wind energy systems mase 5705 spring 2014, feb. 11-13, l7 + l8

1

Wind Energy Systems MASE 5705

Spring 2014, Feb. 11-13, L7 + L8

1. Key points of L5+L6

2. Turbulence ( pp. 39 – 52)

2.1 Atmospheric boundary layer (ABL) and

surface layer turbulence

2.2 Turbulence intensity (TI)

2.3 Autocorrelation and Power Spectral

Density (psd)

2.4 Preview of later lectures on turbulence

2.5 Recommended reading

2

L7 and L8

3. Ch. 3 (Aerodynamics of Wind Turbines)

3.1 One-dimensional momentum theory and Betz limit

4. Project 2

3

1. Key points of L5+L6

4

1. Bin width, bin probability or frequency and the

corresponding pdf

For arbitrary values of bin width ΔU,

2. Estimation of AEO (Annual Energy Output) of a

proposed site

a) When bin database is available;

That is, (Umax, Umin, no. of hours in each bin)

b) When only is known U

Key Points of L5+L6

iUU

i f(U)pdfingcorrespondΔUwidthBin

ffrequency Bin

For case (2a), we use the method of bins.

For case (2b) , we create our own bins; then we

estimate the number of hours in each bin by the

Rayleigh estimation.

3. Estimation of AEO of a specific wind turbine in a

proposed site

a) Specific wind turbine ≡ a wind turbine with

specified Uci, Uco and power curve.

b) power curve ≡ an official document that gives

measured power output vs wind speed (other

details are similar to (2a) and (2b))

For example see next table and figure, and also

consider the method of bins:

7

Measurements of site Bin Data and WT power

output data for Mod-2 2.5 MW WT

* Reasons for such a wide bin width are not known. Lower values are more

likely than higher values. Take Ū18= 17.25 m/s

*

(Measured) Power Curve for Mod-2 2.5 MW HAWT

(Uci = 6.25 m/s , Uco = 22.4 m/s)

8

BINBin Avg. Speed

(Ui)

Power output

(kW)

1 3.125 0

2 6.5 175

3 7 318

4 7.5 460

5 8 603

6 8.5 745

7 9 949

8 9.5 1153

9 10 1316

10 10.5 1479

11 11 1642

12 11.5 1805

13 12 1968

14 12.5 2120

15 13 2263

16 13.5 2385

17 14 2500

18 17.25 2500

0

500

1000

1500

2000

2500

3000

6 7 8 9 10 11 12 13 14 15 16 17 18

Po

we

r o

utp

ut

(kW

)

Wind Speed U

Power output (kW)

9

Method of Bins

2. Turbulence (pp. 39-53)

10

"Turbulence is a dangerous topic which is often at

the origin of serious fights in the scientific meetings

devoted to it since it represents extremely different

points of view, all of which have in common their

complexity, as well as an inability to solve the

problem. It is even difficult to agree on what

exactly is the problem to be solved.”

Marcel Lesieur, Turbulence in Fluids , Springer

2008, The Netherlands

11

One is usually at a loss to calculate with

greater accuracy, say ±50%"

Lumley, AIAA Journal, 1998.

Valid even today

12

2.1 Atmospheric boundary layer (ABL) and

surface layer turbulence

13

2.1 (Ambient Boundary Layer (ABL)

and Surface Layer Turbulence

Of concern to wind turbines is turbulence in the lowest

levels of the ABL, also referred to as surface-layer

turbulence. The ABL thickness is not a precisely defined

quantity; it varies “from a few hundred meters to several

kilometers.” For practical purposes, surface layer

encompasses operational heights up to 200 ft, say 10% of

the boundary layer, and the rest is called outside layer.

14

Surface turbulence has been an actively

researched area of the past 30 years by

meteorologists, and engineers associated with

dynamic loading on exposed structures and wind

turbines. As for modeling, “ classical turbulence

theory” is keyed to surface layer conditions on the

basis both of phenomenological and analytical

considerations as well as other guidelines such as

continually updated Engineering Science Data Unit

(ESDU) series.

