Economics of Wind Generated Energy, Arthur Zatarain, Artzat Consulting, ASME

[Note: this text version is only for web crawler.

Click HERE: PUBLICATIONS to access high quality PDF version]

Economics of Wind Generated Power

A. M. ZATARAIN

El"lglneer.

Otlshore DMsion,

Shel Od Company,

New Orleans, La

W. S. JANNA

AsSIStant Professor,

School of Mechancal Engineering,

Urw. of New Orleans,

New Orleans, La.

The method explained In the paper considers the vanous economic and engineering

variables that afe necessary 10 properly plan a natural energy system The goal of the

methOd is to produce an opbmum size range lor the syslem and to provide e maximum

aJlowable cost based on area 01 the collector that can be spent while still retaining an

ecooomtC advantage The method uses data on the power requirement of the Inslallatlon as

well as estimates oi the IUlance cost and expected Increases In the cost of conventionalluel

The technique includes systems operating In parallel with conventional power sources and

excludes any systems WIth energy storage requirements.

COIIlrlb.,N br l'-e PetnMw .. DMskMI O(T1Ie AlMricu Soclely 0( Med,.nk.1 Ew&inHn: r~ ... .,,-..'.lioII

., !be .:-rv TtchnoiOC COllf~e ., .~ hibil~n. llouslon. Te .... !!, No~_'" ~9. I97&. M"lISl'ripl

~ed .1 ASME lleadqUllrten s.e .. ber 11. 1m.

Copie will be .y.il.bIe until A'*CUS1 I. 1979.

ntI AIIIJUCAJI IOCIITY OF MeCHANICAL ENGINeUs. UNITED ENGINEERING CENTER, 345 EAST .nth STREET, NEW YORK, N.Y. 10017

Economics of Wind Generated Power

A. M. ZATARAIN w. S. JANNA

INTRODUCTION

The applioati on or wind, Bolar, and other

natural energy forms to fl11 our energy needs

1s a matter of both engineering and economics .

Too little emphasis 1s placed on the economics

of such systems in an effort to achieve low

cost energy with maximum engineering efficienoy,

The end result may be workable from a technical

aspeot but may be too costly to be truly practi cal

. Therefore , there exists a need tor a sys tematic

planning approach to estimate the opti mum

Size and cost for a given installation given

basic engineering and economic data .

Such a planning method should predict the

largest area of energy collector that should be

constructed and should also predict the maximum

cost per unit area that can be expended in order

for the system to pay for itself over a given

period or time . Since most natural energy col lectors

, such as wind turbines , solar panels ,

or hydro generators , can be priced at a cost

per area figure , the method should be applicable

to a wide variety of energy sources . Any user

interested in applying such a system to fill a

particular requirement should use a balanced

planning method to avoid construoting too large

a system at a cost that is too h1gh to prodUce

any saving in energy cost.

The method explained in th1s report will

relate the value of the natural resource system

in dollars per area to the cost of conventional

energy sources in dollars per kWh . Only Wind

energy will be considered here, although the

logio is the same for any system whose cost and

annual energy output is related to its area .

A PPLICABLE EQUATIONS

The basic equation is that representing

the total cost of a Wind energy conversion system

(WECS) in terms of total fixed cost and

operating expenses . With time not considered ,

this 1$

Th1s equati on Is valid for only one year beoause

no consideration is made for equipment financing

or increases in operating cost <!ue to inflation.

Factors must be added to these terms to give a

true cost over n years in te~s of present day

dollars .

The expense of financing can be handled

using the standard formUla for payments on a

loan ot n periods at interest rate 1m (SUbscript

m for money). The payment formUla is (1):1

(2)

The value of PV is the present day cost

of the equipment to be financed. Thus , the total

amount spent on the equipnent is the periodio

payment times the number of periods . For a oost

per unit area , Fw (subscript w for wind) , the

total cost of eqUipment including financing becomes:

(})

The term grouped as AI is the tactor that will

increase the value of Fw to reflect the additional

cost of financing .

The periodic operating cost per unit

area (Pw) will also require adjustment to convert

the total maintenance and other annual

costs to be spent over n years to present- day --, Underlined numbers in

des1gnate References at end of

parentheses

paper .

values . This cost is the sum of each annual

cost over the total period with each year taking

an increase of ~ percent (subscript i for inflation)

. The total operating cost becomes:

I")

This situation is s~lar to a sinking fUnd in

which a. fued amount of money is deposited each

period at a tixed rate ot interest to end up

with a predeterc1ned amount at the end of n

years . In this case , the money to be spent in

the fUture is inflating rather than gaining

interest; the end result is the same . Therefore ,

the sinking fUnd formula can be used to account

for the effect of inflation on the periodio operating

cost per unit area (~) .

