Tong W. Wind Power Generation and Wind Turbine Design

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Power Curves for Wind Turbines 605

in such a state, this means that no deterministic drift will occur (see also Fig. 7 ).

(To separate stable from unstable ﬁ xed points, also the slope of D

(1)

( P ) has to be

considered [ 16 ].) D

(1)

( P ) can be interpreted as an average slope of the power signal

P ( t ), depending on the power value. For the stable ﬁ xed points this drift function

vanishes because for constant u the power would also be constant, and thus the

average slope of the power signal would be zero. A simple functional ansatz for

(1)

would be D

(1)

( P ) = k [ P

( u ) − P ( t )], where P

( u ) is a point on the ideal power

curve (see the explanation of P

below).

Using their deﬁ nition [ 15 ], the drift and diffusion functions can be derived

directly from measurement data as conditional moments (called Kramers-Moyal

coefﬁ cients):

(

)

()

() lim ( ) () () ,

D P Pt Pt Pt P

→

=+−=

(9 )

where n = 1 for the drift and n = 2 for the diffusion function. The average 〈 · 〉 is performed

over t . The condition inside the brackets means that the difference [ P ( t + t ) − P ( t )] is

only considered for those times for which P ( t ) = P . This ensures that averaging is

done separately not only for each wind speed bin u

but also for each level of the

power P . If one considers the state of the power conversion system as deﬁ ned by

u and P , one could speak of a “state-based” averaging in contrast to the temporal

averaging performed in [ 3 ].

Using the mathematical framework of eqns ( 8 ) and ( 9 ), also uncertainty estima-

tions can be performed for the ﬁ xed points. For details of the derivation, the reader

is kindly referred to [ 16 , 17 ]. Figure 8 presents the dynamical power characteristic

Figure 7: Illustration of the concept of ﬁ xed points. For constant wind speed u ,

the power would relax to the stable value P

( u ). The deterministic drift

(1)

( P ), denoted by vertical arrows, drives the system towards this ﬁ xed

point (see text). The sketch was taken from [ 17 ].

606 Wind Power Generation and Wind Turbine Design

of a multi-MW turbine [ 11 , 12 ] which has been derived following the procedure

outlined above, including error bars. It can be seen that for most wind speeds the

power characteristic has very little uncertainty. Nevertheless, in the region where

rated power is approached larger uncertainties occur. Here it can be assumed that

the state of the power conversion is close to stability over a range of power values.

These larger uncertainties of the ﬁ xed points can thus be interpreted as a conse-

quence of the changing control strategy of the wind turbine from partial to full

load range. It is of great interest how different turbines behave here, and may be

more or less power efﬁ cient.

It is important to note that in [ 11 ] dynamical power characteristics have been

calculated using wind measurements taken by cup, ultrasonic, and LIDAR ane-

mometers. All three power characteristics were identical within measurement

uncertainty, showing that this approach appears quite robust concerning the wind

measurements.

2.4.2 Summary

The Langevin equation ( 8 ) clearly is a simplifying model of the complex power

conversion process. On the other hand, the drift function D

(1)

( P ), see eqn ( 9 ), is

well deﬁ ned for a large class of stochastic processes and not restricted to those

which obey the Langevin equation.

A central feature of the dynamical approach is the use of high frequency mea-

surement data as shown in Fig. 6 , which enables the analysis of the short-time

dynamics of the power conversion process. The usage of the drift function eliminates

systematical errors caused by temporal averaging combined with the non-linear

Figure 8 : Dynamical power characteristic of a multi-MW wind turbine. Wind

speed u

is normalized to equal power at standard conditions, and power

is normalized to rated power, both according to IEC 61400-12-1 [ 3 ]. For

a description of the measurements, see [ 11 , 12 ].

