Performance prediction of active pitch-regulated wind turbine with short duration variations in source wind

Applied Energy 114 (2014) 700–708

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Applied Energy journal homepage: www.elsevier.com/locate/apenergy

Performance prediction of active pitch-regulated wind turbine with short duration variations in source wind Sanjoy Roy ⇑ Department of Electrical Engineering, Indian Institute of Technology, Ropar, Pb 140001, India

h i g h l i g h t s  Uses turbulence intensity as a parametric measure of short duration wind variations.  Derives statistical expression for the short duration output power curve for a WECS.  Derives statistical expression for the short duration output power covariance for a WECS.  Establishes algorithm for computation of short duration output power variability.  Compares statistical estimates according to IEC 1400-1 with empirical data.

a r t i c l e

i n f o

Article history: Received 15 March 2013 Received in revised form 19 July 2013 Accepted 6 October 2013

Keywords: Wind energy Weibull statistics Pitch-regulated turbines Short duration power distortion

a b s t r a c t Short duration wind variations affect real time performance of active pitch-regulated wind turbines in two ways as evident from reported experimental and empirical studies. First the mean output power, which may be referred to as the short duration output power, differs significantly from the corresponding zero-turbulence value obtained with ideal source wind streamlines. Second, random variation of output around the mean value appears with a significant standard deviation; the normalised value of which is referred to as the short duration variability. In this paper, analytical interpretation of both metrics is presented under assumption of two-parameter Weibull statistics for short duration wind variations. Statistical estimates for the metrics are presented for conditions described by the well known IEC 61400-1 Standards. Finally the statistical estimation procedure is applied to a Vestas V90 3 MW zero-turbulence output curve as an illustrative application example. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction 1.1. Challenges posed by short duration wind variations Randomness inherent in source wind is an unavoidable challenge for large scale integration of wind energy conversion systems (WECS) in conventional power networks [1,2]. By current concepts, this phenomenon is clearly demarcated between long and short duration wind variations. The former type of variations are noticeable in speed data recorded at hourly or half-hourly time steps, and result in real time operational problems of network congestion and unreliable supply [3–8]. Spatial distribution of wind turbines as well as temporal speed swings are equally important in deciding the impact of long duration wind variations on WECS operation and control. While reliable and thorough on-site records of wind speed are requisite inputs for the studies in question; over long time horizons these are often found to be imprecise representation of wind at the turbine hub. As an alternative, wind speed ⇑ Tel.: +91 1881 24 2174; fax: +91 1881 22 3395. E-mail address: [email protected] 0306-2619/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.apenergy.2013.10.009

records over shorter time horizons at nearby measuring stations are processed by data mining so as to generate long-term correlated time series applicable at specific WECS installation sites [9]. A second popular alternative is to employ reduced order numerical weather prediction models (often referred to as mesoscale models) for generation of long duration wind speed patterns. A comprehensive discussion on a range of such developments is presented in [10]. Short duration wind variations on the other hand, are too rapid for the consequent changes in turbine output to be detected as deterministic time series [11]. More than ten swings of wind speed between 7.0 m/s and 10.5 m/s over time spans as short as 140 ms have sometimes been reported [12]. Physically, short duration wind variations include turbulence and gusts, with components that follow from one or more of the following origins [13–15]:     

Ambient turbulence. Vertical wind shear. Wake development behind swept area of turbines. Mutual wakes due to proximity between turbines. Complexity of terrain or surrounding vegetation.

S. Roy / Applied Energy 114 (2014) 700–708

Analysis of the consequent ten-minute variations of turbine output power is complicated due to three reasons: a. Problem of deterministic description [16,17] owing to the rapidity of wind variations. b. Nonlinearity of the WECS power output curve that ‘‘converts’’ wind speed to output power [16]. c. Automated disconnection and reconnection of the WECS unit [18], as governed by variation of wind speed around the designed cut-out value. In summary, the phenomena of short duration wind variations as well as the consequent operational effects on WECS units are subject of much debate, leading to extremely diverse opinion between those interested [19,20].

1.2. Quantification metrics Performance evaluation of WECS typically involves two metrics that are ‘‘averaged estimates’’ of real time variables affected by short duration wind variations:  Short duration output power: the mean output power from a WECS unit; utility interest being focused on its deviation from the corresponding ideal zero-turbulence value [16,17,21].  Output power variability: the standard deviation of power output by a WECS unit, normalised by its rated capacity; this being a measure of randomness of power ‘‘around’’ the short duration output power [22–24]. Perhaps somewhat expectedly, short duration output power has largely been of interest to research groups that assess ‘‘distortion’’ of output power curve due to wind variations. Simulations and experiments of interest are often performed for single WECS units, as references cited above indicate. Though interest in output power variability is not rare among technology research groups, power utilities and associated consultants have shown concerted effort in real-time evaluation of this metric [22–24]. The effort stems from the operational need to ‘‘match’’ random power output by generators to random demand from system loads, which poses a challenge to large scale integration of WECS units in conventional power networks. For example, if the short duration standard deviation of connected load is represented by wL and that of the i-th wind power source is denoted by wi (each distribution assumed to be approximately normal and uncorrelated with others), then the overall standard deviation inclusive of load and WECS units (indexed as i = 1,. . ., N) is [22]

