SAND84-2530 Unlimited Release

Printed 1985

ESTIMATING WIND SPEED AS A FUNCTION OF HEIGHT ABOVE GROUND: AN ANALYSIS OF DATA OBTAINED AT THE

SOUTHWEST RESIDENTIAL EXPERIMENT STATION, LAS CRUCES, NEW MEXICO

D. F. Menicucci and I. J. Hall Sandia National Laboratories

Albuquerque, New Mexico 87185

Abstract

Thermal modelling of photovoltaic arrays requires accurate estimates of wind speed at the level of the arrays. Wind speed data are normally recorded by an anemometer at the 30-foot level. To estimate the relationship between wind speed and height, wind speed data were acquired at various heights in the range of 10 to 30 feet and a nonlinear regres­sion analysis was performed. This analysis shows that, over the range of conditions covered by the present data, the predictive relationship given by the Mechanical Engineer~s Handbook3 is quite accurate; wind velocity can generally be estimated to within! 1.5 mph.

ESTIMATING WIND SPEED AS A FUNCTION OF HEIGHT ABOVE GROUND: AN ANALYSIS OF DATA OBTAINED AT THE

SOUTHWEST RESIDENTIAL EXPERIMENT STATION, LAS CRUCES, NEW MEXICO

Introduction

Sandia National Laboratories, Albuquerque (SNLA), New Mexico, and the

Jet Propulsion Laboratory (JPL), Pasadena, California, use data collected

at various field sites to study the thermal behavior of photovoltaic (PV)

arrays. Because the conversion efficiency of a PV cell decreases as the cell

temperature increases and because the temperature of the cell is strongly

affected by wind speed, it is important that the wind speed be accurately

determined. At most sites, the wind speed is measured by an anemometer

fixed to a tower at a standard height of 30 feet above the ground. The

surfaces of the PV arrays are between 5 and 10 feet. In the past, the

estimation of wind speed at the array has been based upon the assumption

that wind speed is proportional to the 1/7 power of the height. Results

obtained from thermal modelling, using this assumption, indicated that wind

speed was not being accurately estimated. Therefore, an experiment was run

at the Southwest Residential Experiment Station to investigate the relation­

ship between wind speed and height. An analysis of the data indicated that

the 1/7 power assumption is not accurate for predicting wind speeds. An

alternate mathematical model was derived. The results of the analysis of

the experimental data and a description of the alternate model are reported

in this document.

Data Co 11 ecti I)n

In the experiment, wind speed and direction were recorded at 6-minute

intervals from two anemometers. One anemometer was fixed at the 30-foot

level of the tower; the other anemometer was placed for periods of 4 to 6

days at heights ranging from 5 to 30 feet. The placement schedule is shown

Table 1

Schedule for Placement of Anemometer Heights

Anemometer Heights Date ( ft) (1983)

30 vs. 30 Sept. 1-6 30 vs. 25 Sept. 8-12 30 vs. 20 Sept. 14-19 30 vs. 15 Sept. 21-27 30 vs. 10 Sept. 29-0ct. 5 30 vs. 5 Oct. 7-12

The records acquired during the test (experiment) consisted of the wind

speed and direction measurements as sensed by both anemometers at 6-minute

intervals. The entire data record was reviewed to detect and eliminate those

records that showed obvi ous incorrect measuY'ements. Some errors occurred

from faulty computer processing of the wind direction data and others from

faulty sensors. The data affected by these errors--about 4% of the total-­

were deleted from the data record.

Wind speeds were predominantly light during this sampling period. There

were many days with wind speeds no greater than 8 mph. Only a few hours had

wind speeds faster than 12 mph at the 30-foot level. The frequency distribution

of wind velocity recorded at the 30-foot level is shown in Figure 1.

-2-

>,

85

80

70

60

g 50 ell ~ c:r ell

Lt 40

30

20

10

o 2 4 6 8 10 12 14 16

Wind velocity at 30-ft level, mi/h

Fi gure 1. Frequency distribution of the wind velocity at the 30-ft level.

-3-

20

Our analysis did not use all the data associated with Figure 1. An

examination of the entire data record showed many instances in which the

wind velocity at height H(YH) was greater than the wind velocity at height

30 feet (Y30). In all these instances, Y30 was less than 5 mph. This anomaly

was interpreted as turbulence resulting from thermal gradients. To avoid

trying to model "noise" of this type, we deleted data points that had

V30 < 5 mph. The resulting data set is listed in the Appendix, Table A1.

