NBSIR 75-712

Solar Heating and Cooling In

Buildings: Methods of Economic

Evaluation

Rosalie T. Ruegg

Building Economics Section

Center for Building Technology

Institute for Applied Technology

National Bureau of Standards

Washington. D. C 20234

July 1975

Final Report

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

NBSIR 75-712

SOLAR HEATING AND COOLING IN BUILDINGS: METHODS OF ECONOMIC

EVALUATION

Rosalie T. Ruegg

Building Economics Section

Center for Building Technology

Institute for Applied Technology

National Bureau of Standards

Washington, D. C. 20234

July 1975

Final Report

U.S. DEPARTMENT OF COMMERCE, Rogers C.B. Morton. Secretary

NATIONAL BUREAU OF STANDARDS. Richard W. Roberts. Director

PREFACE

This research was begun as a supporting economics effort for The Office

of Housing Technology in the Center for Building Technology, Institute for

Applied Technology, National Bureau of Standards. This report, prepared by

the Building Economics Section, was written to summarize the findings of that

research and to provide a background document that researchers and analysts

can use for economic evaluations of solar heating and cooling systems in buildings. Comments on this presentation are invited. Input from the reader will be

useful in establishing needs and objectives of future publications on the topic

of life-cycle costs of space conditioning equipment.

Appreciation is extended to the solar energy consultants and those members

of the CBT staff who reviewed the paper. Special appreciation is extended to

Dr. Harold E. Marshall, Building Economics Section, for his valuable assistance

throughout preparation of the paper.

i
ABSTRACT

This report addresses economic issues important to the design, acquisition,

and evaluation of the costs to consumers, of solar heating and cooling systems

in buildings. It explains and illustrates with simple, but realistic examples

the use of life-cycle cost analysis and benefit-cost analysis to evaluate and

compare the economic efficiency of solar and conventional energy systems. It

also explains the conditions for making cost-effective tradeoffs in solar system/

building design. By presenting the basic methods and assessing the appropriateness of alternative assumptions, the paper provides a resource document for re- searchers and analysts.

ii
TABLE OF CONTENTS

Page

PREFACE i

ABSTRACT 11

LIST OF FIGURES Iv

LIST OF TABLES . . . Iv

1. INTRODUCTION 1

1.1 BACKGROUND 1

1.2 PURPOSE 1

1.3 SCOPE AND ORGANIZATION 1

2. LIFE-CYCLE COST EVALUATION OF SOLAR HEATING AND COOLING SYSTEMS .... 3

2.1 RATIONALE FOR A LIFE-CYCLE EVALUATION OF PRIVATE COSTS 3

2.2 LIFE-CYCLE COST ANALYSIS: AN OVERVIEW 4

2.3 OBJECTIVES, CONSTRAINTS, AND ALTERNATIVE SOLUTIONS 5

2.4 RELEVANT COSTS 5

2.5 CASH FLOWS AND LIFE-CYCLE COSTS 7

2.5.1 CASH FLOWS 7

2.5.2 DISCOUNTING OF COSTS 7

2.5.3 LIFE-CYCLE COST MODELS 12

2.5.4 TREATMENT OF TAXES, INSURANCE, AND GOVERNMENTAL

INCENTIVES 15

2.5.4.1 TAXES 15

2.5.4.2 INSURANCE

...

22

2.5.4.3 GOVERNMENTAL INCENTIVES 23

2.6 COMPARISON OF ALTERNATIVES 23

2.7 UNCERTAINTY IN COST EVALUATIONS 31

3. CONDITIONS FOR ECONOMIC OPTIMIZATION OF THE SOLAR HVAC SYSTEM AND THE

BUILDING ENVELOPE 35

4. SUMMARY AND SUGGESTIONS FOR FURTHER RESEARCH 40

SELECTED BIBLIOGRAPHY 41

lii
LIST OF FIGURES

Least-Cost Combination

LIST OF TABLES

Cost Data Requirements

Discount Formulas

Discount Factors

Cost Analysis: A Hypothetical Example

Part A General Assumptions

Part B Assumed Additional Capital (Including

Installation) and Maintenance Costs for

Solar System

Part C Energy Costs

Part D Net Benefits Calculation, Assuming No Real

Escalation in Fuel Prices

Part E Net Benefits Calculation, Assuming a 4% Real

Increase in Fuel Prices

Influence of Assumptions on Economic Efficiency of

Solar Energy Systems

Illustrative Assumptions for Evaluating Solar Energy

Systems

iv
1. INTRODUCTION

1. 1 Background

The widespread use of solar heating and cooling systems in buildings hinges

in large part on their economic performance relative to conventional heating and

cooling systems. Economic evaluations and comparisons of alternative solar and

conventional energy systems will be required by research analysts, by builders,

homeowners, architects, lenders, manufacturers of solar energy equipment, government policy makers, and others in order to determine the economic merits of solar

energy systems. To make these evaluations, reliable and consistent procedures

are needed for the collection and analysis of economic costs and benefits associated with the various systems.

1.2 Purpose

The purpose of this paper is (1) to explain and illustrate with simple, but

realistic examples some basic concepts and techniques for evaluating and com- paring the economic efficiency of solar and conventional heating and cooling

systems for buildings, (2) to assess the appropriateness of alternative assumptions and procedures which might be employed in the economic evaluation of alternative systems, and (3) to set forth the necessary economic conditions for determining efficient tradeoffs in system/building design.

The paper is intended primarily as a resource document for researchers and

analysts who are concerned with the design and evaluation of solar heating and

cooling systems for buildings. Although not definitive in all areas, it aims

at providing sufficient background to facilitate later development of a simplified, consumer-oriented handbook to assist homeowners, builders, and others

in evaluating the private costs of alternative heating and cooling systems.

1.3 Scope and Organization

The economic evaluation of solar heating and cooling systems, examined in

Section 2, is approached from the standpoint of life-cycle cost analysis. Solar

and conventional HVAC (Heating, Ventilating, and Air Conditioning) systems are viewed as alternative means of maintaining a dwelling at a specified temperature,

and the focus is on the acquisition, maintenance, and operation costs over the

life of a solar heating and cooling system as compared with a counterpart con- ventional system. Other possible cost differences in solar and conventional

systems, such as in their pollution effects, are recognized as important, but

are not considered here. The relevant cost elements for life-cycle costing are

set forth, the minimum data requirements are identified, and several alternative

approaches (equal in their results) are described for evaluating the efficiency

of comparable HVAC systems. These approaches include life-cycle cost analysis

and benefit-cost analysis. The appropriate assumptions regarding period of

analysis, discount rate, and escalation of costs over time are discussed, and

a tentative set of assumptions is set forth. The focus is on solar heating

and cooling of new buildings, but the same techniques and procedures would

also apply to the application of solar energy systems to existing buildings.

The economics of applying solar systems to both residential and commercial

buildings is considered.

1
Section 3 of the paper is concerned with the problem of making cost-effective

tradeoffs at the design state—including tradeoffs among components of solar

heating and cooling systems, such as thermal storage capacity versus collector

size; tradeoffs in the proportion of heating and cooling to be provided by the

solar system and by the auxiliary support system; and tradeoffs between energy

conserving measures and heating and cooling loads. For purpose of illustration,

the focus of this part is on the latter tradeoff, i.e., finding the economically

efficient combination of building envelope and internal space conditioning system which will satisfy user heating and cooling requirements. The necessary

economic conditions for achieving the efficient combination are identified.

Both Sections 2 and 3 of the paper consider costs in a life-cycle context, but their focus is different. Section 2 outlines the data and methods

required for the life-cycle cost evaluation of given solar system designs.

Section 3 focusses on system evaluation during the conceptual design stages of

the total solar dwelling; life-cycle costing is assumed in the analysis of design alternatives for the combination of the HVAC system and the building envelope.

Section 4 summarizes the paper briefly. It also suggests areas for further

research.

2
2. LIFE-CYCLE COST EVALUATION OF SOLAR HEATING

AND COOLING SYSTEMS

This section outlines the use of life-cycle cost analysis to evaluate solar

heating and cooling systems. The emphasis is on specifying the kinds of private

costs which should be taken into account by owners of buildings, and the general

method for deriving and comparing costs of a proposed or constructed solar energy

system with a counterpart conventional system. The purpose is to develop guidelines for evaluating the economic feasibility of solar energy systems.

Before discussing the methodology of life-cycle costing, let us consider

briefly the rationale for focusing on the life-cycle costs of solar energy systems to consumers, i.e., private costs.

2.1 Rationale for a Life-Cycle Evaluation of Private Costs

As was noted in the Introduction, this analysis is confined to an examination of direct outlays by the purchaser of a heating and cooling system for a building; social costs or social benefits from external diseconomies or economies,

such as air pollution from fossil fuels, are not included. The reason for limiting

the examination to private costs is that they are the relevant factor in the

widespread adoption or rejection of solar energy systems for heating and cooling

residences and commercial buildings. Private decision making, such as selection

of an HVAC system, does not generally take into account all social costs or social benefits.

The reason for emphasizing costs, as opposed to comfort benefits,"'' in this

paper is twofold: (1) Most importantly, there will probably not be important

differences in the comfort performance (i.e., in private benefits) of alternative

heating and cooling systems, in most cases. For systems which are about equal

in their comfort performance and in their satisfaction of constraints, a comparison of costs alone is adequate to determine the more efficient system. (2) Secondly, the differences which are perceived in benefits may be difficult to

quantify. For example, it would be difficult to place a value on the "novelty

appeal" which a consumer might derive from having a non-conventional system. In

any case, use of a life-cycle cost model does not preclude subjective evaluation

of nonquantifiable benefit attributes. Furthermore, differences in system performance can be handled in ^ cost formulation simply by treating positive differences as negative costs. (This technique is used in Section 2.5.4.1 to

account for differences in rental income for solar-equipped versus conventionallyequipped commercial buildings.) In most cases a comparison of explicit costs of

alternative systems will be the easiest evaluation approach and will probably

suffice in view of a number of other inaccuracies.

Section 2.6 presents a benefit-cost comparison of alternatives, but "benefits" is there defined as the fuel cost savings from a solar energy system,

rather than as a difference in the level of comfort performance provided by the

alternative systems.

2

In evaluating HVAC systems, the distinction between system costs and benefits is not always clear, and often differences may be treated as either. For

example, greater reliability of HVAC system A compared with system B may be

treated in an economic analysis as a benefit for A, as a negative cost for A, or as a positive cost for system B.