Power Curves for Wind Turbines 607

dependency of the power on the wind speed. The results are therefore site indepen-

dent because effects of turbulence have no inﬂ uence on the dynamical power

curve. An interesting feature of this approach is the ability to show also additional

characteristics of the investigated system. Examples are regions where the system

is close to stability, as mentioned above, or multiple stable states, see also [ 17 , 18 ].

Because of these features the dynamical power curve is a promising tool for mon-

itoring the power output of wind turbines.

3 P erspectives

Different tools have been deﬁ ned in the previous section to estimate power perfor-

mance. The IEC power curve, in spite of being a good introduction to the topic,

cannot characterize the conversion process of a wind turbine objectively, i.e. the

result depends on the wind condition. The Langevin power curve, on the other

hand, provides robust results that can be applied to determine and monitor the

dynamical behavior of a wind turbine. Rather than competing against each other,

these two power curves, when plotted together, can quickly bring useful insights

on the health of any (horizontal-axis) wind turbine.

An overview of the available applications will be presented in this section.

3.1 Characterizing wind turbines

A striking feature is that the Langevin power curve offers new, complementary

information to the IEC power curve. The two power curves are presented together

in Fig. 9 .

Figure 9: Langevin power curve (dots + error bars) and IEC power curve (line) for

a multi-MW wind turbine (same as Fig. 8 ). The power output is normalized

by its rated value P

608 Wind Power Generation and Wind Turbine Design

Indeed, a striking limitation of the IEC method lies in the way it discretizes the

domain { u , P }. As detailed in the Section 2.3, the IEC method averages all data in

speed bins of size d u = 0.5 m/s. The domain is discretized only for the wind speed,

resulting in a unique point every 0.5 m/s for the IEC power curve. The IEC power

curve is one-dimensional, it is the line P

IEC

( u ) (as represented in Fig. 9 ).

The dynamical method, however, discretizes the domain { u , P } on both wind

speed and power output. The resulting power curve, derived from the drift ﬁ eld

(1)

( P ; u ), is a two-dimensional quantity. As shown in Fig. 10 , each point in the

domain displays a vector indicating how fast (length of the vectors) and in which

direction the system wants to evolve.

Obviously in low power but high wind regions, the vectors point upwards to

higher power values. Correspondingly, high power but low wind regions vectors

point downwards to smaller power values. This is shown in Fig. 10 .

This mathematical framework is necessary to observe the dynamics of a wind

turbine. This point is crucial to characterize power performance. Thanks to the

dynamical method, multi-stable behaviors can be observed, and a greater insight

can be reached. This is easily seen in Fig. 10 , where multiple ﬁ xed points appear

in several speed bins. Multi-stable behavior appears in the slow region ( u ≈ u

cutin

4 m/s) and in the fast region ( u ≈ u

= 13 m/s), where complex dynamics take place.

In the slow region, the turbine transits between the rest and the partial load modes.

Figure 10 : Drift ﬁ eld D

(1)

( P ; u ) (arrows), Langevin power curve (crosses) and

IEC power curve (background line) for a multi-MW wind turbine.

The intensity of the drift ﬁ eld is proportional to the length of the

arrows. The power output is normalized by its rated value P

. This ﬁ gure

represents the same wind turbine as Fig. 4 .

Power Curves for Wind Turbines 609

In the fast region, the transition happens between the partial load and the full load

modes. The dynamics of these transitions can be observed in Fig. 10 . In both regions,

the two different modes of operation are clearly separated by the Langevin power

curve, while the IEC power curve averages both modes into an intermediate value.

The two different wind turbines represented in Figs 9 and 10 were very well

characterized by the Langevin power curve. The method applies to all (horizontal-

axis) wind turbines, even when presenting complex dynamics. Wind turbines

equipped with multiple gears were characterized successfully using this method,

when the IEC method revealed its limitations. The Langevin power curve is a

powerful tool to visualize and quantify power performance.

3.2 Monitoring wind turbines

Monitoring is closely related to the idea of characterization (introduced in the

previous section), as it follows the evolution in time of the power characteristics.

