Payback of solar systems

Solar Energy, VoL 20, pp. 225-232, Pergamon Press 1978. Printed in Greal Britain

PAYBACK OF SOLAR SYSTEMSt K. W. BOER Institute of Energy Conversion, University of Delaware and SES, Incorporated, Newark, DE 19711, U.S,A.

Received 30 July 1976: in revisedform 19 April 1977: receivedfor publication 1 September 1977) Abstract--A variety of solar conversion systems is studied in a dynamic economical model in which the real cost of energy inflates. Payback times and dates of probable market entries are estimated. A distributed system to convert solar energy into heat and electricity in direct proximity to the consumer (Solar One system) is economically attractive even for solar cells with well below 10 per cent conversion efficiency when these can be installed in flat plate collectors for less than $301mz, in addition to the collector cost. I. INTRODUCTION Solar energy conversion systems usually require a substantial initial investment, compared to conventional energy conversion equipment. In order to decide whether such an investment is economically justified, one usually compares the return on investment for a solar system with those in the conventional capital market. Usually for a return on investment analysis a present value approach is used whereby initial investment, annual expenses and annual savings are discounted to their present value and the sum of the first two is compared to the latter. This computation contains as a hidden ingredient the life of the equipment. However, for solar installations this life is often unknown, and an error in its assumption could produce misleading results. It is therefore, suggested to use for solar economic evaluation the payback time analysis, which reminds the user of the importance of the life time of his system when analysing the results of the economic evaluation. Usually the payback time is computed as the time at which first cost and annual expenses with compounded interest equal the total savings of energy cost with compounded interest. The value of the investment, C, at the end of the payback time is A = C(I + r)" (1)

From the condition for the payback time

A+B=D

one obtains an expression for the ratio of annually displaced energy cost to first cost E

r

~ = m + l - ( 1 + r)-"

(5)

a ratio which approaches m + r for large values of n and is substantially larger than m + r for less than 20 yr (See Fig. 1). The payback time is longer the higher the interest rate: It takes longer to compensate via annually saved energy cost for the more rapid appreciation of the first invested capital. From Fig. 1, family A, one sees that for reasonable interest rates and payback times of about I0 yr, a ratio of EIC of about 0.2 is required. The E/C ratio for most solar energy systems is usually less than 20 per cent, hence payback times substantially in excess of 10 yr are expected for such systems in the given economic model. Moreover, the payback time increases very steeply with decreasing E/C. These long payback times and their great sensitivity to small errors in the economic analysis are usually not attractive to stimulate a large market. However, the payback time decreases considerably

with r the annual interest rate.~ The yearly expenses (such as taxes, insurance, replacement,§ and maintenance), often referred to as annual fixed charge rate mC, accumulates (as annuity) to

B=mC(l+r)"-I r

(4)

0.4

~[L

~

|

i

i

(2) I

0.3

while the income from such a system, the accumulated annuity of the annually displaced conventional energy E is

D=E(I+r)"-I r

•

0.I

(3)

4

fWork sponsored by SES, Inc. :~If the capital is borrowed, then r is the cost of capital, a fraction of which, or all may be tax exempt. Since usually alevelized interest and amortization method is used, a somewhat modified computation results. §For continuous operation complete replacement is assumed at the end of the systems lifetime. 225

0

i

2

6

~ I I0

J 14

18

YEARS

Fig. I. Required annual savings (Cost of displaced energy). E, per first Cost C ratio for: (A) m =0.05, i= e =0: (B) m =0.05, i = 0.05, e = 0.09: (C) m = 0, i = 0.05, e = 0.09. For interest rates 1:0: 1: 2:0.085: 3:0.07: 4:0.055.

226

K.W. B6ER

when inflation is considered. This could improve the solar market expectation substantially, as shown below.