Data Analysis

The following nomenclature will be used:

H = Height in feet at which wind speed is measured

VH = Wind speed 1n miles per hour (mph) at height H.

An established relationshi p1,2 for "effective" wind speed as a function

of height for computing wind loads on buildin9s is:

where

VH = effective wind speed at H ft above ground

V30 = reference wind speed at 30 feet above ground

(1)

and the values of the parameters 8 and HG are selected according to the terrain

characteristics

Terrain

Open Suburb City

7 4.5 3

-4-

900 ft 1200 ft 1500 ft

The parameter HG is the gradient height, above which the obstructions on the

surface (e.g., suburban dwellings, city buildings) no longer affect the wind

velocity. For the terrain categories "suburb" and "city", the effective wind

speed yielded by Equation 1 for heights less than 30 feet is not equivalent to

the actual wind speed and is not suitable for use in analysing the SWRES data.

Another relationship, given in the Mechanical Engineer~s Handbook3, is

where

S = 7, for V30 > 35

S = 5, for: 5..; V30 ..; 35

S = 2, for: V30 < 5

(2)

Note that Equations 1 and 2 are identical if the requirements for open terrain

and for V30 greater than 35 mph are both satisfied.

Using Equations 1 and 2 as models, the following equation was selected

as the basis for a regression analysis:

where m = lis and e denotes a random deviation from the expected

relationship.

(3)

Equation 3 has the same form as Equations 1 and 2. The unknown parame­

ters A and m must be estimated from the data. A nonlinear least-squares

computer routine from the SAS library was used to obtain the estimates. The

results are given in Table 2. Statisticians refer to this table as an Analysis

-5-

of Variance Table (see e.g., p. 20 of reference (4». Table 2 also contains

estimates of A and m and the standard errors of these estimates.

Table 2

Statistical Results for Fitting Expression (3).

Analysis of Variance

Source

Model

Resi dual

Degrees of Freedom

2

265

Parameter

A

m

A

Sum of Squares

11762.6

122.4

Mean Square

5881.3

.46

Estimate Std. Error of Estimate

.963

.184

.008

.011

The estima,te of VH, say VH, given values of Hand V30, is thus:

A model that has more intuitive appeal than (3) is (2)

(4)

(5)

where m = liS. This model has the property that VH = V30 when H = 30, as

one would like. The above-mentioned nonlinear routine gives the results in

Table 3.

-6-

Table 3

Statistical Results for Fitting Expression (5).

Analysis of Variance

Source Degrees of Freedom Sum of Squares Mean Square

Model

Resi dual

Parameter

m

1

266

The estimate of VH using (5) is

Estimate

.219

11754.0

131.0

11754.0

Std. Error of Estimate

.008

.49

The fit using (5) is almost as good as the fit using (3): Residual MS = .49

versus .46. Note also that this fit is quite close to that in reference 3

for the range of V30 values, 5 ~ V30 ~ 35, which covers our data. The differ­

ence between observed and fitted values of VH, under (6), exceeded 1.5 mph

for only eight of the 267 data points.

Figures 2-7 are scatterplots of VH versus V30. Equation (6) is also

plotted on the figures. Overall, these figures show that equation (6) fits

the data quite well. Only at H = 10 is there visible evidence of lack-of-fit.

-7-

.c

....... -E .. :I

: •

>

co

•

11

. A

t A

T".

----

-.

10 9 8 71

- 6, 1:;1

-~

I

4~

3~

2 I 4

A

/B

A

r A

/ A

/C

c

A

A

B

. A

A

6

8

B

A

A

/'

A

Leg

end:

10

V30

, m

i/h

A

A

Pre

dict

ed b

y eq

uati

on

(6)

A =

lOB

S,

B =

2 O

BS,

etc.

12

14

Figu

re 2

. S

catt

er p

lot

of

V H ve

rsus

V 3

e fo

r H

• 5

ft.

16

.r::.

....... .... E .. x

I >

1.

0 I

11 ~I--------'---------'--------'--------'---------'--------''--------'--------'

10

A

9 8 7 6L

~

'--

Pre

dict

ed b

y C

eq

uati

on

(6)

5 4 3 A

L

egen

d:

A =

1 0

8S,

8 =

2 0

8S,

etc.