3
Now let us consider the usefulness of a life-cycle cost evaluation. Apart

from novelty appeal or the desire of building operators to insure satisfaction

of their heating and cooling needs in face of the vagaries of the international

oil market, there are at least three conditions under which solar energy systems

would tend to be used: (1) if solar systems are economically competitive with

conventional systems under competitive market conditions; (2) if a system of

governmental incentives for solar energy (or. penalties for conventional energy)

made solar more attractive than conventional energy systems; (3) if "energy

moratoriums" were imposed, prohibiting or limiting the use of nonrenewable energy

sources, such that construction of new buildings would require provision of a nondepletable energy source such as solar energy. These latter two situations

might arise for reasons of national defense, environmental considerations, or from the actual unavailability of fossil fuels. In any of these cases—free

market or the constrained market situations-—economic efficiency requires adoption of the least-cost system available which will satisfy constraints and heating and cooling demands. Thus, in any of these cases, an important attribute of

a solar energy system is its life-cycle cost. In the competitive market context,

the critical factor is the life-cycle costs of solar systems relative to con- ventional systems; whereas, in the constrained market context, the relevant lifecycle cost comparison might well be among alternative solar energy technologies.

It may be argued, however, that life-cycle costs have not been a guiding

factor in building decisions in the competitive market—that architects, builders,

buyers, and the financial community have been generally more concerned with the

first cost, the size of the down payment, and the monthly mortgage payment than

with the total effective costs of a residence to its owner over its life. And

to a large extent, this observation is valid. But lack of attention in the past

does not change the fact that the life-time cost effectiveness of alternative

building systems will probably become a guiding factor in investment and purchasing decisions of the future. There are indications of more attention to

residential fuel and utilities costs as these costs have risen.

2.2 Life-Cycle Cost Analysis: An Overview

The technique of life-cycle cost analysis considers total relevant costs over the life of a system, including costs of acquisition, maintenance, operation, and where applicable, disposal. It is a useful approach both in the com- parative analysis of design or ownership alternatives and in the collection of

data for purpose of future analysis. The life-cycle cost concept is an appropriate approach to cost analysis in both the Federal and private sectors.

The major steps in performing life-cycle cost analysis are the following:

1. Specification of Objectives and Constraints

2. Identification of Alternative Solutions

3. Identification of Relevant Cost Items for Each Alternative

4. Determination of Amounts and Timing of Cash Flows

5. Calculation of Life-Cycle Costs

6. Comparison of Costs for Alternatives

This list shall be used to guide the discussion of life-cycle cost analysis of

solar energy systems; each step will be treated in turn.

4
2 . 3 Objectives, Constraints, and Alternative Solutions

The relevant life-cycle cost objective is to achieve a desired level of

thermal comfort in the home, in terms of temperature, humidity, and other related

attributes, at lowest cost, while also meeting possible constraints, such as safety or aesthetics.

There are a number of possible alternative approaches to this objective.

They would include use of conventional heating and/or cooling systems, such as

a natural gas system, a propane system, an oil system, an electric resistance

system, or an electric heat pump system. The alternatives would also include

solar energy systems of varying design, as well as energy conservation investments to reduce heat loss or gain to the residence and thereby reduce the re- quired capacity and level of operation of the heating and cooling system. Generally, the alternatives are different combinations of solar and conventional

energy systems and energy conservation. (See Section 3 for a discussion of tradeoffs among alternatives.)

A given solar heating and cooling system would be evaluated against these

other alternatives to determine the least-cost means of accomplishing the comfort objective. The alternatives may differ substantially in their comparative

costs, but the direction and size of the differences over the life of the building or the period of use may not be apparent without an explicit cost analysis.

2 . 4 Relevant Costs

As noted above, life-cycle costing takes into account costs over the life

of the system, rather than first costs only. Thus, for the purchaser, lifecycle costing of solar and conventional heating and cooling systems requires

assessment of the following kinds of costs: (1) system acquisition costs, including search costs, purchase prices, delivery costs, and installation costs;

(2) system repair and replacement costs; (3) maintenance costs; (4) operating

costs, comprising mainly energy cost; (5) insurance; (6) taxes, and (7) salvage

values, net of removal and disposal costs.

These costs are required for all parts of the system being costed. The

principal solar subsystems for which costs would be collected are the following:

(1) solar collector, (2) thermal storage, (3) domestic hot water system, (4)

air conditioning components (e.g., absorption system), (5) auxiliary energy subsystems (may include heat pump), (6) heating and cooling distribution subsystems,

and (7) the control subsystem. Motors, pumps, fans, blowers, wiring, and tubing

are included in these subsystems.

For purpose of comparison, cost of acquisition, repair and replacement,

operation, maintenance, insurance, taxes, and salvage values would also be re- quired for the heating and cooling units and the distribution and other applicable subsystems for the counterpart conventional system(s).

In addition, the HVAC system (solar or conventional) is an integral part

of the total dwelling system, and may thereby affect costs of the building

envelope and other building subsystems. For example, use of a solar energy

5
system may impose special structural requirements on the building envelope, such

as additional roof supports to bear the weight of the collector, or special

siting requirements to enable efficient operation of the solar energy system.

Necessary alterations may increase usual building costs or may be cost-reducing,

as in the case of a reduction in the cost of conventional roofing that results

from replacing it in part by the solar collector component. Also, the optimal

expenditure for certain envelope features (e.g., insulation, roof overhangs for

solar shading, and storm doors and windows) may be different among different

types of energy systems. Neglect of these differences in related building costs

would distort the comparison of solar and conventional systems.

An additional cost, not included in the above listing of costs because of

its difficulty to quantify, is the cost of future system modification, which

might be undertaken, for example, for purpose of modernization or expansion.

If it is an important selection criterion, differences in the relative flexibility

or adaptability of alternative HVAC systems can be taken into account in system

comparisons by assigning either a cost value for inflexibility or a benefit value

for flexibility of one system relative to another.

A cost item included in the above listing which may warrant explanation, is

"search cost." Search cost refers to the cost to the purchaser of obtaining the

information necessary to consider use of a solar system. It would include the

cost (in time and direct money outlay) of determining the technical suitability

of a solar system for a given building, of determining the availability of solar

designs, of identifying the availability of maintenance service for solar systems, etc. It is, in short, a "nuisance" cost arising from the typical builder's

and consumer's lack of experience with and knowledge of solar systems. This cost

may not be easily quantifiable; it is variable by purchaser; and it will tend to

change over time. For most purposes, it will be sufficient to treat this cost sub

jectively, noting, for example, that if the costs^f a solar and a conventional

energy system are identical in all respects except search costs, that the solar

system will be disadvantaged in this respect. As these costs are reduced by

greater consumer knowledge and/or are incorporated into purchase price, it will

become unnecessary to consider them separately from purchase and installation

price.

A life-cycle cost comparison of alternative systems may be based on total

costs for each system or on the differences between systems. It is, however,

the cost differences which are critical to determining efficient choices between

alternatives. Cost items which are identical for the alternatives can be con- veniently omitted from a comparison without changing the outcome of the analysis.

For example, if the auxiliary heating system of a solar heating system is identical (or nearly identical) to the conventional system which would be used alone,

omitting the first cost of the conventional system and of the auxiliary system

from the life-cycle cost comparison would not matter, since the addition of an equal sum to each alternative's cost would not alter the difference between them.

Government programs (such as the Solar Demonstration Program authorized by

the Solar Heating and Cooling Demonstration Act of 1974) which increase knowledge

and information regarding solar energy systems, tend to reduce search costs or

at least shift them from the direct consumer to the general taxpayer. And, as manufacturers of solar energy systems provide more information through advertising

service contracts, and warranties, search costs will tend to become reflected in

the supply price of solar energy systems.

6
Collection of total cost data (as opposed to cost differences), however,

is generally preferable when the analysis is not being made "on the spot" or by

an analyst directly familiar with system costs. This recommendation would apply,

for example, to the analysis of government sponsored solar heating and cooling

projects. The requirement of total cost data would help to prevent the omission of relevant costs, and would allow the analyst greater flexibility in the

method of analysis. Furthermore, the collection of total life-cycle costs, rather

than cost differences, is preferable for the purpose of developing a historical

data file which may be used for other analyses.

Table 1 depicts the principal kinds of cost data which would be used to

evaluate solar and conventional energy systems. At a minimum, acquisition,

maintenance, replacement, and operating costs are required for each of the subsystems listed. For more extensive analysis of costs by subsystem, additional

detailing of costs within subsystems would be necessary. (Insurance and tax

costs are not shown in Table 1; assessment of insurance and tax impacts on costs

are discussed in some detail in a later section.)

In evaluating the expected future cost effectiveness of an experimental

solar system which has not yet been produced in quantity, the analyst may wish

to project future costs of the system, in addition to measuring costs of the

prototype system. In this case, data requirements will encompass costs of the

prototype system, as well as projections of future costs for the system, based

on a set of assumptions regarding technological change and production volume.

2 . 5 Cash Flows and Life-Cycle Costs

2.5.1 Cash Flows

After selection of the conventional alternatives to which solar heating

and cooling is to be compared and identification of all the relevant costs for

each alternative, the next step in the analysis is to determine the amount and

timing of positive and negative cash flows associated with each alternative.

The costs and their time of occurrence can be conveniently summarized by using

cash flow diagrams. It is necessary to take account of the timing of cash flows

because money has a time value, and, therefore, equal expenditures made at different times do not have the same value.

2.5.2 Discounting of Costs

In order to compare systems, it is necessary to convert the expenditures

for each system to an equivalent base. This is done by applying appropriate

discounting formulas to costs to convert them all to either a present value

basis or an annual cost basis. There are six basic formulas which are used to

move values in time so that they may be compared on an equivalent basis with

The time value of money reflects the opportunity for investment which will

yield a real return; it is a consideration apart from inflation.

7
H

!z; w

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s is

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« u

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s

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8
values associated with other systems. These formulas are shown in Table 2,

together with their standard nomenclature.

Tabular solutions (called discount^factors) for these formulas are available

in most engineering economics textbooks for a range of values of the parameters

i and N. Use of these tables can greatly simplify the discounting of costs. For

purpose of illustration, tabulated values for the uniform capital recovery

formula and the single present value formula are shown in Table 3. Discount

factors from both of these tables are used in the life-cycle cost models developed in the following section. By multiplying a given dollar value by the appropriate discount factor for a selected time and discount rate, one obtains the same solution as would be obtained by applying the corresponding discount formula

to the given value. For example, with a discount rate of 10%, the present value

of a $500 cost to be incurred 10 years from now may be calculated by applying

the single present worth discount formula to the future amount (i.e., P = $500

1 = $198) , or by multiplying $500 by the single present worth discount

(1 + .10)^^

factor for 10 years at 10% (i.e., P = $500 (.3855) = $198).