The term grouped as BI is the factor that increases

the value of Pw to account tor infla tion

. This same reasoning wi ll later be used

to account tor fUture rising fuel costs per

unit of energy .

Now that the time inclUSive factors have

been presented, the total cost per area of the

WECS over n years in present -day dollars can be

expr essed by:

or , in shorter notation

r ".,,1 ·'l

.... [ It :J

:l: .. •. ,t.' .......

16)

16.)

other necessary equations are those relating

the energy available per area ot the wind

at the site in question . Golding (!) discusses

this in regard to wind turbines and evaluates

the effect of Wind data on total energy estimates

. It was concluded that accurate information

is difficult to obtain ~thout lengthy,

precise measurements. However , for estimation

purposes , a corrected average wind speed and

the number of hours per year that speed is

available can serve as an approximation of the

total energy avail able at a given location .

ThiS corrected wind speed is necesaary

because the energy available varies with the

cube of Wind velocity (1,1 . Thus, a doubling ot

wind speed represents an eight fold increase in

energy . The m1n1m\lJll information necessary to

estimate available energy is the average ~nd

speed and its annual duration in hours . A bet ter

estimate can be obtained with information

on velocity distribution over a long period of

tine . This in/oreation, however, will generally

not be available (!) .

Using only average Wind information, the

energy can be approximated using the following

formula (£, 1,):

17 )

where:

Av _ corrected wind velocity (usually taken as

1 .15 x mean wind speed unless Dore de tailed

information is available)

Hw - hours per year mean wind is available

K - conversion constant (values for K in

various units given at the end of this

report) for power in kw and area 1n ft2

K .. 5 .3 x 10-6

The constant of O . ~ is necessary to derate

the total wind energy present to a value that

can be extracted under actual conai tions . It is

the product of the mechanical eff iciency of the

WECS (75 percent) (~) and the Betz coefficient

of 0 .593 (£) . The Betz coefficient is the

theoretical maximum traction of power that can

be extracted by a wind turbine under ideal conditions

•

Now that the amount of energy per area is

known, the dollar value of this energy can be

determined by comparing it to the cost ot conventional

energy over same period of time. The

conventional source can be a power company or

private generation equipnent . The dollar value

of the WECS energy is the kWh obtained in equation

(7) times the energy oost per kwh of the

conventional source . If VE 1s the value of the

energy obtained per unit area , 1t is given by

18 )

where Fe is the fuel cost or energy cost of the

conventional SOUrce in $/kwh. The units of VE

are '/Area . Equation (8) 1s only valid for the

first year, and a total value over n years is

necessary to determine the true value of the

WECS . It is , therefore, necessary to add a

factor to the fuel cost. FC, to account for the

number of years and the increase in fuel cost

It (subscript f for fuel) over n years . This is

CENI'lt.AL CtTRVE OF

KAXD1UH F., VERSUS Tna:

'. ! 2 i "

)

Tille n Yur.

Fig . 1

similar to the adjUstment made to the operating

cost mentioned earlier . Thus , the total value

of WECS energy over n years becomes:

II.E" .. (9)

The term grouped as Cl is the factor that

accounts for increases in fuel costs over n

years . For simplicity, let Kl equal the energy

per unit area defined as ;

(10)

so that YEn is defined as;

(11)

In order for the WECS to be economically

feasible, the total cost of purchase and operation

must equal to or lower than the equivalent

value of the energy it produces . Th1s is ex pressed

as :

r ... ,,,') . P" ' (I ' )~ ';"(PC) " C ' ) (l2)

where AI, BI , and Cl are the time variant fac tors

presented in the foregoing ,

Pw is normally expre~sed as a percentage

P percent of Fw ( ~) ; taking this into account

and solving for Pw produces:

(l3 )

CENERAL CURVES or SAVINGS

VERSUS TIKE fOR V#JUOllS

FRACTIONS OP P., KAJl.IMUM

S11I1n&11

Area

IrHlkeven

Polnu

! Tarset n

•

Fig . 2

The Fw of equation (l ' ) 1110 the maximum

that can be spent on

order fer the system

saved over n years .

shown in Fig. 1 .

the WECS per unit area in

to pay for itself in fuel

A graph of Fw versus n is

A system bUilt at the maximum allowable

Pw will break even in n years . ThiS is because

the total cost of the WECS will have equaled

the value of conventional energy it replaced .