0.3

2. PAYBACK IN A DYNAMIC ECONOMIC ENVIRONMENT

The annual savings from displaced conventional energy are expected to continue to increase as the cost of energy is expected to increase. Assuming a constant inflation rate for energy, one obtains the savings in year l as E(I + e)t, which yield in n years when invested E(I + e)t(1 + r) "-~. Consequently, one obtains for the accumulated savings D' = (1 + r)" - (1 + e)" E

0,4

(3a)

'm

~ '

'

0.2 0.02

0.1 0

2

L

I

I

6

I0 YEARS

14

18

Fig. l(a). As Fig. 1 for r = 0.065, i = 0.05, e = 0.09 and given m.

r-e

a value which is substantially larger than without inflation (eqn 3). The annual expenses, however, will probably also be increased by the inflation, but usually with a different inflation rate, i. Consequently, the accumulated expenses would be modified in a similar fashion yielding

0.4 0"5

\\

'

'

\\

' r

0.2

g B' = ( l + r)" - ( 1 - i)" r- i mC.

(2a)

The modified payback condition A + B ' = D' yields now a considerably reduced EIC ratio (compared to eqn 5) when e is larger than i: E ct(r-e)+ r-era -C= fl r - i

O. I

0 2

0.065

8

14 20 26 YEARS Fig. l(b). As Fig. 1 for m = 0.05, i = 0.05, e = 0,09 and given r.

(5a) 0.2

with a =

1 (1 +e~" l-\l+r ]

(5b)

0.15 I 2 3 4

0.1

,\

and (l+i~" 1-\l+r ] /3 =

(1 + e~" 1- \iT;r/

O.O5 (5c)

a ratio which approaches r + [ ( r - e)l(r- i)]m - e for large values of n. Assuming an inflation rate of energy of 9 per cent, one obtains the family B in Fig. 1, which shows the resulting decrease in E/C or in n. For both family A and family B, the initial yearly expenses are assumed to be 5 per cent of the first cost. From eqns (5a--c), one sees that the inflation rate i has only a minor influence through the second term relating to m (see Fig. 2) while the inflation rate of energy e is of major influence (Fig. 3). For systems with negligible annual cost, the family C of Fig. I gives an example of further substantially reduced payback times. It is presently believed that most components of a solar conversion system can be made for a life expectancy in excess of 10 yr (a 20-yr life is often suggested). Hence, a payback in less than 10 yr seems to be attrac-

L

0 2

6

I

L

to YEARS

14

18

Fig. 2. ~m(r - e)/(r - i) for m = 0.05, i = 0.05, e = 0.09 and r as a family parameter 1: 0.09; 2: 0.075; 3: 0.06; 4: 0.055; 5 : 0.04,

0.4

0.3 0.2 0"1 0

I

6

I

tO YEARS

A

14

18

Fig, 3. As Fig, 1 for r = 0.065, i = 0.05, m = 0.05 and e as a family parameter 1:0.04; 2:0.06; 3:0.08; 4:0.1; 5:0.12.

Payback of solar systems tive, provided that the solar system appreciates the value of the total installation (e.g. a building) and does not significantly disturb the original buildings cost/depreciation schedule. After a lapse of the payback time for the length of its life, the solar system continues to deliver energy, however, now only for the annual recurring cost, hence with profit. In the following sections, we will discuss a few simple examples to indicate the impact on typical systems. For interest and inflation, we will use in the following text the above given standard values, i = 0.05, e = 0.09 and r = 0.065 if not otherwise indicated, and m = 0.02 for thermal and m = 0.05 for electrical system.t"

(6)

which is plotted for the standard economical parameters, as defined above, in Fig. 4. Here the inflation rate of energy is family parameter. It is assumed that the cost differential between a conventional and a solar heating plant is $8000, and that it produces approximately 25 x 10 6 kcal during a heating season (with a collector surface of = 100 m2). It is evident that only the more costly fuels may presently be replaced by solar energy if a payback in less than 10 yr is required, and if the system can be built with a $8000 cost increment. 16

64

4

e~

12

48

~

64

,

e%

t

16

4

48

-12

6.