The discounting of costs requires selection of a discount rate. In general,

the appropriate rate for discounting costs is the rate of opportunity cost to

the investor; i.e., the rate of return foregone on the next best alternative

investment

.

Discount rates may be expressed in nominal (market) terms or in real terms.

Nominal rates include an inflation factor, whereas real rates are net of inflation. Either expression may be used as long as costs to be discounted are ex- pressed in corresponding terms. That is, if a market rate of interest is used

to discount costs, future costs should be estimated to include inflationary price

changes; if a real rate of interest is used to discount costs, future costs should

be estimated net of inflation. To simplify the analysis and avoid unnecessary

computations, a customary practice is to assume—in absence of strong evidence

to the contrary—that all prices, costs, and incomes inflate or deflate at the

Variations of these basic formulas are often used in life-cycle cost analysis. For examples, the uniform present worth formula may also be written as A[l - 1 J ; (1 + 1)^

and the single present worth factor may be used with a price escalation factor

(e) in the numerator to calculate present value of a yearly cost (Y^) escalating

at a fixed rate, i.e..

See, for example, Gerald W. Smith, Engineering Economy: Analysis of Cap - ital Expenditures , 2nd ed. (Ames, Iowa: The Iowa State University Press, 1973);

and Eugene L. Grant and W. Grant Ireson, Principles of Engineering Economy , 5th ed. (New York: The Ronald Press Co., 1970).

9
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10
Table 3

DISCOUNT FACTORS

Capital Recoverv Factors for Intkrest Hates from 0% to 25%

07c 2^0 4% 0% s% 10% ]2% 15% 20% 25%

1 1.000(K) 1 .02'K)0 l.OKKM) l.O'JOtH) l.OMMK) l.UWH) 1.12000 1.1.5IKJ0 1 20000 1 .2.5000 2 cioono 0.51."iO.". 0..J302U 0..')4.544 0.50077 0.57010 0.5'. a 70 0.01512 0.0-54.55 0.00444

8 o..'j;i3.i.i 0.3 1075 O.30()35 0..i741

1

0.3,S.S0.3 0. 10211 0. iii,:i5 0.43708 0.47473 0.51230

4 0.2.',()(K) 0.202(i2 0.27.510 0.2S.S50 0.30102 0.31.547 O.32023 0.3.5027 0.38020 0.42344

6 0.20000 0.21210 0.22403 0.23740 0.250-l(; 0.20380 0.27741 0.20832 0.33438 0.37184

6 o.ior.c? 0.17S.".3 0.1007i, 0.20330 0.21032 0.22001 0.24323 0.20124 0..30071 0.33SS2

7 0.1 12>:(i 0.1.5451 O.liiOOI 0.17014 0.10207 0.20541 0.21012 0.24030 0.27742 0.310.34 8 O.ll'.'i(X) O.I3i;51 0.1 1S53 O.KilO 1 0.17401 0.1874 1 0.20130 0.22285 0.200<)1 0.30040 9 0.11111 0.12252 0.13140 0.14702 0.10<)f)S 0.17.3(V1 0.1870.S 0.2(KI.57 0.2l8C»ft 0.2S87O

10 o.loooo 0.1 1133 0.12320 0.13567 0.1 1U03 0.10275 0.17008 0.10025 0.23852 0.28007

11 0.00001 0.1021S 0.11415 0.12070 0.1 lOOS 0.1.5.300 0.10S42 0.10107 0.23110 0.27340

12 0.0«.i:i 0.00 150 0.10055 0.1102S 0.1.3270 0.14i\70 0.10144 0.1SU8 0.22.520 0.20.S45 13 0.07002 0.0S.S12 0.10014 0.11200 0.120.52 0.1 I07S 0.15508 0.17011 0.221X12 0.2tV45t 1« 0.071 13 0.()s2ii() O.Wl 107 0.1075S 0.12130 0.13.575 0.1.5087 0.17400 0.210.S9 0.20 1.50 16 0.00007 0.077S3 O.OSOOl 0.10200 0.11GS3 0.13147 0.140S2 0.17102 0.21388 0.2.5012

16 0.0C2.-.0 0.07305 O.0S5S2 0.O0S05 0.1120S 0.127S2 0.14330 0.10705 0.21144 0.25721

17 C.05S.><2 0.0<i007 0.0^220 0.00.544 0.10003 0.12400 0.14040 0.1(>.537 0.20044 0.25570

18 0.0o.->.>0 0.00070 0.07S00 0.00230 0.10070 0.1 2103 0.13704 0.10310 0.20781 0.25450

19 0.0.52(;:! 0.0i;37S 0.07i-,14 0.08902 0.10113 0.110.55 0.13.570 0.10134 0.2LH^4G 0.25300

<a 0.05000 0.00110 0.0735S 0.0o718 0.10185 0.11740 0.13388 0.15070 0.20530 0.25292

S6 0.04000 0.05122 o.noioi 0.07.S23 o.oo.3as 0.11017 0.12750 0.15470 0.20212 0.25095 30 o.os.m 0.04405 0.057S;3 0.072G5 0.(W,S,S3 O.IOOOS 0.12414 0.1.5230 0.20085 0.25031 40 0.02.-)00 0.03i;5r. 0.05052 O.OOOlO O.OK3S0 0.10221; 0.121.30 0.15050 0.20014 0.25003

60 0.02000 0.031M2 n.ftli)55 0.00344 0.08171 o.iiiom; 0.12012 0.15014 0.20002 0.25000 100 0.01000 0.02320 o.aiobi O.OOOIS 0.08004 0.10001 0. 12000 o.isoai 0.20000 0.25000 00 0.U2000 0.04000 0.00000 O.OSOOO O.IOOOO 0.12000 0.15000 0.20000 0.25000

Present Worth Factors for Interest R.\tes from 0% to 25%

V /o 2% 4% 6% 8% 10% 12% 15% 20% 25%

1 1.0000 0.9804 0.0615 0,0434 0.0259 0.0001 0.8029 0.8006 0.8333 0.8000

1.0000 0.0GI2 0 024G 0.8000 0.8473 0.S204 0.7072 0.7.501 0.0044 0.6400

3 i.oono 0.9423 0 SSOO 0.8.300 0.7938 0 7513 0.7118 0.0575 0.5787 0.5120

4 1.0000 0.9238 0.8.548 0.7021 0.7,350 0.0.S30 0.C355 0.5718 0.4823 0.4006 6 1.0000 0.9057 0.8210 0.7473 0.G800 0.0200 0.5074 0.4972 0.4019 0.3277

6 l.OOflO 0.88S0 0.7003 0.7050 0.0302 0.5('.45 0.5006 0.4323 0.3349 0.2621 7 ; 1.0000 0.8700 0.7.500 0.0051 0..>835 0.5132 0.4.523 0.3759 0.2701 0.2007 8 1.0000 0.8535 0.7.307 0.0274 0.5403 0.4005 0.4039 0.3209 0.2326 0.1678

9 1.0000 0.8308 0.702G 0 .5919 0.5002 0.4241 0..3C06 0 2843 0.1938 0.1342 10 1. 0000 0.8203 0.0750 0.5584 0.4032 0.3.855 0.3220 0.2472 0.1G15 0.1074

11 1.0000 0.8043 0.G406 0.5208 0.42S0 0..3505 0.2875 0.21J9 0.1346 0.0.<<.59 12 1.0000 0.7SS5 0.0240 0.4070 0.:!071 0..31S0 0.2507 0.1S09 0.1122 0.00S7 13 1.0000 0.7730 0.0000 0.4GSS 0.3077 0.2S07 0.2202 0.1025 00935 0 05:rf) 14 1.0000 0.7570 0.5775 0.4123 0.3405 0.2033 0.2040 0.1413 0.0779 0.0440 16 1.0000 0.7430 0.5553 0.4173 0.3152 0.2304 0.1827 0.1229 0.0049 0.03-J

16 1.0000 0.7284 0.53.39 0.3936 0.2919 0.2I"76 0.1631 0.1069 0.0.541 0.0281 17 1 Olino 0.7142 0.5134 0.3714 0 2703 0 r.i78 0.14.50 0 0<>29 0.0451 00225

18 1.00(H) 0.7002 0.4030 0.3.503 0.2502 0.1700 0.1300 0 080S 0.0376 0.0 ISO 19 1.0(MH) 0.f.Sil4 0.4740 0..3.iO5 0.2317 0.10.!5 0.1101 0.0703 0.0313 0.0144 30 1.0000 0.0730 0.45^1 0.3118 0.2145 0.1480 0.1037 O.OOll 0.0201 0.0115

26 1. 0000 0.0095 0.3751 0.23.30 0.1400 0.0023 005SS 0.0304 0.0105 0.0038 30 1 (HXU) 0.5521 0 .3(183 0 1741 0.0004 0 (1573 0.0.134 00151 0(HM2 0 (K)12 40 l.oooo 0.4520 0 2(is:t 0 0'.I72 0 OHIO 0 (1221 0.0107 0.0037 0 0007 O.OOOl •0 1 (H)00 0.3715 0.M()7 0.0543 0.0213 0 1 H 185 0.0035 0.0009 0 0001 100 1 0000 0.13*0 0.0193 0.0029 0.0005 O.lVKJl ...

Source: Grant, Eugene L. Principles of Engineering

Economy (New York: The Ronald Press Compai

1950).

11
same rate, such that the changes are offsetting and need not be Included in the

analysis. On the other hand, if price changes for specific items of cost, e.g.,

fuel, are expected to differ from the general level of price changes and can be

forecasted with some confidence, they should be included in the analysis.

Opinion is mixed regarding the appropriate real discount rate to evaluate

the cost effectiveness of building systems. The historical real rate of return

on business investments has generally been estimated at about 3-4%. Agencies of

the Federal Government are directed to use a 10% real rate to evaluate government

investments.^ It has been suggested that the real aftertax opportunity cost of

the "typical" homeowner is quite low—as low as 1% or 2%. The rate on second

mortgages, as an indication of the homeowner's cost of borrowing, suggests a slightly higher real rate.

Given the uncertainty of an appropriate rate, it is suggested that the evaluation of alternatives be made with several interest rates to test the sensitivity

of the analysis to the discount rate. Evaluating costs on basis of a low rate

equal to, say, 2% and a high rate equal to, say, 15% in real terms, would provide a reasonable range of rates for a sensitivity check.

2.5.3 Life-Cycle Cost Models

Life-cycle costs of a solar HVAC system can be computed with either a present value or an annual cost model. Both approaches take into account the

changing real value of money over time. In the present value model, all costs

and salvage values are forecasted over the period of analysis and then discounted

to an equivalent single cost today. In the annual cost model all costs and salvage values are forecasted over the period of analysis and then are divided into

uniform annual costs by discounting. Present value costs can be easily converted

to an annual cost basis, and vice versa.