To obtain a saving in cost per kwh over n years ,

the cost of the WECS must be less than the maximum

of equation (1') .

The savings over n years is the difference

in the total WECS cost and the value of the energy

it produces . 'l'h1s is given by;

St;! :g· ~ ~'.( n: )'(C·) _ ' ... ' (11') _ p", ' (. '\ (14)

A graph of the savings at various Fw for

a general case is shown in Pig . 2 . Note that

for Fw less than the maximum , the break even

point is earlier than the target date at n years

energy produced between the break even point and

the target date is essentially free . All energy

produced after the break even point is free ex cept

for maintenance charges as long as the unit

can be operational.

taken as the 11fe of

However, n will usually be

the WECS to minimize yearly

expenditures by spreading the capital investment

over as long a period as possible. Therefore ,

an Fw greater than the maximum will always produce

a system that cannot pay for itself over

its usefUl litetime .

What size unit to build is more a matter

ot energy required rather than economics . A

given installat10n will have an average power

requirement (PRI that represents the typical

load placed on any source . If the m.ax1mwn power

requirement is not much above the average and is

tairly infreqUent , it is usually beneficial to

design the WEes to till this average requirement

(PR) rather than the max1Jlnun. A WEeS designed

to deliver a peak output much above PR will waste

power capabllIt, and the money spent to build it

(1) . Excess power that cannot be used cannot

be used to calculate fuel savings although the

cost or bUilding this capabilIty i s included in

Fw and Pw' This will reduce the savings in

equation (l~) and will lengthen the break-even

period. It is better to design tor the average

and obtain additional power trom other sources

when it is needed .

The area ot the rotor (Aw) is given by:

....... --.;;-

... ,£ -..-.....-.

1151

The Aw ot equation (15) will produce power

equal to PR tor Hw hours and decreasing amounts

ot power the remainder ot the year .

ASSUMPTIONS

In applying this method , several assunp tions

must be made .

I All energy produced by the WEes is immediately

uaable by the installation . The addi tional

cost of energy stor.ge greatly increases

Pw to the point where long periods of t~e are

necessary to reach a break even point . The

ettect on the method is to limit its applica tion

to installations where energy uae is tair ly

constant over all hours ot the day.

2 The VECS is acting in parallel as a

supplement to a conventional source that supplies

po .... er dUring low winds and calm periods.

Most installations require power on deaand and

cannot wait tor tavorable weather . The storage

system excluded in assunption 1 connected

to a ~ch larger WECS would be necessary If the

WECS were to prodUoe all necessary energy .

This would further increase the capltal investment

and other eosts and would increase the

payback period beyond a reasonable number. The

ettect is to limit the method to applications

Where

stant

the WECS is not the only source ot energy .

3 The values ot 1m ' Ii ' and It are conwith

time . The loan interest rate will

almost always be a known constant at n _ 0 , but

the other rates must be estimated at a constant

rate for the perlod ot tiDe to be considered .

The ettect is to decrease reliability of the

method with increasing n . The value of It is

especially important and will probably be the

most difficult to estimate tor long periods ot

time . Government sources should be able to

provide established tigures tor at least 10

years .

4 The value of n does not exceed the

expected litetlme ot the WECS . It the unit is

unoperational tor the later periods ot n, it

cannot produoe any energy to oft set the capital

payments that would still be in progress . A

11fetiDe less than n would give an incorrect

Fw that is too high . A core reliable approach

is to choose n less than the lifetime of the

unit so that the 'lECS will have additiona1 t1tae

to produce free energy .tter it ie paid for and

only operational expenses remain . Also, a

shorter n will increase reliability of the

interest rates and will leave the WECS with a

salvage value at the end of n years.

METHOD OF APPLICATlOO

Anyone interested in applYing wind power

to provide an alternate energy source and to

lower energy cost can apply this method with a

Din1m\lr.l of input c:lata .

The first value to be determined is A , w

the naximum area that can be utilized. ~h1s

reqUires basic infol'r.!ation on the installationl s

power reQuireoent and the local wind data ae

discussed earlier . :he various tactors A', Bl.

and C' are then calculated tor the length ot ttoe

to be considered , usually the '. .l EeS l1fet1zee .