~32

a r~

16

Solar heating systems have been discussed extensively in respect to size optimization, collection expectation in different climates and certain systems aspects[I-4]. The system can be characterized by its average solar utilization capacity during a heating season, i.e. the total annual amount of Btu's, Q, obtained from solar input, and by its first cost, E. Consequently, on~ can estimate the minimum starting energy cost K (in S/Million kcal) for which amortization takes place in a desired period r-e

However, the payback time is substantially influenced by the cost. If 10 yr is required, then lower cost fuels can be replaced if the cost differential drops below $6000, as shown in Fig. 5 (parameters are the same as above). Conversely one would need to replace at least $45/Mkcal fuel considering an energy inflation rate of 9 per cent, if the cost of the system exceeds $10,000.

,o

3. SOLARHEATINGSYSTEMSOF HOUSES*

C

227

4

L

i

4

o

I

6

8

I0

0

COST ($000)

Fig. 5. Required cost of unit of energy as function of the first cost differential of the solar space heating system for 10yr payback time: otherwise as Fig. 4. 4. HEATINGSYSTEMFOR DOMESTICWATER Somewhat more optimistic is the situation for hot water systems, mainly due to an increased utilization factor and a decreased cost differential: The hot water system can be used during the entire year, while solar space heating in climates which may justify solar installation is at best during half a year in operation. Also, the water tank which is needed also in the conventional system acts as storage, hence does not get added to the cost differential of the solar system, unlikely in the solar space heating system, where the storage subsystem increases the cost differential appreciably. With an average size of 2001~. per day per family of hot water at 70°C, a collector of approximately 6 m 2 is sufficient. Using a 400 1. storage tank, the system may deliver more than 3 Mkcal/yr in over 50 per cent of the continental U.S.A. In Fig. 6, the pay-back time is plotted as function of the cost of energy with the first cost differential as family parameters (other parameters as above). It is evident that the investment for a solar water heater instead of an electric water heater is probably amortized in

32 64 / ~

'

]

'

16

$ 600

16

14 1

2

6

t

L

I0

14

18

YEARS

,8 \ \ \ \ j ~ 4 o o 2,6 \ \ ~ , ~ /

,2

,200 ,8

,3oo

Fig. 4. Required cost of unit of energy as function of pay-back time for r = 0.065, i = 0.05, m = 0.02 and e as family parameter. $8000 first cost differential and 25 Mkcal annual solar conversion assumed. 0

tWith a life expectancy of more than 20 or 40 years for the electrical or thermal system respectively. :~Here, and in the following sections only a limited analysis is presented and such features as load leveling for power utilities, diversity analysis, utility rate structure etc. are not included.

' 6

' I0 YEARS

' 14

0

18

Fig. 6. Required cost of unit of energy as function of payback time for solar domestic water heater with first cost differential as family parameter with r = 0.065, i = 0.05, e = 0.09 and m = 0.02. Thermal energy harvesting assumed 3 Mkcal annual.

228

K. W. B6ER

less than ten years. With an estimated cost differential of $600, even with oil as fuel, the first cost can be paid back in less than 13 yr. When such first cost differentials can be achieved, this could make the solar water heating system economically attractive with a substantial market in the U.S.A. 5. HEAT STORAGE