wTotal

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N u X ¼ tw2L þ w2i

ð1Þ

i¼1

The standard deviation of load net of wind generation is useful to the utility in deciding the regulating reserve. A comprehensive presentation on operational and economic aspects of the problem is available as [25]. The distinct importance of the two metrics follows immediately from the above discussion. Short duration output power can be simply viewed as the average output from an active-pitch controlled WECS. The concept should therefore allow one to ‘‘correct’’ ideal zero-turbulence WECS output curves for short duration wind variations. On the other hand output power variability allows assessment of overall randomness in utility power inclusive of connected load, and thereby facilitates plans for regulating reserve.

701

Empirical estimation of either metric involves recording of real-time data ensembles followed by ensemble averaging. With implicit assumption of ergodicity, accuracy of computation depends on the recording time horizon [22–24]. Such estimates are referred to as ensemble estimates in the rest of this paper. 1.3. Focus and objective Accuracy of empirical ensemble estimates is governed by the estimation time horizon and the samples within. Additionally if wind speed data is approximately ergodic, then ensemble estimate of short duration output power would approach the mean output power at a specific mean wind speed. Similarly, ensemble estimate of output power variability would approximate the normalised standard deviation of short duration power  given the mean wind speed. If available as an alternative, a direct computational approach to statistical estimation of the metrics may be expected to be accompanied by the following advantages: 1. A significant reduction in computational effort is expected to follow, since data recordings over limited time horizons are not a primary requirement for statistical estimates. 2. Unlike ensemble estimates, statistical estimates have little dependence on the ergodicity assumption; that is, wind speed data distribution over time need not be identical to probability distribution of wind speed at an instant. 3. Statistical estimates are in general expected to be free of recording errors. 4. Most importantly, formulations may include suitable parametric representation of turbulence in source wind, with consequent computational convenience. This paper presents closed form analytical expressions for short duration output power and output power variability applicable to horizontal axis WECS with active-pitch control. The aim of the work is to explain and corroborate empirically observed trends for both metrics by the analysis introduced. The formulations are based two fundamental assumptions. First, a suitable statistical distribution is assumed for short duration wind variations; with parametric representation for the phenomenon of turbulence. Second, a suitable definition is assumed for the ideal zero-turbulence output curve of a horizontal axis WECS with active-pitch control. Section 2 provides a detailed review of relevant observations from simulations and field studies, as reported in existing literature. Section 3 formulates closed form expressions for short duration output power and output power variability of a WECS unit with active pitch-regulation. Section 4 applies the derived expressions to calculate statistical estimates for both metrics under realistic wind conditions at a typical WECS turbine hub, inclusive of short duration variations. Section 5 justifies statistical estimates corresponding to wind conditions according to the IEC 61400-1 Standards, against observed trends of ensemble estimates described in Section 2. It concludes by a computation example involving the well known Vestas V-90 3 MW WECS, so as to illustrate application of the concepts introduced in Section 3. 2. Reported data and existing practices 2.1. The concept of turbulence intensity If complexity of deterministic bifurcation and chaos theory is to be avoided [26], then the alternative option of statistical analysis allows representation of short duration wind variations in terms

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of a simple parameter, namely the turbulence intensity s. With  respecstandard deviation and mean of wind speed as r and u tively, s is given by [12]

s,r=u

ð2Þ

In practice, several useful features may have contributed to the popularity of s as a representative parameter:

in the form of Table 1, shows approximate values for negative and positive power deviations around rated and cut-in conditions, respectively. As in other observations reported in [16,17], the data in Table 1 shows increasing magnitudes of power deviation for higher turbulence intensity, both around uin and urat. 2.3. Experience with ensemble estimates of output power variability

 An easy to interpret, dimensionless quantification for randomness of source wind.  [12],  Ease of ten-minute ensemble estimation for r and u and hence s.  Viable sites for wind based generation are found to have s in the range of 0.1–0.4, with preferred levels of 0.2 or less.  Convenient quantification of wind turbine design conditions by the International Electrotechnical Commission (the IEC 1400-1 Safety Requirements [27]), in terms of s.  s is representative of cumulative turbulence experienced by a WECS [13], and may combine a variety of turbulence components of different origin (refer Section 1.1).