Once a machine was characterized using the dynamical approach, it becomes pos-

sible to compare and monitor power performance on a regular basis. Dynamical

anomalies can be rapidly brought to light when deviations appear on two consecu-

tive Langevin power curves. The precision of the method allows localizing the

anomaly in the domain { u ; P }, giving more insight towards the deﬁ cient com-

ponent of the wind turbine. Applied on a monthly (or even weekly) basis, such

monitoring can prevent anomalies from limiting the power production, or worse,

damaging other components of the wind turbine.

3.3 Power modeling and prediction

Once a machine was characterized using the dynamical approach, basically once

the drift ﬁ eld D

(1)

( P ; u ) and the diffusion coefﬁ cient D

(2)

( P ; u ) were computed, it

becomes possible to model the power output P ( t ) from any input wind speed time

series u ( t ). The Langevin equation [see eqn ( 8 )] can be solved knowing D

(1)

( P ; u )

and D

(2)

( P ; u ) to generate a realistic power output P ( t ).

A simple, artiﬁ cial case was created in Fig. 11 . In this ﬁ gure, one can see the

real power output P ( t ) of a wind turbine, then the same quantity modeled in good

running condition, and ﬁ nally modeled with an artiﬁ cial anomaly (that limits the

power extraction to roughly 45% of the rated power P

). A comparison of the ﬁ rst

and second graph shows that through a simple model, it is possible to estimate the

power output P ( t ) of a wind turbine knowing only D

(1)

( P ; u ) and D

(2)

( P ; u ). This

model can be applied to any wind situation u ( t ), as an effective way to study the

behavior of a wind turbine in different wind conditions.

This model shows great potential in the continuing evolution of current methods,

principally in the prediction of power production. When coupled with a meteoro-

logical wind forecast, the model could be used to generate the power output of a

wind turbine (whose power performance has been characterized). In addition to

providing quantitative power production estimates, power quality, i.e. ﬂ uctuations

in power, stability, and regularity of the high frequency power output P ( t ) too will

610 Wind Power Generation and Wind Turbine Design

be assessable. Predictions of power quantity, and especially power quality are of

major importance for a large integration of wind energy into the electrical net-

works. However, important developments towards a high frequency wind fore-

cast u ( t ) need to be performed ﬁ rst. Efforts towards such developments are

being made.

4 Conclusions

Power curves for wind turbines establish a relation between wind speed and

electrical power output. This relation is essential for project planning and opera-

tion of wind turbines. Also for the monitoring of turbines, concerning proper

operation and detection of possible misconﬁ guration or failures, the power curve

is an important tool. It can therefore be considered as a central characteristic of

a wind turbine.

Figure 11 : Time series for the power output P ( t ). The upper graph shows the real

power output measured on a multi-MW wind turbine. The graph in

the middle represents the power output P

model

( t ) modeled for the same

turbine. The lower graph shows the power output P

anomaly

( t ) modeled

for the same turbine, but spoiled by an artiﬁ cial anomaly. A horizontal

line represents the artiﬁ cial limitation in power due to the anomaly.

The power output is normalized by its rated value P

. The three graphs

are given for the same time window of 24 h.

Power Curves for Wind Turbines 611

The current industry standard IEC 61400-12-1 [ 3 ] deﬁ nes, among others, a

uniform procedure for the measurement of power curves. This deﬁ nition relies on

temporal averaging of wind speed and power output. Due to the turbulent nature of

the wind and the non-linear dependency of power on wind speed, this power curve

combines the characteristic of the turbine together with the statistical features of

the wind at the special site under investigation. This combination makes the esti-

mation of the annual energy yield at a certain site especially easy. On the other

hand, systematic averaging errors are introduced through the mentioned non-

linearity, and the power characteristic of the turbine cannot be separated from the

site effects. These weaknesses are well known, and several corrections have been

proposed, e.g. [ 19 ].