Heat storage has the characteristic of extending the useful capability of a solar system by extending its utilized capacity, since charging of a storage bin usually occurs only when the load does not demand heat. Proper determination of the annual capacity of a storage bin is a task which requires an incremental analysis of solar harvesting potential, demand, available storage capacity, and losses during an entire heating season, There are a number of computer programs developed with the aim to optimize the seize relation between collector, bin and load for different climates and systems parameters. From such programs, the utilization factor of a storage bin can be obtained and used for a separate cost analysis. For a mid-U.S.A, belt and for a conventional solar heating system (as used in Section 3), such utilization factor is approximately 75, i.e. during a heating season approximately 50-90 times the maximum storage capacity is cycled through the bin[2]. Assuming a cost of $1000 for a 125,000kcal bin, one obtains the curves of Fig. 7 for different utilization factors as family parameter (other parameters as above). It is remarkable how attractive the payback values are for a storage bin of the given cost and capacity. Even with a utilization factor of only 60 equivalent full cycles per year, the payback time is less than 10 yr.t One of the major reasons is the low cost of the subsystem per unit of energy, as compared to the collection and distribution system. However, one may not be misled by this analysis and plan to increase the storage capacity with insufficient consideration of the total system, since an increase in capacity near the optimum point usually will result in a decrease of the utilization factor. Again, a proper sys-

terns analysis is essential to obtain highest economical return. 6. ELECTRICALSTORAGE Electric storage batteries provide another rather transparent example. The payback depends critically on the number N of equivalent full storage-discharge cycles per year. Since these batteries have a limited cycle life, it is advantageous to plot the payback as function of the cycle life (v) during each cycle only a fraction "O(usually 75 per cent) of energy is recovered. Hence, one obtains as payback equation

{ (l+i~v/N~ r-e+m 1-\]--~] 3 r+e E=C (l+i]'u~ . r+i"

n~'{1+\F+-;'/ 1

Figure 8 shows a family of curves for different battery costs as family parameter. One of the curves (for a cost of $40/kVAh is also shown for different numbers of equivalent cycles per year (from 175 to 225) to indicate the economical sensitivity to the cycling frequency. For instance, a lead acid battery with a price tag of $40 used for 200 equivalent full cycles a year and 25% per cent loss per cycle delivers a kWh for approximately 2.5¢ for a cycle life of 700. However, Fig. 8 also shows the critical dependence of the payback on cycle life. Cells with a cycle life of less than 400 and a first cost of $40/kVAh yield a cost in excess of 7c/kWh for stored electric energy.

~/kVAh 9 6

60

~o"

20

3

2 2 5 - ~ ~

0 64

~

,

,

16

75~ I00

400

700

I000

130(

CYCLELIFE

UTILIZATION FACTOR

~\\\~ ,8

,

(7)

)\\\\\\\\2o, 8 84x

16 -

Fig. 8. Price increment of a kWh after passing through a battery at 75 per cent efficiencyas a function of battery cycle life for first cost as family parameter and 200 cycles/year. Subfamily for $40/kVAh shows spread for 175-225 cycles/year (m =0.05, i = 0.05, r = 0.065 and e = 0.09).

4 7. COMBINED ELECTRICAL AND THERMAL SOLAR SYSTEM

0

~

I

2

6

I

I0 YEARS

I

14

18

O

Fig. 7. Required cost of unit of energy as function of payback time for thermal storage system with the annual utilization factor as family parameter, using r = 0.065, i = 0.05, e = 0.09 and rn = 0.02 and assuming a first cost of $1000 for the storage bin. tWhen replacing fuel at $15/Mkcal.

A combined solar electrical and thermal conversion system, as built with the Solar One house of the University of Delaware, gives an example which may become an economically attractive conversion system[5]. One way regard this system as a solar thermal converter for house heating as given in Section 3, however, with the addition of a photovoltaic collector plate plus electrical power processing equipment. In doing so, one has separated the payback analyses

Payback of solar systems for the heating subsystem and can now investigate under which conditions the added electrical subsystem increases or decreases the total payback time. As the electrical subsystem, we will define the photovoltaic collectorplate which will replace the passive black sheet inside a flatplate collector, the wiring to the electric power processor, the processor itself, which consists of an invertert and possibly a transfer switch,t and a small battery to store electric energy for a few hours to bridge transients and provide capacity for power surges. The cost per rated~: kW of the electrical subsystem Co can be estimated using a formula§ developed by DeMeo and Spencer [6]. (8)

Co = 4AC[rI~,,, + Cp,,

with A C the cost of the installed array plus wiring per m 2 and rl~e. the measured solar cell conversion efficiency (Cp,, = cost of power processing/kW). The system delivers an average of 13.5 kWh per day for every rated kW installed, or =4900 kWh/year with a collector surface of a (-~ 100 m2 in the example in Section 3), the rated electric power of the system is given by Pr = 0.25 ~ce,,a

(9)

with a total cost of the electric subsystem (10)

C = ( A C + 0.25Ct, onceH)a.

