AiiiDE mofl^a NBS

Economic Evaluation of

Solar Energy Systems in

Commercial Buildings

Methodology and Case Studies

NBSIR 82-2540

Economic Evaluation of Solar Energy Systems

in Commercial Buildings: Methodology and Case Studies

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AUG 1 9 1982

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Rosalie T. Ruegg

G. Thomas Sav

Jeanne W. Powell

E. Thomas Pierce

U S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Building Technology

Washington, DC 20234

July 1982

Prepared for; U S. Department of Energy

Office of Solar Heat Technologies

Active Heating and Cooling Division

Washington, D.C. 20585

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director
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ABSTRACT

This report develops a comprehensive economic optimization model for evaluating

the economic feasibility of active solar energy systems to provide service hot

water and combined space heating/service hot water in commercial buildings.

The model is demonstrated in a number of case studies for office buildings and

retail stores. Data and assumptions for use in the model are compiled for the

selected case studies. Using these data, the model is applied to estimate

present value net savings (or net losses) of the solar energy systems over a 20-year life cycle. Break-even values for hot water loads, solar energy system

costs, and current and future energy prices are also calculated to determine

the minimum conditions under which the solar energy systems become cost effective for the selected buildings. Economic optimization paths which show the

optimal solar collector areas and the corresponding present value of net savings

(or net losses) associated with a range of hot water loads are developed in

the case studies. Sensitivity analysis is conducted for key variables. The

relationship between total life-cycle costs and the solar fraction is tested

for selected cities to demonstrate how net savings (net losses) change as the

solar fraction is increased. In its approach, this report is of interest to

solar analysts; in its results, to the solar policy, research, and building

communities

Key words: building economics; commercial buildings; economic analysis;

energy economics; life-cycle costing; solar energy

iii
PREFACE

The study on which this report is based was conducted by the Applied Economics

Group, Center for Building Technology, National Bureau of Standards (NBS), for

the Office of Solar Heat Technologies, Active Heating & Cooling Division,

Washington, D.C. 20585, U.S. Department of Energy (DoE) , under Interagency

Agreement No, E (49-1) - 3800, EA-77-A-01-6100, Task 9.

This report is one of several related documents prepared under this contract.

It is the technical analysis report which details the economic model and the

data, assumptions, and findings of selected case studies of solar economic

feasibility. It is addressed primarily to the research and policy communities.

Additional products include papers and articles [1-8] which were prepared in

full or in part from the technical work described herein.

It is to be stressed that, while the model is widely applicable to the

evaluation of solar energy for commercial buildings, the feasibility results

for the case studies are founded on a host of assumptions which are variable

among buildings of like size, function, and geographic location, not to mention

buildings that are different from those studied. The reader should use caution

and judgment in extending results or conclusions contained in this report to

other buildings.

iv
ACKNOWLEDGMENTS

The authors wish to thank all of those persons who contributed to the study,

particularly Dr. Roger Bezdek, formerly Chief of the Barriers and Incentives

Branch of the U.S. Dept, of Energy, for his direction and support of the

project.

Special appreciation is extended to Dr. Harold Marshall, Leader of the NBS

Applied Economics Group, for his continuing guidance and contributions to the

project, to Mr. James Gross, Division Chief, and to Mr. Robert Dikkers, Leader

of the NBS Solar Technology Group, for their valuable advice and administrative

support throughout the project. Credit and appreciation are due to Mr. Joel

Levy of the Applied Economics Group for his extensive work on revising,

extending, and documenting the computer program used to apply the model in the case studies of economic feasibility.

Helpful editorial and technical assistance was provided by Dr. Kent Reed,

Dr. Samuel Silberstein, Mr. Larry Salomone, and Ms. Patricia Christopher of the

NBS Center for Building Technology, and by Mr. Robert Atkins of the NBS Institute for Computer Sciences and Technology. Credit is also due Mr. Philip

Decker, NBS student worker, who assisted with the computer analyses,

Ms. Sieglinde Fuller who assisted with the manuscript preparation, and Mrs.

Brenda Thompson who typed the report.

V
TABLE OF CONTENTS

Page

ABSTRACT ill

PREFACE Iv

ACKNOWLEDGMENTS v LIST OF FIGURES lx

LIST OF TABLES xi

EXECUTIVE SUMMARY xiii

1. INTRODUCTION 1 1.1 Background 1 1.2 Purpose and Scope 2 1.3 Approach 2 1.4 Organization 3

2. ECONOMIC MODELING 4

2.1 Model Requirements 4

2.2 Basic Framet^ork 4

2.3 Algebraic Model 15

2.4 Using the Model to Determine Net Life-Cycle Savings,

Optimal System Size, Break-Even Values, Years to Payback,

and Sensitivity 22

2.4.1 Net Life-Cycle Savings and Optimal System Size 22

2.4.2 Break-even Values 22

2.4.3 Years to Payback 23

2.4.4 Sensitivity Analysis 23

2.5 Simplified Model for Solar Hot Water Analysis 24

2.5.1 A New Concept 24

2.5.2 Universal Economic Optimization Paths 24

2.6 Integrated Optimization of Solar Energy and Energy

Conservation: A Theoretical Framework 27

2.6.1 Economic Foundations 27

2.6.2 Optimization Model 30

3. CASES SELECTED FOR STUDY: BUILDINGS, BUILDING SYSTEMS, AND

GEOGRAPHICAL LOCATIONS 41

3.1 Buildings 41

3.1.1 Office Buildings 41

3.1.2 Retail Stores 44

3.2 Air Handling Systems 44

3.2.1 Office Buildings 52

3.2.2 Retail Stores 53

3.3 Heating Plants 53

3.4 Geographical Locations and Data 58

3.4.1 Selection Cities 58

3.4.2 Geographical Data 58

4. DATA AND ASSUMPTIONS FOR CASE STUDIES 64

4.1

Heating and Hot Water Requirements 64

vl
TABLE OF CONTENTS (Continued)

Page

4.1.1 BLAST Transient Load Model—Space Heating 64

4.1.2 Results of BLAST Analyses—Space Heating 65

, 4,1.3 Building Loads Model—Service Hot Water 67

4.2 Solar Energy System Performance 69

4.3 Purchase and Installation Costs of the Solar

Energy System 70

4.3.1 Estimating Fixed and Variable System Costs 70

4.3.2 Reviewing the Cost Estimates 73

4.4 Energy Prices 82

4.5 Financial Variables—Discount, Borrowing, and Inflation Rates .. 82

4.5.1 Discount Rate and Inflation 82

4.5.2 Financing Terms 89

4.6 Taxes and Incentives 90

4.7 Other Data and Assumptions 95

4.7.1 Operation, Maintenance and Repair Costs 95

4.7.2 Solar Energy System Life and Salvage Value 95

4.8 Summary of Economic Parameters Common to All Cities 95

5. CASE STUDY RESULTS 97

5.1 Case Study Results for Solar Service Hot Water Heating Systems

Under Base-Case Assumptions 97

5.1.1 Summary Overview and Comparisons 97

5.1.2 Detailed Results for Selected Case Studies 104

5.2 Case Study Results for Solar Combined Space Heating and

Service Water Heating Systems 112

5.2.1 Results Compared for Two Sets of Assumptions 112

5.2.2 Break-even Analysis 112

5.2.3 Effects of Financing, Taxes, Incentives, and

Discounting 116

5.3 Sensitivity Analysis 116

5.3.1 Sensitivity of Solar Cost Effectiveness to System Size .. 116

5.3.2 Sensitivity of Solar Life-Cycle Costs to Selected

Economics Assumptions: Phoenix Case Study for New

Office Buildings 120

5.4 The Changing Status of Solar Economic Feasibility 120

REFERENCES 127

APPENDIX A - Mathematical Programming and Stochastic Modeling:

Exploratory Research A-1

A, 1 - The Importance of Multivariate Analysis in

Solar Design A-1

A. 2 - Classification Framework for Solar Design

Variables A-5

A. 3 - Mathematical Programming as a Potential Tool for

Solar Analysis A-5

vii
TABLE OF CONTENTS (Continued)

Page

A. 4 - The McGarity-Revelle Solar Performance Model A-7

A. 5 - Solution Properties of the McGarity-Revelle Linear

Program A-16

A. 6 - Mathematical Relationships Employed in the Solar

Energy Linear Program and a Listing of Mathematical

Notations A-18

A. 6.1 - Component Performance Relationships A-18

A. 6.2 - Combination of Performance Formulas for Linear Program

Format A-20

A. 6. 3 - Mathematical Notation A-22

A. 7 - Derivations of Solution Properties of the McGarityRevelle Linear Programming Model A-24

A. 8 - Test Results: The Linear Program Compared with

TRNSYS A-31

A. 9 - Subsequent Research A-32

A. 10- Stochastic Modeling of Weather Variables A-32

APPENDIX B - Computer Program Used to Apply the Economic Evaluation

Model to Selected Case Studies for Combined Space Heating

and Hot Water B-1

viii
LIST OF FIGURES

Page

Figure 2,1 LCC Trade-Off Between Solar Energy and Auxiliary Energy 9

Figure 2.2 Determining the Cost-Minimizing Size of the Combined Solar/

Auxiliary Energy System: U-shaped TLCC^, g Curve 11

Figure 2.3 Determining the Cost-Minimizing Size of the Combined Solar/

Auxiliary Energy System: Monotonically Rising Curve 12

Figure 2.4 Determining the Cost-Effective Solar Investment 13

Figure 2,5 Minimizing Losses from a Solar Investment 14

Figure 2.6 An Uneconomical Solution for a Solar Energy System 16

Figure 2,7 Universal Economic Optimization Path: Generic Form 28

Figure 2,8 Integrated Optimization Procedure: Load Reduction 32

Figure 2,9 Integrated Optimization Procedure: Solving for the

Combination of Conservation, Solar, and Nonsolar Inputs to

Thermal Comfort 33

Figure 2.10 Integrated Optimization Procedure: Compressed Model 34

Figure 3.1 Schemata of Office Buildings 46

Figure 3.2 Office Building Load Schedules 47

Figure 3,3 Schemata of Retail Stores 49

Figure 3,4 Retail Building Load Schedules 50

Figure 3,5 Annual Heating and Cooling Energy Consumption of a Building

with Alternative Mechanical Systems in Four Geographical

Locations 51

Figure 3.6 Solar Collector 55

Figure 3,7 Schematic Diagram of a Combined Solar Service Hot Water

System for a Commercial Building 56

Figure 3.8 Schematic Diagram of a Solar Space Heating/Hot Water System

for a Commercial Building 57

Figure 3.9 Climatic Regions and Cities Selected for Study 59

Figure 4.1 Solar Universal Performance Curves for Estimating the

Performance of a Combined Solar Space Heating and Service

Hot Water System for a Commercial Building 71

Figure 4.2 Solar Universal Performance Curves for Estimating the

Performance of a Solar Service Hot Water System for a Commercial Building 72

Figure 4,3 Comparison of Two Cost Functions: The NBS-Estimated Cost

Function for a Combined Space Heating/Hot Water System

for New Construction Versus the Regression-fitted Cost

Function for 43 Sample Demonstration Projects 78

Figure 4.4 World Oil Prices: Recent History and Alternative Future

Projections, 1960-1995 80

Figure 5.1 Break-even Results: Minimum Hot Water Usage Rates (Gallons

Per Minute) for Solar Energy System Cost Effectiveness 101

Figure 5.2 Economic Optimization Paths: Boston 108

Figure 5.3 Economic Optimization Paths for Phoenix Ill

lx
LIST OF FIGURES (Continued)

Page

Figure 5.4 Impact of Financing, Taxes, Incentives, and Discounting on Solar Cost Effectiveness: New Office in Phoenix, Oil

Auxiliary System 117

Figure 5.5 Solar Cost Effectiveness as a Function of the Solar

Fraction: Phoenix 118

Figure 5.6 Solar Cost Effectiveness as a Function of the Solar

Fraction: Bismarck 119

Figure 5.7 Life Cycle Costs of the Solar Energy System as a Function

of the Size of the Down Pa3mient 121

Figure 5.8 Life Cycle Costs of the Solar Energy System as a Function

of the Size of the Energy Tax Credit 122

Figure 5.9 Life Cycle Costs of the Solar Energy System as a Function

of the Length of the Depreciation Period 123

Figure 5.10 Life Cycle Costs of the Solar Energy System as a Function

of the Purchase and Installation Cost 124

Figure A. 1 Sensitivity of Solar Acquisition Cost to Storage Volume/

Collector Area Ratio for Given Solar Fractions and

"Typical" Storage Cost A-3

Figure A. 2 Sensitivity of Solar Acquisition Cost to Storage Volume/

Collector Area Ratio for Given Solar Fractions and

High Storage Cost A-4

Figure A. 3 Diagram of a Solar Energy System with Designation of

Component Constraints A-9

Figure A. 4 Solar Water and Space Heating System Diagram Showing Energy

Flow and Temperature Parameters A-23

Figure A. 5 Comparison of Performance Test Results for a Commercial

Solar Water Heating System in Boulder, Colorado: Linear

Program Versus TRNSYS A-33

Figure A. 6 Comparison of Performance Test Results for a Residential

Solar Water and Space Heating System in Madison, Wisconsin:

Linear Program Versus TRNSYS A-34

x
LIST OF TABLES

Page

Table 2.1 Definition of Symbols Used in Section 2 36

Table 3.1 Types of Commercial Buildings 42

Table 3.2 Estimated Floor Space by Type of Nonresldential Buildings ... 43

Table 3.3 Office Building Construction Characteristics 45

Table 3.4 Retail Store Construction Cliaracteristics 48

Table 3.5 Annual Insolation Values for Selected Cities 61

Table 3.6 Water Supply Temperatures Used in Analysis of Service Hot

Water Requirements 62

Table 3.7 Ground Temperatures Used in Analyses of Space Heating

Requirements 63

Table 4.1 Estimated Annual Space Heating Requirements of the Selected

Buildings 66

Table 4.2 Estimated Annual Hot Water Requirements of the Selected

Buildings 68

Table 4.3 Solar Energy System Cost Equations 74

Table 4.4 Average Labor Adjustment Factors 75

Table 4.5 Acquisition Costs for 262 Nonresldential Solar

Demonstration Projects, By Type of System 81

Table 4.6 1978 Energy Prices and Projected Rates of Changes 83

Table 4.7 Early-1980 Energy Prices and Projected Rates of Changes 84

Table 4.8 Mid-1980 Energy Prices and Projected Rates of Change 85

Table 4.9 Total Present Value of 1 Million Btu (1 GJ) of Energy

Purchased Each Year Over 20 Years 86

Table 4.10 Cost of Capital — Average Returns on Debt and Equity in

Nonfinanclal Corporations 88

Table 4.11 Rates of Sales Tax, Property Tax, and State Income Tax by

City 91

Table 4.12 Economic Parameters Common to All Cities: Base Case 96

Table 5.1 Economic Analyses of Solar Hot Water for a New Office

Building with an Electric Resistance Backup System: Case

Study Results for 13 Cities 98

Table 5.2 Economic Analyses of Solar Hot Water for a New Retail Store

with an Electric Resistance Backup System: Case Study

Results for 13 Cities 99

Table 5.3 Economic Parameters Specific to Boston 105

Table 5.4 Break-even Results: Boston, Hot Water Only 106

Table 5.5 Economic Parameters Specific to Phoenix 109

Table 5.6 Break-even Results: Phoenix, Hot Water Only 110

Table 5.7 Economic Evaluation Results for Four Selected Cities Based

on Early-1980 and Mid-1980 Energy Prices and Federal Tax

Credit 113

Table 5.8 Break-even Analysis of Base-Year Fuel Prices and Fuel

Escalation Rates for Selected Cases Based on Early-1980

Assumptions .114

xl
LIST OF TABLES (Continued)

Page

Table 5.9 Break-even Analysis of System Cost for Selected Cases

Based on Early-1980 Assumptions 115

Table 5.10 The Changing Status of Solar Economic Feasibility 125

Table A. 1 Classification of Solar Design Variables A-6

Table A. 2 Summary of Observations on the Solution Properties of the

McGarity-Revelle Linear Program A-19

Table B.l City File for Boston B-17

Table B.2 Sample Program Printout B-18

xii
EXECUTIVE SUMMARY

The total energy use of comraercial buildings in the U.S. for space heating and

service water heating is nearly half that of residential buildings. However,

they account for a substantially smaller part of the use of solar energy systems. Space heating and service water heating for commercial buildings is a large, but for the most part unrealized, potential market for solar energy.

This report is a resource document for investigating the economic feasibility

of comraercial building applications of solar energy. It documents in detail

an economic model for evaluating commercially used solar energy systems, and

guides in the use of the model for economic feasibility studies. It demonstrates the model in a number of case studies which (1) assess the cost effectiveness of selected solar energy systems under representative conditions, (2)

Investigate through break-even analysis the necessary conditions for minimum

cost effectiveness, and (3) analyze future prospects under alternative

conditions

The economic evaluation model takes a life-cycle costing approach and includes

full treatment of taxes and Federal and State incentives. It is used to determine the optimally sized solar energy system and the present value net dollar

savings or losses for the building type, location, auxiliary fuel type, and

other specified data and assumptions. Break-even values of key parameters are calculated to determine the necessary conditions for solar energy to be equal

in dollar cost effectiveness to the energy systems with which it is compared.