2 L

I _

__

__

__

_ -L

__

__

__

__

-L

__

__

__

__

~ _

__

__

__

_ ~ _

__

__

__

_ ~~ _

__

__

__

_ ~ _

__

__

__

_ ~ _

__

__

_ ~

10

12

4 6

8

VlO

• m

i /h

Figu

re 3

. S

catt

er p

lot

of

V H ve

rsus

V

30

fo

r H

• 10

ft.

18~1 ~--~~-r~--r-~~~--r-~~~--~-r~

16

14 iT

A

~A

.c

....... :~

lO~

A

I

/~ Pr

edic

ted

by

......

0 I

equa

tion

(6

) Br

-A

~

D/,A

61

-B

D

BA

/

I/C

Le

gend

: A

= lO

BS

, B

= 2

OBS

, et

c.

4r-

B

2 I

I

4 6

8 10

12

14

16

18

20

V'JO

, m

i /h

Figu

re 4

. S

catt

er p

lot

of V

H ve

rsus

V

I' fo

r H

z 15

ft.

16rr----.-----.-----.-----.-----.-----.-----~----r-----r-----r---~----~

14

12

1-

A""

A

.&:. .....

..- E ..

I >

x

--'

--'

I /~:"

'--

Pre

dict

ed b

eq

uati

on

(6)

8 6 L

egen

d:

A =

1 O

BS,

B =

2 O

BS,

etc.

4'

c--~----L---~---L--~--~--~~--L---~--~--~

4 6

8 10

12

14

16

V

u,

mf/

h

Figu

re 5

. S

catt

er p

lot

of

V H ve

rsus

V

iO

for

H •

20 f

t.

14

12 II

/" A

A

A/.

.s:

: .....

.. ..- E

8

r .

I ..

/. ~ P

redi

cted

by

-' ~

equa

tion

(6

) N

I

6r

0/

E

/.

41

-B

L

egen

d:

A =

lOB

S,

B =

2 O

BS,

etc.

2~1 _

__

__

_ ~ _

__

__

_ ~ _

__

__

_ ~ _

__

__

_ ~ _

__

__

_ ~ _

__

__

_ ~ _

__

__

_ ~ _

__

__

_ ~ _

__

__

_ ~ _

__

__

_ ~

4 6

8 10

12

14

V30

, m

1/h

Fig

ure

6.

Sca

tter

plo

t o

f V H

vers

us V

IO

for

H 3

25 ft

.

15

A

13 111

/A

..c

.....

.. ..- E

l-

/'

A

I .

--'

x W

I >

9L

/'

"-

Pre

dict

ed b

y A

C

eq

uati

on

(6)

/E

5'

,/c/:

4

U

M

6--:-----l--8~ _

_ L

_ _

__

L_

7

Leg

end:

A

=

1 O

BS,

B =

2

OBS

, et

c.

10

12

14

VU

t m

1/h

Figu

re

7.

Sca

tter

plo

t of

VH

vers

us

V30

for

H •

30 f

t.

One use of equation (6) is to convert il sequence of V30 measurements to

a sequence of VH estimates. The error in such estimates is described by

statistical prediction limits. In particuhr, approximately ninety-five A

percent statistical prediction limits for Vii are given by: VH + Bnd , where 1\

VH is given in (6) and

A The RMS in Table 3 provides Var(e) = .49. As shown in the appendix, over the

AA ~ range of our data, Var(VH) is always less than .03. Thus, Bnd ::! 1.96 i.52

= 1.4 yields negligibly conservative prediction intervals.

Example:

Suppose H = 12 ft. and V30 = 15 mph. Then, from (6),

A V12 = 15 (12/30)·219 = 12.3 mph

and the prediction limits are:

12.3 + 1.4 = (10.9, 13.7)

Thus, observing V30 = 15 mph leads to a prediction at 12 feet of 12.3 mph and

we can be about 95% confident that V12 will be between 10.9 and 13.7 mph.

Summary

The SWRES wind data were fitted with an equation of the form, VH = V30 (H/30)m.

The value of m obtained is consistent with the Mechanical Engineer~s Handbook

and yields predictions of VH usually within 1.5 mph of the measured value.

7heseresults pertain to a single set of data obtained at a single site, so

-14-

we cannot claim that they apply elsewhere. Their consistency with the Mechanical

EngineerJ,s Handbook, though, is encouraging. If a model of this form is used,

for other locations, the model should be verified by similarly collecting and

analyzing wind data.