As an alternative, recently a different approach has been proposed to obtain the

power characteristic of wind turbines [ 16 , 17 ], the Langevin power curve, which

relies on high frequency measurement data (approximately 1 Hz). Inspired from

dynamical systems theory, the power conversion process is regarded as a relax-

ation process, driven by the turbulently ﬂ uctuating wind speed. The power charac-

teristic can then be obtained for every wind speed as the stable ﬁ xed points of this

process. Averaging errors and inﬂ uence of turbulence are thus avoided. Possible

multiple stable states are also captured, allowing deeper insight in the dynamics of

the power conversion. These features make the dynamical power characteristic

especially interesting as a monitoring tool for wind turbines.

As a work in progress, the simulation of high frequency power output signals

based on eqn ( 8 ) is currently developed. One application of this procedure will be

the prediction of energy yields for speciﬁ c wind turbines under speciﬁ c wind

conditions.

References

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output with special respect to turbulent dynamics. J. Phys: Conf Ser , 75 ,

pp. 012045, 2007.

[2] Burton, T., Sharpe, D., Jenkins, N. & Bossanyi, E., Wind Energy Handbook ,

Wiley: New York, 2001.

[3] IEC. Wind turbine generator systems, Part 12: Wind turbine power performance

testing, International Standard 61400-12-1, International Electrotechnical

Commission, 2005.

[4] Böttcher, F., Barth, S. & Peinke, J., Small and large ﬂ uctuations in atmospheric

wind speeds. Stochastic Environmental Research and Risk Assessment , 2 1,

pp. 299–308, 2007.

[5] Betz, A., Die Windmühlen im Lichte neuerer Forschung. Die Naturwissen-

schaften , 15 , pp. 46, 1927.

[6] Rauh, A. & Seelert, W., The Betz optimum efﬁ ciency for windmills. Applied

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[7] Rauh, A., On the relevance of basic hydrodynamics to wind energy technology.

Nonlinear Phenomena in Complex Systems , 11 (2), pp. 158–163, 2008.

612 Wind Power Generation and Wind Turbine Design

[8] Bianchi, F., De Battista, H. & Mantz, R., Wind Turbine Control Systems , 2nd

ed., Springer: Berlin, 2006.

[9] Hanse, A., Jauch, C., Soerense, P., Iov, F. & Blaabjerg, F., Dynamic wind

turbine models in power system simulation tool DIgSILENT, Risø Report

Risø-R-1400(EN), Risø National Laboratory, 2003.

[ 10] Böttcher, F., Peinke, J., Kleinhans, D. & Friedrich, R., Handling systems

driven by different noise sources – Implications for power estimations. Wind

Energy , Springer: Berlin, pp. 179–182, 2007.

[ 11] Wächter, M., Gottschall, J., Rettenmeier, A. & Peinke, J., Dynamical power

curve estimation using different anemometer types. Proc. of DEWEK , Bremen,

Germany, 2008.

[ 12] Rettenmeier, A., Kühn, M., Wächter, M., Rahm, S., Mellinghoff, H.,

Siegmeier, B. & Reeder, L., Development of LiDAR measurements for the

German offshore test site. IOP Conference Series: Earth and Environmental

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[ 13] Rosen, A. & Sheinman, Y., The average power output of a wind turbine in

turbulent wind. Journal of Wind Engineering and Industrial Aerodynamics ,

51 , pp. 287–302, 1994.

[ 14] Rauh, A. & Peinke, J., A phenomenological model for the dynamic response of

wind turbines to turbulent wind. Journal of Wind Engineering and Industrial

Aerodynamics , 92 (2), pp. 159–183, 2004.

[15] Risken, H., The Fokker-Planck Equation , Springer: Berlin, 1984.

[ 16] Gottschall, J. & Peinke, J., How to improve the estimation of power curves for

wind turbines. Environmental Research Letters , 3 (1), pp. 015005 (7 pages),

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[17] Anahua, E., Barth, S. & Peinke, J., Markovian power curves for wind

turbines. Wind Energy , 11 (3), pp. 219–232, 2008.

