed as

πm =NmN

, (23)

where Nm is the number of seconds with m operationalturbines, and N = 2 · 30 · 24 · 60 · 60 s, the number of secondsin 2 months—which is the duration of the observed period.The results are plotted in Fig. 3, and from (10), we get theexpected number of operational turbines, ξ = 62.7461.

In the next step, a Weibull distribution is fitted to the distri-bution of wind speed utilizing MATLAB R©’s dfittool function.The resulting scale and shape parameters are a = 6.54818 m/s,and b = 2.35952, respectively. Figure 4 plots the Weibull fitand the distribution computed from the raw wind speeds. Theresults show excellent agreement.

Next, we determine the p − v characteristic of the windfarm. Figure 5 plots the p− v characteristic of the wind farmcomputed from (6)-(8) and it also plots the raw power data asa function of wind speed. Finally, we follow along the steps inAlgorithm 1 to determine the distribution fP (p) and comparethe results with the field data. Results plotted in Fig. 6 showgood agreement over a wide power range. The mean powerfrom the field data is equal to 0.2673 p.u., while that computedusing the proposed method is 0.2374 p.u.

B. Computing Wind Generation Indices

In this case study, we demonstrate the applicability of theproposed framework in computing the wind generation indiceslisted in Section II-F. Consider a wind farm comprised of n =50 wind turbines with specifications in Table I. We will inves-tigate wind-generation indices for wind regimes characterizedby the Weibull distributions in Fig. 7. The installed wind power

can be computed from (17) as IWP = n · Pr = 50 × 2 =100 MW. Similarly, the installed wind energy can be computedfrom (18) as IWE = IWP× 8760 = 8.76× 105 MWhr.

Figure 7. Representative Weibull distributions (a varied, b = 5) to modelwind speeds.

Figure 8. Power distribution computed for the wind-speed distributions inFig. 7

Figure 9. Wind generation indices as a function of the scale parameter.

Figure 8 depicts the pdfs fP (p) computed for wind speedsmodeled by Weibull distributions in Fig. 7 assuming no wind-turbine failures. The base power corresponding to 1 p.u. is 100MW, the IWP. Figure 9 depicts the expected available windenergy (EAWE), expected generated wind energy (EGWE),capacity factor (CF), and wind generation availability factor(WGAF) as a function of the scale parameter a, assuming ξ =0.9 × n, i.e., 45 turbines are expected to operate on average.These indices are computed from (19)-(22).

IV. CONCLUDING REMARKS AND FUTURE WORK

This work proposes a numerical method to obtain pdfs forwind-farm output power, given the pdf of wind speed at thesite. The main advantage of the proposed method is that itprovides a unified way to integrate wind-farm availability,and is agnostic to the distribution adopted to model the windspeed. The suggested method can be applied in generationcapacity adequacy and other similar power-system reliabilitystudies. Additionally, the proposed method could be utilized byplanners/developers to explore the impact of different repairstrategies, wind turbines, and locations on the power outputof the wind farm. Future work could investigate a Markovreliability model to capture the availability of the turbines inthe farm. Interference effects, wind direction, etc., could alsobe incorporated into the framework.

ACKNOWLEDGMENTS

This work was supported by the National Science Foun-dation under Career Award ECCS-CAR-0954420 and undergrant ECCS-0925754.

REFERENCES

[1] N. Negra, O. Holmstrom, B. Bak-Jensen, and P. Sorensen, “Aspects ofrelevance in offshore wind farm reliability assessment,” IEEE Transac-tions on Energy Conversion, vol. 22, no. 1, pp. 159–166, March 2007.

[2] C. Singh and A. Lago-Gonzalez, “Reliability modeling of generationsystems including unconventional energy sources,” IEEE Transactionson Power Apparatus and Systems, vol. PAS-104, no. 5, pp. 1049–1056,May 1985.

[3] E. Byon, L. Ntaimo, C. Singh, and Y. Ding, “Wind energy facilityreliability and maintenance,” Handbook of Wind Power Systems: Op-timization, Modeling, Simulation and Economic Aspects, in review.

[4] I. Abouzahr and R. Ramakumar, “Loss of power supply probability ofstand-alone wind electric conversion systems: a closed form solutionapproach,” IEEE Transactions on Energy Conversion, vol. 5, no. 3, pp.445–452, September 1990.

[5] J. Choi, J. Park, K. Cho, T. Oh, and M. Shahidehpour, “Probabilisticreliability evaluation of composite power systems including wind turbinegenerators,” in Proc. IEEE International Conference on ProbabilisticMethods Applied to Power Systems (PMAPS), June 2010, pp. 802–807.

[6] M.-R. Haghifam and S. Soltani, “Reliability models for wind farms ingeneration system planning,” in Proc. IEEE International Conference onProbabilistic Methods Applied to Power Systems (PMAPS), June 2010,pp. 436–441.

[7] A. Mehrtash, P. Wang, and L. Goel, “Reliability evaluation of re-structured power systems with wind farms using equivalent multi-state models,” in Proc. IEEE International Conference on ProbabilisticMethods Applied to Power Systems (PMAPS), June 2010, pp. 716–721.

[8] L. Wu, J. Park, J. Choi, A. El-Keib, M. Shahidehpour, and R. Billinton,“Probabilistic reliability evaluation of power systems including windturbine generators using a simplified multi-state model: A case study,”in Power Energy Society General Meeting, 2009. PES ’09. IEEE, July2009, pp. 1–6.

[9] M. Mohanpurkar and R. Ramakumar, “Probability density functions forpower output of wind electric conversion systems,” in IEEE Power andEnergy Society General Meeting, July 2010, pp. 1–7.

[10] S. Karaki, R. Chedid, and R. Ramadan, “Probabilistic performanceassessment of autonomous solar-wind energy conversion systems,” IEEETransactions on Energy Conversion, vol. 14, no. 3, pp. 766–772,September 1999.

[11] A. P. Leite, C. L. T. Borges, and D. M. Falcão, “Probabilistic WindFarms Generation Model for Reliability Studies Applied to BrazilianSites,” IEEE Transactions on Power Systems, vol. 21, no. 4, pp. 1493–1501, November 2008.

[12] F. Castro Sayas and R. Allan, “Generation availability assessment ofwind farms,” IEE Processdings-Generation, Transmission and Distribu-tion, vol. 143, no. 5, pp. 507–518, September 1996.

[13] G. M. Masters, Renewable and Efficient Electric Power Systems. Hobo-ken, NJ: Wiley Interscience, 2004.

[14] Class notes for ENG571, “Theory energy and sustainable engineering,”University of Illinois, 2010.

[15] Vestas, “V90-1.8/2.0 MW data sheet,” 2011. [Online]. Available:"http://www.vestas.com"

[16] H. Stark and J. W. Woods, Probability, Random Processes, and Esti-mation Theory for Engineers. Englewood Cliffs, NJ: Prentice Hall,1994.

[17] T. Ackermann, Wind Power in Power Systems. West Sussex, England:John Wiley and Sons, 2005.