Life-cycle net savings or losses corresponding to a wide range of solar fractions can be calculated to determine the sensitivity of costs to system size.

Data and assumptions can be modified to reflect changes in tax laws, financing

terms, depreciation rules, and other conditions affecting economic feasibility.

(The computer program used to apply the model is presented as appendix B to

the report.)

A simplified version of the model for the evaluation of solar hot water systems

is presented. The simplified model reduces the search for the optimally sized

solar hot water system to a single deterministic equation. Analysis of this

equation supports the construction of "universal economic optimization paths"

which can show the optimal solar energy system size and corresponding net

savings or losses for a range of hot water loads. This model can be used to

extend evaluation results to a range of hot water loads.

The evaluation of solar energy systems by mathematical programming techniques

is explored. Although this approach is not sufficiently developed to apply as

a principal evaluation tool for a feasibility study, the theoretical approach

is significantly extended.

The case studies focus on commercial buildings representative of a large share

of the commercial building stock, and on 13 city locations accounting for

major variations in U.S. solar radiation and temperature characteristics and

also reflecting regional diversity in taxes, incentives, and other economic

parameters

xiii
The selected buildings are not buildings specifically designed to optimize

solar gain or daylighting. They are (1) an existing three-story office

building with metal curtain-wall construction, an occupancy of 300 people,

single-glazed windows, uninsulated walls and roof, a floor area of 2,700 m^

(30,000 ft^), equipped with two constant-volume air-handling systems; (2) a new three-story office building identical to the existing building except

that it conforms to ASHRAE 90-75 energy conservation standards and is equipped

with two variable-volume air-hatidllng systems and an energy economizer cycle;

(3) an existing one-story retail store with metal curtain wall construction, a floor area of 460 m^ (5,000 ft^), an occupancy of 100 people, uninsulated

walls and roof, and equipped with a single constant-volume air-handling system;

and (4) a new one-story retail store identical to the existing store except

that, like the new office building, it conforms to the ASHRAE 90-75 standard

for the building envelope and is equipped with an energy-efficient variablevolume air-handling system. The nonsolar heating plants for these buildings

are assumed to be conventional hot water boilers fired alternatively by

electricity, natural gas, and distillate fuel oil.

The cities selected for the regional case studies are Apalachicola, FL;

Bismarck, ND; Boston, MA; Charleston, SC; Fort Worth, TX; Los Angeles, CA;

Madison, WS; Miami, FL; Nashville, TN; Omaha, NE; Phoenix, AZ; Seattle, WA; and

Washington, DC. The economic evaluation is based on a detailed analysis of the

energy requirements of the selected buildings in the different locations estimated by the BLAST loads analysis program, as well as on other region-specific

data.

The solar energy systems are assumed to be high quality liquid systems with

double-glazed, flat-plate collectors. The hot water system for the retail

stores is assumed to be an "off-the-shelf" system. The hot water system for

the office buildings and the combined space heating/hot water system for the

retail stores and the office buildings are assumed to be custom designed and

engineered. Solar energy system performance is estimated by the LASL Solar/

Building Load Ratio Method, which is incorporated in the economic evaluation

model.

The reader is cautioned that the general conclusions that can be drawn about

solar cost effectiveness on the basis of case study results are limited. There

is no single answer. Results are strongly dependent on a number of factors

that are either variable over place and/or time, or uncertain, or both, such as climatic conditions; building and user characteristics; system cost, performance, durability, maintainability, and operating energy requirements; and

current and projected prices of other energy sources. Furthermore, it should

be realized that economic feasibility may be dependent on decision criteria

other than the quantitative cost-effectiveness data. Reflecting the difficulty

to generalize results, the case study results are secondary, and the development and illustration of a methodology that can be used to address the dynamics

of solar economic performance over time are primary,

A constrained optimization analysis was used in the case studies to determine

the size of the solar energy system which will maximize net dollar savings or minimize net dollar losses. Reflecting the relative prices of the different

xiv
energy sources, the optimal solar fractions and system sizes (constrained to

be greater than 0) were found to be substantially larger—and solar economic

greater—when the auxiliary system was assumed to be an electric resistance

system rather than an oil-fired furnace. In turn, the oil-furnace resulted in

larger solar fractions and system sizes than did the gas furnace.

A comparison of case study results for the combined solar space heating/hot

water system for the office buildings and retail stores showed a slightly better

economic performance for the systems applied to the office buildings, other

things being equal. A comparison of the results for the new and existing

buildings showed a better economic performance for systems applied to the new

buildings, other things again being equal.

An analysis of the relationship between net savings and solar fraction was made

in the case studies to determine the impact of system size on economic performance. By providing a measure of the dollar consequences of increasing or decreasing system size, this kind of information is particularly valuable in

those cases where factors other than direct energy-related costs are important

to the decision to choose solar (e.g., the need for an alternative energy

supply to operate the building during fuel curtailments). In some of the case

studies, net dollar savings or losses were found to change little over a wide

range of sizes of the solar energy system, indicating only a small cost penalty

for selecting a solar energy system larger than the economically optimal size

based on direct life-cycle costs. In the Phoenix case study, for example,

economic performance of the combined system in office buildings was best for a system supplying between 40 percent and 50 percent of the load, but there was little change up to a size providing a solar fraction of about 75 percent. In some of the case studies, however, substantial dollar penalties were estimated

for choosing the size of the solar energy system without regard to economic

optimization. For example, in the Bismarck case study, net losses were found

to increase rapidly as the solar fraction was raised beyond the level determined

to be economically optimal.

The break-even analysis allows the reader to compare current values of solar

energy system costs and energy prices with the minimum calculated values necessary to move solar from a net loss to a net savings outcome for the case study

applications examined. Again taking the Bismarck case study as an example,

the breakeven analyses showed that the price of oil for heating in the new

office building would have to be about twice that in mid-1980 in order for

heating by solar to be as cost effective as oil heating (given other data and

assumptions of the case and taking into account deductions of fuel costs

from taxable income). Alternatively, future escalation in the price of oil

would have to be 1.8 times the projected rate as of mid-1980.

Based on the initially assumed data and assumptions (1978 energy prices for

water heating and early-1980 energy prices for combined systems), solar space

heating and water heating were found not to be cost effective in the case studies examined. Sensitivity analysis was used to investigate the importance

of the parameters and the changes in the data and assumptions necessary for

cost effectiveness. For example, the cost effectiveness of the solar hot water

systems was found to be critically sensitive to the amount of hot water used.

XV
With higher hot water usage rates than those initially assumed, the solar water

heating moved towards cost effectiveness against electric resistance water

heating in some of the cities, principally Phoenix, Boston, Bismarck, Nashville,

and, for small scale applications in Los Angeles, even with the understated 1978

energy prices and escalation rates.

In addition to changing the parametric values for the purpose of conducting

sensitivity analysis, the data were updated during the course of the study to

reflect changes in projected energy prices, the size of the Federal tax credit,

and projected financing terms and depreciation rules. These changes affected

the results dramatically, substantially improving the economic performance of

the case study systems.

The table below gives comparative examples of results based on the initial set

of data and assumptions for the combined system analysis and for the revised

data and assumptions, showing in some cases changes from large net losses to

significant net savings. The cost effectiveness of solar energy is, in other

words, shown by the case studies to change over time and events, and to vary

with location and application. Periodic reassessment is necessary for a valid

appraisal of its current economic status.

Changing Solar Cost Effectiveness Over Time

City Fuel Type

Early-1980

Energy Prices

10% Federal

Tax Credit

Less Favorable

Financing Terms

($1,000)

Mid-1980

Energy Prices

15% Federal

Tax Credit

More Favorable

Financing Terms

($1,000)

Bismarck, ND Electricity -15.8 +25.2

Oil -17.3 +10.1

Phoenix, AZ Electricity -13.5 + 8.1

Oil -18.8 - 2.3

Boston, MA Electricity -18.7 +13.2

Oil -22.0 - 7.5

Charleston, SC Electricity -18.0 - 5.7

Oil -20.4 - 8.0

* Based on case study results for new 3-story office building and current and

projected energy prices and Federal tax credits existing at two points in

time (early 1980 and mid 1980), and projected financing terms under the Solar

Energy and Energy Conservation Bank (which were not realized).

xvi
1 . INTRODUCTION

1 . 1 BACKGROUND

Total energy demands for space heating and service hot water in conmierclal

buildings in the United States are about 40 percent as great as in residential

buildings [9, 10]*. Yet of the approximately 100,000 solar energy systems now

installed on buildings in this country, about 98,000 are estimated to be on residential buildings and less than 2,000 are on commercial buildings [11].

Of the total sales of solar collectors (in terms of collector surface area)

in a recent year, 82 percent went for residential applications and only 18

percent for nonresidential applications [12]. While commercial buildings

provide a large potential market for solar energy systems, that market is as yet largely untapped.

Paralleling the small demand for solar energy systems for commercial buildings

is a relatively small body of economic literature on the topic. Residential

use of solar energy for space heating and hot water appears to have received

considerably more attention by economists and other analysts than has commercial use.l Few economic studies have been made of solar energy systems for

commercial buildings. Studies that have been conducted looked mainly at rental

single-family housing, at small nonresidential buildings whose energy requirements are dominated by the building envelope, and at a single, particular

comraercial building.

Two factors which have probably combined to dampen the interest of the typical

commercial building owner in solar energy are the following: (1) The size and

complexity of many commercial buildings necessitate the use of specially

designed and engineered solar energy systems that typically cost substantially

more to purchase and install than most systems for residential use. (2) Considerable uncertainty usually exists concerning the short- and long-term performance of large commercial building systems and the effects of part-loading

on the operation of a conventional backup energy system.

The limited experience with solar energy systems for commercial buildings has

meant less documentation of the thermal performance of these systems than for

residential systems. Moreover, the great diversity of commercial buildings by

type and energy requirement has made it much more difficult to develop generalized profiles of cost and performance than has been possible for residential

buildings.

Numbers in brackets designate references listed at the end of the paper.

(References 1 through 8 were cited in the Preface.)

^ For examples of economic studies of solar energy systems for residential

buildings, see [13-18].

2 For examples of economic studies of solar energy systems for commercial

buildings, see [19,20].

1
Another reason for the greater emphasis on residential use of solar energy is

the difference in the Federal tax treatment of residential and commercial energy

operating costs. To a commercial building ovraer, these costs are deductible as

a business expense, but they are not deductible to the ovmer/occupant of a house. Hence, a dollar of fuel cost reduction constitutes a dollar of savings

to a homeowner, but only about fifty cents of after-tax savings to many commercial building owners.

Despite the limited activity in this area, our national energy goal of reducing

reliance on foreign oil, together with the spiralling costs of operating com- mercial buildings, make it important to examine the economics of solar energy

systems for commercial buildings. To make well-informed Investment decisions,

to estimate the potential of solar energy in meeting the nation's energy needs,

and to formulate energy policy, it is important to know if solar energy systems

for commercial buildings are cost effective under existing market conditions,

and, if not, under what conditions they might be cost effective.

1.2 PURPOSE AND SCOPE

This report is intended to broaden understanding of the economics of solar

energy for commercial buildings. The evaluation model that is developed provides a tool for addressing many economic questions facing investors and policy

makers. The compilation of data, discussion of assumptions, and applications

of the model in case studies serve as guides for others who wish to make similar studies. The results and conclusions of the case studies provide indications of the current status and future potential of solar economic feasibility

for an Important share of the commercial building market—office buildings and

retail stores of a design representative of much of the existing building

stock.

1 . 3 APPROACH

A comprehensive economic optimization model employing a life-cycle costing

approach is developed as the tool of analysis. This model allows the determination of the least-cost combination of solar energy and nonrenewable energy

necessary to meet a commercial building's energy requirements, the economic

performance of the optimally sized solar energy system, the minimum conditions

for system cost effectiveness, and the sensitivity of the results to values

of key parameters.

1 At the time this study was undertaken, there were no other comprehensive

economic optimization models available for the analysis of commercially

applied solar energy systems, and no detailed assessment and comparison of

the economic analysis models contained in the larger computer simulation

programs had been made. Now a number of the computer programs for analyzing

the thermal performance of entire building energy systems or of solar energy

systems alone contain relatively comprehensive subroutines for carrying out

an economic analysis. These include SOLCOST, F-Chart, BLAST, and DOE-2.

However, a recent evaluation of the economic models of these programs showed

that none of them provided all of the features desired for performing this

study. (See Ref. 54, p. 69 for a list of the economic evaluation features

not provided by these other models.)

2
Case studies of economic feasibility are performed by applying the model to

four types of solar-equipped commercial buildings—a new and an existing office

building and a new and an existing retail store—representative of a large

share of the commercial building inventory. Thirteen city locations accounting

for major variations in U.S. solar radiation and temperature characteristics

are selected as the basis for the regional case studies. The envelopes and

mechanical systems of these hypothetical, representative buildings are characterized in detail, and a comprehensive building energy load analysis program is

used to estimate their energy requirements in each of the 13 different cities.

Solar energy systems are characterized for the four types of buildings and their

cost functions are estimated. The solar load ratio performance model is used to

estimate the thermal contribution of the solar energy systems.

Data and assumptions needed to use the model are compiled for the selected

buildings, systems, and locations; and factors to consider in setting parametric

values are discussed. Regional variations in the costs of energy and solar

energy equipment, tax rates, and incentives are taken into account.

Economic feasibility results are given for the case studies based on alternative

conditions. Comparisons are made among various locations, buildings, and

systems

1.4 ORGANIZATION

The report is organized into five sections and two appendices. Section 2 contains the economic modeling. Section 3 describes the buildings, building

systems, geographical locations, and environmental data selected for the case

studies. Section 4 specifies the data and assumptions used in the case studies.

It describes the thermal analyses of the buildings and the performance model for

evaluating the solar energy systems. It defines the cost-estimating functions

developed for the case studies, provides tables of data for present and projected future energy prices, and describes the determination of a discount rate,

borrowing rate, inflation rate, investment life, nonfuel operating and maintenance cost, and salvage value for use in applying the economic model in the case

studies. The applicable tax rates and governmental incentives are also provided.

Section 5 describes the case study results. First, the results are given for

base-case data and assumptions. An analysis is provided of the different

factors contributing to cost-effectiveness results and comparisons are made for

the different building types, different conventional fuel alternatives, and

different regions. The scope of the analysis is then extended by relaxing the

base-case conditions and testing the outcomes to other conditions. The impact

on economic feasibility of revised energy price projections, more favorable

depreciation rules, increased incentives, improvements in financing terms, and

changes in the inflation rate are assessed.

Appendix A describes a mathematical programming approach to solar optimization.

Appendix B lists and describes the computer program used to apply the economic

evaluation model in the case studies.

3
2. ECONOMIC MODELINGI

First, for perspective, the major requirements for a comprehensive economic

feasibility model are discussed. Then, the basic framework of the model is

depicted graphically and explained. This is followed by the algebraic formulations. Special features and variations of the basic model are then presented.

Preliminary modeling to Include trade-offs between solar energy and energy con- servation is shown. Potential limitations of the model are discussed. In

appendix A, the approach and results of an exploratory effort to develop mathematical programming and stochastic models as alternative tools for solar

economic optimization are described.

2.1 MODEL REQUIREMENTS

A model with trade-off capability is needed to identify and evaluate the

economically efficient solar energy system. The model should account for the

significant factors affecting the economic performance of each alternative

considered. It should account for the significant components of both shortterm and long-term costs and savings, and should treat taxes and other factors

of concern to commercial building owners. In this study, this requirement is

met by using a comprehensive life-cycle costing model which incorporates the

major parameters affecting the cost effectiveness of commercial building

investments

Modeling capability is needed to evaluate different types of solar energy

systems as compared with different types of nonsolar energy systems. In this

study, this requirement is met by incorporating into the model the necessary

technical performance models and data, the appropriate environmental data, and

the cost models and price data for each type of system to be considered.

As a tool of policy analysis, the model should facilitate the Impact assessment

of alternative policies and events on solar economic feasibility. This study

provides this feature on a microeconomic level by Incorporating into the model

the capability for sensitivity analysis and break-even analysis. It does not

include a market penetration component to estimate aggregate commercial use of

solar energy under different scenarios.