But in fact, the assumption that if inflation affects costs and revenues equally it does not affect the outcome of the analysis, is valid only if the HVAC

system is purchased completely with equity funds (i.e., unborrowed funds), and

if the system is not depreciated for tax purposes. With debt financing of the

system, the payments of the purchaser tend to be fixed in amount, and their present value declines with inflation. In the case of depreciation of the system

for tax purposes, which is allowable for commercial applications, inflation

causes the depreciation expenses to fall in real value, since depreciation is

based on the initial purchase price of the system. In addition, there are other effects of inflation beyond these two. For evaluation of solar systems

in residences, these effects are not likely to be important, but in evaluating

commercial applications, they may warrant attention.

''Off ice of Management and Budget. Executive Office of the President,

Circular No. A-94 (Revised), March 27, 1972.

3

Stephen R. Petersen, Retrofitting Existing Housing for Energy Conservation ; An Economic Analysis , National Bureau of Standards, Building Science Series 64,

December 1974, pp. 19-21.

4

For further general discussion of present value and annual cost models,

see any of the following: Gerald W. Smith, Engineering Economy ; Eugene L. Grant

and W. Grant Ireson, Principles of Engineering Economy ; E. Paul DeGarmo, Engineering Economy , 4th ed. (New York: The Macmillan Co., 1969).

12
Apa^-c from taxes, the basic formula for computing the present value of an HVAC system can be developed by applying to the cost items from Table 1, the

appropriate discounting formulas from Table 2. (The effect of taxes on costs

will be deferred to Section 2.5.4.) The following formula includes terms for

the basic kinds of costs, i.e., acquisition, replacement, maintenance, and

operating costs, and would be suitable to compute the costs of either a solar

(plus conventional auxiliary) system or a conventional system alone.

N (R. - "ST.) PV = I (1 + i)^ j=i (1 + i)^ id + i)N ° j=iVi + i /

N /I + e^ \

^ + Z I -] + B + Q

j=l\l + i /

(1)

where

PV = present value cost of the HVAC system over period N,

I = initial investment costs, including costs of acquisition, delivery,

and installation of the heating and cooling system(s),

S = remaining value of the HVAC system(s) at the end of the period of

analysis

,

i = annual discount rate in real terms.

N = period of analysis in years, (may be the life of the building or a shorter designated period)

,

= Replacement and repair costs in year j at present prices, including

costs or replacing or repairing any part of the system,

= salvage value in year j, where j < N, at present prices, of replaced

parts,

R- - S. = net replacement and repair costs in year j,

•J 3M = estimated annual maintenance cost at present prices, assumed here to

be constant over the life of the system. (Alternatively, these costs

might be assumed variable from year to year, in which case they could

be included in the repair and replacement term; or they might be assumed

to escalate at a constant rate or amount over time, in which case they

could be treated as fuel costs are treated above or discounted by use

of gradient series interest formulas-*-, respectively.)

Interest computations for periodic sums changing by an equal amount each

period or by an equal rate each period are explained in Gerald W. Grant, Engineering Economy , pp. 47-52.

13
estimated annual energy cost at present prices; subscripts indicate

different sources of energy, e.g., Fq might indicate #2 heating oil

and F might indicate electricity cost.

e = annual rate of change in real price of energy, where subscripts

indicate different sources of energy,

B = initial investment costs for HVAC - related building modifications

(if modifications are cost-reducing, B will be subtracted from costs).

Q = value of building space occupied by HVAC system components, evaluated

as building cost/sq. ft. x number sq. ft. occupied.

Investment costs are entered in the equation without discounting, because

these costs, as first costs, are already in present value terms. The remaining

value (salvage) of the system when use has terminated or when the defined period

of analysis has ended, is converted to present value by use of the single present worth formula, and is deducted from investment cost because it represents

investment costs not actually incurred. Cost of replacing parts of the system,

net of the salvage value of the old parts, are discounted from the year they are expected to be incurred to present value, summed, and added to other costs. Annual

maintenance and repair costs might generally be assumed constant in real terms,

and, if so, would be discounted to present value by use of the uniform present

value formula.

Two terms are included for energy costs to indicate that several sources

of energy of varying price might be used. The annual expenditure on each energy

source should be escalated if real increases in price are expected. The escalated

annual costs are then discounted to present value and summed. The last two terms,

cost of building modifications and cost of building space occupied, are incurred

initially, and, therefore, are already in present value equivalents.

The basic cost elements in the annual cost formula, equation (2), are identical to those in the present value equation. The only difference is in the

discounting procedures. There are several ways to formulate the annual cost

equation. One way is simply to apply a capital recovery factor to the present

value costs as expressed in equation (1) to convert them to an equivalent uniform

stream. This is essentially what has been done in the following annual cost model

for computing life-cycle costs of an HVAC system, with the exception that annual

maintenance cost, M, is entered directly, without discounting, since it is already in appropriate form.

AC = I -

(1 + i)^ +

(R. s.)

+ F. j=l (1 + i)J

+ e + F- + i

N

Z j=l

ll +

+ B + Q

1 + i

fid +i)_ . L(l + i)^-l + M,

(2)

where the variables are as defined previously.

14
Generally, in the cost analysis of building subsystems, the present value

of costs would be computed for the life of the building; i.e., N would be defined as building life (usually 30 to 40 years or greater). Alternatively, N

may be defined as "the period of analysis," which may be equal to, or much

shorter than building life, say 20 years. Reasons for limiting the period of

analysis to a shorter period than building life might be that forecasts of

energy availability and conditions in the housing market become increasingly

uncertain at farther points in time, or that intended use of the building is

limited and resale uncertain. Other things equal, if a very low discount rate were used, the results would tend to be quite sensitive to whether the choice

of periods were relatively short or long; but if a relatively high discount

rate were used, the results would be much less sensitive to the length of the

period.

2.5.4 Treatment of Taxes, Insurance, and Governmental Incentives

Thus far, only the more obvious elements of life-cycle costs have been

treated, the effects of various institutional arrangements which may alter the

effective costs of solar energy systems to the purchaser have not yet been

taken into account. Taxation, for example is one institutional effect which

frequently alters the cost or profitability of investment decisions. Insurance

is another. In addition, state, local, or Federal programs of incentives or penalties to encourage or discourage respectively the choice of certain HVAC

systems may change life-cycle costs.

Since widespread adoption of solar energy systems in buildings depends

upon the effective dollar costs to the owners and users, these effects should

be considered. The remainder of Section 2.5.4 discusses how the life-cycle

cost formulas given earlier, equations (1) and (2) could be modified or expanded

to account for the effects of taxation, insurance, and special government programs on the life-cycle costs of owning a building. It also assesses in general

terms the probable direction of impact of taxation on an owner's costs.

2.5.4.1 Taxes

The effect of taxation on costs of a solar system to the owner may be con- sidered for two main cases: (1) for the owner-occupied solar residence, and

(2) for the rental solar residence or other solar commercial building. For

both cases, taxes impact on costs in several ways, in some instances raising

and in other instances lowering life-cycle costs of the solar HVAC system relative to its conventional counterpart. Let us examine the two cases in turn

for tax implications.

For the owner-occupied residence, the primary tax effects are from the

property tax and, indirectly, from the income tax. The particular effect of

either of these taxes could be expected to vary considerably among individual

solar residences, depending upon local property tax rates, property assessment

practices, and the income tax bracket of the homeowner. The focus here is on the

nature of the effects and on the method of including them in the life-cycle cost

analysis

.

15
The property tax , which is levied as a percentage of a share of the market value of a building, would tend to raise the life-cycle costs of a solar

HVAC system relative to a conventional system. Life-cycle costs would be

raised because, other things equal, the greater first cost of solar HVAC equipment would be reflected in a higher market value for the residence, and, hence

in a larger assessed value for the solar residence than for a conventional re- sidence with its lower first-cost HVAC system. Thus, the capital intensiveness of an HVAC system influences the amount of property taxes levied on a residence, and thereby, alters the life-cycle cost of the HVAC system.

As a simple example, let us compare the property tax on a $60,000 solar

residence, of which $8,000 is attributable to additional cost of the solar

HVAC system, with a counterpart conventionally heated and cooled home valued

at $52,000 (i.e.. $60,000 - $8,000). Given a typical tax rate of 4.50% of 50%

of market value, the $60,000 solar residence would be assessed at $30,000 and

taxed $1,350. The counterpart conventional home would be assessed $26,000

(i.e., 50% of $60,000 - $8,000) and taxed $1,170. For purpose of illustration

further assume a real discount rate of 2%, a constant real property value (including a constant real value for the solar system with replacements made as needed), and a constant property tax rate over a 20 year period of evaluation,

(the assumption of a constant real property value and a constant property tax

rate means that even though the nominal, or market, property value changes,

the yearly property tax remains constant in terms of present prices.) Over 20

years, the property tax on the solar residence would amount in present value

terms to $22,072 (i.e., $1,350 (1 + .02)^^ - 1 = $22,072); the present value of

.02 (1 + .02)^^

the property tax on the conventional residence would amount to $19,130 (i.e.,

$1,170 (1 + .02)20 _ I = $19,130). Thus, the effect ofTzhe property tax in

.02 (1 + .02)20

the above example is to raise the life-cycle cost of the solar residence relative to the conventional system by nearly $3,000. This simple illustration

suggests that the property tax provides a disincentive for choosing solar HVAC

systems.

In the example, a constant real property value was assumed for ease of

illustration. This assumption implies no real depreciation of the system over

time during which necessary replacement of parts is made; i.e., the salvage

value, in real terms, is assumed equal to the original first cost. In some cases this assumption might be reasonable. An alternative assumption is that

the HVAC system (with parts replacements) depreciates in real terms from the

time of purchase, such that little or no real value remains after, say, 20

years.

From sample census data for single-family houses in a number of U.S.

cities, assessment values appear to range from about 8% to 93% of market value;

nominal property tax rates from a low of about 1% to more than 25%; and the

effective tax rate from only about 1% to 4% of market value. A typical property tax might be 4.50% of 50% of the market value, i.e., an effective rate

of 2.25% of the market value of a residence.

16
^ ria + 1)" 1 . (4)

L(i + 1)" - J

A general expression of the present value of property taxes attributable

to the HVAC system (PVt) is

PV = z

'

, j=l (1 + i)J (3)

where

t = the property tax rate,

G- = the assessed value of the HVAC system in year j, in present dollars.

This formula would cover both the case of a constant real assessed value for

the HVAC system and the case of changing real assessed values over time.