The cost of annually displaced energy by such a system is given by

(I l)

E = 4900P, p

with p the price per kWh. Using eqns (5) and (9)-(11), applied for the electric subsystem, one obtains for the permissible kWh price

p=2

f37~_i

1 (12)

Presently the cost of solar cells is by far too high to warrant a meaningful payback analysis. However, assuming that their cost will continue to come down, finally to between $10 and $40/m 2, and assuming a pay-

229

back time of 10 yr and as cost/6] of the power processing equipment Cp,, = $265/rated kW, one obtains Fig. 9 which shows the required cell efficiency for a given price of the replaced energy unit. Family parameter is the cost/m z of the installed array. .

,% 2

2O 1

5

,

$/rnz

~

I0 0

I

,02

I

I

.05 .08 .11 CELL- EFFICIENCY

.14

Fig. 9. Price of electric energy collected from solar panels as a

function of cell efficiency with deployed panel cost as family parameter (payback time 10 yr). m = 0.05, i = 0.05 and r = 0.065, cost of power processing equipment $265/kWr). There are several estimates made to indicate that $15/m 2 for CdS/Cu2S and $40/m = for Si may be achievable,~ with 7 and 14 per cent efficiency respectively. One obtains from Fig. 9 that the permitted cost of replaced energy for the CdS/Cu2S array is 3g/kWh and for the Si array is 4e/kWh. This installation assumes consumer-oriented economics. If electric power utilities are involved to subsidize or to own such a system, a considerably longer payback time may be permissable,tfWe have here assumed a return on investment of r = 10 per cent and r = 20 per cent (other parameters as above) and shown in Figs. 10(a, b), respectively, the price to the consumer at which the harvested energy could be sold for different efficiencies as family parameter ($15/m 2 and $45/m = array costs are assumed for CdS/Cu2S and Si, respectively). One sees that with 7 per cent cells and 24 yr payback, a price of 2e/kWh can be achieved at 10 per cent and 3.8¢ at 20 per cent return on investment. For a cost of $45/m 2, one would need 12 per cent cell efficiency to achieve 3e/kWh with 25 yr payback, and 18 per cent cells are needed for 8

i

i

$15

v

$/m2

6 -

$45 tHowever, one may use a phase-locked inverter with the

capability of feeding electric energy into the grid. ~:DeMeo and Spencer/6] assume a 13.5h/day average systems operation, hence a peak power times average daily hour output/13.5 h; this obviously includes a modest storage capacity). §The numerical factor (4) depends on climate and systems efficiency,and is given as ratio of l/systems efficiency(exclusive cell efficiencyhere assumed 0.65) times average rated isolation in KW/mz over rated output period (here estimated 5.4/13.5= 0.4). ~Assuming 20¢/W and 30c/Watt deployed full panel cost for CdS/Cu2S and Si respectively. IIprovided that the life expectancy of the systems components is high enough, or replacement cost is included in m (here assumed to be 5 per cent). SE, Vol.20,No. 3-.-.('

0 I0

i 15

t 20 YEARS

24°/o i 25

30

Fig. 10(a). Price of electric energy from solar panels as a function of payback time with cell efficiency as family parameter for two typical deployment costs (m =0.05, i=0.05, r=0.1 and e = 0.09).

K. W. B6ER

230 I

!

,,5

approximately 8 yr. However, for 4e/kWh and $16/Million kcal, the payback time is 13.5 yr. With a slightly reduced first cost differential, the payback terms look even more attractive (Fig. 12), with a reduction of the payback time below 10 yr for first cost differential below $10,000 at cost of less than $18/Million kcal (a cost of 4e/kWh is assumed).