2.2 BASIC FRAMEWORK2

The economic evaluation model calculates the net life-cycle dollar savings or losses attributable to solar energy and determines the combination of solar

1 A list of the symbols used in section 2 is provided at the end of section 2,

2 The description of the model presumes familiarity of the reader with the

fundamentals of benefit-cost and life-cycle cost analysis. For a general

reference to these techniques, see a benefit-cost or engineering economics

textbook [21, 22]. For a simplified guide to these techniques applied to

solar energy and energy conservation decisions, see [23]. For a more detailed treatment of the same topic, see [24, 25],

4
energy and conventional energy that will maximize net savings or minimize net

losses from solar. The model does this by finding the difference (TLCS) between

the life-cycle costs of a nonsolar energy system used alone (TLCC^qq%(,, where

subscript c denotes the nonsolar energy system) and a combined solar/auxiliary

energy system (TLCC(, g, where subscripts c, s denote the combined solar/nonsolarauxiliary energy system). That is,

TLCS = TLCCjqq%(, — TLCCQ^g (1)

where

TLCS = the total life-cycle savings attributable to the solar energy

system;

TLCCioo%c ~ total life-cycle costs of meeting all the energy demands

with the nonsolar energy system alone; and

TLCC(, g = the total life-cycle costs of meeting all the energy demands

with a combined solar/auxiliary energy system.

The model is developed for the case that the nonsolar energy system is a conventional type system such as a gas or oil furnace, electric resistance

system or heat pump. To use the model to evaluate solar energy relative to, or in combination with, other renewable energy systems that are, like solar,

eligible for special tax credits and other incentives, requires modification

of the model to add those special incentives to the calculation of TLCCigo%c»

By applying the model iteratively to successively larger sizes of the solar

energy system being considered, the trade-offs between solar and auxiliary

energy costs can be determined. This enables the identification of the

economically optimal size of the solar energy system.

The life-cycle costs each of the solar energy system, the auxiliary energy

system, and the nonsolar energy system used alone consist of system acquisition

costs, operating and maintenance costs, repair and replacement costs, salvage

or resale values, financing costs, taking into account the time value of money.

Additionally, the auxiliary energy system and the nonsolar energy system entail

fuel costs. If the auxiliary and the nonsolar energy systems are assumed identical in that only fuel costs differ between them, like costs will cancel out

of equation 1 and can be omitted from the evaluation. The costs are adjusted

for the financial effects of taxes and government Incentives at the local,

state, and Federal levels.

Simplifications employed in the model include the following: (1) The size of

the solar energy collector array is assumed to be the key variable in the optimization procedure. (2) The type, capacity, and nonenergy costs of the auxiliary energy system are assumed to be constant regardless of the size of the

solar energy system. (3) Energy loads of the building are Inputs to the model

(l.e., trade-offs between energy conservation and energy supply are not internal to the model). (4) Average meteorological data are used to predict the

economic performance of the solar energy system (i.e., the stochastic nature

5
of meteorological data Is not taken into account). (5) The time of the initial

solar investment is given (i.e., the dynamics of optimal Investment timing are

not incorporated into the model). Each of these assumptions is discussed

briefly below.

The design of a solar energy system involves a large number of variables, such

as storage type and volume, heat exchanger size, collector type, number of

glazings and surface type, collector tilt, and collector size. Using an iterative computer search procedure, such as that used in the case studies, to optimize solar economic performance across all design variables would require massive computations. To reduce computational requirements, a convention widely

used in solar optimization studies is to optimize with respect to a single

design parameter, collector size. Using this approach, the variable costs of

storage volume and other system components are assumed to change in direct

proportion to the change in collector costs as the collector array is increased.

Previous studies have investigated at least six design parameters to determine

their effects on optimal system design and life-cycle costs. Lof and Tybout,

in their 1970 and 1973 articles, "Solar House Heating" [26] and "Cost of House

Heating with Solar Energy" [13], discussed the effects of the following parameters on system performance and cost; heat transfer coefficient of insulation

on the storage tank, heat capacity of the collector, angle of the collector

tilt, number of glass plates, heat storage, and collector area.

Lof and Tybout varied these six parameters to determine the least-cost

combination for providing solar heating in each of eight cities in different

climate regions. Each parameter was varied while the others were kept constant

at levels thought at the time to be near their optimal economic values. Lof

and Tybout found the cost of delivering energy to be more sensitive to collector area than to other variables. Also, they verified that storage tank insulation and collector heat capacity have little impact on system performance and

costs, and that collector tilt relative to latitude has only a small effect

over a wide range of tilt angles and a wide range of latitudes. A constant

relationship of latitude plus 10° to 20° was shown to be an appropriate tilt

assumption in estimating solar performance. In addition, they described the

trade-off between optical and thermal losses with double-glazed versus singleglazed collectors, showing that atmospheric conditions appear to justify

Increased optical loss to reduce thermal loss.

In their examination of the relationship between thermal storage capacity and

cost, Lof and Tybout found only a moderate dependency over a wide range of

storage sizes. However, they found that varying storage size above and below

the optimal size has a greater effect on cost than varying the tilt angle.

Their test results showed the optimal range for water storage to be 49-73

kg/m^ (10-15 Ibs/ft^) of collector in the three cities tested.

This work by Lof and Tybout has been an Important source of evidence for

modeling solar economic performance as a single variable optimization problem.

Since Lof and Tybout found that collector area has a greater effect on the

cost of providing heat than other variables, collector area has commonly been

the only design variable subjected to the optimization routine in determining

6
minimum cost design. A linear relationship between storage and collector area

in solving solar system design problems has usually been assumed. For systems

using water as the storage medium, values in the range of 49-73 kg/m^ (10-15

Ibs/ft^ of collector area have typically been used in sizing storage and in

estimating the costs and performance of the systems.

The results obtained by Klein, Beckman, and Duffie [27] from simulations for

several storage capacities in Madison generally agreed with those of Lof and

Tybout . They used a storage capacity of approximately 73 kg/m^ (15 Ib/ft^)

to design solar systems and to predict solar performance. There is, however,

also some evidence that the relationships among storage volume, collector area,

and costs have not been adequately explored and that more attention to storage

volume/collector area trade-offs may be warranted under certain conditions.

This issue is discussed in appendix A, and mathematical programming is explored

as a tool for optimizing system design with respect to both collector area and

storage volume.

A second simplification to the model is the assumption that the costs of

equipment, repair, and maintenance for the auxiliary system are constant regardless of the use of solar energy in combination with it. The assumption of constant equipment costs for the auxiliary system is consistent with the currently

prevailing practice of providing a full capacity conventional backup system

capable of meeting the entire energy load, in order to compensate for the

stochastic nature of solar energy. If the solar/auxlllary system is then compared with a nonsolar energy system identical to the auxiliary system, the

nonenergy cost elements of both systems cancel out of the model. This simplification is used in the case studies presented in section 5. However, it cannot be used if the purpose is to compare plant capacity trade-offs, in addition

to energy cost trade-offs.

A third simplifying assumption is that the energy loads of the buildings are constant, based on an existing or planned level of energy conservation. That

is, the model lacks the capability of analyzing in an integrated context the

potential trade-offs between supplying energy (via either solar or auxiliary

means) and reducing the energy requirements through investments in energy

conservation.

This limitation is to some extent compensated for in the case studies by (1)

examining the selected buildings for two levels of energy conservation as represented by the new (energy conserving) and the existing (less conserving)

versions of the buildings, and (2) assuming a night-time temperature setback

and a reduction in fan operation at night for all buildings examined.

^ While the versions of the model and computer programs used for the case studies do not provide for energy supply/energy conservation trade-offs, a theoretical framework for making these trade offs was developed in con- junction with this study and is described in section 2.6. Further work in

this area is underway.

7
A fourth simplifying assumption is that stochastic variations in weather data

can be ignored in evaluating solar economic performance. The significance of

this simplification — which produces deterministic rather than probabilistic

estimates of solar performance — depends chiefly on the sensitivity of the

building owner to failure of the solar energy system to meet consistently a given percentage of his or her energy needs and costs. For example, in some circumstances, a building owner's objective may be a solar energy system sized

to meet a minimum fraction of the load with a given level of confidence, rather

than a system sized to minimize building life-cycle costs. (Stochastic

modeling of solar energy is discussed in appendix A.) On the other hand, the

nonstochastic model used here has the advantages of less extensive data and

computational requirements.

A fifth simplifying assumption, that of a static time frame, also offers the

advantage of greater computational ease. Its disadvantage is the failure to

consider the economic consequence of alternative starting times for the solar

investment. For example, a solar project may show a negative cash flow over

part of the life cycle, and a positive cash flow over the remainder, in which

case, it might be economically efficient to defer the project even if net

life-cycle savings are currently positive.

It would be economically efficient to defer the project to avoid the Initial

losses, unless project deferral would raise initial investment costs sufficiently to Increase the initial losses rather than diminish them. (This might

occur, for example, if the choices were between investing earlier and including

solar in a new building at a relatively low installation cost versus waiting

and having the project be a retrofit application to the existing building at a relatively high installation cost.)

The trade-off between solar energy system costs and auxiliary energy costs is

depicted graphically in figure 2.1, where collector area (A) is measured along

the horizontal axis, and life-cycle ($) costs are measured along the vertical

axis. The curve labeled LCCg illustrates the costs of purchasing, installing,

maintaining, and operating the solar energy system. The LCCg curve is assumed

to increase linearly with collector area; that is, each additional unit of collector is assumed to cause a uniform Increase in system cost. The LCCg curve intersects the cost axis above the origin, reflecting the assumption of "fixed

costs" (Fj^), i.e., costs of components of the solar energy system that are minimum prerequisites for the functioning of even the smallest collector area.

The curve labeled LCC^ depicts the costs of the auxiliary energy system. Its

shape reflects the assumption that the amount of reduction in conventional

energy costs associated with a given incremental change in the size of the solar

collector declines in size as the percentage of the load met by solar increases.

This would be particularly descriptive of solar space heating systems or cooling

systems which are subject to changing seasonal demands, and, hence, exhibit

significant excess capacity during parts of the year if they are sized to meet

a large part of the yearly load. The LCC^, curve is shown to "bottom out" — not

declining to zero — because of the assumption that equipment and maintenance

costs of the auxiliary system remain constant over all sizes of the solar

energy system.

8
Figure 2.1 LCC Trade off between Solar Energy and Auxiliary Energy

9
The net cost impact of increasing solar collector area depends on the relative

slopes of the two curves. If, for a given increase in collector area, auxiliary energy costs fall faster than solar energy costs rise, total life-cycle

costs for the combined solar/auxiliary system will fall. As long as this relationship holds, total life-cycle costs of the combined energy system can be

reduced by expanding the size of the solar energy system. But if, as collector

area is increased, solar costs rise faster than auxiliary energy costs fall,

total life-cycle costs will rise.

This effect can be seen by summing the two curves vertically as is shown in

figures 2.2 and 2.3. In figure 2.2, TLCC^. g is a U-shaped curve. It decreases

initially, reaches a minimum value at A*, and thereafter increases. Beyond A*,

solar energy costs rise faster than auxiliary energy costs fall, causing total

life-cycle costs to rise. At point A* the decrease in auxiliary energy costs

just balances the Increase in solar energy costs associated with an additional

unit of collector area.

In figure 2.3, TLCC(, g is an increasing function of collector area. This curve describes the situation where solar energy costs continuously rise faster than

auxiliary energy costs fall as the size of the solar energy system is expanded.

The size of the solar energy system for which life-cycle costs are minimized is zero

The cost effectiveness of the combined system relative to a nonsolar energy

system used alone can be depicted graphically by comparing the TLCC(, g curve with a curve showing the total life-cycle costs of a nonsolar energy system

(TLCCioo%c)* *3iven the assumption that the nonsolar system used alone is the same as the solar auxiliary system, and that the equipment and nonenergy costs

of the auxiliary system are constant with respect to the size of the solar component, the life-cycle cost of the nonsolar energy system can be illustrated by

a horizontal line, TLCC^qq^^* intersecting the vertical axis at the same point

as the LCCj. curve. A life-cycle net savings (or net losses) curve for solar,

TLCS, can be derived by taking the difference between the TLCCc s TLCCioO%c

curves

Figure 2.4 illustrates one of the possible shapes the TLCS curve may take.

TLCS, the difference between TLCC^qq%(, and TLCC^. g, first increases from a negative value to a positive value, reaches a positive maximum at A*, and thereafter

decreases, again becoming negative. In the illustration, the combined solar/

auxiliary energy system is shown conceptually to be cost effective relative to

the nonsolar energy system over a range of solar energy system sizes. Moreover,

it is shown that the size of the solar energy system represented by A* of collector area is economically desirable in that it maximizes TLCS. Any other

system size will provide less than maximum savings over the designated life.

Because it maximizes TLCS, A* is referred to as the "economically optimal size".

Note that the TLCS curve reaches its maximum value at A* coincident with the

minimum value of TLCC^, g. That is, maximizing TLCS is equivalent to minimizing

TLCC(|. g

.

In figure 2.5, TLCC^, g attains the same generic shape as illustrated in figure

2.4, but unlike figure 2.4, it always remains above the TLCC^qq^^ curve. This

10
Figure 2.2 Determining the Cost-Minimizing Size of the Combined Solar/

Auxiliary Energy System: U-shaped TLCCc^g Curve

11
Figure 2.3 Determining the Cost-Minimizing Size of the Combined Solar/

Auxiliary Energy System; Monotonically Rising Curve

12
+

o

Figure 2.4 Determining the Cost-Effective Solar Investment

13
Figure 2.5 Minimizing Losses from a Solar Investment

14
illustrates the case in which TLCS is negative for all system sizes, and the

investment in solar energy does not pay based on the costs included in the

model. (There may be Important, but dif f icult-to-quantify effects omitted

from the life-cycle cost model, whose consideration would affect the investment

decision.) If, in the case depicted, a solar energy system were to be installed

despite its apparent lack of cost effectiveness, the collector area designated

A* is economically optimal in the sense that it minimizes the quantified losses

associated with having the system.

In figure 2.6, which corresponds to figure 2.3, the TLCC^, g curve is

continuously increasing and always remains above the TLCCioO%c curve. As a result, TLCS is again negative for all system sizes, and becomes increasingly

negative as the system becomes larger. In this case, solar energy is uneconomical based on the costs included in the model, and, furthermore, it is not

possible to minimize losses with a solar energy system size larger than zero.^

2.3 ALGEBRAIC MODEL

Using present value analysis, and taking into account capital, maintenance

and operating costs of the solar energy system. State and Federal income tax

deductions for business expenses related to the solar and auxiliary energy

systems, tax credits and other incentives for utilization of solar energy in

commercial buildings, property and sales taxes attributable to ownership of

the solar energy system, salvage value and energy costs of the auxiliary

energy system, equation 1 may be restated as follows;

TLCS =E-[In-V-S-TC-D-W+G+P+M+R+(l-F)*E] (2)

equivalently, or,

TLCS = [E*F] -[In-V-S-TC-D-W+G+P+M+R], (3)

where

E = present dollar value, after income taxes, of energy costs for the nonsolar

energy system to be used in lieu of the combined solar/auxiliary energy

system (since the nonsolar energy system is assumed identical to the

auxiliary system, E = TLCCjqq^^^, and

1 In FEDSOL, a computer code developed by the National Bureau of Standards for

the National Bureau of Standards economic optimization analysis of solar

energy systems for Federal buildings, a minimum size constraint can be imposed

on system size to prevent a solar optimization solution of zero. The Department of Energy's "Solar in Federal Buildings" program, for which the computer

code was prepared, does not require system cost effectiveness, but does require

that system economics be considered [28].

2 If more than one type of fuel is affected by the user of solar, equation 2 or

3 can be expanded to repeat the energy cost calculation for each fuel type

and the results summed.

15
Figure 2.6 An Uneconomical Solution for Solar

Life-Cycle

Costs

$

16
E = P, UPW, Q • (1-t), ( 4 ) d.nj.ej

P(, = the present price (including sales tax) per unit of fuel, stated in terras

of the same physical unit of measure as 0.

UPWd n e ~ ^ uniform present worth factor based on a selected nominal

’ j’ j discount rate, d, for n years (the length of of the study

period), and modified to include j=l to k projected rates of

energy price escalation, occurring in consecutive time intervals, where nj represents the number of years contained con- tained in the interval over which the j escalation rate

occurs. For a constant rate of price escalation (ej) over nj

years

,

/ 1+e^J

^

(1+ej^) (l+d)^l - (l+ej)*^!

j=l U+d ' (d-ej) (l+d)^l

( 5 )

For changing rates of price escalation (ej, 62, and e3) over three intervals

(nj

,

n2 , and n3)

,

UPwl „ ^ = UPWj d.Til,2,3»ei,2,3 d -UPWd,n2, S 2

( 6 ) /l+e,\’^l /l+eo\’^2 * + 1 • 1 ] • upwt

\ 1+d ) \ 1+d / d , n3

,

63

where each UPW. „ o calculated according to the equation for the above

’ j’ j * constant rate case, UPW. ’ d,n,e

(The model allows for three rates of three periods of price projections

provided by DoE’s Energy Information Administration, the source of the energy

price data used to apply the model. If desired, the discount rate could also

be allowed to vary.)

0 = total annual quantity of nonsolar energy purchased, taking into account the

relevant energy load(s) (L) of the building and the efficiency of the

nonsolar energy equipment (y)» i.e., Q = L/y,

(If quantity is to be expressed in sales units of energy such as liters

(gallons) of oil, it is, of course, also necessary to take into account the

energy content (u) per sales unit of energy, i.e., 0 = L/y»u.

t = the building owner’s composite Federal, state and local marginal income

tax rate.