To account for the property tax effect in the present value formula

equation (1) , the above term would simply be added to equation (1) . In the case of the annual cost formula, it would be necessary first to convert the

present value to an annual cost by applying the capital recovery discount formula; that is,

N (t • G ) AC = Z j=l (1 +

The above term would be added to equation (2)

.

The income tax , in contrast to the property tax, would tend to reduce the

life-cycle cost of a solar vis-a-vis conventional residence in two ways. For

one thing, the homeowner is able to deduct his mortgage interest payments from

taxable income. The higher first cost of the solar HVAC system, by increasing

the size of the mortgage to be amortized, raises interest payments; the higher

interest payments can then be deducted from income for purpose of computing

income tax. The value of the tax deduction to the homeowner depends on his

personal income tax bracket. Consider, for purpose of illustration, the case

of a solar HVAC system whose first cost of, say, $8,000 comprises part of the

homeowner's mortgage. With a 10% market rate of interest on the residential

mortgage, the $8,000 amortized over 20 years would add approximately $940 per

year to the mortgage payment. (For simplicity let us assume yearly mortgage

payments rather than monthly payments.) The addition to the yearly payment is

fixed at $940 over the 20 years, and interest comprises a declining portion of

the payment over time. In the first year, interest amounts to $8000 (i.e.,

$8,000 X .10 = $800), and the principal is reduced by $140 (i.e., $940 - $800 = $140). In the second year interest is $786 [i.e., ($8,000 - $140) x .10 = $786], and the principal is reduced by $154 (i.e., $940 - $786 = $154); etc.

Thus in the first year, the solar system would result in additional interest

deductions from taxable income of $800, and, if the homeowner is in a 25% in- come tax racket, the end-of-year value of the deduction would be $200 (i.e.,

$800 X .25 = $200). At a 2% real discount rate the present value of this

savings would be $196 (i.e., $200 = $196). In the second year, the

(1 + .02)^

present value return from the income tax deduction would be $188 (i.e..

^The use of a 10% market rate of interest and a 2% real discount rate

implies an inflation rate of approximately 8% annually.

17
$786 X .25 = $188). Over the full twenty years, the present value of the in-

(1 + .02)^

terest deductions (PV^) would be calculated as

N 7 (L • m)

PV-r = Z j , (5)

^ j=l (1 + i)J

where

t = personal income tax rate

L. = the additional mortgage loan principal outstanding in period j, i.e.,

that part associated with the HVAC system

m = the market rate of interest on the mortgage.

(During a period of price change, it would be necessary to apply a price index

to the amount of the tax deductions to convert them to present prices. This

conversion of current dollars to real terms would be necessary because the tax

deductions are fixed, and do not reflect changing prices.) The above term

would be subtracted from equation (1) to account for the effect of tax deductions of interest on the present value of the homeowner's life-cycle costs.

To account for this tax effect in the annual cost formula, equation (2),

it is necessary to convert the above present value expression to an annual

cost, i.e.

,

N t (L. • m)

AC^ = E 1 i (1 + i)^

(1 + i)N -

(6)

j=l (1 + i)J

This term would then be subtracted from equation (2).

XT In addition to the deduction of interest pajmients, the homeowner would

also be able to deduct payments of property taxes from taxable income. Thus,

a part of the Increase in life-cycle cost of a solar HVAC system resulting

from higher property tax on the solar residence would be offset by related

income tax deductions. The present value of the property tax deduction (PVp) can be calculated as _

N t (t • GJ

PV = E j=l (1 + i)J

where

t = the personal income tax rate,

t = the property tax rate.

, (7)

G = the assessed value of the HVAC system in year j, in present dollars.

This term would be subtracted from equation (1) to take into account the effect

of income tax deductions of property taxes on present-value costs. The annual

cost formula would be adjusted for this tax effect by converting the net present value to the annual cost equivalent (as was done in the preceding case)

and subtracting this from equation (2).

18
Alternatively, the net present value effect of the property tax and the

related income tax deductions (PV ) could be included in the present value

formula by adding the following single term which can be derived from equations

(3) and (7). _ N (t • G ) PV ^ = (1 - t) Z i- . (8)

j=l (1 + i)j

In conclusion, property taxes tend to increase the homeowner's cost for

a solar HVAC relative to a counterpart conventional system due to the greater

capital- intensiveness of the typical solar system. Income tax effects on the

other hand, tend to reduce the relative costs of the typical solar system,

principally due to deduction from the homeowner's taxable income of interest

payments which are larger for more capital-intensive systems. With these tax

effects included (and without simplifying the equation) , the formula to derive

present-value life-cycle costs of an HVAC system to the homeowner is the

following:

Maintenance &

Acquisition Occup. Space Replacements Repairs Energy Costs

Bldg. Modif. N (R - S ) ^ N

PV= I + B + Q + E j i_ + M

"*" vf" + F E / iJ_i

° j=l (1 + i)J id + i)^ j=l U + i

Property Tax, Net of Tax Deduction of Mortgage

Tax Deduction Interest Payments _ N (t • G ) N r (L • M) + (1 - t) E - Z 2 . , (9)

j=l (1 + i)J j=l (1 + i)^

where PV = present value cost to the homeowner of an HVAC system over period N,

with tax°effects included, and other terms are as previously defined.

Let us now consider the tax effects on the life-cycle cost of a commercial building equipped with a solar HVAC system, as compared with a commercial

building equipped with a conventional HVAC system. In the case of commercial

use of solar systems the previously described property tax and income tax

effects would also apply. There are in addition, other income tax deductible

expenses to consider in evaluating the commerically used system, such as depreciation deductions and deductions of operation and maintenance expenses.

Also, after-tax rental income of commercial buildings may be influenced by the

choice between solar and conventional HVAC systems, and therefore, may need to

be considered.

The larger capitalized value of the solar HVAC system would result in

increased deductions of depreciation from taxable income. For example, using

a straight-line method of depreciation and assuming a first cost of $8,000 for

the solar system, a 20 year life, and no salvage value, the annual depreciation

1

The institutional treatment of property tax and interest charges would

be somewhat different for commercial buildings than for owner-occupied houses

in that these items of cost would be deductible as business expenses. The

effect on costs, however, would be described by the same mathematical expressions as developed above.

19
would be $400. The present value to the building owner of the $400 depreciation in a given year may be found by applying his income tax rate to the $400,

and discounting that amount to the present. ^ Alternatively, a depreciation

method might be used which does not yield equal yearly amounts in either real or nominal terms (e.g., a declining balance method). A general expression of the

present value of the tax deduction resulting from depreciation (PV^) is

N (D • t") PV = E -_J ^ , (10)

j=l (1 + i)J

where

D = depreciation in year j, in present dollars,

t = building owner's income tax rate.

This term would be subtracted from equation (1) , reducing the present value

cost of the solar HVAC system. For the annual cost equation, the above ex- pression would be converted to an annual basis, i.e.,

ACq = N (DI. _1

t)

i (1 + i)

N

(1 + i)N - 1

(31)

j=l (1 + i)J

and this value would be subtracted from equation (2).

On the other hand, a solar HVAC system would generally involve lower

operating (fuel) costs than its conventional counterpart. Tax deductions for

operating costs would, therefore, tend to be lower for a solar system than

for its conventional counterpart.

Because of the time value of money, the presejit value of depreciation ex- penses are less than the present value of the capital expenses upon which they

are based. In contrast, the present value of the deductible operating expenses

are approximately equal to the corresponding operating expenses incurred. Consequently, if present value capital costs are substituted (traded off) for

present value operating costs on a dollar-f or-dollar before-tax basis, there

will not be a corresponding dollar-for-dollar tradeoff on an after-tax basis.

Rather, the present value of after-tax capital costs will increase relatively more than operating costs decline, and after-tax total costs will, therefore,

rise as a result of the more capital-intensive system. Hence, the fact that

operating costs are fully deductible as a current business expense, while

capital costs are deductible only as a depreciation expense may in some cases bias building owners towards relatively less capital-intensive conventional

HVAC systems over solar HVAC systems. In the case of systems sized for small

buildings, the biasing effect will probably be inconsequential, but for large

buildings, it may significantly discourage the selection of solar HVAC systems.

'For purpose of this illustration, future price change is not considered,

and the yearly depreciation found by dividing the first cost of the solar HVAC

system (V) by the number of years of its assumed life (n) , is taken as the

annual depreciation cost. In fact, however, an amount of depreciation fixed

in nominal dollars would become a decreasing real amount during a period of

inflation. To take account of changes in the real value of annual depreciation,

it would be necessary to apply a projected price index for each future year

to V_, thereby converting depreciation to present dollar terms, n

20
Thus far the cost evaluation formulations have been based on the assumption of equal private benefits for the solar HVAC system and the conventional

counterpart to which it is compared. However, in the case of some rental properties, particularly low density rental residences, it may be necessary to

take into account possible differences in rental revenue. I^Jhere conventionally

provided utilities are paid by the tenant, rental revenue could be expected,

other things equal, to be higher on a solar residence than on a comparable

conventionally-equipped residence. That is, the owner of a solar rental re- sidence would incur the costs of solar equipment that would be reflected in

higher rent but lower utility bills to the tenant. The owner of the rental

solar residence would require higher rental payments to cover his higher capital costs. Assuming other things equal and a well functioning market, tenants

should be willing to pay an additional amount of rent up to the amount of the

additional utilities outlay which they would incur in a counterpart conventionallyequipped residence (i.e., an amount sufficient to equalize the life-cycle costs

to the tenants of counterpart solar and conventional rental units)

.

Different amounts of benefits (i.e., rental income) for buildings equipped

with different HVAC systems means that benefits of the alternative systems

are unequal. To compare the alternative systems, differences in their benefits

as well as in their costs should be evaluated. Inequalities in benefits can be treated in the present value and annual cost equations as negative costs,

by entering, in this case, as a negative cost any additional after-tax rental

income generated by the rental solar residence over the conventional counterpart. Annual after-tax rental income (Yr^) would be expressed as

= (1 - t) Y, (12)

where

Y = additional annual gross rental revenue for a solar residence over its

counterpart conventional residence, and

1 - t = the factor applied to obtain after-tax income.

This additional amount of annual income, i.e., (1 - t) Y, would be subtracted

from equation (2) to adjust the annual cost formula. To adjust the present

value formula, the term would be converted to present value equivalence and

subtracted from the equation; i.e., the expression

PV^ = (1 - t) Y

would be subtracted from equation (1)

(1 + i)^ -—1

i (1 + i)

(13)

Taking into account tax effects, the formula to derive present-value lifecycle costs of an HVAC system to the owner of a commercial property (PV^^) is the

following:

21
Replacements Maintenance & Acquisition Occup. Space Repairs Energy Costs

Bldg. Modif.