This enables a ~ax~ F~ to be calculated and

compared to information on current construction

coats tor year zero . Only ~~en the unit can be

bUilt tor less than the l!:aJtirlum Fw will the W:ECS

be teasible from an econamic standpoint . A look

at Pig . 1 shows that increasing time allo~s

higher 'v which means 1II0re expenSive units are

practical . TOO long of a period may be avoided ,

particularly on large installations , due to uncertainties

about the estimate ot It . If a

construction coat leas than Pw maximum can be

realized , the payback period can be round by

plottins the sav1ngs of equation (14) against

time . The point where the ourve is the break·

even point , and. energy produced atter that point

will be essentially cost tree . AS can be seen

in Fig . 2, a longer operating period. greatly increased

total sav1ngs although the system may

operate at a loss tor a number of year3 .

CONCLUSION

The optimum Size and cost of a WECS 1s a

funct10n of many variablos . The effects of the

important variables on F are summarized in the

following table:

EHooct 00\ ' ... t U9t..~ 01: I '-·~

~ lnc;~ .... !)ME ....

'. I I

" I I

" r I .' r I

~ r I

" t

The optimum situation is a construction

cost far below the maximum Fw in a geological

area With a higher AV and Hw' Lower financing

rates , as with government bonds. and lower ex pected

inflation rates against higher fuel costs

and Inde~ are also encouraging to WECS construction

The method is easily converted t o other

natural energy rorms by changing the input data

to KI. This variable 1s the energy per area per

year that can practically be collected at the

site. The appropriate informat1on on yearly

sunlight or water tlow would be necessary to

convert Kr for solar or hydro collect10n. The

remainder of the equations remain unohanged for

other energy torms .

EXAMPLES

Consider a homeo ....' ner who desires to add

the cost of a wind turbine to a new home mortgage.

~he yearly energy require~ent of the

home 1s 26,000 kwh (~) . An average load of 3

kw is expected and the mortgage is to be for

20 years . The local average wind is 13 mph for

4200 hr per year (2,) . The load interest rate

is 8 percent , and the inflation rate and tuel

increase index are taken as the published figures

of 4.5 and 15 percent, respectively (~) .

Haintenance is expected to be } percent or the

fixed cost (~) . current electricity cost is

4.5 ./kwh I>') ,

First determine the area to be considered .

Prom equations (lO) and (15):

Il' .. ~.' · (5.J" lO·'I'(1.U· UI'· (UDD)

.... It.n ~.,,~

Next, calculate AI, 5 1, and C* for n-20 years

" ... 20 · G D.n l .. l.D'

1· 1l.000-:ZUJ

" 0 Il.OU~2D_l 0 n.n

0.0 $

c' 0 11.U~20.1 0 102 .44

O. S

The maximUm cost per area at 20 years is then

calculated from equation (I}):

, ..:. {o.OHl 129.151 pOl.HI

.. - 2.d' b.b]!! Li1J

' .. ~ 41.DO Jt2

This is a fairly high figure due primarily to

the long term of 20 years. However, a WECS

that Will last 20 years will be more expensive

than one that w111 last only 10 years or less .

Many of smaller units available today (1977)

are advertised as having a lifetime of 10 years .

Even the best units are only 30 to 35 $/ft2 installed,

so this homeowner can afford the best

unit up to his maximum area ot 5}8 tt 2 • His

saving over 20 years from equation (14) 1s

u;~;t.. (U. HI' [0. OU) · (IH. H) ·117, 5)' U.Q()· (D. Ol)' US . D) ' (ll.l')

For a total or 425 rt2 , the total savings over

the life of the unit (2O years) is

25 . 15 " us .. HO,l75

This may seeo like a large figure, but it must

be realized that one kwh will cost much more in

20 years . In this case , the cost per kwh in 20

years is $.65 if the 15 percent increase is ac curate

. At that rate, $10.775 is not 80 large

when cOlllpared to the energy it actually repre sents

. Over 20 years, the total energy produced

by the WECS 1s

•

The home would have used 26 ,000 x 20 or 520,000

kwh, so the WECS provides ~9 per cent of the

homes energy and saves money as well .

As a second example , consider a chemical

plant desiring wind energy to supplement electric

power in process heat generation . The

yearly energy requirement is 1 x 106 kwh with

a typical load of 100 kw . LOcal wi nds are 17

mph for 4900 hr per year . Maintenance is expected

to be 4 percent of the fixed cost with

electricity currently costing 2.75 ¢/kwh . Use

the Bame economic information given in the previ ous

example with n - 15 years .