As explained in Section 1.2, output power variability of WECS units attracts considerable interest of power utilities. The investigations reported below are sourced accordingly. While confirming the importance of output power variability as observed through North American wind integration studies across 2007–09, [22] reports some typical observations based on three years’ of actual scenario profile data (ten-minute averages) as part of the NREL-EnerNex Eastern Wind Integration and Transmission Study:  For any specific WECS cluster, the output power variability monotonically increases from very negligible values corresponding to small fractions of rated output (around cut-in conditions) to a maximum. Following the maxima, once again values of estimated variability progressively drop to negligible levels at full rated output (around base wind speed conditions). To summarise, variability exhibits a clear monotonic trend with a maxima between zero and rated values of output power.  The monotonic trend described above for output power variability is asymmetric  the maxima occurs at output power levels stated in Table 2 (and not necessarily at half the rated value).  For larger WECS clusters, as the overall power rating increases, the per-unit output power variability generally drops as shown in Table 2. This is to be expected due to significant ‘‘mutual smoothing’’ of short duration power variations between WECS units.

The last mentioned feature establishes the strength of turbulence intensity as a unique parametric consolidation of short duration wind variations. Following the treatment presented in [13], the effective value of s suitably compounds all components of short duration speed variations into a single quantity. The natural question to follow is therefore about how WECS output power is affected by short duration wind variations; and to what extent can the effect be possibly interpreted in terms of overall turbulence intensity? As an attempt to address these questions, several established studies are cited in the following subsections. 2.2. Experience with ensemble estimates of short duration output power Ref. [16] as well as previous work by the same authors, have reported on the basis of ten-minute ensembles, ‘‘... A shift of measured power curves to higher power outputs where the power curve is left curved (at lower wind speeds around maximum Cp) and to lower power outputs where the power curve is right curved (at wind speeds in the transition region to rated power)...’’. Around the point of maximum power coefficient (Cp), [16] observes an increase of power output by 1–2% corresponding to each percent increase in turbulence intensity. The study, though an early one, notes that ‘‘...the observed effect of turbulence intensity is significantly larger than the effect of 10-min averaging’’ [16]. This supports the premise that ensemble estimates may be prone to significant measurement and computational error, so that reliable statistical estimates are desirable as well as convenient A range of simulation results that corroborate deviation of short duration output power from corresponding zero-turbulence values have been reported in [17]. The reference attempts to judiciously distinguish between the effects of wind shear and turbulence through two distinct sets of aerodynamic simulation studies. The first of these focuses on a 3.6 MW Siemens turbine unit, as simulated on Risø-DTU’s HAWC2Aero software. The second uses a Vestas VTS simulation of a generic wind turbine with 80 m hub-height and 100 m rotor diameter. Both sets of studies yield scatters of short duration output power against wind speed, which indicate clear deviations from zero-turbulence output power curves similar to those reported in [16]. More usefully, [17] provides representative values of normalised power deviations for a typical turbine (rated or base wind speed urat = 12 m/s) when operating with different values of average wind speed at hub and turbulence intensity. The data summarised below

The WECS cluster integration studies presented in [23] span over several North American states with three types of footprint scenarios: i. In-area scenarios, each involving wind power resources of a single state with assumed zero inter-state exchange. ii. Local priority scenarios involving renewable resources across multiple states, but with a ten percent advantage on capital cost for generators local to a state. iii. Mega project scenarios that assume unrestricted power exchange between states without any economic advantage assigned to local resources.

Table 1 Percent deviation of output power from ideal zero-turbulence power curve [17]. (A dash ‘–’ is indicative of a negligibly small value.) Wind speed at hub (m/s)

5 6 7 8 9 10 11 12

Turbulence intensity (%) 0

10

15

20

+6.67 +1.11 – – – – – 2.77

+27.78 +7.22 +2.22 +1.11 – – – 5.00

+55.00 +17.22 +6.67 +3.89 +3.89 +2.22 1.11 7.78

+85.56 +32.22 +13.33 +8.33 +6.67 +5.83 2.77 10.00

S. Roy / Applied Energy 114 (2014) 700–708

choice. Though one may be tempted to choose urat for the purpose, the choice need not be universal since the speed variables relate mutually in (5) as a ratio. Eqs. (4), (5) essentially define a conditional steady state description for power output by a WECS; the ‘‘condition’’ being absence of short duration wind variations. In reality, the power output would respond to the wind variations as decided by dynamics of the WECS. Under the four assumptions that follow, (4), (5) can be extended as a quasi-steady state description to such short duration dynamic conditions:

Table 2 Observations on normalised ten-minute variability from EWITS studies [22]. Rated output of WECS cluster (MW)

Maximum variability

Output at max. variability (pu Prat)

500 5000 15,000 40,000 85,000

0.067 0.026 0.021 0.017 0.019

0.433 0.417 0.416 0.400 0.458

Only in mega project case studies, it is found that output power variability depends on short duration output power in a manner similar to individual WECS units. This is to be expected due to significant ‘‘mutual smoothing’’ of turbulence effects as mentioned above. [23] goes on to examine various regulating reserve rules and practices as decided by output power variability; and shows how distinct reserve rules must apply if mega project type scenarios are to eventually evolve. The obvious importance of turbulence intensity as a critical determinant of wind based generation follows. Similar to [23], Ref. [24] presents output power variability against levels of aggregate output power under different conditions of wind persistence. It goes onto establish that the reserve regulation requirement can be augmented according to