F = the annual fraction of the building's energy requirements met by the solar

energy system, as estimated or predicted by a solar performance model.

17
Using the solar load ratio method, the relationship between collector area (A)

and the annual solar fraction (F) in a given climate and for a given building

load is predicted by the following equations :

12

F = (a«A/L) Z Ij for 0 ^ A»Ij -Lj _< Z^, (7)

j=l

and

12

F = 1 - b/L Z Ij exp (-C‘A«Ij/Lj) for Zj _< A«Ij/Lj _< (8)

j=l

where A = solar collector area; Ij = solar radiation incident on a tilted

collector surface in month j ; Lj = domestic hot water load or combined space

heating and domestic hot water load in month j; a, b, and c are parameters

which depend on the type of collector and the hot water or heating coll design

temperature; and exp is notation for the constant e, (equal approximately to

2.718). Zj and Z^ are the endpoints of the intervals within which the approximations of equations 7 and 8 are correct; their values differ according to

whether the system is for domestic hot water only or combined space heating and

domestic hot water. (Note: the value of F is assumed in the model to remain

constant over the life of the system.)

(The economic model can be modified to operate with other solar performance

models. Alternatively, the economic model can be separated from the solar

performance model and modified to receive precalculated solar performance data

as direct input.)

In = the present value of capital Investment costs associated with designing,

purchasing, and installing a solar energy system, including financing

costs, i.e..

In = (Fx + vA)(a + (l-a)(UCR_i ^

2m)(UPWd ^m) ) , (9)

12

where F^ = fixed cost and v*A = variable cost times collector area for labor

and materials associated with acquiring the solar energy system, including

costs of design, purchase, and installation, i.e.,

Fx + v«A = (Fxl + Vl*A) (1 + r) (1 + tgj^) + (Fxm + vm*A)( 1 + tg^^), (10)

where Fxl and Fxm represent the fixed costs associated with labor and materials,

respectively; v^^ and v^ represent the variable costs proportional to collector

area; r is a regional labor cost adjustment factor (a single, national market

is assumed to exist for materials); tgL and tgf^ represent State sales tax rates

applicable to labor services and materials; and A = collector area.

a = fraction of the system contract cost (Fx + v»A) placed as an initial

down pa5rraent.

18
V =

UCR_i,12Tn = uniforni capital recovery factor for calculating monthly

12 pa3mients of principal and Interest based on an annual loan

rate, i, and a loan life of m years.

UPWd m

= uniform present worth factor based on a discount rate, d, and a loan life of m years.

the present value of resale, net of disposal costs, remaining at the end

of the evaluation period of n years. Expressed as a function of the

initial contract cost.

V = s(Fx + v.A)(SPWd,n). ( 11 )

S =

where s is the nominal resale value as a fraction of the initial contract

cost remaining in n years.

the present value of the decrease in income taxes due to State sales tax

deductions from Federal income tax. (Because this deduction is assumed to accrue close to the time the system is purchased, no discounting is

Included in the formulation.)

TC =

^ ^ ttgL(FxL vl«A)( 1 + r) + tg^(FxM + Vj^»A)].

present value of Federal and State government tax credits.

TC = (Fx + v.A)[(CF)(SPWd j) + (Cg) (UPW^ ^g) ] ,

( 12 )

(13)

D =

where Cp = Federal solar tax credit as a percent of system acquisition

costs; SPWjj^j = single present worth factor for discount rate, d, and the

year, j, in which the credit is taken; Cg = effective State solar tax

credit, net of Federal income tax adjustments (see section 4.6 for further

explanation of the treatment of State tax credits); and UPW^j

g

= the uniform present worth factor for g, the number of years over which the State

credit is taken.

the present value of the decrease in income taxes owed due to capital

depreciation deductions from taxable Income. Depreciation may be

modeled in several different ways. Using the straight-line method.

D = l(Fjj + vA) - S/t](l - ° -1 . UPWj j (14)

(1+z)"

where Z = the depreciation period, n = the length of the study period,

and z = the rate of general price inflation.

Using the declining balance method with switchover to straight-line

depreciation when the annual straight-line amount exceeds the annual

declining balance amount.

19
where y

= the depreciation rate (e.g., 1.75) and d = the discount rate.

(At the time the case studies were performed, the maximum accelerated

rate on new nonresidential real property was 150 percent. For used

nonresidential real property, the straight-line or other IRS approved

method giving "a reasonable allowance" could be used. New residential

rental property could, under specified conditions, be depreciated using

the 200 percent declining balance method, while used residential property

was subject to the straightline method or the 125 percent declining balance method if the building’s useful life was 20 years or more. Business

equipment qualifying as IRS Code Section 1245 property could be depreciated under the 200 percent declining balance method. Coraponente depreciation of real property could be used to reduce the depreciation period,

though not generally to change the depreciation method. The depreciation

provisions were changed by the Economic Recovery Tax Act of 1981. A

revised version of the computer code now in preparation will incorporate

the changes. For a description of past and current depreciation

procedures, see [29, 30].)

W = the present value of the decrease in income taxes due to loan interest

deductions, l.e., for d^i,

W = t(l-a)(Fx + vA) [12UCR i^i2m * UPWd,m+<|)(l-ll UCR 12m)] . 12 i 12

where

/ ^ m ^ \12(m+l)

( i+d)^[( i+d) - (i+yI] ]

(16)

and

(17)

and

m = the life of the loan, 12m = the number of mortgage payments, and

i = the annual loan rate.

G = the present value of capital gains taxes due at the end of the study

period, i.e..

G = s(l - -J ) (F + vA)

(l+z)n X eg

SPW

d ,n (18)

where = combined State and Federal capital gains tax, and other terms are as defined above.

20
(Capital gains taxes are modeled according to provisions in effect prior

to passage of the Economic Recovery Tax Act of 1981 ; for revised rules

regarding capital gains and depreciation recapture tax, see [30].)

P the present value, after income taxes of property tax pa3mients

attributable to the solar energy system. Expressed as a rate applied to

the initial contract cost.

P = n+l-f

\l+dj ] .

[ n(l+d)

]

1+z-zn+nd

(19)

where tp = the effective property tax rate, n = length of study period,

0) = system deterioration or obsolescence rate, i.e..

(1) = n[ 1-s

1+z

1/n

»

where s = nominal resale value as a fraction of initial cost, z = inflation rate), and f = number of years for which property taxes are deferred (a=l, if no deferral).

M = the present value, after income taxes, of yearly recurring costs of

operating, maintaining and repairing the solar energy system (Including

any insurance costs net of reimbursables) . Expressed as a fraction, m,

of the initial contract costs,

M = m(F^ + v«A(UPW*^^^^)(l - t), (20)

where UPW^

n z ” ^ modified uniform present worth factor based on a discount rate, d, n years, and a general price Inflation rate, z.

(Energy costs (electricity) required for operating the motor-driven parts

of the solar energy system are not separately identified from nonfuel

operating and maintenance costs of the system. If adequate estimates of

electricity operating costs are available, this element of cost can be

separately modeled, using the UPW, factor to account for

escalation.)

R = the present value, after income taxes, of all maintenance, repair, and

replacement costs that do not recur annually.

n R = E R.(SPW^ . (21)

j=l

21
where is the cost of maintenance, repair, and replacement in year j; and

SPW(j ^^^2 ~ ^ modified single present worth factor based on discount rate, d,

year j, and a general price inflation rate, z.

2.4 USING THE MODEL TO DETERMINE NET LIFE-CYCLE SAVINGS, OPTIMAL SYSTEM SIZE,

BREAK-EVEN VALUES, YEARS TO PAYBACK, AND SENSITIVITY

2.4.1 Net Life-Cycle Savings and Optimal System Size

The basic model, as represented by equations (2) through (21), calculates the

net present value savings of a solar energy project of a given type and size.

To determine which size of a given type of system is optimal in the sense of

maximizing TLCS (or minimizing net losses if TLCS is negative), the collector

area for a given type of system can be incremented over a wide range of values

and the corresponding TLCS’s calculated. The system size. A*, for which TLCS

is maximum (-TLCS is minimum) can then be identified. This operation is facili

tated by computer search. (The computer code developed to Implement the optimization analysis employs dynamic programming search techniques to reduce

computing time. See appendix B.)

The TLCS results are estimates of the economic feasibility of the solar energy

project built to different scales, stated in terms of present value dollars.

The TIXS for system size A* is an estimate of the long-term cost effectiveness

of the project to the commercial investor if it is built to the "optimal" scale

A positive value for TLCS means that the solar investment is estimated to earn

a return over and above the "opportunity cost of money" as indicated by the

value of the discount rate used in the equations. (The concept of opportunity

cost and the discount rate are discussed briefly in section 4.5. For additional explanation, see [21-25].) A negative value of TLCS indicates that the

solar Investment is estimated to be uneconomical in terms of the direct savings

to the commercial investor.

By inspecting the entire schedule of TLCS values and collector areas (as

represented graphically by the TLCS curve in figures 2.4 through 2.6), the

economic penalty, in terms of decreased net savings or increased net losses,

from installing a system of "nonoptimal" size can be determined. This penalty

can be balanced against the possible benefits of a larger- or smaller-thanoptimal system that are not captured by the life-cycle cost model. For example

expanding the solar energy system beyond the optimal size may decrease dependence on conventional fuel supplies and reduce the vulnerability to fuel shortages and the threat of unscheduled shut-down. On the other hand, constraints

on collector size such as a limited available space for installation or a limited capital budget, may dictate a size smaller than that indicated by the

model

2.4.2 Break-even Values

The basic model set forth above is extended in the computer code (appendix B)

to provide additional analytical capability. One extension is to compute the

break-even value of key parameters whenever the TLCS for the optimally sized

system is negative. The parameters subjected to the break-even analysis are

22
(1) solar investment costs (divided into fixed and variable cost components)

(2) base-year fuel prices, and (3) future fuel price escalation rates. The

break-even analysis identifies the change in the values of these parameters that are necessary to achieve a TLCS of zero, that is, a break-even point for the

solar energy investment whereby the optimally sized solar energy system in com- bination with the auxiliary system will cost the same over the life cycle as the

nonsolar energy system.

This is done for each of the above three cost parameters by changing the value

of the designated parameter in steps and reoptimizing the collector area at each

step until TLCS approaches zero. The break-even value of the parameter and

corresponding solar fraction and collector area are thereby obtained.

Two interesting characteristics of the break-even solutions may be noted:

Given the TLCS formulation in section 2.3, the factor by which fixed and variable investment costs must be reduced in order for the investment to break even

is the inverse of the factor by which the base-year fuel price must be

increased. Secondly, the optimal collector area and the solar fraction will be

larger under the break-even cost conditions than under the initial conditions

upon which a negative TLCS is obtained.

The break-even analysis serves two purposes: (1) It establishes minimum

conditions for an investment in solar to be cost effective, and thereby provides a broader basis for extrapolating study results to other cases and (2) it

provides information that may be useful for formulating solar policy.

2.4.3 Years to Payback

The basic algebraic model is further extended in the computer code (appendix B)

to determine the elapsed time until the investment pays for itself. This payback computation takes into account estimated escalation in energy prices, the

opportunity cost of capital through use of the discount rate, and all other

parameters included in the basis model. It is a "discounted payback" measure,

in contrast to a "simple payback" measure. Payback is, in fact, another application of break-even analysis where time is the parameter evaluated for its

minimum value.

The payback measure is provided because it is popular among business investors

and provides useful information to those who are particularly concerned about

the turnover rate of investment funds. It should be cautioned, however, that

the payback measure is an unreliable and incomplete indicator of an investment's

economic profitability, and an Investment with a longer payback period may be more profitable than an investment with a shorter payback period, depending on comparative net returns after payback is achieved,

2.4.4 Sensitivity Analysis

By using alternative values for the key parameters in the model, it is possible

to evaluate the sensitivity of the solar Investment to variations in those

parameters. This process, called "sensitivity analysis," can be used to identify variables of particular economic significance, to test results for the

23
consequence of estimating errors due to uncertainty regarding data and

assumptions, and to assess the effects of alternative policies and changing

conditions. The computer program developed to implement this model (appendix

B) provides a sensitivity test of TLCS to solar collector size.

2.5 SIMPLIFIED MODEL FOR SOLAR HOT WATER ANALYSIS

2.5.1 A New Concept

A simplified version of the basic economic evaluation model described in

sections 2.2 to 2.4 was developed to facilitate the solar economic analysis for

a range of energy requirements for hot water. The simplified version of the

model incorporates all of the elements specified in section 2.3, but reduces

the search for the economically optimal size of the solar hot water system to a single deterministic equation.

This modified solar hot water model, therefore, makes it possible to identify

the economically optimal solar energy system size without using the iterative

computer search procedure. Moreover, an analysis of the resulting equation

indicates that the solar collector area which maximizes total life-cycle savings

(TLCS) is a linear function of the annual hot water load, and the economically

optimal fraction of the hot water load supplied by solar is constant over a range of loads. This finding suggests that the results of system optimization

based on a given hot water load can be extended to a range of loads. This

approach is used in the case studies to construct "universal economic optimization paths" which generalize, to some extent, the results to similar buildings

with larger and smaller hot water loads. The derivation and use of the paths

are explained further below.

2.5.2 Universal Economic Optimization Paths

A universal economic optimization path is defined as a locus of points

describing the solar collector areas which maximize total life-cycle savings

for various levels of commercial hot water loads. Graphically, the path shows

the economically optimal collector area as a function of the annual hot water

load. Because geographical location affects both the thermal and economic

performance of a solar energy system, a different path exists for different

geographical locations. In addition, for a given climate region, an entire

family of paths can be derived based on different economic assumptions. The

results constitute sensitivity analyses. Families of curves can be used to

conduct quantitative assessments of the impact of alternative governmental

actions on the economic feasibility of solar energy.

Universal economic optimization paths are developed by combining the equation

from the solar hot water performance model with the total life-cycle savings

equation, optimizing for collector area, and substituting the optimal collector

area back into the system performance equation to find the optimal solar fraction. The model is dependent upon the use of annual values of domestic hot

water loads and solar radiation, in place of monthly values, in the solar performance equation. This is a suitable approach when monthly thermal loads are equal or show little variation. Water supply temperatures vary over a much

24
narrower range than air temperatures. Los Alamos Scientific Laboratory (LASL)

has confirmed that annual data can be used for analyzing systems for domestic

hot water, without substantial loss of accuracy [40].

Using annual values for hot water loads and solar radiation, TLCS may be

expressed as follows

TLCS = X*L(l-b exp(-c*A»l/L))-(Fx+v»A)Y - R, (22)

where

X = (1-t)

Y

UPWd.nj.ej (23)

TC-D-W+G+M

+ vA

(24)

and all terms are as defined in section 2.3, except that, for simplicity, S,

tax savings due to sales tax deductions, is redefined as Sg, with the distinction dropped between the sales tax due on labor versus materials; i.e., Sg =

^ * *^s(^x v*A), where tg is the State sales tax rate; and + v»A is assumed

to aggregate labor and materials costs, preadjusted for the regional labor

adjustment factor and sales tax.

Note that only the nonlinear range of the solar load ratio function (i.e., A

0.8 L/I) is relevant in the TLCS maximization problem. If solar energy is cost

effective, it will not pay to stop the collector area. A, short of the nonlinear

range; that is, the maximum point on the TLCS curve will always occur in the

nonlinear range or at A=0.

The economically optimal collector area (A*) can be determined by differentiating

equation 22 with respect to A and setting the result equal to zero. The solution

for A* can then be expressed as follows

A* = L_ In (bj c»I*X) (25)

c»I v»Y

The economically optimal annual fraction (F*) of hot water load met by solar is

obtained by substituting A* above into the solar performance model as represented

by the following equation;

F = 1 - b exp (-c*A»I/L), (26)

12

where L is the annual hot water load, i.e., L = S L^; and I is the annual

j=l

12

incident solar radiation on a tilted surface, i.e., I = E I-j

j = l

25
Hence

F* = 1 - b exp (-c«A**l/L). (27)

By applying the economic concept of elasticity to the formulations for optimal

collector area (eq. 25) and optimal solar fraction (eq. 27), one can observe

the important relationships among the optimal values of these variables that

enable the optimization results for one load to be generalized to different

loads

The elasticity of A* with respect to L (e^* defined as the percentage

change in A* resulting from a given percentage change in L, expressed as a ratio to the percentage change in L.

Algebraically,

e A*»L

= % A in A* = ^

% A in L A* 9L

’ (28)

where 3A*/9L represents the first partial derivative of A* with respect to L.