N (R - S ) N N j

PV =I + B + Q + Z —j i_ + M (1 + ^) + F E P

+ ^\

j=l (1 + i)^ i (1 + i) j=lU + ±j

Property Tax, Net of Tax Deduction of Other Tax Deductions

Tax Deduction Mortgage Interest

Pa3niients _ N (t • G_) N t"(L. • M) N (E, • t") + (1 - t) E L - E j - E —j ^ j=l (1 + i)J j=l (1 + i)j j=l (1 + i)J

Additional After-Tax Income

2.5.4.2 Insurance

There are several reasons why insurance costs might differ for a solarequipped residence as compared with one conventionally equipped. For one thing,

the homeowner might insure his larger capital investment from damages, e.g.,

the solar collector from breakage by natural or human forces. For another, the

fire insurance rate for a solar HVAC system may be_ less than that for a conventional

system used alone, because of such factors as a smaller use of a fired furnace. Differences in probability of fire occurrence may become reflected in differential in- surance rates.

Since insurance costs are one of the costs of operating a residence, they

should be considered in the life-cycle cost analysis. In so doing, it is important to remember that insurance costs represent a tradeoff to the homeowner for

incurring damage costs. It is the net cost to the homeowner of damage (1^^)

which is relevant; that is, the cost of insurance (insurance premiums) plus

damage losses, net of insurance reimbursements collected. The net cost in annual

cost terms is as follows:

I = I + L - C, (15) n where

I = annual insurance premiums,

L = annual damage loss,

C = Annual insurance reimbursements.

E^ = Depreciation and other deductibles in year j, at present prices.

22
Alternately, M in both equations (1) and (2) could be redefined ag net

annual maintenance and repair costs, where repair costs are adjusted to account

for annual insurance outlays and receipts.

2.5.4.3 Governmental Incentives

Effective life-cycle costs to owners and users of solar-equipped buildings

may be further changed by governmental programs designed to encourage adoption

of solar HVAC systems. These programs might offer special incentives for solar

systems in the form of tax credits, low interest loans, or direct grants or subsidies to manufacturers and/or buyers of solar HVAC systems. Alternatively,

incentives for solar systems might be provided in the form of penalties applied

to conventional HVAC systems, such as by taxing conventional HVAC systems more

severely than solar systems. In either case, if the comparative cost to the

homeowner is altered by special programs, the cost evaluation should reflect

the induced changes.

The method of treating the cost effects of such programs would vary. Subsidies to producers of solar systems, for example, might be reflected in the

lower purchase price of the systems, and no additional expression need be introduced into the life-cycle cost model in order to assess this effect. On

the other hand, a subsidy to the purchaser of a solar HVAC system, say, in the

form of a low interest loan for the purchase of a solar home, might require

specific evaluation of the interest subsidy, including income tax effects.

Some programs intended to provide incentives for purchase of solar energy

systems may do this by reducing previously existing disincentives for solar

energy. For example, some states and localities are exempting from the property tax some part of the first cost of a solar HVAC system (e.g., in Indiana,

a 1974 law requires that county property assessors exempt up to $2,000 of the

cost of a solar heating or cooling system installed in a residential or commercial building form the real property tax)

.

A comprehensive evaluation of the effects of government incentives on lifecycle costs of solar versus conventional HVAC systems is beyond the scope of this

paper. However, a general treatment of the expected impact of alternative incentive techniques would be a useful background to the cost evaluation of

specific solar residences and commercial buildings.

2. 6 Comparison of Alternatives

To compare solar and conventional systems, several approaches can be used.

One approach is to calculate the life-cycle costs of each system with either a present value or annual cost formula as depicted by equations (1) and (2) . Cost

differences between systems may then be found simply by subtracting the lifecycle cost equation for one system from that of another, term by term if desired. In this way, the system with the lowest life-cycle costs can be identified.

This discussion of governmental incentives is not intended to recommend

incentives; the purpose is only to identify all the factors which should be

considered in a comprehensive analysis of solar HVAC systems.

23
There are a number of additional methods for analyzing investment decisions,

such as benefit-cost analysis, Internal jate of return or yield method, payback

method, and return-on-investment method. These methods are used to analyze

conventional investment problems, which involve both cash outlays and cash

inflows.

For heating and cooling systems, cash flows are generally negative. It is

possible, however, to structure this evaluation problem in conventional investment

terms, by focusing on the incremental investment cost which is typically required

for a solar HVAC system over a conventional system, and on the reduction in energy

costs resulting from the higher investment cost. Converting the problem into one which involves costs and savings (reductions in costs) allows us to use these

additional methods of analysis.

The payback method and the return-on-investment method are popular methods

of analysis because they are easily calculated and readily understandable. However, they characteristically have two important weaknesses which may seriously

distort the evaluation, and, therefore, they are not recommended as principal

methods for evaluating alternative systems. First, they do not (as usually

calculated) take into account the timing of cash flows. Second, they do not

take into account the magnitude of total benefits (i.e., total energy cost

reductions over the period of analysis), or total costs. The payback method

ignores benefits which accrue after the payback date; the return-on-investment

method focuses on average returns. These methods, nevertheless, may be useful

in providing supportive evaluation Information.

Neither is the internal rate of return, or yield, method recommended, although it usually gives the correct solution. Shortcomings of the internal

rate of return method are that (1) it is more cumbersome to calculate, and (2)

indeterminant solutions may result under some circumstances.

A net present value benefits method is in most cases a reliable approach to

evaluating a capital investment. This technique can be used to convert a problem

in which all cash flows are negative into a conventional investment problem involving cash outlays and cash benefits. This is a suitable method for evaluating

solar HVAC systems vis-a-vis conventional systems, because solar systems generally

involve a larger investment cost than conventional systems, but give rise to

less operating costs than conventional systems. That is, they result in savings

(i.e., benefits) in the form of reduced energy costs.

2 Using this approach, the costs are defined as the present value of the

extra costs of owning and maintaining a solar system and the benefits as the present value of savings in energy costs for a solar system as compared with a conventional system.-^ The difference between these costs and benefits, i.e., net

benefits, is the measure of the efficiency of the investment in a solar system as compared with a conventional system.

For comprehensive descriptions of these techniques, see Gerald W. Smith,

Engineering Economy , pp. 87-190, and Eugene L. Grant and W. Grant Ireson,

Principles of Engineering Economy , pp. 66-147.

2The analysis could be stated in terms of uniform annual value rather than

present value, in which case net annual benefits would be calculated.

3

As used here this approach assumes that the benefits of the alternative

systems in terms of heating and cooling performances are equal, and that the

systems will differ only in their costs.

24
Positive net benefits indicate that the solar system is more efficient, i.e., more cost effective, than the conventional system. Negative net benefits indicate that the conventional system is more efficient.''"

The net benefits of the extra Investment required for the solar system

are calculated as follows:

= - ) - (C3 - ), (16)

where

= Net benefits of a solar system as compared with a conventional system, in present value terms over N years,

= Energy cost for the conventional system, in present value terms over

N years,

F^ = Energy cost for the solar system, in present value terms over N years,

F^ - Fg = Energy cost savings in present value terms over N years,

Cg = Capital and maintenance cost for the solar system in present value

terms over N years,

= Capital and maintenance cost for the conventional system in present

value terms over N years,

Cg - C^ = Extra investment for the solar system in present value terms over

N years.

Table 4, which is in 5 parts (A through E) , is an illustration of a benefitcost comparison of a solar heating system and a conventional heating system. It

is based on hypothetical data, for a specific set of assumptions. (Note that

the purpose of this example is to demonstrate the methods of evaluation, and

not to present evaluation results of actual systems. Therefore, undue attention should not be given to the numerical outcome.)

Part A of Table 4 shows the general assumptions upon which the example is

based, including the heating load, type of systems costed, period of analysis,

discount rate, and two rates of cost escalation. Part B shows the additional

costs of acquisition and maintenance for the solar system over the counterpart

conventional system. It is assumed that the solar system requires a conventional

backup (auxiliary) system of size and with acquisition and maintenance costs

equal to the conventional system used alone. This assumption reflects the need

to ensure adequate heating during long periods of cloudy weather when the solar

system is unable to contribute to heating requirements.

Part C shows the assumed annual energy costs for both systems and the annual

energy cost savings (benefits) for the solar system in present prices without

price escalation. Energy costs for the solar system ($194) consist of electricity costs to power motors and pumps ($20) both for the solar components and the

Alternatively a benef its-to-costs ratio could be computed, in which a ratio greater than 1 indicates that the extra investment costs of the solar

system is economically worthwhile, and a ratio less than 1 indicates that the

investment in the solar system is not efficient.

25
Table 4

PART A

COST ANALYSIS: A HYPOTHETICAL EXAMPLE^

General Assumptions

° AVERAGE ANNUAL TOTAL HEATING LOAD: 84,000,000 BTU'S

° SOLAR SYSTEM: SOLAR ^ 60% LOAD

CONVENTIONAL AUXILIARY ^ 40% LOAD

EFFICIENCY, AUXILIARY ^ 55%

FUEL -> #2 HEATING OIL

° CONVENTIONAL SYSTEM: 100% LOAD

EFFICIENCY 60%

FUEL ^ #2 HEATING OIL

° PERIOD OF ANALYSIS: 20 YEARS

° DISCOUNT RATE: 2% REAL RATE

° COST ESCALATION: (1) NO REAL CHANGE

(2) 4% REAL INCREASE IN FUEL COSTS ONLY, COMPOUNDED

ANNUALLY

^In view of the presently accepted practice for building technology in this

country, common U.S. units of measurement have been used in this example. (To

convert Btu/hr^ ft to W/m^, multiply by 3.152; to convert ft to , multiply

by 9.290 x 10"*^; to convert Ibm. to Kg, multiply by 4.535 x 10. The reader

interested in using the system of SI units more extensively is referred to

American Society for Testing and Materials, Metric Practice Guide , ASTM No.

#380-72, 1972.

26
Table 4

PART B

Assumed Additional Capital (Including Installation) and

Maintenance Costs for Solar System

COLLECTOR

THERMAL STORAGE TANK

PIPES,

FITTINGS

MOTORS, PUMPS

HEAT EXCHANGER

SYSTEMS' CONTROL

BUILDING

MODIFICATIONS

ROOF

INSULATION

BASEMENT SPACE

AUXILIARY

HEATING

UNIT

CAPITAL COSTS

$7,200/20 YRS.

(800 FT.^ I? $9.00 PER FT.^

$400/20 YRS.

(8000 LBM. FLUID)

$200/20 YRS.