15

Basically, the von Karman turbulence

model with empirically adjusted parameters

correlate well with test data and it is widely

used. Therefore in this lecture, we follow the

text and briefly describe turbulence intensity

and von Karman power spectral density

(PSD), which is a common frequency-

domain description of turbulence, Equation

(2.27), p. 43). This PSD is usually referred to

as von Karman turbulence model.

16

2.2 Turbulence intensity (TI)

17

In this lecture, we follow the text (Ch. 2):

Just mention the basic equations and apply

them. We address turbulence in some detail,

well beyond the text, after covering Ch. 3

(Aerodynamics of Wind Turbines) and Ch. 4

( Mechanics and Dynamics).

18

U = Ū + u

Ū = wind speed (x-direction)

u = fluctuating (random) component in the x- direction (at present we neglect lateral and vertical components)

19

20

z(vertical)

x

disk plane

x

w)v,(u,)u,u,(uu

(U,0,0)U

uUU

321

In General,

For the present, we follow the text:

Wind speed in the x direction, which is

perpendicular to the disk (γ=0)

uUU

Turbulence Intensity TI

(2.23 p. 40)

22

static loads in the system.

23

u

u

Ū Ū

Text,

p. 41

Sample wind data (fig, 2.14 , 41)

24

U

25

(y, z) ≡ rotor plane

x ≡ +ve in the downwind direction

26

Wind Fluctuations and Average Power

U = Ū + u

27

?

28

29

E(u) = 0

E(un) = 0 for n odd (why?)

E(u2) = σU2

30

31

32

33

34

35

(2.54) p. 58 and (2.63) p. 60 in 2nd Edition

36

(2.23) p. 40 2nd Edition

No U-bar^2

37

38

p. 61, 2nd Edition

Table 2.4

P. 61

k Ke

1.2 0.837 3.99

2 0.523 1.91

3 0.363 1.40

5 0.229 1.15

39

Variation of parameters with

Weibull k shape factor

Compare Ke = 1+3(TI)2 = 1+3(.837)2 = 3.10

1+3(.523)2 = 1.82

1+3(.363)2 = 1.40

1+3(.229)2 = 1.16

Random Data, Analysis of Measurement

Procedures (3rd Edition, 2000), Wiley

J.S. Bendat and A.G. Piersol

A thorough and easy-to-read account of

random processes.

41

For details, see

Each Ui(t) represents a unique set of

measurements (not likely to be repeated)

Ui(t) = sample function or a random signal

Ui(t) = wind speed (in our case)

{Ui(t)} ≡ {U(t)}

The ensemble {U(t)} describes a random or

stochastic process of wind speed, and the

sample function Ui(t) belongs to this

process.

42

43

UN(t)

U3(t)

U2(t)

U1(t)

2.3 Autocorrelation and Power Spectral Density

44

Autocorrelation

We consider a stochastic process {U(t)} and

consider two time instances ‘t’ and ‘t + ’.

Given {U(t)}, we cannot predict {U(t + )}.

But {U(t)} and {U(t + )} are ‘related’ or

correlated. This correlation is described by

the autocorrelation function RUU(t,)

RUU(t, t + ) = E[U(t) U(t + )]

45

46

Wind Speed measurements belong to a

Stationary random process.

That is,

Ū(t) = constant, E[U2(t)] = constant

Autocorrelation function t2 – t1 depends only

on time difference:

RUU(t1 , t2) = RUU(t2 – t1)

RUU() = E[U(t)U(t+)]

While the autocorrelation function

characterizes turbulence as a function of

time or time lag in the time domain, the

corresponding description in the frequency

domain leads us to the power spectral

density function.

That is, the power spectral density function

characterizes turbulence as a function of

frequency in the frequency domain.