From equations 25 and 28, we find that L

“ Hence, for any area of

equivalent Incidence of solar radiation (i.e., holding I constant), a given

percentage change in the annual hot water load (L) produces an equal percentage

change in the economically optimal collector area (A*), all other factors being

equal

The ratio A*«l/L remains unchanged for variations in L, and, because A*»l/L is

constant for different hot water loads, F* is Independent of L. As illustrated

by again using the elasticity concept,

e = ^ A In F* = L = 0 . (29)

.1- % A In L F* 3L

Since a given percentage change in the load causes no change in solar fraction,

the economically optimal fraction is unique and independent of the hot water

load for a given set of economic parameters and radiation values.^

Economic optimization paths showing the optimal collector size (A*) for

different hot-water loads (L) can be established on the basis of the results

presented above. The optimization paths are applicable only to solar hot water

systems in geographical locations which have the same amount of annual solar

radiation and to which the same technical and economic assumptions would apply.

Figure 2.7 illustrates a universal economic optimization path. The optimal

collector area (A*) is measured on the horizontal axis, the annual hot-water

load (L), on the vertical axis in the upper part of the figure. As was demonstrated above. A* is a linear function of L, and F* is independent of

L; therefore, F* is constant along the optimization path.

^ F* is Independent only of those changes in the hot water load that alter

neither the assumed cost estimating equations nor the technical performance

relationships

26
TLCS is measured on the vertical axis of the lower part of figure 2.7. The curve for total life-cycle savings (TLCS) increases with increasing loads (L)

and the corresponding increasing optimal collector areas (A*).

Larger hot water loads imply that larger absolute savings can he realized by

installing a solar system. As the load and corresponding optimal collector

area become smaller, savings decline and at some critical level (L^ in

fig. 2.7), TLCS reaches zero. Below the critical level, TLCS is negative.

The critical value of L, L^,, is determined as follows:

Lc + R

v«Y \ _ V »Y c^T-X"^ cl

ln( b «c »I «X \ V »Y

(30)

where all parameters are as previously defined.

Equation (30) indicates that the value of L^, in figure 2.7 depends directly upon

the fixed cost components of the solar energy system, other things being equal.

As fixed costs decline, Lq declines.

Several immediately practical uses of this universal path methodology are suggested. First, it could be an extremely valuable tool for developing

regional guidance to the commercial building community on the optimal sizing

and cost effectiveness of solar hot water systems. Second, the methodology

could be used to conduct a comprehensive analysis of the sensitivity of optimal

design with respect to selected parameters. Third, the methodology could be a useful tool for facilitating the analysis and development of efficient Federal

and State solar incentive programs.^

2.6 INTEGRATED OPTIMIZATION OF SOLAR ENERGY AND ENERGY CONSERVATION: A

THEORETICAL FRAMEWORK

This section describes an extension of the economic optimization framework to

incorporate the analysis of trade-offs between solar energy and energy conservation in buildings. It draws upon the findings of the preceding section

regarding the unitary elasticity of A* with respect to L and the independence

of F* with respect to L, to simplify the Integrated optimization procedure.

2.6.1 Economic Foundations

The primary economic objective in designing or retrofitting the heating/cooling

components of a building is to provide at the lowest possible life-cycle cost

a desired level of thermal comfort, comprising temperature, humidity, and

other related attributes (and taking into account related factors such as lighting). The minimum cost search considers the technical substitution

^ A more thorough treatment of the model and examples of its use for

sensitivity analysis and policy analysis is provided by Sav in "Universal

Economic Optimization Paths for Solar Heat Water Systems in Commercial

Buildings," Energy [31].

27
Figure 2.7 Universal Economic Optimization Path: Generic Form

28
between inputs and the relative prices of inputs. Holding all inputs constant

except those concerning the building envelope and the energy systera(s), tradeoffs exist between (1) energy conservation alternatives which improve the

thermal integrity of the building envelope, and therefore reduce the building's

thermal load, and (2) energy system alternatives, e.g., nonsolar and solar

energy systems, which satisfy given thermal loads. Economic optimality is

attained when the marginal dollar expenditure for each input per marginal unit

of thermal comfort obtained from that input is equal for all inputs. This

optimality condition may be stated algebraically as follows

MCLR

MPLR

(31)

where MC represents marginal cost, MP marginal product, and subscripts c,

s, and LR represent, respectively, nonsolar energy inputs, solar energy

Inputs, and load reduction (energy conservation) inputs.

In other words, the search is for a given combination of energy conservation

inputs (load reduction options) and energy system Inputs which will minimize

the total life-cycle costs of achieving a desired level of thermal comfort (or

maximize the net life-cycle savings from the total investment).

Holding all factors constant except thermal load (L) as determined by the

thermal integrity of the building envelope, and noting that alternative energy

systems can satisfy L, thermal comfort, k, can be expressed as follows:

k = k (qc, qg, q^R), (32)

where q^ represents a quantity of nonsolar energy input, qg a quantity

of solar energy input, qL^ a quantity of load reduction obtained by

upgrading the thermal integrity of the building envelope, and all inputs

are expressed in a common unit measure such as the Joule (Btu).

The cost (C) of achieving various levels of k can be described by a family of

isocost curves:

C = Pc(qc) • ^c + Ps^^s) • ^s + Plr(<?Lr) * ^^3)

where Pi(q^), subscripted c, s, or LR, is an expression which gives the

price of the l^h input as a function of the level of the i^h input used.

Once a target level of thermal comfort (say k) is determined, the economic

objective is to minimize C subject to the constraint of k = k. The minimum

cost combination of inputs q^, qg, and q^]:^ can be obtained by using the

technique of the Lagrange multiplier:

rain C = Pc(qc) * ^c + Ps(qs) * ^s + PLr(<1LR> * ^LR " “ k(qc,qs>qLR) 1 (34)

where X is the Lagrange multiplier and is interpreted as the marginal cost

of producing thermal comfort (k).

29
From equation (34), the optimality condition is found to be the following:

Pc^_c

k''

+ P

"a,"

k'

PlR PlR^LR

LR

( 35 )

The numerator of each ratio in equation 35 is the marginal cost (MC) of the

respective input. The denominator is the marginal product (NP) of the input

in producing thermal comfort (k). Hence, this is consistent with the

optimality condition stated at the outset in equation 31.

Since initially all Inputs except thermal load were held constant, it follows

that marginal products are all expressed in a common unit, e.g.. Joule (Btu).

The ratios of the marginal products are therefore unity and the optimality

condition can be restated in the following familiar form;

MCc = MCg = MClr (36)

Economic optimality is attained when the marginal costs of all inputs are equal.

2.6.2 Optimization Model

When the optimality condition stated in equation 36 is achieved, total

life-cycle costs of maintaining a given level of thermal comfort are minimized

or alternatively net life-cycle savings from the investment in thermal comfort

are maximized. These alternative criteria provide the Impetus for an empirically workable optimization model and are discussed below. The first criterion

is achieved by minimizing the following equation;

TLCCLj = LCCLj + LCCLj + LCCLj (37)

c s LR

where TLCC represents total life-cycle costs, and LCC subscripted c, s,

and LR represents the life-cycle cost of nonsolar energy inputs, solar

energy inputs, and load reduction inputs, respectively. The superscript

Lj represents alternative loads.

Equation 37 describes a family of TLCC curves, each corresponding to a different load, Lj . The empirical form of equation 37 differs depending on the

nature of economic and technical trade-offs among the size (capacity) of the

solar energy system, the size of the nonsolar energy system, and the level of

energy conservation. If, for simplicity, it is assumed that there are no capacity reductions in the nonsolar energy system as the size of the solar energy

system is expanded, LCC^, consists only of conventional fuel costs. bCC^j^ presents the most difficulty for empirical investigation. It is building-design

design specific and depends upon the initial design load against which the

costs of load reductions are to be evaluated.

The alternative to the TLCC minimization criterion, the TLCS maximization

criterion, is achieved by maximizing the value of the following equation;

30
TLCSLj = LCCLo _ (LCcLj - LCC^j - LCC^j ) ( 38 )

,

c c s LR

where LCC^® represents the sum of the initial space heating (L^ ) and hot

water (Ly) loads from which load reduction options are evaluated. (Note that

Lj = Lyj + Ly, where Ly represents a fixed hot water load which is unaffected

by energy conservative design in the building envelope.)

Maximization of equation 38 is equivalent to the minimization of equation 37.

Both are consistent with the optimality condition of equating the marginal

costs of alternative inputs given by equation 36.

Figures 2.8 through 2.10 graphically depict the integrated optimization

procedure. Figure 2.8 shows a generic LCC^r function. The horizontal axis measures the fraction (3) by which the initial space heating load (L^^q) is

reduced. LCC^r for various load reductions are measured along the vertical

axis. The lower scale in Figure 2.8 shows the combined space heating and hot

water load (Lj = + Ly) corresponding to each 3 .

Figure 2.9 shows in the upper portion a family of TLCC curves. In the middle

portion are TLCS curves; and in the lower portion, solar performance curves for

alternative loads (Lj’s). The figure also shows, a life-cycle cost curve

(LCC^°) for a nonsolar energy system providing all of the energy to meet the

initial load of Lq, used j^o derive the TLSC curves. For any given load, the

optimal collector area (A-), which is Indicated on the horizontal axis, and

the fraction of load supplied by solar (F.), which is indicated on the lower

segment of the vertical axis, occurs at the minimum point on the TLCC^J curve

or the corresponding maximum point og the TLCS^J curve. The optimal amount of

nonsolar energy input is simply (l-Fj), The optimal combination of all inputs,

including load reduction options, occurs at the minimum point on the lowest

TLCC^J curve among the family of TLCC^ curves or the corresponding maximum

point on the highest TLCS^J curve. For example, the optimal combination of all

inputs in figure 2.9 occurs at the minimum point on the TLCC^^ curve, which is

th^ same as the maximum point gn the TLCS^^ curve. The optimum is to be

A, of solar collector area, Fj fraction of th^^load by solar, 1-F fraction

or the load met by nonsolar energy input, and 3j^ of load reduction,

corresponding to a combined thermal load of Lj = L|jj + Ly.

As reported in gection 2.5, preliminary investigations indicate that the

^ elasticity of A with respect to Lj (defined as the percentage change in A per

percentage change in Lj ) may be approximately unity under certain conditions.

Since solar performance as measured^by the solar fraction (F) is a function of

the ratio of A to L- , changes in F would be nearly invariant with respect to

changes in thermal loads (Lj)* If the elasticity of F with respect to Lj were approximately zero, there would be little or no movement in Fj for different

Lj's in the southeast quadrant of figure 2.9.

Based on these concepts, the optimization procedure can be considerably

compressed according to the graph shown in figure 2.10. Various combined space

heating and hot water loads (L-;) are shown on the upper portion of the vertical

axis; optimal collector areas TA*) corresponding to each Lj are shown on the

31
Figure 2.8 Integrated Optimization Procedure; Load Reductions

32
Figure 2.9 Integrated Optimization Procedure: Solving for

the Combination of Conservation, Solar, and

Nonsolar Inputs to Thermal Comfort

TLCC-1

33
Figure 2.10 Integrated Optimization Procedure; Compressed Model

34
horizontal axis. Lq is the initial load. The constant F** path relates Lj’s to

A*'s. The F** path is constant assuming that F** is uniquely determined and

independent of Lj . In the lower part of the figure, the locus of points corre- sponding to the maximum values of the family of TLCSy curves is plotted against

the optimal collector areas (A*). The "global optimum" in figure 2.10 occurs at

the maximum point on this curve. Thus, figure 2.10 indicates all the relevant

optimization Information in an extremely compressed form: A**, L**, F**, and,

by derivation, 1-F**.l

These findings regarding the elasticity of A* with respect to L. and the

relationship between F* and Lj are based on the analysis of solar service

water heating. They can be extended to the analysis of space heating only if

the annual solar fraction can be expressed in a single equation as a function

of collector area and load. The problem of jointly optimizing the building

envelope, the mechanical system, and the solar energy system is a topic of

current research by NBS Applied Economics Group, Center for Applied

Mathematics.

35
A

A**

a

a

6

B**

C

Cp

Cs

D

E

F

FxL

^xM

F*

G

Table 2.1 Definitions of Symbols Used in Section 2

= solar collector surface area

= economically optimal size of collector area, in terms of maximizing

TLCX, based on a given energy load j

= "global" optimal solar collector area, in terms of maximizing TLCS,

taking into account energy conservation investments

= number of years by which property taxes are deferred

= fraction of the solar system contract as (F^ + vA) placed as an initial downpayment

= fraction by which the initial space heating load (L^o) is reduced by

energy conservation

= the "global" optimal fraction of space heating load reduction taking

into account solar and nonsolar energy

= cost of achieving various levels of thermal comfort

= Federal solar tax credit as a percent of system acquisition costs

= effective state solar tax credit net of Federal income tax adjustment

= the present value of the decrease in income taxes owed due to capital

depreciation deductions from taxable income

= present dollar value, after income taxes, of energy costs for the

nonsolar energy system

= the annual solar fraction of a building’s energy requirements met by a solar energy system

= fixed cost for labor and materials associated with acquiring the solar

energy system, including cost of design, purchase, and installation

= fixed labor cost of acquiring the solar energy system

= fixed materials cost of acquiring the solar energy system

= the optimal solar fraction based on a given energy load

= the "global" optimal solar fraction taking into account energy

conservation investments yielding alternative energy loads denoted

by subscript j

= present value of capital gains taxes due at the end of the study

period

36
Table 2.1 - (continued)

Y

= efficiency coefficient of nonsolar energy equipment

Ij = solar radiation incident on a tilted collector surface in month j

In = the present value of capital investment costs associated with designing,

purchasing, and installing a solar energy system, including financing

costs

k = level of thermal comfort

k = target level of thermal comfort

X = Lagrange multiplier

L = annual hot water load

L(, = critical hot water load below which the solar energy system is not

cost effective

Lj = space heating and domestic hot water load associated with a level of

energy conservation denoted by the subscript j

Lq = initial domestic hot water load or combined Initial space heating and

domestic hot water load prior to energy conservation

Ly = hot water load

= fixed hot water load unaffected by energy conservation design in the

building envelope

Ljj = space heating load associated with a level of energy conservation

J denoted by the subscript j

Ly = initial space heating load prior to energy conservation

o

LCC(, = life-cycle costs of the auxiliary energy system

LCCg = life-cycle costs of purchasing, installing, maintaining and operating

the solar energy system (but not Including energy costs for the

auxiliary energy system)

LCC(. = life-cycle costs of nonsolar energy Inputs given load j

LCCg = life-cycle costs of solar energy inputs given load j

37
Table 2.1 (continued)

LCClr

Z

M

MCc

mclr

MCg

MPc

MPg

0)

p

Pc

PLR

Ps

Q

*?c

«1LR

'Is

= life-cycle costs of load reduction inputs given load j

= length of capital depreciation period

= the present value, after income taxes, of yearly recurring costs of

operating, maintaining, and repairing the solar energy system (including

any insurance costs net of relmbursables and not including auxiliary

energy costs)

= marginal cost of the nonsolar energy system

= marginal cost of load reduction

= marginal

= marginal

= marginal

= marginal

cost of solar energy system

product of the nonsolar energy system

product of load reduction

product of the solar energy system

= the present value of yearly recurring operation, maintenance and repair

costs of the solar energy system, expressed as a fraction of Initial

contract costs

= system deterioration or obsolescence rate

= the present value, after income taxes, of property tax pa3nnents

attributable to the solar energy system

= the present price (including sales tax) per unit of fuel

= price of unit of input of nonsolar energy

= price of a unit of load reduction input

= price of a unit of input of solar energy

= total annual quantity of fuel purchased, taking into account the

relevant energy load of the building and the efficiency of the

nonsolar energy equipment

= quantity of nonsolar energy input

= quantity of load reduction input

= quantity of solar energy input

38
Table 2.1 (continued)

R

r

S

SPWdj

SPWd,n

s

TC

TLCCq

'I^CCioo%c

L,

TLCC 3

TLCC^°

TLCS

t

teg

tSL

tSM

UPWd,g

= the present value, after income taxes, of all maintenance, repair and

replacement costs that do not recur annually

= the cost of maintenance, repair and replacement in year j

= regional labor cost adjustment factor

= the present value of the decrease in income taxes due to state sales

tax deductions from Federal income tax distinguishing between tax on labor and tax on materials)

= present value of tax savings due to sales tax deductions (not

distinguishing between sales tax on labor and sales tax on materials

= single present worth factor based on discount rate, d and the year,

j, and a general price inflation rate, z

= single present worth factor based on discount rate, d, and n years

= nominal resale value as a fraction of the Initial contract cost

= present value of Federal and State government tax credits

= total life-cycle cost associated with a combined solar/auxiliary

energy system

= total life-cycle costs of meeting all energy demands with a nonsolar

energy system alone

= total life-cycle costs for energy-related building components,

given load j

= total life-cycle costs of a nonsolar energy system providing all of

the energy to meet the initial load, Lq

= life-cycle net savings (or, if negative, net losses) for solar

= the building owner’s composite Federal and state marginal income tax

rate

= effective property tax rate

= combined state & Federal capital gains tax

= state sales tax rate applicable to labor services

= state sales tax rate applicable to materials

= uniform present worth factor based on discount rate, d, and the

number of years, g, over which the state tax credit is taken.