$200/10 YRS.

$100/20 YRS.

$150/20 YRS.

$100/20 YRS.

$75/20 YRS.

$125

(25 FT.^ (3 $5.00 PER FT.^)

(SAME AS CONVENTIONAL

SYSTEM)

MAINTENANCE COSTS

$25/5 YRS.^

$25/YR.

$8,550

$8,714

$470

$9,184

ADDITIONAL FIRST COST FOR THE SOLAR SYSTEM

PRESENT VALUE OF ADDITIONAL

CAPITAL COST INCLUDING PARTS RE- PLACEMENTS OVER 20 YEARS

PRESENT VALUE OF ADDITIONAL MAIN- TENANCE COST (I.E., $25 EVERY 5 YEARS PLUS $25 EVERY YEAR, FOR 20 YEARS)

PRESENT VALUE OF TOTAL ADDITIONAL

CAPITAL AND MAINTENANCE COST FOR THE SOLAR SYSTEM

3THE NOTATION $/YRS. INDICATES THE COST AND FREQUENCY OF OCCURRENCE; E.G.,

$25/5 YRS. INDICATES AN EXPENDITURE OF $25 WHICH WILL HAVE TO BE DUPLICATED

EVERY 5 YEARS.

'^THIS ESTIMATE CONSISTS OF $5.50 PER FT.^ FOR MATERIALS AND $3.50 PER FT.^

FOR INSTALLATION, AND WAS SUGGESTED BY EXPERTS IN THE SOLAR ENERGY FIELD. IT FALLS IN THE LOWER HALF OF THE RANGE OF PRICES QUOTED AT A RECENT SOLAR INDUSTRY

TRADE SHOW, WHERE PRICES GIVEN BY MAJOR PRODUCERS OF SOLAR COLLECTORS RANGED

FROM $3.50 TO ABOUT $25.00 PER FT. 2 INSTALLED. (SOLAR ENERGY INDUSTRIES ASSOCIATION INDUSTRY CONFERENCE AND TRADE SHOW, SHERATON PARK HOTEL, WASHINGTON, D.C., MAY 27 - 29, 1975.)
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29
Table 4

PART E

Net Benefits Calculation, Assuming a 4% Real Increase in Fuel Prices

(i = .02; e = .04)

Bn = \jc - ^] - [c s - Cc]

20

n j = -, ($410 - $194) / 1 + .04 \

V 1 + .02/

"1 $9,184

1

= $5,330 - $9,184

= -$3,855

30
auxiliary components, and fuel oil costs ($174) for the auxiliary system to

provide 40% of required heating demands. Energy costs for the conventional

system ($410) consist of electricity costs to power motors and pumps ($10)

,

and fuel oil costs ($400) to provide 100% of required heating demand. The

annual energy savings amounts to $216, i.e., $410 - 194.

Parts D and E of Table 4 show the calculation of net benefits of the solar

system as compared with the conventional system, under two assumptions for

energy cost escalation. Alternative assumptions are employed in the example in

order to illustrate the sensitivity of the results to different rates of energy

cost escalation (see Sect-ion 2.7 for a discusssion of sensitivity analysis).

Part D calculates net benefits based on present estimated energy costs, with no escalation in real costs over time. Based on the assumptions of this hypothetical example, net benefits are shown to be a negative $5,653, and, accordingly,

the life-cycle cost of the illustrative solar system exceeds that of the con- ventional system by this amount.

Part E of the exhibit calculates net benefits based on an increase in the

real price of energy of 4%, compounded annually. The effect of this assumed

escalation in energy costs is to raise substantially the energy savings of the

solar system. However, the life-cycle cost of the solar system still exceeds

that of the cbnventional system by $3,855.

Again, the reader is cautioned against placing undue emphasis on the

numerical outcome of the case example. For a different set of assumptions,

such as higher energy prices and/or lower cost of solar components, the outcome

would be different. As research and development work proceeds, the costs of

solar components may fall significantly from present levels.

2 . 7 Uncertainty in Cost Evaluations

As may be seen by the preceding example, evaluation results can be quite

sensitive to both the data estimates and assumptions employed in the analysis.

Factors affecting the outcome include (1) the discount rate used to convert

future costs to an equivalent base; (2) the investment costs necessary to

implement the HVAC systems; (3) the lives of the systems, their salvage values,

and the length of the period over which the systems are compared; (4) the projections of future costs of maintenance and repair; (5) the rate of real price

escalation in energy sources; (6) and the proportion of the total energy load

provided by the different systems. The general direction of influence of these

variables is summarized in Table 5.

The preceding discussions of evaluation methods have not dealt with the

problem of uncertainty regarding the estimated values of costs and benefits

to be used in the analysis. In fact however, alternative systems will involve

varying degrees of uncertainty or risk as to what will be their actual costs

of acquisition and operation. Uncertainty regarding values for the solar system arises principally because it is a "new" technology. Uncertainty arises

for the conventional system primarily because future availability and prices

"hrhere are technical differences in the meaning of risk and uncertainty,

but for the purpose here they can be treated as one, in that they both can cause results to vary from predictions. For a survey discussion of risk and uncertainty, see William Fellner, Probability and Profit: A Study of Economic

Behavior Along Bayesian Lines (Homewood, Illinois: Richard D. Irwin, Inc.,

1965, pp. 25-34).

31
Table 5

INFLUENCE OF ASSUMPTIONS ON ECONOMIC EFFICIENCY OF SOLAR ENERGY SYSTEMS

DISCOUNT RATE

HIGHER RATE -> FEWER NET BENEFITS

PERIOD OF ANALYSIS

LONGER PERIOD -> LARGER NET BENEFITS

FUEL COST ESCALATION

LARGER REAL COST RISE ^ LARGER NET BENEFITS

SOLAR MATERIALS AND LABOR

COST REDUCTIONS IN PRODUCTION, INSTALLATION, AND MAINTENANCE ^

LARGER NET BENEFITS

32
of fossil fuels are unknown. Thus, there are varying degrees of uncertainty

attached to the estimated values of the cost items listed in Table 1.

One method of handling uncertainty in cost evaluations is to express costs

and benefits as "expected values." This is done by multiplying the probability

of an expected occurrence by the dollar value if the event does occur. For

example, the probability of failure of the motors which circulate the collection

medium through the collector plates might be multiplied by the cost of repairing

the motors in order to find the expected cost of repairs.

Probability analysis requires determination of the probabilities attached

to the various variables of each system—a difficult and uncertain effort in

itself. Broad practical experience with solar systems or systematic research

is needed to provide informed bases for estimating system probabilities. At

present, the "best effort" will likely to be rely upon performance information

provided by manufacturers, considered judgment of the analyst, and the use of

break-even and sensitivity analysis to investigate the impact on costs of

possible variations in the values of the determinant parameters.

Break-even analysis focuses on a single key variable which is regarded as

a "risk factor." This form of analysis identifies the minimum (or maximum)

value of the risk factor for which the alternatives would be equal, or which is

required to achieve a targeted outcome. For example, one might solve for the

break-even rate of escalation in fuel prices which would equate the life-cycle

costs of a solar and a conventional system, given other costs. Alternatively,

one might solve for the break-even collector plate price per sq. ft., or for

the break-even number of years of use (i.e., the payback period). The analyst

can then assess the likelihood of achieving less than or more than the breakeven value for that factor.

Sensitivity analysis allows the analyst to determine the effect on the outcome of variation in one or more factors. A matrix can be developed to show the

results of various combinations of assumed values for the determinant factors.

Those factors which have a large impact on the outcome can thereby be identified

and subjected to further study. Sensitivity analysis was used in the illustrative cost comparison of Section 2.6 to test the impact of alternative future

changes in fuel prices on solar net benefits.

The appropriateness of assumptions to be used in evaluating alternative

systems will vary depending upon the nature of use; for example, it might be

reasonable to use different discount rates in evaluating HVAC systems used on commercial versus noncommercial buildings. However, it is important that assumptions be uniformly applied in making comparisons among systems. Table 6 shows

an illustrative set of assumptions which might be used to evaluate life-cycle

costs of alternative systems. A range is given for each variable to allow

sensitivity assessments.

33
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34
3. CONDITIONS FOR ECONOMIC OPTIMIZATION OF THE SOLAR

HVAC SYSTEM AND THE BUILDING ENVELOPE

Section 2 presented methods of measuring the costs and comparing the efficiency of solar and conventional HVAC systems. These same methods of analysis

can be applied to a number of tradeoffs in the design and operation of an HVAC

system within a building. For example, at the system design level, life-cycle

cost or benefit-cost analysis may be used to evaluate the efficiency of substituting among the various components and subsystems, such as increasing

thermal storage capacity and decreasing collector area, or vice versa. In

choosing among HVAC systems for a particular dwelling, evaluation may be made

of the relative efficiency of various combinations of solar and auxiliary con- ventional systems. In designing or adapting a building for the use of a particular HVAC system, analysis techniques can be used to determine the efficiency of

investing in energy conservation to reduce heat loads as compared with substituting a larger supply of heated or cooled air to the building. Analysis may

also be made of tradeoffs among energy conservation techniques in order to determine the optimal combination of techniques to accomplish a given reduction in

heating and cooling loads.-'- For example, different combinations of attic, wall,

and floor insulation; storm windows and doors; insulating glasses; weather

stripping; as well as building-site orientation and solar shading may be evaluated.

Since the alternatives will usually differ in their relative costs, the

life-cycle costs of achieving a given heating and cooling objective may vary

greatly depending upon the particular choices which are made. The evaluation

techniques discussed in Section 2 can be used to evaluate the costs and benefits

associated with incremental changes in the size or design of a system, and,

therefore, are useful in optimizing the HVAC system within the building envelope.

The basic evaluation techniques having been treated, the focus of Section

3 is on setting forth and illustrating the conditions which are necessary for

optimization, in the context of solar system/building design. These conditions

and the basic cost-minimization approach which are outlined below could be

applied to any of the tradeoff problems mentioned above. For purpose of illustrating the approach, however, emphasis is given in the discussion and examples

to the problem of determining the least-cost combination of HVAC capacity and

energy conservation measures which will meet a target level of comfort.

Two basic approaches to the problem of optimization are (1) minimization

of total costs for a given output requirement, and (2) maximization of net

benefits (i.e., finding the greatest difference between total benefits and total

costs.) Where benefits are fixed, the two approaches give the same result and

the choice between them is chiefly a matter of convenience given the nature of

the particular problem. Since, for the problem at hand, all cash flows are negative and a level of comfort performance is given, the cost minimization

approach will be discussed.^

^For a di scussion and analysis of these tradeoffs, see, Steve Petersen,

Retrofitting Existing Housing for Energy Conservation: An Economic Analysis .