47

It is proved that (Wiener-Khinchine

relation) autocorrelation function and

power spectral density function are a

pair of Fourier transform.

(N. Wiener in the United States and

A.I. Khinchine in the USSR).

48

49

UU

UU

UU

UU

50

For future reference we also define the

Gaussian pdf (2.25), p. 42 2nd Edition

51

0

pdf

Turbulent Wind. Velocity about Mean -4

-1 1

Fig. 2.15

p. 41

4

52

53

Turbulence Modeling Two important parameters in turbulence

modeling

54

p. 43, 2nd Edition

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(2.27) p. 43, 2nd Edition

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57

58

59

, Fig. 2.16, p. 42

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Power spectral density functions, Fig. 2.`17, p. 44

von Karman Turbulence Model

Based on first principles;

No empirical constants;

For above about 150 m (height z > 150 m);

good correlation with test data;

For z < 150 m, some terrain-sensitive

deficiencies.

61

B. 2.13 (p. 620)

Power Spectral Density Estimation Similar to Equation 2.27 in the Text (p. 43) , the following empirical

expression has been used to determine the power spectral density

(psd) of the wind speed at a wind turbine site with a hub height of z. The frequency is f(Hz), and n (n = f z / Ū ) is a non-dimensional

frequency.

Where

Plot the power spectral density of the wind at a site where the surface

roughness is 0.05 m (z0) and the hub height is 30 m (z), and the mean wind speed Ū is 7.5 m/s.

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Table 2.2 (p. 46)

Values of surface roughness length for various types of terrain

Terrain Description Zo (mm)

Very smooth, ice or mud 0.01

Calm open sea 0.20

Blown sea 0.50

Snow surface 3.00

Lawn grass 8.00

Rough pasture 10.00

hallow field 30.00

Crops 50.00

Few trees 100.00

Many trees, hedges, few buildings 250.00

Forest and woodlands 500.00

Suburbs 1500.00

Centers of cities with tall buildings 3000.00

next:

b) Plot of S(f) vs f

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─ 10

─ 1

─ 0.1

─ 0.01

─ 0.001

─1

0

─1

─0

.1

─0

.01

─0

.00

1

f(hz)

S(f

) m

s2 /hz

(Observe the rapid decrease in turbulence energy with increasing

frequency.)

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bending

An example:

68

bending

Around 4 Hz

The turbulence has little energy!

69

─ 100

─ 10

─ 1

─ 0.1

─ 0.01

─ 0.001

─1

0

─1

─0

.1

─0

.01

─0

.00

1

f(Hz)

S(f

) m

/s2 /H

z

fn= 4 Hz

b ) Why is it that blades are sensitive to gust

(turbulence) excitation?

The answer is keyed to the difference between

turbulence PSD in space-fixed non-rotating

coordinates (hub element) and turbulence psd in

blade-fixed rotating coordinates (blade element).

Hub element and blade element feel turbulence

differently.

See next section on preview of later lectures on

turbulence. We will study this difference in detail

under rotational sampling. pp. 143, 330.)

70

2.4 Preview of later lectures on

turbulence

Ch. 3 (aerodynamics of wind turbines, p. 143)

and Ch. 7 (wind turbine design, p. 330)

71

In later lectures on turbulence we cover the

following:

1.Classical theory of turbulence, that is,

theory of homogeneous and isotropic

turbulence.

2.Revisiting von Karman model and

extensions

3.Rotational sampling, that is, why

turbulence seen by the hub element differs

from that seen by a blade element.

72

For later reference:

Homogeneous and isotropic

turbulence

73

Turbulence is homogenous. That

is, its statistical properties do not

vary form point to point in the field,

and thus all the functions described

are independent of the location of

the origin in the field.

74

The concept of isotropy simplifies

the description of turbulence even

further. If a turbulence field is

isotropic, its statistical properties

are independent of direction in the

field, and thus they do not change

with rotation of the coordinate axes.