39
Table 2.1 (continued)

UPW(j j, = uniform present worth factor based on discount rate, d, and the

length, of the capital depreciation period

IQ = uniform present worth factor based on a discount rate, d, and a loan life of m years

UPW^,^ ^ = uniform present worth factor based on discount rate, d, for n

J’ j years (the length of the study period) and modified to include j=l

to k projected rates of energy price escalation, occurring in

consecutive time intervals, where nj represents the number of years

contained in the interval over which the j escalation rate occurs

UPWd,n z “ ^ modified uniform present worth factor based on a discount rate d, n years, and a general price inflation rate, z

u = energy content per sales unit of energy

V = the present value of resale, net of disposal costs, remaining at

the end of the evaluation period

V = variable cost proportional to collector area

VL

VM

W

y

= variable labor proportional to collector area

= variable materials cost proportional to collector area

= the present value of the decrease in income taxes due to loan

interest deductions

= capital depreciation rate

40
3. CASES SELECTED FOR STUDY: BUILDINGS, BUILDING SYSTEMS, AND GEOGRAPHICAL

LOCATIONS

3.1 BUILDINGS

Case studies were performed for several selected types of commercial buildings.

The population of commercial buildings from which the selection was made is

illustrated in table 3.1. This list, taken from a building construction

valuation manual, outlines seven major categories composed of 23 subcategories

of commercial buildings [34]

Office buildings and retail stores were selected for case study because they are major users of energy [35]. Their selection is also supported by a com- parison of building types by floor space, as shown in table 3.2. Of the

total floor space of the nonresidential buildings shown, offices and retail/

wholesale buildings account for the largest share—about one third.

The case study buildings are hypothetical, but selected to be representative of

their class. They are a three-story office building and a single-story retail

store—each with two alternative envelope designs. ^ For each type of building,

one envelope design, designated "conventional," is intended to represent much

of the existing inventory that has been constructed with little attention to

energy conservation. The other design for each building type, designated

"energy conserving," represents conventional-type buildings constructed in

accordance with ASHRAE 90-75 [39]. The conventional designs were used to assess

the feasibility of retrofitting solar energy systems to existing office buildings

and retail stores; the energy conserving designs were used to assess the use of

solar energy in new office buildings and retail stores. It should be noted

that none of the building designs were specified to take optimal advantage of

solar energy nor of daylighting.

3.1.1 Office Buildings

The three-story office building is 30 m (100 ft) long by 30 m (100 ft) wide,

with metal curtain-wall construction. The new building design has double-glazed

windows and insulated exterior wall and roof. The existing building design has

single-glazed windows and uninsulated exterior walls and roof. There are 15

thermal zones within the building , modeled in the thermal analysis as 10 zones

,

with the first and second floor zones combined vertically. Both the new and

the existing versions of the office building were assumed to have an average

occupancy during peak operating hours of 300 people, with occupancy extending

1 In establishing the scope of this study, it was jointly agreed by the U.S.

Department of Energy (DoE) and the U.S. Department of Housing and Urban

Development (HUD) that this study, under the sponsorship of DoE, would not

consider apartment buildings.

2 Honeywell, Inc., Energy Resources Center, a consulting engineering firm,

provided descriptions of the buildings and system specifications under

contract to NBS [36]

41
Table 3.1 Types of Commercial Buildings^

MAJOR CATEGORIES SUBCATEGORIES

Mercantile Banks, Drive-ins, Laundries, Office Building,

Stores, Supermarkets

Medical Clinics, Hospitals, Nurses’ Residences

Entertainment Bowling Alleys, Clubs, Theaters

Garages Parking, Sales & Service

Service Stations Stations

Residential Apartments , Hotels , Motels

Miscellaneous Grain Elevators, Lumber Storage, Trucking

Terminals , Warehouses

^ This classification excludes "industrial buildings," such as mills,

factories, and plants; "institutional buildings," such as dormitories,

gymnasiums, and libraries; and "agricultural structures."

Source: The American Appraisal Company, Boeckh Building Valuation Manual , Vol . II, Commercial [34].

42
Table

3.2

Estimated

Floor

Space

by

Type

of

Nonresidential

Building'

(Billions

of

ft^)*^

Total

Noiiresidential'- (13) o

CMO

CM

CO

C3N <r

CM

22,246 1

ri. ON

CM

CM

23,474 1124,236 1125,035 1125,909 1126,853 1 AO

00

CM 1

28,325<i !

COdo

d>d

O 00 CM cn NO o nO

A

A

cO ^ in 00

A

A o A CM 00

iM CMd r>^ oo ON o o <—* CM

AA

rM

d)

W CM CM CM CM CMAA

AA

AA

O

CO s

COdo m .-H <}• NO <f C7N nO CM 00

A

•M /-V 00 CM

A

O o CM

A

A r*«. 00

O

bO

—

•-H CM CM CM

AA

AAA

A <r

i-H d> Qd

1-^cO

J-) ^ CO CM S> -cr 00

A

OO CMO

•H o •H Si ri. •<r

OO nO

CL ^ sT <rAA NO r-«. ri* 00 ON ON o

CO

w

O CM do

•M

O ON 00 ON

A CM <r CM o <r 00

r*-* ON CM

A 00

O

A NO o A NO

d c^ 00 00 ON ON ON <r> o o r>H CM d wCL •-H •—

f-Htodo ON on ON

AA nO ON ON NO >3* •M -sT lA vO A A 00 CM A A NO NO

iJ CO

O CM NO 00 CJN CM

A ri

A

to N-' o m lA

AAAA NO NO nO nO NO

d

t3U

rM

d> M

d

O CO CO to r-«.O ON 00 r«-

AAA s ^ r*-. C^ A NO NO r*» O CM nO

r«- CM CM

A

AAA

A •o <f

<u uod

d)

COd CO vO

A

O <r ON CM MT <r o ^ 00 lA CM O o 00 NO <X) CM ON d d CO ON A NO r-* r««. 00 ON CM CM

<U

wLi CM CM CM

CO 3

(UbO

to ^ uoA NO •<rA ON 00

Lim r-.

O

A NO o A

A (X)OO

cOw CO

AAAA nO NO nO

O

fO

•<r lAA o 'cr nO CM MT pi. s>

CO ^ 00A <r o A CJN

O r*i pi CM

iJ CM o CMA 00 CM

A 00A r*i

O

w

H 11 Si vO vO NO r*-. r*-. p*- 00 ON ON ON

>odJ t f-H 1—i cO CO 00 vO NO 00

A r*i 00

i-i CO /-s s> CM ON r>. ON 00 r-.

AA 00 -3-

CO 0) cn CO nO OO o CM

A 00

n CM

M ^ w d) o CO CO

AAA -d- *0AA cii d3

d)O r>v r-'.

A CM <f On

O CO

O

•M X-N lA lA

m NOA nO 00 00

CM CM 00 ON

O f—

A •«d- nO r-. ON o CM s_^ o CM CM A A A A A A A <r

Li lA sO <X) ON

O CMA <3-

A

CO ''-V vOd Si nO nO fi. r-«. P-1 pi Pi

d) -H ON ON ON ON CJN CJN CjN ON On CJN

>-l

W -H c

Q> CO

to si H u

CO

bO 0)u

0)O

u

B

iJ CO a (u

0) TJ3

C3

a o

d)

J.J to a\ a

(OCNJ

tO 1^

CO ^ CDbO a c O 1-1 U t3

y-C r-<

0) •-< U cO

cO 1^

CU iJ

CO

d

<i) Li -d OO CO rH 0) CM Ud

-H

O

0)d

rH d)

(U o

4J to o a

.d CO d L> •H oo

0)

00 y-i cO

0) ^

Li to

0)

d; CO d CO 4J d)

Li

u o

to

d

d iH

-d Q>

d>d

a <>-> d u

COd

CO CM

CO

CO iH 5 * d)

Li

d)d

M bO

tOa y-<

CO

d> tobO

d) d> M Li

CO bO

bO

CO to • d> ^ M J-> 00 d d

bO <U ^ •M O

IM d) ON

Li r>.

d) I CO d 00 H S

43
over 10 hours per day, five days per week. Table 3.3 gives further description

of the construction characteristics of the new and existing office buildings.

Figure 3.1 shows the office building layout and the exterior faces. Figure 3.2

shows the office building load schedules.

3.1.2 Retail Stores

The one-story retail store is a rectangular building, 29 m (95 ft) long by 16 ra (53 ft) wide, of metal curtain-wall construction. It is assumed to have an average occupancy extending over 12 hours per day, 6 days per week.

The new version of the store was assumed to be less energy conserving than the new office building, though representative of new stores constructed in accor- dance with ASHRAE 90-75. (R-values of exterior walls are set at 1.14 m2*K/W

(6.45) in the new retail store, as compared with 1.84 m^*K/W (10.45) in the new office building.) The existing retail store has an envelope with thermal

characteristics like those of the existing office building. Table 3.4 gives

further description of the construction characteristics of the new and existing

retail stores. Figure 3.3 shows the layout and outside faces. Figure 3.4

shows the building load schedules. The retail store was assumed to have three

thermal zones (see figure 3.3).

3.2 AIR HANDLING SYSTEMS

The energy requirements of commercial buildings are considerably affected by the

type of heating and cooling systems used. Hence, it is important in estimating

the energy requirements of a building to specify the characteristics of the

mechanical systems.

The importance of the mechanical system on building energy use is demonstrated

by figure 3.5, which shows the simulated energy requirements of a two-story

office building fitted with different mechanical systems. For each of four

different cities in the U.S., runs were repeated for the building with from six

to eight different mechanical heating and cooling systems. Figure 3.5 shows

the building's energy consumption in each location for different mechanical

systems. The amount of energy required is shown to vary by a factor as high as about seven, depending upon the design and operation of the mechanical system

[40].

The study referenced above looked at eight different systems; there are, however,

approximately 25 different types of heating and air conditioning systems—with

additional subclassifications within many of the types —identified in the ASHRAE

Handbook and Product Directory [41]. Many different types of systems are in use

in existing buildings, and different types continue to be installed in new

buildings. Like the diversity of building types, the diversity of mechanical

systems in these buildings makes it difficult to generalize case study results

to other buildings.

44
Table 3,3 Office Building Construction Characteristics

Surface

Description New Office Building Existing Office

Windows* Insulating glass

[1.3 cm (1/2") air space]

R=(.36m2»)K/W (2.04)

0.6 cm (1/4") single pane glass

R=(.15 m2»K)/W (.88)

Exterior Doors* Insulating glass

[1.3 cm (1/2") air space]

R=(.36 m2*K)/W (2.04)

0.6 cm (1/4") single pane glass

R=(.15 m2»K)/W (.88)

Interior Walls’^ 1.3 cm (1/2") gypsum board

10,2 cm (4") metal frame

1.3 cm (1/2") gypsum board

R=(.32 m2*K)/W (1.83)

1.3 cm (1/2") gypsum board

10.2 cm (4") metal frame

1.3 cm (1/2") gypsum board

R=(.32 m2*K)/W (1.83)

Exterior Walls'*' 5.1 cm (2") metal curtain wall

10,2 cm (4") metal frame

5.1 cm (2") rigid insulation

1.3 cm (1/2") gypsum board

R=(1.84 m2.K)/W (10.45)

5,1 cm (2") metal curtain wall

10.2 cm (4") metal frame

No Insulation

R=(.35 m2-K)/W (2.00)

Upper Floors Carpet - pad

7.6 cm (3") concrete reinforced

floor

[1.1 ra (3.5*) air space]

R=(.59 m2.K)/W (3.37)

Carpet - pad

7.6 cm (3") concrete reinforced

floor

[1.1 m (3.5*) air space]

R=(.59 m2*K)/W (3.37)

Roof (Flat) 1.0 cm (3/8") built-up roofing

5.1 cm (2") roof insulation

Metal deck

[1.1 m (3.5*) air space]

R=(1.20 m2-K)/W (6.82)

1.0 cm (3/8") built-up roofing

No insulation

Metal deck

[1.1 m (3.5*) air space]

R=(.22 m2-K)/W (1.26

Ceiling 1.3 cm (1/2") ceiling panel

R=(.22 m2*K)/W (1.25)

1.3 cm (1/2") ceiling panel

R=(.22 m2-K)/W (1.25)

Main Floor 10,2 cm (4") concrete slab

No insulation

R=(.08 m2*K)/W (.48)

10.2 cm (4") concrete slab

No insulation

R=(.08 m2*K)/W (.48)

Source: Honeywell, Inc., Energy Resources Center [36],

* R-values Include surface resistances.

+ Studs spaced 61 cm (24’’).

45
Figure 3.1 Schemata of Office Buildings

NORTH FACE

Interior Layout

Outside View of East and West Faces

Tp

IJcIpI

Outside View of North and South Faces

4

1 WINDOW S 1 o

THIRD FLOOR

1 WINDOW ^ 1 4p

SECOND Floor t 1 1 1

1/ 35' i

1 WINDOW 5 1 p

FIRST floor >19/ ?

SOIL GRADE

^

! SOILGR

Source: Honeywell , Inc , Energy Resources Center [ 36 ].

46
Figure 3.2 Office Building Load Schedules

Source

Hour of the Day

Office Building Light Scheduie

[Light at Peak Load = 26.9 W/m^

(2.5 W/ft2)l CT 100 C

<Do 80 %a 60

•o

vTJ O

40

20 (0

0) a 0

Office Buiiding People Load Schedule

[Average Number of People During Peak Hours

of Operation = 300]

1 I I i I

I I I I 1—

2 4 6 8 10 12 2 4 6 8 10 12 Hour of the Day

Office Building Hot Water Schedule [Maximum Hot Water Demand = 568l7hror

1.577 X 10-4 m3/s)(150 gai/hr)]

Honeywell, Inc., Energy Resources Center [36].

47
Table 3.4 Retail Store Construction Characteristics

Surface

Description New Retail Store Existing Retail Store

Windows^ Insulating glass

[1.3 cm (1/2") air space]

R=( .36m2*K)/W (2.04)

0.6 cm (1/4") single pane glass

R=(.15 m2*K)/W (.88)

Exterior Doors^ Insulating glass

[1.3 cm (1/2") air space]

R=(.36 m2*K)/W (2.04)

0.6 cm (1/4") single pane glass

R=(.15 m2*K)/W (.88)

Interior Walls 1.3 cm (1/2") gypsum board

10.2 cm (4") metal frame

1.3 cm (1/2") gypsum board

R=(.32 m2*K)/W (1.83)

1.3 cm (1/2") gypsum board

10.2 cm (4") metal frame

1.3 cm (1/2") g)rpsum board

R=(.32 m2.R /W (1.83)

Exterior Walls 5.1 cm (2") metal curtain wall

10.2 cm (4") metal frame

2.5 cm (1") rigid insulation

1.3 cm (1/2") gypsum board

R=(1.14 m2«K)/W (6.45)

5.1 cm (2") metal curtain wall

10.2 cm (4") metal frame

No insulation

R=(.35 m2-K)/W (2.00)

Roof (Flat) 1.0 cm (3/8") built-up roofing

5.1 cm (2") roof insulation

Metal deck

0.8 m (2.5') air space

R=(1.20 m2«K)/W (6.82)

1.0 cm (3/8") built-up roofing

No insulation

Metal deck

0.8 m (2.5*) air space

R=(.22 m2-K)/W (1.26)

Ceiling 1.3 cm (1/2") false ceiling

R=(.22 m2«K)/W (1.25)

1.3 cm (1/2") false ceiling

R=(.22 m2-K)/W (1.25)

Main Floor 10.2 cm (4") concrete slab

No insulation

R=.08 m2-K)/W (.48)

10.2 cm (4") concrete slab

No insulation

R=(.08 m2-K)/W (.48)

Source: Honeywell, Inc., Energy Resources Center [36].

^ R-values include surface resistances.

48
Source

Figure 3.3 Schemata of Retail Stores

NORTH FACE

ri>

ZONE1

“RETAJUAREA" T

21 . 7

-

I KoH

r~‘s"

if

T 1 - r I

I—

3

SOLTTH FACE

Interior Layout

[tfir

Outside View of South Face

SOIL GRADE SOIL GRACE Outside View of East Face

SOIL GRADE SOIL GRADE Outside View of West Face Outside View of North Face

Honeywell , Inc .

,

Energy Resources Center [36]

49
Figure 3.4 Retail Building Load Schedules

Office Building People Load Schedule

[Average Number of People During Peak Hours

of Operation = 100]

Retail Building Light Use Schedule

[Maximum Lights = 32.3 W/m2 (3.0 W/ft2)j

8 10 12 2 4

Hour of the Day

Retail Building Hot Water Schedule

[Maximum Hot Water Demand = 6.11 I/hr or

1 . 698 X 10-6 m3/s)(1.615gal/hr)]

Source: Honeywell, Inc., Energy Resources Center [36].

50
ro

0) 3

•r^

o o o

o o o £ o o o cn (N CO>

CO

(qOl./nia) Lue;sAs OVAH 9M1 uoudiunsuoo ASjaug lenuuv

51

Source:

Douglas

C.