Regulation requirement with wind sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 Load regulation 2 ¼3 þ ðVariabilityÞ 3

703

ð3Þ

i. Only wind streamlines perpendicular to turbine swept area contribute to power output by a WECS; effect of any other streamline being negligible. ii. Movement of the turbine nacelle and blades does not affect wind streamlines appreciably. iii. Cumulative response speed of pitch angle controller, turbine blades, nacelle, and gear train is very fast in comparison to short duration wind speed variations. iv. Dynamics of turbine inertia and generator it drives are very fast, again in comparison to short duration variations of wind speed. The treatment presented in the next subsection assume strict validity of ‘‘i–iv’’. The extent of their validity for a practical WECS unit would decide applicability of the formulations. 3.2. The probability distribution of wind speed in short duration

for large scale wind resource integration in a power network.

Any acceptable probability density function p(u) that describes short duration wind variations must necessarily satisfy

3. Theory

Z

Manufacturers of commercial pitch angle controlled (PAC) WECS units provide the customer utility with an output curve, which is essentially a plot of PAC output power P(u) as a function of steady source wind speed u at hub. Given the specifications of cut-in (uin), rated (urat), and cut-out (uout) wind speed at hub [28] and power rating Prat of the WECS, P(u) can be expressed by a fractional ideal zero-turbulence output coefficient l(u) as

ð4Þ

where l(u) can be conveniently described as [28]:

8 0; > > > < ðuv  uv Þ=ðuv  uv Þ; rat in in mðuÞ ¼ > 1; > > : 0;

if u 6 uin if uin < u 6 urat if urat < u 6 uout

pðuÞ  du ¼ 1

ð6Þ

0

3.1. The WECS output curve with steady streamlined source wind

PðuÞ ¼ lðuÞ  Prat

1

ð5Þ

if u > uout

The index m typically assumes values in the range of one to three, depending on the drive-train and generator in a specific configuration. For turbines with axial induction factor of 0.5 or less driving synchronous generators, m is close to a value of three. Induction generator based units usually have m k 1.0, though those with doubly-fed induction generator approximately follow a cubic function. For a detailed discussion on this aspect, the reader may refer to [12]. At this juncture it is useful to contemplate a convenient perunit system for variables that occur in (4), (5) and subsequent analytical treatment. For all power variables, (4) suggests a base value equal to the rated capacity Prat. It immediately follows from (5) that the ideal zero-turbulence output coefficient l(u) can be alternately interpreted as the ideal zero-turbulence per-unit power output. A base value for the speed variable u is relatively open to

Statistical description of wind behaviour was originally motivated by a need to model wind load on buildings  an interest that gained prominence in the 1960s. Over the subsequent decade, the two-parameter Weibull distribution found application in the domain of wind power [29] as an outcome of several encouraging observations:  The need for an acceptable probability density function between cut-in and rated values of wind speed for the PAC turbine  a purpose for which the two-parameter Weibull is a suitable candidate.  The parameters C and j that describe the Weibull distribution can be easily estimated from wind speed data to good degree of accuracy.  When validated against field recordings, the two-parameter Weibull description result in very low rms errors in comparison to other candidate descriptions (such as the log-normal distribution [29]). Recent appreciation of progressively precise concepts (the distinction between short and long duration variations, for example) has led to investigation of alternative statistical descriptions. A comprehensive review reported in [30] concludes by recognising the two-parameter Weibull as the most appropriate choice on grounds of flexibility, a convenient and continuous functional description, and adequate justification by goodness-of-fit tests. To put the discussion in fair perspective, the two-parameter Weibull is found to be an unsatisfactory description of wind speed data in cases of (a) high incidence of null wind speed, and (b) bimodal distributions [31]. In such situations, it is essential to use case specific statistical descriptions despite possible computational burden and practical inconvenience.

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S. Roy / Applied Energy 114 (2014) 700–708

Accepting the two-parameter Weibull distribution as the description for wind speed statistics at turbine hub, its probability density function is given in terms of short duration scale factor C and shape factor j [32] as

pðuÞ ¼

j uj1 C C

  j  u  exp  C

ð7Þ

It follows from (6), (7) that C must have the same units (or the same per-unit system) as u, while j is always a dimensionless parameter.  is then obtained as The short duration stable mean wind speed u

  1  ¼CC 1þ u

ð8Þ

j

where C() is the complete gamma function for the independent index parameter (1 + 1/j) [32]. An established empirical approximation for j [33], within the loose range of 1 [ j [ 10 is

j  ðr=uÞ

1:086

¼s

1:086

ð9Þ

From the turbulence intensity s at the turbine hub, Weibull parameters j and C can be computed corresponding to any short duration  (9), (8). Thus (7) can be fully defined if stable mean wind speed u (l, r) are known and is therefore, better rewritten as