References

1. "Lecture Notes for Designing for Wind," Institute for Disaster Research, Texas Tech University, Lubbock, Texas, August 1-3, 1977.

2. American National Standard-Building Code Re~uirements for Minimum Design Loads in Buildings and Other Structures, AN I-A58.1-1972, National Bureau of Standards, Washington, D.C., 1972.

3. Lionel S. Marks, ed, Mechanical Engineer-Ls Handbook (New York: McGraw-Hill Book Company, Inc., 1951).

4. Draper, N. and Smith, H., Applied Regression Analysis, John Wiley & Sons, 1981.

-15-

APPENDIX

Wind Speed Data From the Southwest Residential Experiment Station

Wind Speed Data

In Table A1, V30 is the wind speed recorded by the fixed anemometer

at the 30-foot level. The value of n is the number of records for a given

value of V30. The height at which the second anemometer is placed is H, and

the wind speed recorded by the second anemometer at height H is VH. The

frequency f is the number of records for which V30, H, and VH have the indicated

values. For example, the heading of the first column indicates that there are

64 records (n = 64) in which the wind speed recorded by the fixed anemometer

at the 30-foot level had a value of 5 mph. Of these 64 records, there is one

record (f = 1) for which the second anemometer was at the 5-foot level (H = 5)

and recorded a wind speed of 2 mph (VH = 2).

ApproximatePredictiorJ Limits for VH

1\

The prediction error for VH has a variance of Var(VH) + Var(e). A

The RMS in Table 3 provides an estimate of Var(e), namely Var(e) = .49. A

The variance of VH, from (5), is:

1\ A . Var(VH) = Var[V30 (H/30)m]

An estimate of this variance of is obtained as follows (by Taylor series

approximation).

-16-

Table A1

Wind Speed Versus Height Data from SWRES

V30 = 5 V30 = 6 V30 = 7 n = 64 n = 72 n = 39

f H -1lL f H -1lL f H -1lL 1 5 2 1 5 2 3 5 4 1 5 3 2 5 3 3 5 5 5 5 4 3 5 4 1 10 4 1 10 3 1 5 5 3 10 5 5 10 4 6 10 4 3 10 6 4 10 5 4 10 5 4 15 6 2 15 4 3 15 5 4 15 7 9 15 5 2 15 6 2 20 6 3 20 4 6 20 5 1 20 7

10 20 5 6 20 6 5 25 6 2 25 4 4 25 5 2 30 6 9 25 5 4 25 6 8 30 7

12 30 5 13 30 5 17 30 6

V30 = 8 V30 = 9 V30 = 10 n = 32 n = 14 n = 14

f H ~ f H ~ f H ~

1 5 4 1 5 4 1 5 6 1 5 5 2 5 6 1 5 8 1 5 6 5 10 7 3 15 8 1 5 7 1 15 6 1 15 10 1 10 4 1 15 7 2 20 9 1 10 5 1 15 8 1 20 10 2 10 6 1 20 8 2 25 9 2 15 6 1 20 9 3 30 9 6 15 7 1 25 9 1 15 8 1 20 6 1 20 8 5 25 7 2 25 8 5 30 8 1 30 9

-17-

f

1 2 1 1 3 1 1 1 1 1

f

1 1 1 1 1

V30 = 11 n = 13

H

5 5

10 10 15 20 20 25 30 30

V30 = 14 n = 5

H

5 15 20 25 30

V30 = 20 n = 1

~

8 9 7 9

10 9

10 10 10 11

~

11 13 13 13 14

f H ~

1 15 17

f

1 1 1 1 2

f

1 1

Table Al

(Continued)

V30 = 12 V30 = 13 n = 6 n = 5

H ~ f H ~

5 7 1 5 10 10 8 1 15 11 10 10 1 15 12 25 10 1 20 12 25 11

V30 = 15 V30 = 16 n = 2 n = 1

H ~ f H ~

5 11 1 15 13 20 12

-18-

so,

(Al)

AA The quantity Var(m) = (.008)2 from Table 3.

AA A The Var(VH) expression in (Al) is a function of Hand VH; it decreases

A. as H increases to 30 feet and increases as VH increases. The minimum H in

our data was 5 feet and the maximum VH observed was 17 mph. Using these

values in (Al) gives:

" A Max(Var(VH)) = .027

which is quite small when compared to .49. Thus, over the range of our data,

a prediction error variance of .49 + .027 = .52 can be used with negligible

conservatism.

-19-

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