HiCtle

and

Dale

L.

Herron,

"Simulation

of

the

Performance

of

M^ltizone

and

Variable

Volume

HVAC

Systems

in

Four

Geographical

Locations

"

[

40

].
The mechanical systems were specified in detail for the four buildings used for

the case studies in this report.^ In keeping with current trends in system

design, energy-conserving variable air volume (VAV) systems (of the single-duct

reheat type) were assumed for the new buildings; less energy efficient constant

volume (CV) systems were assumed for the existing buildings.

Constant and variable volume air systems differ considerably in the energy

loads they impose upon a building. Typically, in both systems, central supply

air is cooled to the estimated lowest required temperature over the year. (A

15,6°C (60°F)) cold deck temperature was assumed in this report). Once the air

is cooled, the air distribution system regulates heat delivery to the building

so that the desired temperature is maintained. The CV system does this by

changing the supply air temperature, while keeping the same air flow rate. The

VAV system maintains a constant heating or cooling coil temperature but changes

the supply air flow rate. A VAV box at the entrance to each building zone limits the air being reheated and supplied to the building space to the minimum

required at any given time. The VAV system considerably reduces the total

energy requirement for heating and cooling as compared to the CV system.

Night-time temperature setbacks and reduction in operation of fans at night were assumed for all buildings because these simple measures for energy conservation

are now widely undertaken in both new and existing buildings. An economizer

cycle was specified for all but the existing retail store.

An economizer cycle used in conjunction with either a VAV or CV system controls

the Inflow of outside air to the air system. Outside air mixed with recirculated return air is supplied to the preheat and cooling coils. By increasing

the inflow of outside air during in-between seasons and cold seasons, and

reducing it in hot or humid months, the economizer cycle serves to reduce, or even eliminate, the load on the cooling coil,

3.2,1 Office Buildings

Two VAV systems were assumed for the new office building: one for the interior

zones and one for the exterior zones. System capacities were specified for each

building and location based on an analysis of the sensible and latent loads

estimated for each building and location. (System capacities were established

using the loads portion of the BLAST loads analyzer program — see section 4.1.)

For the interior zones, the supply air capacity is set to meet the annual zone- peak latent cooling load, with the VAV system supplying its minimum air fraction of 10 percent. For exterior zones, supply air capacity is set for the

annual zone-peak sensible cooling load, with the VAV system supplying its maximum or capacity air flow. During operating hours, a minimum of 5 percent outside air, or one outside air change per hour, whichever was greater, was assumed

to be maintained.

1 The selection of mechanical systems and specification of their characteristics

were tasks assisted by Honeywell, Inc., Energy Resources Center, under

contract to NBS [36].

52
The capacity of an air system is generally sized to meet peak-load conditions

(temperature and humidity) whether or not these conditions occur simultaneously

in all zones. The annual zone-peak sensible cooling load, i.e., the energy

required to maintain a desired temperature during the non-heating season,

usually dictates the capacity of the system. (For buildings with a large

percentage of exterior walls and roofs relative to interior space, sensible

energy gains through the building envelope and from people and lights generate

the major portion of the energy load.) However, in the interior zones of a multi-story building, the sensible loads tend to be smaller than the latent

cooling loads (caused by the humidity from human breathing). A VAV system

sized for peak sensible loads may not be sufficient to meet this load. Furthermore, because the VAV system is controlled on the basis of dry-bulb temperature,

i.e., sensible loads only, the control system may dictate a low supply air

fraction when sensible loads are low but latent loads are high. Thus in establishing a capacity of the system, it is necessary to consider the latent loads

being generated while the VAV system is operating at a low supply air fraction

as well as at capacity. In the test runs, it was established that the capacity

would be based on the latent load at the VAV minimum. (A CV system sized for

the peak sensible load supplies sufficient air to satisfy the latent load.)

Two CV air systems were assumed for the existing office building, each with a preheat coll in the mixed air duct. The supply air volume for each zone was set

to meet the annual zone-peak sensible cooling load. A minimum of 10 percent of

the supply air was outside air, with a minimum of six air changes per hour. In

both buildings, heat was supplied to the building space by perimeter radiation

units (exterior zones) and by terminal reheat coils (interior zones). An

enthalpy controlled economizer cycle (which adjusts for humidity as well as for

temperature conditions) was assumed for both office buildings.

3.2.2 Retail Stores

A single VAV system was assumed for the new retail store building. This system

had a minimum supply air fraction of 20 percent of the system capacity, as compared with 10 percent in the new office building. Supply air capacities were

set to meet the annual zone-peak sensible cooling loads with the VAV system

operating at full capacity. Outside air was set at a minimum of 5 percent of

the supply air volume, or one air change per hour, whichever was greater. A

temperature-only type economizer cycle was assumed for the new retail store.

The CV system for the existing retail store had no preheat coll and no economizer

cycle. Ventilation outside air was 10 percent of the supply air volume,

3.3 HEATING PLANTS

The heating plants for the buildings consisted of a solar energy system used in

combination with a conventional hot water boiler, fired alternatively by

electricity, natural gas, or distillate fuel oil. The gas or oil-fired boilers

in both of the existing buildings were assumed to be 60 percent efficient, while

the gas and oil-fired boilers in the new buildings were assumed to meet the

ASHRAE 90-75 standard of 75 percent efficiency. The electric resistance systems

for all buildings were assumed to be 100 percent efficient.

\ 53
These assumed boiler efficiencies of 60 and 75 percent are considerably higher

than would normally be reasonable to assume for the heating system in its

entirety, including the distribution system. Efficiency ratings of the

delivery system were not taken into account in evaluating the economic feasibility of the solar energy systems because it was assumed that the solar energy

would be subject to the same inefficiencies in the delivery system as the con- ventional energy source, thus cancelling out that efficiency effect. Possible

cost of modification to the mechanical system to accommodate solar energy,

and the efficiency effects thereof, were not taken into account. Boiler or furnace efficiency is a critical factor to consider in evaluating solar energy

because it is a key determinant of the quantity of conventional energy saved.

For both the new and existing buildings, it was assumed that without solar, a conventional system identical to the backup conventional system in type, size,

operating efficiency, and maintenance and repair costs would be used alone.

This assumption may be deficient in that it neglects possible effects of part

loading on boiler efficiencies. Other things equal, the lower the efficiency

of the conventional furnace, the higher the net benefits of a solar energy system. However, if the efficiency of the conventional furnace drops substantially in direct response to the displacement of conventional energy by solar

energy, the lowered conventional boiler efficiency offsets part, or all of the

solar benefits. Although there is little hard measurement of the part loading

effects it would appear, a priori , that a substantial solar-induced decline in

plant efficiency would tend to drive solar towards either end of the sizing

extremes. It is expected, however, that the part loading effects will

generally not be drastic, particularly in newer systems.

Temperature settings of the heating systems were specified to be 20 °C (68 °F)

during occupied periods of the heating season, 26 “C (78°F) during occupied

periods of cooling season, and 17 °C (62°F) at night during the heating season. The cooling systems were assumed to be set at 37 °F (99°F) during unoccupied

periods.

The solar energy equipment was assumed to include high quality, double-glazed

flat-plate collectors with selective absorber coatings like that illustrated

in figure 3.6. The systems were assumed to use liquid as the heat transfer

storage medium. ^ An off-the-shelf solar energy system was assumed for the

retail store's hot water system, and custom designed and engineered systems

for all of the other applications.

Figures 3.7 and 3.8 show illustrative schematic diagrams of solar energy

systems like those assumed for this study. Figure 3.7 shows a service hot

water system only, and figure 3.8, a combined space heatlng/hot water system.

^ The solar energy systems were characterized by Honeywell, Inc., Energy

Resources Center, under contract to NBS [36]. The collector design is

representative of those installed in commercial buildings under the Solar

Heating and Cooling Commercial Demonstration Program.

54
Figure 3.6 Solar Collector^

Glass Covers (LSC18-1S Collectors

Source: Honeywell, Inc., Energy Resources Center [36].

^ The collector covers illustrated are two sheets of .3 cm (1/8") tempered,

antireflection low iron glass, with a gasket seal. The solar absorber is an assembly of parallel copper tubes bonded to a steel plate of black chrome on nickel. The collector has 8.9 cm (3 1/2") of fiberglass Insulation beneath

the absorber plate and 2.5 cm (1") around the sides of the collector enclosure. The enclosure is constructed of corrosion-resistant heavy gauge galvolume steel with a special electro-process paint finish.

55
Figure

3.7

Schematic

Diagram

ofa

Solar

Service

Hot

Water

System

fora

Commercial

Building

O

Z)CD

56

Source:

Honeywell,

Inc.,

Energy

Resources

Center

[36].
Figure

3.8

Schematic

Diagram

ofa

Combined

Solar

Space

Heating/Hot

Water

System

fora

Commercial

Building

57

Source:

Honeywell,

Inc.,

Energy

Resources

Center

[36].
3.4 GEOGRAPHICAL LOCATIONS AND DATA

3.4.1 Selection of Cities

The following four selection criteria were applied to determine regions and

cities for study:

1. Coverage of major climate zones in the continental U.S.;

2. Coverage of solar radiation variability;

3. Representation of major population centers;

4. Availability of hourly weather data and solar radiation data.

A survey was made of the major regions designated by other studies of solar

energy, in particular the three "Phase 0" reports by General Electric [42], TRW

[43], and Westlnghouse [44]; two reports by the Mitre Corporation [45, 46];

and a report by the Intertechnology Corporation [47].

A comparison of these classifications with the Trewartha climate typology

indicated that the twelve climatological regions identified by General Electric

and later adopted by Intertechnology represented nine of the 11 climatic types

designated by Trewartha. This is a broader climatic coverage than is provided

by the other regional classifications examined. On the basis of its broad

coverage, the General Electric regional classification was adopted for this

study.

To select specific cities within each of the 12 regions, the total of 26 cities

identified by the six reports listed above were located on a map divided into

the 12 climate regions to determine their relative placements within each

region. Cities were then checked against the availability of hourly weather

data tapes.

Applying the selection criteria, it was determined that the cities identified

by the General Electric study were also suitable for this study. However,

Apalachicola, Florida was added to the sample to represent the Gulf Coast area. Figure 3.9 shows the final delineation of climatic regions and the specific

cities selected for study.

3.4.2 Geographical Data

Weather data for evaluating the energy requirements of buildings in the 13

cities are from the Test Reference Year (TRY) series of weather tapes furnished

by the National Climatic Center, Ashville, NC. The TRY tapes contain hourly

data for dry-bulb temperature, wet-bulb temperature, dew-point temperature,

wind direction, wind speed, barometric pressure at the station, weather (precipitation, fog, haze, dust), total sky cover, cloud amount (4 layers), type

of cloud, and height of base of each cloud layer. (New Orleans weather tapes are used for the thermal analyses of buildings in Apalachicola, due to the

lack of TRY tapes for that city.)

Monthly values for solar radiation incident on the solar collector were calculated using average daily radiation values reported by Liu and Jordan [48]

58
Figure

3.9

Climatic

Regions

and

Cities

Selected

for

Study

z d

O d

59
and procedures for adjusting for collector tilt developed by Klein [49]. (These

procedures are incorporated within the economic optimization computer model

in appendix B.) Annual insolation values derived using these data and procedures

are given in table 3.5,

The analyses of energy requirements also required two measures of underground

temperature: 1) the temperature of the water supplied through city mains for

service use, and 2) the temperature of the earth around the building perimeter.

Water supply temperatures were assumed to approximate deep-ground [1,2-3. 7 m

(4*-6’)j average soil temperatures reported by Kusuda [50], (Kusuda reports

data for 60 cities. Data for the cities nearest those selected for study were

used .

)

Arens and Carroll have published procedures for correcting the deep-soil

temperatures to reflect temperatures 15 cm (6") below the ground surface. Semiannual data corrected according to these procedures were used to measure thermal

losses through the building floors [51], These data are shown in tables 3.6

and 3.7, respectively.

60
Table 3.5 Annual Insolation Values for Selected Cities^

Cities

Hot Water

Systems

Space & Water

Heating Systems

MJ/m2 Btu/f t^ MJ/m^ Btu/f t^

Boston 5,458.1 480,614.1 5,238.9 461,308.2

Washington, D.C. 6,148.3 541,392.2 5,924.6 521,688.9

Nashville 6,599.4 581,108.6 6,324.1 556,871.1

Charleston 6,917.5 609,119.9 6,717.3 591,490.5

Miami 7,485.7 659,151.4 7,295.9 642,441.0

Apalachicola 7,531.9 663,222.6 7,317.0 644,302.3

Bismarck 6,101.6 537,275.3 5,872.4 517,092.7

Madison 7,198.4 633,858.6 6,942.4 612,196.0

Omaha 6,740.8 593,560.1 6,526.0 574,647.4

Fort Worth 7,629.6 671,812.5 7,385.6 650,347.5

Phoenix 9.164.0 806,936.7 8,897.6 783,480.4

Seattle 4,623.7 407,136.6 4,380.5 385,728.0

Los Angeles 7,769.5 684,144.0 7,522.9 662,432.5

Source of radiation data; B.Y.H. Liu and R. C. Jordan, "A Rational

Procedure for Predicting the Long-Term Average Performance of FlatPlate Solar Energy Collectors" [48].

^ Corrected for collector tilt.

61
Table 3.6 Water Supply Temperatures Used in Analysis

of Service Hot Water Requirements

City Selected

for Study

Nearest City

for which

Data is Available

Water Supply Temperature

Degrees Centigrade (Fahrenheit)

Jan . -Mar

April--June July--Sep

Oct-Dec

Boston Ithaca, NY 5 (41) 5 (41) 14 (58) 12 (54)

Washington, D.C. Upper Marlboro, MD 7 (44) 9 (49) 19 (66) 18 (64)

Nashville Jackson, TN 10 (50) 13 (55) 21 (69) 18 (64)

Charleston Calhoun, SC 11 (52) 14 (57) 23 (73) 21 (70)

Miami Gainesville, FL 16 (61) 22 (71) 26 (79) 26 (78)

Apalachicola Athens , GA 13 (55) 16 (60) 24 (75) 23 (73)

Madison East Lansing, MI 5 (41) 5 (41) 16 (61) 14 (58)

Bismarck Madison, SD 2 (36) 3 (38) 15 (59) 13 (55)

Omaha Lincoln, NE 4 (40) 7 (44) 18 (65) 16 (61)

Fort Worth Temple, TX 16 (61) 18 (65) 27 (81) 25 (77)

Phoenix Tempe, AZ 14 (58) 17 (63) 25 (77) 23 (74)

Seattle Seattle, WA 8 (46) 10 (50) 15 (59) 13 (56)

Los Angeles Brawley, CA 19 (66) 23 (73) 31 (87) 29 (85)

Source: T. Kusuda, NBSLD, The Computer Program for Heating and Cooling Loads in

Buildings [50]

62
Table 3.7 Ground Temperatures Used in Analyses of

Space Heating Requirements^

Degrees Fahrenheit (Centigrade)

Location October-May June-September

Boston 14 (57) 18 (64)

Washington, D.C. 16 (60) 20 (68)

Nashville 17 (62) 21 (69)

Charleston 17 (63) 22 (72)

Miami 20 (68) 24 (75)

Apalachicola 19 (67) 24 (75)

Bismarck 13 (56) 18 (65)

Madison 13 (55) 18 (64)

Omaha 14 (58) 19 (66)

Fort Worth 19 (67) 24 (75)

Phoenix 19 (67) 23 (73)

Seattle 16 (60) 18 (64)

Los Angeles 20 (68) 24 (75)

Source: Edward A. Arens and William L. Carroll, Geographical

Variation in the Heating and Cooling Requirements of a Typical

Single Family House, and Correlation of These Requirements to

Degree Days [51].

^ The data were determined from averages of the winter and spring

values and summer and fall values reported by Kusuda [50].

These averages were in turn averaged with the indoor-air average

temperature. The actual cities for which the data were prepared

are shown in the second column of table 3.6.

63
4. DATA AND ASSUMPTIONS FOR CASE STUDIES

4.1 HEATING AND HOT WATER REQUIREMENTS

4.1.1 BLAST Transient Load Model

The BLAST (Building Loads Analyses and System Thermodynamics) transient load

model was used to perform the analyses of heating requirements for the selected

buildings in the designated locations.^ The BLAST model accounts for 1) hourly

variations in solar loading on the building envelope, 2) transmission loads due

to the building construction R-value characteristics, 3) Internal loads from

people, lights, and distribution equipment, including multizone variable air

volume systems, 4) infiltration loads, and 5) the thermal capacity of the

building. It has the advantages of a relatively fast running time. Data from

the building loads and distribution systems subprograms are suitable for use with the solar load ratio performance model (section 4.2) and with the NBS

economic analysis model (section 2.3).