75

A Descriptive Account of

Rotational Sampling

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Predicted and Measured Longitudinal Turbulence PSD

78

• Experiment

___ Von Karman

(Radius 24.5 m, Rotation Rate = 0.625 Hz)

Rotational-coordinates

Fixed-coordinates

This figure shows the measured and

predicted psd (von Karman) of longitudinal

turbulence in both space-fixed and blade-

fixed coordinates.

The model refers to a 35-m radius HAWT

and the blade element has a local radius of

24.5 m, 70% of radial location. Two sets of

predictions from von Karman model –with

and without rotational effects– are shown.

79

Since the area under the PSD curve

represents the turbulence energy, which is

the same in both coordinates, the turbulence

energy is transferred from the low-frequency

region (P<1) to the high-frequency region

with PSD peaks at 1P, 2P, etc., where

P=0.625 Hz. The correlation in this graph

brings out two key points of a much broader

significance.

80

First, the classical turbulence

theory that is judiciously modified to

account for the local conditions

provides a means of modeling

surface-layer turbulence seen by a

turbine blade element.

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Second, the von Karman model

in rotating coordinates correlates

qualitatively well with the

measurements.

82

2.5 Recommended Reading

The Nature of Wind R.I. Harris

(It is available on pass-word-protected Telesys.)

Although some 40 years old, this 25 - page article gives a

through description of classical turbulence theory and its

adaptation to modeling turbulence for structural

applications.

83

3. Ch. 3 (Aerodynamics of Wind Turbines)

3.1 One-dimensional momentum theory

and Betz limit

84

The assumptions are

1. Fluid flow is incompressible,

2. No frictional drag,

3. Uniform thrust over entire disk area,

4. Nonrotating wake,

5. Steady-state operation: constant wind

speed over the turbine disk and constant

rotational speed.

85

Momentum or actuator disk

theory of wind turbines

86

87

88

89

U (1)

U - vi U-w

(4) (3)

+ve direction

Force on

fluid

disk

F = force on the

disk = Thrust

= T

(2)

90

U U - vi U-w

+ve direction

91

92

(1) (4)

(3) (2)

- Turbine does negative work on the fluid ≡ extracts energy from the fluid

- Fluid slows down

Tube expands

93

94

95

96

97

Limitations of Betz’s (Momentum)

Theory

Windmill or windmill brake state

98

99

Power is extracted from the fluid by the wind

turbine

Windmill State

Velocity slows down (tube expands) WT brakes

the fluid

Windmill brake state

100

101

(p. 93)

102

Operating parameters for a Betz turbine; U, velocity of undisturbed air; U4 , air

velocity behind the rotor, Cp power coefficient, CT thrust coefficient

(p. 95)

In L9 we will consider an extension of the

momentum theory with rotational effects on

the fluid; p. 96

103

Project 2: Not assigned in 2014

104

105

B.2.11 Actual Data Analysis and Power

Prediction, P. 619

Based on the spreadsheet (MtTomData.xls) which contains

one month of data (mph) from Holoyke, MA. determine:

a) The average wind speed for the month

b) The standard deviation

c) A histogram of the velocity data (via the method of

bins- suggested bin width of 2 mph)

d) From the histogram data develop a velocity-duration

curve

e) From above develop a power-duration curve for a

given 25 kW Turbine at the Holyoke site.

For the wind turbine,

assume:

P = 0 kW

P = U3/ 625 kW

P = 25 kW

P = 0 kW

f) From the power duration curve, determine the energy that would

be produced during this month in kWh.

0<U<6mph)

6<U <25 (mph)

25<U<50 (mph)

50 <U(mph)

107

Verify and explain:

108

109

110

f) The total energy produced can be

determined from integrating the product of

the turbine power and the numbers of hours

of operation at that power level, yielding an

annual energy production of 2474 kWh.

111