 ; rÞ ¼ pðuju

j uj1 C C

  j  u  exp  C

ð10Þ

The output power variability t of a WECS unit is the normalised standard deviation of its short duration output power. Therefore

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1  ; rÞi2  pðuju  ; rÞg  du ) t t  Prat , ½fPðuÞ  hPðu 0

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Z 1 1 2 2   ¼ ½PðuÞ  pðuju; rÞ  du  hPðu; rÞi Prat 0

ð15Þ

 ; rÞi appears in (15) as a known staIt is important to note that hPðu tistical mean value and not a variable. The integral in (15) is iden ; rÞiÞ. tical to the short duration covariance of output power ðhP2 ðu The covariance can once again be evaluated by integration steps detailed in [34] so as to obtain  ; rÞi ¼ P 2rat hP 2 ðu

Z

1

l2 ðuÞ  pðuju; rÞ  du 0

  urat  j 2m uin m  m 2m C ; ðu=CÞj  C ; ðu=CÞj  P2rat  eðuout =C Þ j j C ðumrat  umin Þ j uin   urat  C 2m P2rat 2m uin m  m 2m   C ; ðu=CÞj  C ; ðu=CÞj ð16Þ 2 j m m j j C ðurat  uin Þ uin ¼

C 2m P2rat

2



The last approximation in (16) is valid if the uout is very high in comparison to the short duration scale factor C. The following steps outline a procedure for estimation of output power variability using the expressions formulated thus far:

3.3. Formulations for statistical estimates With valid assumptions ‘‘i–iv’’ of Section 3.1, and with l(u) as defined by (5) the statistical estimate of short duration output power can be computed as

 ; rÞi, hPðu

Z

1

 ; rÞ  du ¼ Prat PðuÞ  pðuju

0

Z

1

lðuÞ  pðuju; rÞ  du

0

ð11Þ By steps detailed in [34], (11) can be solved further to obtain  m i j C m ðm=jÞP rat h  m  ; rÞi ¼ m

C ;ðuin =CÞj  C ; ðurat =CÞj  P rat  eðuout =CÞ hPðu j j urat  umin  m i C m ðm=jÞP rat h  m ð12Þ  m C ;ðuin =CÞj  C ; ðurat =CÞj j j ðurat  umin Þ

 ; rÞ, j and C can be computed by (9) and (8). where for specific ðu The two C(, ) terms in (12) are upper incomplete gamma functions for index parameter (m/j) with lower limits of integration (uin/C)j and (urat/C)j respectively [32]. The last approximation in (12) applies to WECS for which uout is very high in comparison to the short duration scale factor C. e ðu  ; rÞ can be deBy (12), the short duration output coefficient l fined as a ‘‘corrected form’’ of (5) that includes considerations of short duration wind variations:

 m i j C m ðm=jÞ h  m C ; ðuin =CÞj  C ; ðurat =CÞj  eðuout =CÞ m m ðurat  uin Þ j j  m i C m ðm=jÞ h  m  m C ; ðuin =CÞj  C ; ðurat =CÞj ð13Þ m ðurat  uin Þ j j

le ðu; rÞ ¼

so that

 ; rÞi ¼ l e ðu  ; rÞ  Prat hPðu

ð14Þ

Following the per-unit interpretation of l(u) described above, e ðu  ; rÞ can be accordingly interpreted as the short duration per-unit l output power on the same power base Prat. (Notably, the base wind speed continues to be an open choice since each occurrence of u in (13) includes a division by C).

 and turbulence s at the WECS hub, r a. Given the values for u and thereby the parameters C and j can be computed by (8), (9). b. The short duration output power from the WECS can be estimated by (12). c. The short duration covariance of WECS output power can be estimated by (16). d. Finally, the short duration variability can be evaluated by (15). The algorithm ‘‘a–d’’ is convenient to apply over different combina and r, and has no direct requirement of field data. Cumtions of u bersome ensemble recordings as well as associated measurement errors are therefore avoided altogether.