/m2

12%

~

4

10%

~

2 8. TIME TO ENTER THE MARKET

0 I0

15

20 YEARS

30

Fig. 10(b). Same as 10(a), however, for r = 0.2. 2e/kWh and 27 yr payback time (at r = 10 per cent) and 4e/kWh for 20 per cent return on investment. Let us now return to the combined system. From Fig. 4, we see that an $8000 cost differential thermal subsystem has a payback time of 10 yr if it replaces $36 per million kcal. With 7 per cent solar cells at $15/m2, one estimates from eqn 10 a cost of $3140--for the electrical subsystem-for a total systems cost differential of $11,500. The electrical system produces 13,800kWh/yr. Figure 11 shows the payback time of the total system for an energy inflation rate of 9 per cent. In comparison with Fig. 4, one sees that for electric energy cost in excess of 2.5e/kWh, the combined system shows a reduced payback time. With 5e/kWh electric energy cost and $321Miltion kcal thermal energy cost, the system pays itself back in

S4

'~~~~

¢/'kVVh

In the previous sections, we have shown that for energy conversion which has zero fuel cost (such as solar or energy storage), the payback time of the first cost depends critically on the price of displaced energy. In a dynamic economic environment in which the cost of energy increases with a higher rate than the common inflation rate, a time can always be found at which the market entering will be profitable, i.e. at which the payback time drops below a value considered as market threshold, provided the parameters describing the dynamic economic system remain unchanged. We will return to this question in the following section. Appreciable market penetration will start when the EIC ratio drops below a critical level. With time, this ratio changes as

E [E~ (l+e]"

(13)

and is given as function of time for 9 and 5 per cent inflation rates for e and i respectively in Fig. 13.

16 12

48

0.4 0.3

-

0.2

16

4

0

0

0.1 2

6

I0 YEARS

14

18

Fig. 11. Required cost of unit of thermal energy as function of the payback time for a Solar One System with 25 Mkcal and 13,800kWh per year harvesting. Price of displaced electric energy as family parameter (First cost 11,5005,m = 0,02, r = 0.065, e = 0.09 and i = 0.05).

,,t

\\\\\

:,2000 / ,,,.ooo

12

4

6

IO

14

0

5

I0 15 YEARS

20

Fig. 13. Delay time of market entry to reach from a present E/C value the desired entry ratio (Fig. 1).

16

/ 2

0

18

O

YEARS

Fig. 12. Same as Fig. 11, however, for first cost differential of total Solar System as family parameter (¢4 kWh electric energy cost assumed).

The permitted EIC ratio to obtain a desirable payback time can be obtained from Fig. l. Assuming a payback of 10 yr in a system with m = 5 per cent and r = 7 per cent is desired, one needs EIC for entry of approximately 0.15. If the present value of EIC is 0.1, one has to wait 11 yr (Fig, 13) before market entry is advisable. Using Figs. 1 and 13, one can define a probable market entry date for any of the solar energy conversion systems. A number of examples with their respective assumptions are given in Table 1 and are shown in respect to probable entry date, depending on desired payback time assumption (length of arrow) in Fig. i'4.

231

Payback of solar systems Table 1. Solar energy conversion systemst First years annual mean return ($)

First cost differential ($)

Solar conversion systems 1-Space heating 2-Space heating 3-Heating and Cooling 4-Domestic water 5-Domestic water 6-Solar thermal 7-Solar thermal 8-Solar One, CdS 9-Central Sta. Si 10-Central Sta. CdS I 1-Windmill

Return on inv. or int.