[18] Gottschall, J. & Peinke, J., Power curves for wind turbines – a dynamical

approach. Proc. of EWEC 2008 , Brussels, Belgium, 2008.

[ 19] Albers, A., Jakobi, T., Rohden, R. & Stoltenjohannes, J., Inﬂ uence of

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EWEC 2007 , Milan, Italy, 2007.

CHAPTER 19

Wind turbine cooling technologies

Yanlong Jiang

Department of Man-Machine - Environment Engineering ,

Nanjing University of Aeronautics and Astronautics, China.

With the increase of the unit capacity of wind turbines, the heat produced by

different components rise signiﬁ cantly. Effective cooling methods should be

adopted in developing larger power wind turbine. In this chapter, the operating

principle and main structure of wind turbines are ﬁ rstly described, following

with the analysis of heat production mechanisms for different components. On

this basis, current cooling methods in wind turbines are presented. Also, optimal

design of a liquid cooling system for 1 MW range wind turbine is conducted.

Finally, some novel cooling systems are introduced and discussed.

1 Operating principle and structure of wind turbines

In brief, the operating principle of a wind turbine is that rotation of impellors

driven by wind power converts the kinetic energy of wind into mechanical energy

of the impellor shaft, which drives the generator. There are mainly two types of

wind turbine operating modes. One is the independent power-supply system,

which is usually used in the remote areas, where electric network is not available.

The terminal electrical equipments are powered by alternating current, which is

converted by a DC–AC converter from the electricity in a storage battery charged

by small scale wind turbines. Generally, the unit capacity is from 100 W to 10 kW.

Or a hybrid power-supply system comprising a middle scale wind turbine and a

diesel generator or solar cells with capacity, range from 10 to 200 kW, is adequate

to meet the need of a small community. In another wind turbine operating mode,

the wind turbines are used as a power resource of an ordinary power network,

paralleling in the electricity grid system. It is the most economic way to utilize

wind power in a large scale. This mode can synchronize and close with a unit inde-

pendently and also can be made of multiple, or even thousands of wind turbines,

called wind farm [1–3].

614 Wind Power Generation and Wind Turbine Design

As shown in Fig. 1 , a wind turbine working in a parallel operation is mainly

comprised of an impeller, a nacelle, a pylon, a foundation and an electric trans-

former. Among these components, the impeller is wind collecting device, includ-

ing blades and hub. It can convert wind power at a certain height to mechanical

energy, representing as shaft rotation at a low speed but with high torque. The

nacelle, comprised of a gearbox, a generator and control systems, is the core com-

ponent of the wind turbine where the mechanical movement is accelerated, then

converted to electric energy with modulated frequency to meet the demands of

parallel operation. The pylon and foundation are mostly used to support the nacelle

and impeller to a certain height and ensure the safe operation. The function of the

electric transformer is to perform the voltage regulation to the output electricity so

as to transfer power efﬁ ciently.

To sum up, the operating procedure of a wind turbine is as follows: the impeller

rotating under the wind force action drives the main shaft in the nacelle to rotate

simultaneously. This movement is then accelerated in the gearbox, and supplies

the high-speed revolution for the generator rotor by connecting with high-speed

shaft. The rotor cuts the magnetic lines of force, and thus produces electric energy.

With the increasing unit capacity of wind turbines, the length of impeller blades

and the height of pylon are gradually increased for the purpose of capturing more

wind energy.

2 Heat dissipating components and analysis

It is well known from the operation principle mentioned in Section 1, the nacelle

is the core component for a wind generating set and also the concentrated area of

heat production in the operating process. The conﬁ guration of the nacelle is shown

in Fig. 2 , and the mechanisms of heat production for different components are

explained as follows.

Figure 1: Sketch of a wind turbine generator connecting to power system: 1, impeller;

2, nacelle; 3, pylon; 4, foundation; 5, transformer.