4. Calculation Experimental research on output power variability usually report ensemble estimates corresponding to specific levels of per-unit short duration output power [22–24]. The flexibility allowed by (12)–(16) together with algorithm ‘‘a–d’’ of the previous subsection encourages alternate forms of data analysis as well. Indeed, the formulation makes it possible (at least numerically) to obtain estimates of short duration output power and output power variability  and s; and the user may for practically any combination of u choose parameter values for specific cases.  ; rÞ may vary significantly due to arbiIn general, values of ðu trary choices of sites, WECS makes, or their formation at the installation site [13]. Both editions of the IEC 61400-1 standards [35,36] discuss turbulence effects with reference to wind loading on WECS units. Empirical expressions presented in either standards are open to interpretation and have been widely discussed [37,38]. However they represent turbulent wind condition over a wide range of sites and WECS units, and are good choice for the analysis presented in this paper. The 1999 edition of IEC 61400-1 [35] was originally intended for turbines with swept area equal to or greater than 40 m2. The

S. Roy / Applied Energy 114 (2014) 700–708

 and r (both standards introduces the empirical relation between u in units of m/s)

r ¼ s15  ð15 þ a  uÞ=ða þ 1Þ

ð17Þ

where s15 is the turbulence intensity measured at a height of 15 m above ground and ‘‘a’’ is an empirical slope parameter. For high turbulence characteristics of WECS (category A), relevant parameters are s15 = 0.18 and a = 2.0. Corresponding values for low turbulence characteristics (category B) are s15 = 0.16 and a = 3.0, respectively. The 2005 edition of IEC 61400-1 [36] restricts applicability of (17) to small WECS units; while the following alternative is intro and r in duced for a broader gamut of WECS ratings (again both u units of m/s):

r ¼ s15  ð0:75  u þ 5:6Þ

ð18Þ

Eq. (18) predicts r by the 90% quantile of wind speed at WECS hub  , and has been used for studies height for ten-minute averages of u presented in the next section. For this expression, suggested values for s15 [36] are 0.16 for category A representing high turbulence, 0.14 for category B representing medium turbulence, and 0.12 for category C representing low turbulence characteristics. Application of (17), (18) (or any suitably acceptable alternative)  . Statistical estican predict standard deviation r for any value of u mates of both short duration output power and short duration variability can then be obtained by the procedure ‘‘a–d’’ of Section 3.3. 5. Results and discussion 5.1. Statistical estimates of short duration output power For a representative set of WECS, wind speed specifications are assumed as urat = 11 m/s, uin = 0.32 pu and uout = 2.27 pu (all perunit values for this example use urat as the base speed). Some of the commercial makes being currently manufactured and marketed by Suzlon Energy are designed for similar specifications [39]. Values of m = 1 and m = 3 are used to respectively represent variation of output power between cut-in and rated values of wind speed, the engineering implications of which have been discussed in Section 3.1. For both values of m, Fig. 1 displays statistical estimates of short duration output power in per-unit, with Prat as base e ðu  ; rÞ power. As mentioned before, this is identical to a plot of l  with r decided by u  against corresponding mean wind speed u according to (18). The short duration output power curve is found to progressively deviate from the ideal zero-turbulence curve as conditions change from IEC turbulence category C to A. An important aspect common to turbulence index s, short duration shape parameter j, and short duration scale parameter C, should be noted at this stage. The long duration values of j and C sustain over significant time horizons at a WECS installation site, largely decided by seasonal conditions [12.29]. Over the short duration however, empirical relations such as (18) decide the dependence of randomness of wind speed on its mean value, so  . This interesting observathat s, j, and C are together decided by u  , which tion follows from division of (18) by the mean wind speed u leads to the following empirical variation for turbulence intensity

s ¼ s15  ð0:75 þ 5:6=uÞ

705

and it is useful to keep these in view when examining the statistical estimates presented through Figs. 1–4. For all three turbulence categories, significant reduction is noted in the range of wind speed that would allow a WECS to output rated power (Fig. 1). The ‘‘rated power segment’’ of the output curve ‘‘becomes shorter’’ as turbulence in the source wind increases; an observation that agrees with power curve definitions reported in [40] and used widely in case study simulations (for example, [5]). Table 4 quantifies this effect neglecting up to one percent reduction of short duration output power over the nominal value Prat. Notably under severity of category A turbulence, a WECS cannot be expected to output power within one percent of Prat at any acceptable mean wind speed! Category B turbulence allows power output within one percent of Prat across a range of 0.1pu wind speed if m = 3, and 0.2 pu if m = 1. Corresponding wind speed range for category C turbulence are 0.3 pu and 0.4 pu corresponding to m = 3 and m = 1, respectively. It is apparent from Fig. 1 that deviations of short duration output coefficient from corresponding ideal zero-turbulence values depend on changes in slope of l(u) according to (5). The ‘‘sharper’’ a dise ðu  ; rÞ continuity in l(u), the more significant is the deviation of l e ðu  ; rÞ  lðu  Þg at the specific mean wind speed. The differential f l between cut-in and rated speeds, for both values of m, are shown in Fig. 2. With m = 3, the change in slope of l(u) is gentle at cutin, but relatively sharp at rated speed. Accordingly, Fig. 2 shows e ðu  ; rÞ  lðu  Þg to be gradual around cut-in, and the difference f l limited to about 0.05 for either turbulence category. With m = 1, the change in slope is comparable at cut-in and rated wind speed, e ðu  ; rÞ  lð u  Þg at cut-in and this reflects on comparable values of f l and rated speed as seen from the lower plot of Fig. 2. e ðu  ; rÞ from lðu  Þ around cut-out is The substantial deviation of l explained by the automated shutdown or startup by a WECS unit whenever wind varies ‘‘across the cutout point’’ in short duration. Analytically, this is represented by the exponential term involving e ðu  ; rÞ  lðu  Þg uout in (17), (18). Fig. 3 shows that the differential f l can assume magnitudes of up to 0.5 both above and below the cut-out wind speed. The plots included in Fig. 3 show the abrupt