Replaced energy

Desired payback time (yr)

8000

360

160

r = 0.065

Oil

8, 10, 15

8000

1110

160

r = 0.065

Electric

8, 10, 15

10,000

1000

500

r = 0.065

Heat pump

8, 10, 15

400

40

8

r = 0.065

Oil

5, 8, 10

400

135

8

r = 0.065

Electric

800/kW,

4900 kWh 0 2 ¢ = 98¢

40

Rol = 0.2

Electric

15, 20, 30

1200]kWr

98

50

Rol = 0.2

Electric

15, 20, 30

11,500

910

224

r = 0.065

Oil/Elec.

8, 10, 15

750/kW~

1800 kWh @2¢ = 375 = 375 2900 kWh 0 2 ¢ = 585

37

Rol = 0.2

Electric

15, 20, 30

15 30

Rol = 0.2 Rol = 0.2

Electric Electric

15, 20, 30 15, 20, 30

7000 kwh 140$

60

Rol = 0.2

Electric

15, 20, 30

7000 kWh 02¢ = 140t

45

Rol = 0.2

Electric

15, 20, 30

400/kWp 600/kWp

12-Ocean thermal

Estimated maintenance taxes insurances etc. ($)

1200/kW

O2¢ =

13-BTU PlantationS:

320/kW

tThis Table does not include a risk factor to achieve the assumed first cost at the projected market entry time. *Assuming a plantation of $24,000,000 first cost and $9,000,000 annual operating cost delivering 1.5 × 109 kWh/yr plant $200/kW utility conversion equipment (80 per cent Plant Factor).

9. VARIABILITY OF MODEL ASSUMPTIONS

2

6 8

~t ,

1980

i

1990

I

I

2000

Fig. 14. Suggested times of market entry (start of arrow) for assumed* solar systems (for identification of numbers, see Table l) with given payback times (length of arrow).

tPlease note that these solar systems are based on mass fabrication technology and the proposed price reduction may not be reached at the estimated market entry times. :~Fig. 14. shall not be used to judge relative merits of the different systems, since it does not contain the risk factor to achieve the economic parameters assumed in Table 1. §Provided its components are not energy intensive to produce.

From the above description, it may seem that almost any energy conversion scheme,§ no matter how unattractive today, will find its marketplace in the future, provided one waits long enough before entry. This can obviously not be the case, since more economical systems will tend to saturate the market. This will cause the inflation rate of energy to decrease until it equals the general inflation rate. As a consequence, the curves in Fig. 13 will become horizontal again at the end of this economical transition period in which e > L One of the major issues in economical trend prediction should he to determine the length of this transition period. It is determined by a number of factors: (a) The certainty that fossil fuels, except coal, will be phased out; (b) The determination of the best candidate for major energy supply; (c) The availability at sufficient scale; (d) The fair market price per unit of energy from this candidate. If this candidate is coal, then a proper long-term estimate of the cost of its gassified or liquefied product, including all environmental debits, must be made. If it is nuclear fusion, a life cycle cost estimate of the

232

K. W. B6ER

most probable reactor and proper return on investment analysis must be given and properly updated. If it is solar, a more thorough competitive analysis of the different solar options is necessary. Presently, most of these estimates are hesitant, since they involve the high risks of the technical development. However a doubling of energy cost compared to the present normalized consumer price index seems to be a possible development over the next 20 yr. One other road to obtain some insight into possible energy cost developments is the use of the relation between gross national product per capita and energy consumption per capita. Extensive empirical material is available [l l] and suggests a linear relationship for expensive energy forms (e.g. electricity). The gross national product per capita can be described as: g = go+ S E

(14)

with go=0 or =50S/capita and S=0.7$/kWh or 635S/Million kcal for electric and for thermali" energy respectively. Deviations from this straight line occur towards lower energy consumption per capita where energy is expensive and towards higher energy consumption per capita where energy is amply available; however, these deviations are observed within a limited band, obviously dictated by the elasticity of energy economics. Energy conservation (i.e. reduction of energy waste) utilizes this elasticity without reducing the standard of living. The degree of elasticity may indicate the margin to which the compounded energy inflation to general consumer index inflation ratio could be tolerated before a reduction in the standard of living will result. Again, a factor of two~ over the next 20 yr seems to be possible. It seems that the transition from resource (fuel) stimulated economy with "unlimited" fuel supply in the past to a first cost limited energy economy may present a limit to growth. The recognition of this limit and the proper distribution of our present resources to develop the most

tHere a linear approximation is used for first orientation only. ~:Obviouslymuch more research is needed to substantiate this factor.