K·" 0"'1$.). 1O~'> 'U .U . 17) l,. C" OO)

It· • 71.61 _

.... Ly~

"" .. lUO • H OO .. n12 .... 2

77.12

.... . n . 0.0' .. 1. 75

I-H.OI,-U

.... 11.0 ' !>115~1 .. 20 .71

6.0 5

c' .. !1.UlU~l" n .n

O. S

"w<C !o,on5) · f7.n\ . ~ 47 .5 '1 - 1.15 + t .01' i .n,

"", "".25 • y"

Large wind turbines are currently (1977)

being estimated between '5 and 40 dollars per

square foot (~) . This example represents a

marginal installation that is caused by the

relatively low energy cost tor an industrial

user . Note also that n is only 15 years ,

causing a lower index to increase t he presentday

fuel cost which is already low . Although

this installation may not produce direct energy

savings , the alternate energy source it provides

at no increase in cost per kwh may be reason

enough to build it if the future availability

of conventional electricity is a questionable

subJect .

AS a last example consider a power plant

where it is desirable to use ~nd energy as an

electric power source for the utility grid .

The company uses No . 6 fuel all in the steam

generators at a cost of 0 .0169 $/kwh output

(~). The output of the plant is 3 .5 x 109 kwh

per year with a typical load of 400 MW. Local

winds are 18 mph for 4500 hr per year. A lower

finance rate ot 6 percent is available from

government sources , although If and 11 are the

sarne as in the previOUS examples . A lifetime

of 15 years is necessary for govern:nent fi nanc1ng,

and the periodic maintenance is expected

to be 5 percent due to the complex control

systems Buch as grid feeding generators

reqUire . Continuing as before ,

II ' .. 0" ' ('.3.10-'> ' (1,15 . 11)3. (UOO)

" ... ".n JIWII

.... 2_y ~

"" .. 1400 • IOlrl'5001 .. 21.) .1111 .... Pt2

14 ••

This is a huge ar ea . and the maximum Aw would

probabl y not be utilized due to f i nancial as

well as engineering problems ot such a large

installation . The calculation of Fw is:

A' .. ZO.O.I" • I.H

l_(l.n!~'o

.' ll.oH120-,,, lI.)7

0.0 5

c' .. !1.15tlO~1 .. 102.U

O. 5

' .. .t; to.OU') lu.n\ pOl.H)

- 1.14 . b.OI J .'m

,- !: n.t! 1

.. ..~1

An aotual cost less than Fw could probably

be reali zed, especially considering the large

area to be utilized . The 21 million square feet

given as the opt1Jnum is too large to be practi cal,

although several million square feet could

be obtained in several large units ,

suppose that 3 million square feet is to

be built at ,tItO/ft2. This oan be accomplished

by 15 units with rotor diameters of 250 ft.

less than the largest currently considered

feasible from a technical standpoint (!) . This

represents a fixed cost of 120 million dollars ,

a realistic sum for such a proJect . The sav ings

of the venture can be calculated as before:

Savings

Area

Savings

Area

- (84 .62) • (0 .0169) . (102 . 4-4) -40 ' (l. 74) -

1_00) - 100 ) -1'1.'7)

0 20 _67 1 Ft 2

For the 3 million square feet , the total

savings is (3 x 106 ) (26 .67) • 80 million dol lars

. ThiS saving is above the cost of the

wind turbines and is in terms of present -day

dollars .

CLOSING

AS can be seen in the examples. the

actual calculation of the optimum areas and

costs are easy when the information reqUired

is available . The gathering of the input data

is the most difficult part, although increasing

use of wind and other natural energy resources

should provide more accurate data from which

better and more reliable estimates can be made .

For the present, the method should be used in

conjunction with good e~neering and economic

Judgment to detel'llline the best natural energy

conversion syaten for each specific installation

APPENDIX

Values of K for various units (2):

:POwer Area Velocity K

7 - 6 kw mph 5 .3 x 10~

kw ft2 knots 8 .1X10

hp ft2 mph 1.1 x 10·

watt ft2 fp. 1.7 x 10 -3

kw meter2 meter/sec 6.4 x 10-4

kw meter2 kilometer/sec 1.4 x 10~5

REFERENCES

1 HUdson, R. G. , Engineers Manual , Wiley,

1944 .

2 Golding, E. W. , The Generation of

Electricity by Wind Power, E&P Spon, Ltd . ,

London, 1976 .

3 fUtnam . P. C .. Power From the Wind,

Van Nostrand, 1948 .

4 ERDA, proceedings of the Second Alll'Ulal

Conference on Wind Energy, lJSAPO, 1976 .

5 Schuth, H. C" LouiSiana Power and

Light Company , personal communication with

author .

6 u. S. Weather Service, New Orleans

Area Weather History, personal communication

with author .