ð19Þ

The turbulence intensity s obviously assumes very high values at low mean wind speed, while it approaches the limit of 0.75s15 at high speeds. Accordingly by (9), the short duration Weibull shape factor j assumes different range of values at cut-in, rated, and cut-out conditions. The short duration Weibull scale factor C on the other hand (8), increases consistently with mean wind speed. Specific values assumed by each parameter are shown in Table 3,

~ ðu  ; rÞ Fig. 1. Per-unit short duration output power on a base of Prat (identical to l corresponding to IEC 61400-1 turbulence categories A (red), B (green), and C (violet). The zero turbulence per-unit output (l(u)) is plotted in blue. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.).

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Table 3 Short duration wind speed distribution parameters for the example of Section 5.1. IEC turbulence category

A

B

C

Turbulence intensity s by (19):

Cut-in Rated mean speed Cut-out

0.39 0.20 0.16

0.34 0.18 0.14

0.29 0.15 0.12

Weibull shape factor j by (9):

Cut-in Rated mean speed Cut-out

2.77 5.70 7.55

3.20 6.59 8.73

3.78 7.79 10.32

Weibull scale factor C by (8):

Cut-in Rated mean speed Cut-out

0.34 1.08 2.45

0.33 1.07 2.43

0.33 1.06 2.41

Fig. 2. Per-unit deviation of short duration output power from the zero turbulence ~ ðu  ; rÞ  lðuÞ) between cut-in and rated wind speed. The plots value (identical to l correspond to IEC 61400-1 turbulence categories A (red), B (green), and C (violet). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.).

change in l(u) as the dominant factor, so that the differential plot e ðu  ; rÞ  lðu  Þg changes only marginally between IEC turbuof f l lence categories.

increasing levels of short duration output power. After reaching a loosely detectable maximum, the ensemble variability values fall monotonically with increasing output power (up to Prat and beyond). Needless to say that an acceptable analytical explanation for the trend is difficult to attempt. The computed results that follow justify the formulations of Section 3 as analytical explanation for variability trends summarised above. Fig. 4 presents the statistical estimates for output power variability against short duration output power as obtained by steps ‘‘a–d’’ described earlier in Section 3.4, corresponding to all three turbulence categories [36]. A comprehensive view of these reveals strong corroboration with the reported empirical observations [22–24]; details to follow. For all three categories of wind turbulence, WECS units with m = 1 are expected to perform better than those working with m = 3. Since variations in wind converts to larger power swings  Þ ¼ P rat in comparison to lower levels, most variability around Pðu  ; rÞi. The plots in Fig. 4 show an asymmetry towards higher hPðu skewness or asymmetry is more pronounced in the case of cubic transition of output power across the speed range (uin, urat). This is to be expected since for similar variations in wind speed, m = 3 converts to higher levels of power covariance (16) as compared to m = 1. Finally, between the three turbulence categories, A has the  as obtained by (18) and consequently highest r for any value of u higher output power variability.

5.2. Statistical estimates of output power variability 5.3. An application example The popular practice for analysis of ensemble variability at wind installations is to study the data-scatter against short duration output power [22–24]. This follows from the obvious question of expected variability at specific levels of output that is critical for utilities and practicing engineers. Further, such analysis has a strong bearing on dispatch and commitment decisions. Data-scatters presented in [22–24] show commonly observed trends for ensemble variability. In all such scatters, the ensemble variability increases monotonically from near zero values, with

Fig. 3. Per-unit deviation of short duration output power from the zero turbulence ~ ðu  ; rÞ  lðuÞ) around cut-out wind speed. The plots correspond value (identical to l to IEC 61400-1 turbulence categories A (red), B (green), and C (violet). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.).