^For more indepth discussions of economic optimization, see Henderson and

Quandt, Microeconomic Theory: A Mathematical Approach (New York: McGraw-Hill

Book Co., 1958), Chapter 3: and M. M. Bober, Intermediate Price and Income

Theory (New York: W. W. Norton and Co., 1955), Chapter VI.

35
The problem of achieving a target comfort level by trading heating and

cooling for energy conservation"^ is analogous to the classical economics problem of the "widget" manufacturer's need to determine the least-cost combination of labor and capital to produce a given quantity of widgets.

The designer of a solar dwelling may be viewed as the producer of a specified level of thermal comfort for a dwelling, where thermal comfort comprises

temperature, humidity, or other related comfort attributes. The resources

(inputs) to achieve the comfort objective consist of labor and material for the

HVAC system, insulation, storm doors and windows, solar shading, and other

energy conservation techniques, and also include various sources of energy.

The quantity of output, i.e., comfort level, can be expressed as

q = f(Xi, X2, . . . X^) J where X-^, X2, . . • X^ are the quantities of the

variable inputs. A production function could he specified that states the

comfort level obtainable from every possible input combination. Many different

combinations of inputs may be used to produce a given level of comfort. But

the technical relationship between the inputs and the comfort level does not

indicate the optimal combination of inputs to produce a given comfort level;

the economically efficient input combination for the production of a particular

comfort level depends upon the relative prices of the inputs.

A necessary condition for arriving at the minimal cost of producing a given comfort level is that the contribution of inputs towards achieving the

desired comfort level be in the same proportion at the margin as their prices.

This means that each input will be used up to that level at which its additional

contribution to the objective per extra dollar spent is just equal to that for

all other inputs. Assuming continuous and smooth functions, this necessary

condition is expressible mathematically for two inputs as

9Q Aq = £1 , (17)

3Xi / 9X2 P2

where

Q = units of output of comfort level,

X-|^5 X2 = units of inputs 1 and 2, and

P^j P2 ~ cost per unit of inputs 1 and 2 in present value, life-cycle

terms

This expression could be expanded in a different form to accommodate as many

inputs as are relevant.

This necessary condition is derived as follows: The decision maker

attempts to minimize life-cycle costs (C-^) > where

= PiX;l + ^2^2. (18)

iThe higher the thermal resistance of the envelope, the less the change in

the temperature of the interior air, and therefore, the smaller the need for

additional heating or cooling of the space. Energy conservation actions to the

building—by reducing conductive, convective, and radiation heat losses in winter and heat gains in summer—reduce the necessary load capacity and level of use of the HVAC system for a given building.

36
subject to the constraint that a specified level of comfort (Qq) be met.

Making the problem unconstrained by the Lagrange multiplier, we can minimize

the expression

Cl = PlXl + P2X2 + ^[Qq - Q(Xi, X2)]. (19)

By setting the partial derivatives of Cj^ with respect to X-^ and X2 equal

to zero, we obtain

9C.- = Pn - A = 0, and (20)

9x1 ^ ax^

9C — = P2 - A ^ = 0. (21)

9X2 9X2

To derive the optimality condition shown in equation (17) , we rewrite

equations (20) and (21) as 9Q 9Q , divide the first equation by

^1 = ^ 9X3^ and P2 " 9X2

the second, and simplify, thus arriving at

(22)

This necessary condition for the optimal combination of two inputs is

illustrated graphically in Figure 1. Assume that the horizontal axis of Figure 1 measures the BTU load capacity of a given solar HVAC system (X-^) ; the

vertical axis measures the quantity of a given package of energy conservation

techniques (X2). Curve indicates all the combinations of X-^ and X2 which

will yield a given level of comfort Qq (assumed to be the target, i.e., minimum

acceptable comfort level). For example, can be produced by combining the

quantity "op of X^ and oj of X2, or by combining "os of X^ and og of X2 . Moving

from left to right down Q , we can determine on the horizontal axis the increase

in the capacity of the solar HVAC system which would be necessary to offset a given reduction in the quantity of energy conservation techniques in order to

maintain Q level of comfort. The curve shows the technical tradeoffs between

1 • o the two inputs.

The lines C]^, C2, and Co in Figure 1 illustrate three of a family of cost

functions. Each function indicates a specific total cost, and shows the com- binations of inputs 1 and 2 which may be purchased for that total cost. With

the expenditure equal to C2> for example, one could buy either ot of X]^ or o^

of X2 or any combination of X^ and X2 that lies on the cost line C2 • The

slope of the cost line indicates the relative costs of the two inputs, i.e.,

^±/^2' Total costs rise as larger quantities of both inputs are purchased,

that is, C3 > C2 > C]^.

The total benefits of each input must exceed its total costs in order to

insure that the input is economical to add.

37
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38
Given the relative costs of Xi and X2 indicated by the slopes of the cost

curves, the lowest total cost at which comfort level can be produced is

using OS" of X-|^ in combination with "og of X2. For any cost less than C2 (e.g.,

C-j^) the desired comfort level, Q^, could not be achieved. To achieve at any

cost greater than C2 (e.g., C2), would be inefficient since 0^ can be achieved

at the lower cost, C2- Thus the least-cost combination of factors is determined

by the point of tangency between the output curve, Q^, and a cost curve. At

the point of tangency,

/ P

3Q / 9Q _ and the basic optimality rule is met.

9Xj^ / 9X2 P2

In practice, the optimal combination of HVAC/energy conservation inputs

will vary with assumptions and given information. The particular optimal com- bination is dependent upon (1) climate factors, (2) comfort requirements, (3)

functional characteristics of the building, (4) present and future energy costs,

(5) costs of solar and conventional HVAC components, (6) costs of insulation

and other energy conservation techniques, and (7) discount rates.

39
4. SUMMARY AND SUGGESTIONS FOR tWTHER RESEARCH

This paper addresses some economic issues important to the design, acquisition, and evaluation of solar heating and cooling systems. In Section 2, the

paper explains and illustrates methods for evaluating and comparing the economic

efficiency of solar and conventional heating and cooling systems for buildings.

It identifies relevant costs, discusses data collection requirements, illustrates the discounting of costs, develops generalized life-cycle cost and

benefit-cost models, sets forth techniques for developing models for unique problems, and discusses how the effective life-cycle costs to the owner of a solarequipped building would be altered by current tax laws, insurance, and governmental incentives programs. Section 2 also discusses assumptions regarding the

discount rate, the period of analysis, and the rate of price escalation in

nonrenewable energy sources.

Section 3 of the paper sets forth the logic of and identifies an optimality

rule for making cost-effective tradeoffs in the design of solar energy projects.

For clarity and convenience, much of the discussion and illustration of optimality

centers on the optimal tradeoffs between capacity of an HVAC system and investment in energy conservation in the building envelope.

While the paper treats a number of economic issues, it is not an exhaustive

study of the economics of solar heating and coooling systems; nor does it meet

the needs of all parties concerned with solar heating and cooling. In particular,

the following tasks would appear to require further effort:

(1) Systematic investigation into the expected life-cycle costs of operating,

maintaining, and repairing solar energy systems, and the projecting of future

costs of solar system components, based on large scale production.

(2) Application of the evaluation techniques described in this paper to actual

solar energy systems.

(3) A comprehensive analysis of the impact of laws, regulations, codes, zoning

ordinances, and other practices on the costs of solar energy systems.

(4) Preparation of a consumer-oriented handbook to assist the homeowner and

business person in making economic comparisons of alternative HVAC systems.

40
SELECTED BIBLIOGRAPHY

American Society for Testing and Materials. Metric Practice Guide . ASTM No.

E380-72, 1972.

Bober, M. M. , Intermediate Price and Income Theory . New York: W. W. Norton

& Co. , Inc. , 1955.

Davidson, Paul, Falk, H* Lawrence, and Lee, Haesung. "Oil: Its Time Allocation and Project Independence." Brookings Papers on Economic Activity , Vol. 2. Washington, D.C.: The Brookings Institution, 1974.

DeGarmo, E. Paul. Engineering Economy , 4th Ed. New York: The MacMillan Co.,

1967.

Fellner, William. Probability and Profit: A Study of Economic Behavior Along

Bayesian Lines . Homewood, Illinois: Richard D. Irwin, Inc., 1965.

Grant, Eugene L. Principles of Engineering Economy . New York: The Ronald

Press Company, 1950.

Grant, Eugene L., and Ireson, W. Grant. Principles of Engineering Economy , 5th Ed. New York: The Ronald Press Co., 1970.

Henderson, James M. , and Quandt, Richard E. Microeconomic Theory; A Mathmatical

Approach . New York: McGraw-Hill Book Co., 1958.

Nordhaus, William D. "The Allocation of Energy Resources." Brookings Papers

on Economic Activity , Vol. 3. Washington, D.C.: The Brookings Institution,

1973.

Office of Management and Budget. Executive Office of the President. Circular

No. A-94 (Revised), March 27, 1972.

Smith, Gerald W. Engineering Economy: Analysis of Capital Expenditures , 2nd

Ed. Ames, Iowa: The Iowa State University Press, 1973.

U.S. Department of Commerce. National Bureau of Standards. Center for Building

Technology. Retrofitting Existing Housing for Energy Conservation: An

Economic Analysis , by Stephen R. Petersen, Washington, D.C., Building

Science Series 64, December 1974.

USCOMM-NBS-DC 41
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U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUm.K A riON OR KI- POR l' NO. NBSIR 75-712

2. Gov't Aeeession

No.

4. TlTLi: AND SUIVl lTj.l'; Solar Heating and Cooling of Buildings: Methods of

Economic Evaluation

5. Publication Date May 1975

6. Performing Organization C ode 463.02

7. AUTHOR(S)

Rosalie T. Ruegg

8. Performing Organ. Report No. NBSIR 75-7T2

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No.

11. Contract /Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

National Bureau of Standards

Department of Commerce

Washington, D.C. 20234

13. Type of Report & Period Covered

Final

14. Sponsoring Agency (-ode

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.)

This report addresses economic issues important to the design, and evaluation of so] ar

heating and cooling systems in buildings. It explains and illustrates with simple, but

realistic examples the use of life-cycle cost analysis and benefit-cost analysis to

evaluate and compare the economic efficiency of solar and conventional energy systems. It

also explains the conditions for making cost-effective tradeoffs in solar system/building

design. By presenting the basic methods and assessing the appropriateness of alternative

assumptions, the paper provides a resource document for researchers and analysts.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons

)

Solar energy; solar heating and cooling; HVAC systems; life-cycle cost analysis;

economic optimization

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