economic conversion systems is a path essential for early guidance and for avoiding instabilities. 10. SUMMARY It has been shown that the payback times for solar conversion systems are substantially reduced in a dynamic economic environment in which the cost of energy increases faster than other commodities with time. In this transitional period in which the inflation rate of energy lies above the inflation rate of other commodities, a sequence of points in the time scale can be defined for most probable market entries of solar conversion systems with different return to first cost ratio. Markets can be distinguished in respect to maximum desirable payback time for different consumer groups and conversion systems. It is expected that a variety of solar conversion systems will become profitable during the next 20yr, in which a doubling of real cost of conventional energy may be expected. The conversion of solar energy into heat and electricity (Solar One System) is expected to be economically very attractive even for solar cells well below 10 per cent conversion efficiency, provided that cell panels can be installed for less than $30/m ~. Acknowledgement--It is a pleasure to acknowledge discussions

with Dr. Carter Waid. REFERENCES

1. G. O. F. L6f and R. A. Tybout, Solar Energy 14, 253 (1973) and 16, 9 (1974). 2. S. A. Klein, W. A. Beckman and J. A. Duttie, A design procedure for solar heating systems. Solar Energy 18, 113 (1976). 3. Phase Zero Studies, General Electric, T. R. W. and Westinghouse Final Reports NSF/RANN 0974). 4. R. T. Ruegg, NBS-IR 75/72, July 0975). 5. K. W. BiSer, Proc. 9th IEEE Photovoltaic Spec. Conf., p. 8 (1972); Chem. Tech. 3, 394 (1973). 6. E. DeMeo and D. F. Spencer, Proc. Int, Workshop CdS Solar Cells and Other Heterojunctions, p. 109 (1975). 7. K. W. B6er and J. Olson, Proc. lOth IEEE Photovoltaic Spec. Conf., p. 254 (1973). 8. R. J. Mytton, Solar Energy 16, 33 (1974). 9. T. F. Jordan, Proc. Photovoltaic Power Generation, Deutsche Ges. f. Lu[t.und Raum[., p. 221 (1974). 10. T. P. Brody, Proc. Int. Workshop on CdS and other Heterojunctions, p. 499 (1975). II. G. C. Szego, ITS Report C654 to RANN (1971).

Resumen--Se estudia una variedad de sistemas de conversi6n solares en un modelo econ6mico dimlmicoen el que

el costo real de la energia se infla. Los tiempos de amortizaci6n y las probales fechas de entrada en el mercado son estimados. Un sistema distribuido para convertir la energia solar en calory electricidad en proximidad directa al consumidor (Sistema Solar Unico) es atractivo economicamente incluso para c61ulas fotoel6ctricas con rendimientobastante inferior al 10%,cuando estas pueden ser instaladas en colectores pianos pot menos de US 30 por m2, encima de costo del colector. R~umO--On 6tudie un certain nombre de syst•mes de conversion solaire suivant un modOle 6conomique dynamique clanslequel le coOt rOeld'Onergieaugmente. On estime leurs temps d'amortissement et les dates de leurs entrOes probables sur le marchO. Un syst~me distribu6 pour convertir 1'Onergiesolaire en chaleur et 61ectricit6 proximit6 directe du consommateur (syst~me solaire I) est une solution attirante du point de vue 6conomiquem~me pour des cellules solaires d'un rendement de conversion bien infOrieur ~ 10% lorsque celles-ci peuvent ~tre installOesen capteurs plans pour moins de 30 dollars par m2 en plus du coot du capteur.