It is useful to close this section with a ‘‘tutorial’’ illustrative of the steps, by which short duration performance of any commercial make of WECS may be estimated. The popular Vestas V90 3 MW turbine [41] is selected as the make of interest. The zero-turbulence output curve for the V90 is available online in [42]. In two important ways, this example augments the concepts illustrated in earlier subsections. First, [41] provides the rated power for the V90 as 3 MW with characteristic wind speeds at hub as rated 15 m/s, cut-in 3.5 m/s, and cut-out 25 m/s. Therefore, the choice of base power as 3 MW and base wind speed as 15 m/s follow almost directly. However (in contrast to the case presented in Section 5.1), the ideal zero-turbulence output curve for the V90 provided in [42] shows that rated power is output by the V90 at wind speeds of 14 m/s upwards until cut-out, which is 0.93 pu on the 15 m/s base. The rated power being available at wind speeds somewhat below the base value is a feature incorporated in many commercial makes of WECS. Its implications on performance have been presented as part of a sensitivity analysis in [34]. For the present example, expressions (5), (12), (13), and (16) apply equally well to the V90, with urat replaced by 0.93 pu (not 1.0 pu), while uin = 0.23 pu and uout = 1.67 pu on the 15 m/s base. Second, the ideal zero-turbulence output curve (hence output power coefficient l(u)) for the V90 is specified in [41] as a set of

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Fig. 5. Per-unit short duration output power for the Vestas V90 on a base of 3 MW ~ ðu  ; rÞ) corresponding to IEC 61400-1 turbulence categories A (red), B (identical to l (green), and C (violet). The square markers plotted indicate the ideal zeroturbulence per-unit power curve (l(u)) as per specification [42]. The function (19) with m = 1.86 fitted into the data by least-squares error minimisation is plotted in blue. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.).

Fig. 4. Statistical estimates for output power variability against per-unit short duration output power. The plots correspond to IEC 61400-1 turbulence categories A (red), B (green), and C (violet). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.).

Table 4 Per-unit range of mean wind speed corresponding to 99–100% of rated power output. Turbulence category

m=3

m=1

A B C

– 1.6–1.7 1.5–1.8

– 1.5–1.7 1.4–1.8

plotted points indicating output power against wind speed at hub. This form of ideal output curve specification is popular with WECS manufacturers as it is not restricted by predefined mathematical functions, and is convenient for display of experimentally determined data. However, in order to apply the concepts presented in Section 3, it is necessary to have the zero-turbulence output curve in the form defined by (4), (5); that is a definite value is required for the index parameter m. Fig. 5 shows the output curve points for the V90 [41], plotted as the function l(u). In order to obtain an acceptable value of m for the WECS, the function (5) is fitted by least-squares error minimisation within the range [u = 0.23 pu, l(u) = 0] to [u = 0.93 pu, l(u) = 1] so as to obtain m = 1.86. Thus the zero turbulence output power for the V90 is completely defined by (4) in conjunction with

8 0; > > > < ðu1:86  0:231:86 Þ=ð0:931:86  0:231:86 Þ; lðuÞ ¼ > 1; > > : 0;

if u 6 0:23 if 0:23 < u 6 0:93

9 > > > =

if 0:93 < u 6 1:67 > > > ; if u > 1:67 ð20Þ

where the values of wind speed u are in per-unit on the base of 15 m/s. With short duration wind conditions given by (18), Fig. 5 plots e ðu  ; rÞ by (13) for statistical estimates of output power coefficient l the IEC turbulence categories A–C. It is observed that with category C turbulence, the V90 outputs within one percent of Prat only across the mean wind speed range of 1.2–1.3 pu. With category B turbulence, the output just about reaches Prat around 1.2 pu wind speed,

Fig. 6. Statistical estimates for output power variability against per-unit short duration output power. For the Vestas V90 3 MW WECS The plots correspond to IEC 61400-1 turbulence categories A (red), B (green), and C (violet). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.).

while with category A, the short duration output is significantly below rated power at all mean wind speeds. Fig. 6 plots short duration output power variability for the V90 operating in IEC turbulence categories A–C. While the maximum variability is found to be in the range of 0.15–0.25 pu similar to the case of m = 1 of Fig. 4, the plots are asymmetric into the extent comparable to the curves for m = 3. 6. Conclusions This paper has proposed analytical formulations that explain commonly observed trends for short duration output power from a PAC wind turbine, as well as its variability, when operating in presence of short duration wind variations. Together with empirical relations that quantify randomness in wind speed (such as those specified by the IEC), the formulations can be used to generate the short duration output power curve for a given WECS, with statistical estimates of power variability. This provides a route to wind resource assessment at specific installation sites, without having to depend on excessive amount of measured data. The expressions presented can be useful in prediction of the two performance metrics for individual WECS units; and may thereby be used to assess expected cumulative power and its variations across spatially distributed units within a wind based generating station. Since turbulence intensity is a well established and accepted measure for variations in wind speed, statistical estimates of both metrics have obvious value.

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Several specific application areas of the proposed concepts can be visualised as future avenues open to research. First, resource assessment and wind based generation planning can be refined to include the effect of short duration wind variations. This can be naturally followed by better evaluation of effective full-load hours and capacity credit of wind based generation. Second, a proper assessment of short duration output power and its variability allows refinement of pitch-angle control by a ‘‘reverse engineering’’ approach. Specifically, one may thereby attempt evolution of WECS control to reduce short duration distortion of output power curve or to reduce variability, both regardless of wind variations. Finally, with precise estimate of power variability, better assessment of regulating reserve follows almost as a matter of course.

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