Modifications to the BLAST program were required before using it to analyze

the energy requirements of the prototypical building designs and systems in

the 13 different cities. First, it was necessary to develop a methodology for

using BLAST to size the air handling systems specified for the four buildings

and locations and to validate the system control assumptions for each building.

The new BLAST subprogram prepared for these purposes was used to calculate peak

latent and sensible loads for each building and geographical location and to

establish the sensible and latent load combinations at different points in time.

This was done by performing full-year simulation runs to utilize extreme

weather days. Very large system capacities were assumed for these runs to

maintain zone temperatures at their inner limits (upper limit for heating, lower

for cooling). Data obtained from running this subprogram were used to determine

the system capacities for each building and location.

Additional modifications in the BLAST simulation Included the following:

1) A masonry wall located diagonally across the space was simulated for the

retail store zones, because the interior furnishings of the retail store sales area were assumed to have greater thermal storage effects than allowed for in

BLAST.

2) Since only five- and seven-day operation schedules were provided for in BLAST,

the program was modified for the retail stores to allow a six-day schedule.

This modification was done by performing design day runs for five- and seven-day

schedules, and adjusting the weekend schedule to make results close to the

average of five- and seven-day results. Essentially, this entailed having the

store operate half days on both Saturday and Sunday. Although this method may

have altered start-up loads slightly over loads for a six-day week, the resulting

monthly heating requirements appeared to be a good model of a six-day week.

^ For a description of the BLAST computer program, see [52] and [53]. The

version of BLAST used in this study is designated BLAST-1.2.

64
3) To facilitate operation of BLAST, the number of zones in the office building

was reduced. Since the first and second story zones of the office building have

nearly the same geometry and loads, it was possible to combine them in the computer simulation to form 10 thermal zones from the Initial 15 zones. Trial runs showed differences of less than two percent in monthly loads with either the CV

or VAV systems as a result of combining subjacent zones.

4) Because the computer responds more quickly in the BLAST simulation to a control action than would an actual building, control schedules were adjusted so that control actions occur an hour later than in a real building. The control

schedules were set to coincide with the occupancy schedules shown in figures

3 . 2 and 3.4.

The weather files used in the BLAST simulation were prepared using the March 1,

1979 version of WIFE, the weather file encoding program developed by the Construction Engineering Research Laboratory (CERL) of the U.S. Department of the

Army.

4.1.2 Results of BLAST Analyses — Space Heating

With the above modifications, hourly heating and cooling loads for each zone

were calculated with the BLAST Loads subprogram, using the system capacities

generated in the design runs, hourly data from the TRY weather file, and the

building descriptions and operating schedules described in sections 3.1 and 3.2.

The BLAST Loads subprogram accounts for a) transmission loads through walls,

roofs, floors, and windows, b) solar gains through windows, walls, and roofs,

c) internal loads from people, lights, and building equipment, and d)

infiltration loads.

Results from the BLAST Loads subprogram, together with the detailed

specifications of the mechanical distribution system (including ventilation air

requirements, equipment operating schedules, flow rates., fan pressures and motor

efficiencies, heating coil capacities, dry and wet bulb temperatures, water

supply temperatures, water velocity and flow rates, reheat capacities, and

supply air volume) were entered into the systems simulation subprogram in BLAST.

This final production phase of the BLAST program produced monthly requirements

for thermal energy, broken down into space heating and cooling components.

Table 4.1 shows the estimated annual space heating requirements of the new and

existing office buildings and retail stores for each city aggregated from the

monthly space heating requirements. These are the annual loads to be met by

the solar energy auxiliary system or its alternative.^

^ Only the loads and nonsolar simulation portions of the BLAST program were

used in this study. The economics portion of BLAST was not used because of

its limited capability. It does not provide a tax or financing analysis,

allow the input of multiple rates of fuel price escalation, nor calculate

net savings in present value dollars [54].

65
Table 4.1 Estimated Annual Space Heating Requirements of the Selected Buildings^

Location

Office Building Retail Stores

New Existing New Existing

GJ 10*^Btu GJ lO^Btu GJ lO^Btu GJ lO^Btu

Apalachicola 200 190 3,431 3,252 93 88 818 775

Bismarck 1,249 1,184 6,312 5,983 478 453 1,612 1,528

Boston 658 624 4,840 4,588 268 254 1,139 1,080

Charleston 233 221 3,491 3,309 116 110 903 856

Ft. Worth 268 254 3,996 3,788 130 123 1,083 1,027

Los Angeles 138 131 3,825 3,626 51 48 779 738

Madison 896 849 5,623 5,330 350 332 1,354 1,283

Miami 66 63 2,348 2,226 26 25 705 668

Nashville 388 368 4,005 3,796 171 162 1,014 961

Omaha 753 714 5,314 5,037 312 296 1,379 1,307

Phoenix 226 214 4,416 4,186 117 111 1,175 1,114

Seattle 593 562 5,375 5,095 247 234 1,175 1,114

Washington, D.C. 462 438 4,466 4,233 208 197 1,174 1,113

^ These requirements were calculated by applying the BLAST computer program to the

energy analysis of the selected buildings and mechanical systems and by aggregating

the monthly data. Heating requirements are shown before adjustments for boiler

efficiencies were made.

66
Table 4.1 shows considerable variation in the estimated annual heating

requirements between the new and the existing buildings, between the offices and

the retail stores, and among the cities. The differences in estimated energy

requirements between the new and existing buildings appear particularly great.

4,1,3 Building Loads Model—Service Hot Water

The energy requirements for service hot water were estimated separately from the

space heating requirements. Service hot water loads were estimated to be 3.155

X 10“^ m^/s (0.5 gal/mln) for the office building, based on an average of 300

people during 10 operating hours, and 0.095 x 10“^ m^/s (0,015 gal/min) for the

retail store, based on an average of 100 people during 12 operating hours.

These estimated rates of hot water usage were assumed to represent the average

hot water demand for these buildings. The approach to the development of

these estimates is as following: Honeywell, Inc., under contract to NBS,

recommended a maximum hot water demand of 1.577 x 10“^ m^/s (150 gal/hr) for

the office building and 1.698 x 10~^ m^/s (1.615 gal/hr) for the retail

store. These maximum rates were compared with data published by ASHRAE for

office buildings of 1.514 x 10“^ m^/person (.4 gal/person) for a maximum

hour; 7,570 x 10~3 m^/person (2.0 gal/person) for a maximum day; and 3.785

X 10”3 m^/person (1.0 gal/person) for an average day. The ASHRAE average day

rate per person was used directly for the office buildings analyses, resulting

in an estimate of 3.155 x 10~5 m^/s (30 gal/hr or 0.5 gal/minute) for the

building (based on 300 person occupancy over a 10-hour day).

Comparable average usage data were not available for retail stores. Half the

maximum demand of 1,698 x 10~^ m^/s (1.615 gal/hr) recommended by Honeywell

was taken as an estimate of the average rate. From this figure a usage rate

was calculated which was rounded to 0,946 x 10“^ m^/s (.015 gal/min).

Using these water usage rates and the deep ground supply temperatures given in

table 3.6, energy requirements for service water heating are calculated with

the standard static energy balance equation;

Ql = (Tout - Tin). <3«)

where

,

Ql = energy requirements for hot water. J (Btu)

D = quantity of hot water demanded, m^ (gal)

W = density of water, 100 kg/m^ (8.34 Ib/gal)

Cp = specific heat of water, 4,190 x 10^ j/(kg»K)(l Btu/lb/°F)

Tout ~ water supply temperature °C (°F)

Tfn = supply water temperature, °C (”F)

The resulting annual energy requirements for service hot water are shown in

table 4.2. These estimates were used in the evaluation of the solar hot water

systems in section 5.1. They were added to the space heating requirements in

the analyses of combined solar space heating and hot water systems in section 5.2.

67
Table 4.2 Estimated Annual Hot Water Requirements

the Selected Buildings^

Location

Office Building Retail Stores

GJ lO^Btu GJ lO^Btu

Apalachicola 45.3 42.9 2.0 1.9

Bismarck 58.4 55.4 2.5 2.4

Boston 57.4 54.4 2.4 2.3

Charleston 47.2 44.7 2.0 1.9

Ft. Worth 41.6 39.4 1.8 1.7

Los Angeles 36.7 34.8 1.6 1.5

Madison 56.1 53.2 2.4 2.3

Miami 40.6 38.5 1.8 1.7

Nashville 49.6 47.0 2.1 2.0

Omaha 54.5 51.7 2.3 2.2

Phoenix 43.7 41.4 1.9 1.8

Seattle 54.3 51.5 2.3 2.2

Washington, D.C. 52.2 49.5 2.2 2.1

^ See text for estimation procedures.

68
A. 2 SOLAR ENERGY SYSTEM PERFORMANCE

The solar load ratio model for active systems, developed by the Los Alamos

Scientific Laboratory (LASL) was used to estimate the performance of the solar

energy systems [32, 55]. Unlike most other models of solar energy system performance, the solar load ratio model has documented capability for simulating

commercial building systems, a prime factor in its selection for the case

studies.

The solar load ratio model describes the performance of solar energy systems

using general design and sizing curves which represent the empirical results

of hourly simulations of different types of systems in different locations

based on weather data from the 1950-62 period. The curves used in this study

are based on simulations of a "typical" three-story commercial office building

modeled for eight U.S. cities: Bismarck, ND; Fresno, CA; Los Alamos, NM;

Madison, WI; Medford, OR; Miami, FL; Nashville, TN; and New York, NY. Note

that four of these eight cities are among the 13 selected for case study.

"Universal" design and sizing curves have been developed by LASL for both

combined space heating/hot water systems and for hot water systems. The universal curves express the fraction of the monthly thermal energy load met by

solar as a function of collector area, monthly energy load of the building, and

monthly solar radiation incident on the collector surface. Families of curves

are available for three generic types of flat-plate collector and for various

design water temperatures and heating coil supply temperatures.

The curves used for the case studies are based on a system of double-glazed,

non-selective , flat plate collectors, a hot water setting of 54°C (130'*F) for

water heating only systems, a heating coil inlet temperature of 54 °C (130®F)

for combined space heating and hot water systems, and water storage sized for

73 kg/m2 (15 Ibs/ft^) of collector. The design characteristics of the solar

energy components specified for this study are either equal or superior to the

design characteristics upon which the universal curves are based. Universal

curves were not available for double-glazed collectors with selective coatings — the collectors specified for the case studies. However, the performance

differences are not expected to be large. Adding a selective coating to the

absorber surface of double-glazed collectors should improve performance only

slightly.

Of the family of solar universal performance curves for combined space heating

and hot water systems shown in figure 4.1, curve "d" is the one used for the

feasibility study, and of the family of curves for service hot water systems

shown in figure 4.2, curve "b" is the one used.

^ LASL has since revised the universal design curves used in this study,

such that they now predict somewhat lower performance for service hot water

systems. The revisions are contained in an updated edition of the DOE Facilities Solar Design Handbook [32],

69
The computer code developed to apply the economic evaluation model incorporates

the solar load ratio performance model within the economic model. The mathematical equations for the universal curves (shown in box inserts to figures 4.1

and 4.2) are linked with the economic model of section 2.3.

4.3 PURCHASE AND INSTALLATION COSTS OF THE SOLAR ENERGY SYSTEM

4.3.1 Estimating Fixed and Variable System Costs

The costs for purchasing and installing a solar energy system are often quoted

to a buyer either as (1) the total dollar cost of a particular system of given

size, possibly broken down into labor, materials, and overhead (including

profit), or (2) the average total system cost per unit of collector area, based

also on a system of given size.

While either of these types of cost quotations will provide sufficient

information about acquisition costs to determine if a given system is cost

effective, neither gives adequate information to determine the economically

optimal size of the system (l.e., the size that will provide the largest net

dollar savings).

To identify the economically optimal size of a solar energy system, it is

necessary to know the relationship between system size and system costs. For

active solar energy systems there is often a substantial "fixed" element of

acquisition costs that is Independent of the size of the system over a range of

system sizes. There are additional "variable" costs that change as the size of

the system changes. It should be recognized, however, that estimates of fixed

and variable costs may not hold over all sizes of a system to be considered,

due to possible discontinuities in the relationship between collector size (and

costs) and the size (and costs) of other system components as collector size is

expanded. The presence of significant discontinuities in size relationships

may make it necessary to estimate fixed and variable costs for specific collector size intervals. Unfortunately, however, it is often difficult to distinguish the fixed from the variable costs and to develop the necessary functional

costs relationships for appropriate intervals of system size.

To estimate fixed and variable system costs for the case studies of this

report, NBS worked with a private contractor experienced in the commercial

building/solar energy market, Honeywell, Inc., to develop acquisition cost

estimating equations for the solar energy systems characterized in section 3.3

[36]. Fixed and variable costs were estimated for both the solar service hot

water system and the combined solar space heating/hot water system described in

section 3.3, as applied to the buildings described in section 3.1. Estimates were based on the contractor's experience with selling and installing systems

of varying sizes, supported by Internal records and professional judgement.

The resulting cost estimating equations are shown in table 4.3.

The first terra in each equation is the estimated fixed component of system

costs, and the second term, the coefficient on collector area, is the variable

component. There are separate equations in table 4.3 for new and existing

buildings, for labor and materials costs, and for the service hot water and

the combined space heating/hot water systems.

70
Figure

4.1

Solar

Universal

Performance

Curves

for

Estimating

the

Performance

ofa

Combined

Solar

Space

Heating

and

Service

Hot

Water

System

fora

Commercial

Building I

CM

JB|OS UOIPBJJ AmiUOlAJ

O)

CQ

CO SQ,

CD ocDCDO

CD

C

ID

00

CM

CMcn

Co

•H

4-1 CO 4Jo

••

C *0 O CO •H CO O

-O tH

CO

D

C

•u o

PQ -H

4J bO OJ »> CO

to > *0 ^4

CO (U 4J "O O C

CO

C

<U CO <UbO

(U a>

<U 4J bC O CO CO iH Is M CD

CU > 4-» > fl o . CO o; X -rH

II II II II

CM

bO

•H

CO 4J (U a

<l) u

a *-i 3 o CM cO

4-» 4J f—

CO 3 o

t-l 4.4 CO

(U « >1

CX, JO (0 0 * (U

0) T3 3 •H

4-) CO 4J 4J OCM CQ •H

4J 1—1 4J fH

>s

bO

0)cw

4Jc

0)s

4.1

cOa

(U o

CO

t=>

<uo

S-I 3o

CO

71
Figure

4.2

Solar

Universal

Performance

Curves

for

Estimating

the

Performance

ofa

Solar

Service

Hot

Water

System

fora

Commercial

Building^

JB|OS uouoBJj

cc-j

CDCD

_o

re CLDreo

_i

a)c • re rH r—

•H res re

CNJ re 4-1

B re

B

CO

X re re CO o

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X fa CO re CO 4J re X

res <3 o re o re 'O fa CO fa o o re 40 fa

re • re e

re CJ b-,, re c a •> re o reJ 40

re

M CO *-i fa re CO

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<4-1 CO re CO CO c •H re TJ re bO re XI •re CO So •H re 40 1 fa re

CO >> re re

X 40

<U re iH re re 40 cO Q X X o So 15 4J iJ fa bO X

re fa re o X

re

X o fa •re fa o

tH

w s re

X

X o •H bO bO 5

CO 15 re fa re

X

CO fa re CCS CO 4J re o

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•H

X

X re CO CO 4J 40 4J o > re •H 40 > re r—

•re

p So So

X •H re re re O re X u re X o fa o

re o W fa fa

fa fa re 40 CO fa o M re re re w X CJ o CO re o CO r-{ o fa S4 o Q < re fa 40 re re X re CJ > CCS re re fa B *' re 40 So re £ M CO re fa fa re o

re o fa

(U > re S4 re

X c )-i re o re X re fa re 40 fa H a

cj re o

<4-1 o re CO re o re > • X CO re fa fa re X u re o 40 re So c X -H fa re re re <u 4J 4-1 CO re X X e re re re CCS fa iH re re re X o re CO CO

re bO o X

Cu re !u re re

X re

re o o fa CO > Q •H CJ re > o

CO fa re • fa ^4 CO bO CO o

CO > re 40 o X • re 40 fa CO X & U 40 re re 40 T3

re to > re re X re So re re • •• X u re CO X . re 40 CO o fa o re >

re CO

X fa re CCS 3 M fa 40 o So B

r

O

CO re

72
Although materials costs were assumed approximately uniform across the 13 cities,

labor costs were assumed to vary by city. The breakout of labor and materials

in the cost equations facilitated the conversion of these basic equations into

region-specific equations for use in the case studies. To account for this

locational variation, the regional labor cost adjustment factors shovm in table

4.4 were applied to the fixed and variable components of the system labor cost

equations given in table 4.3. Both the labor and materials equations were

further adjusted to fit the analysis for each of the 13 cities, by applying the

appropriate sales tax rate (the city tax rates are given in table 4.11) to materials and/or labor costs. (The resulting 13 sets of cost estimating equations

for the case studies are not shown here, but are included in the data files of

the companion computer program of appendix B.)

Each of the cost equations in table 4.3 describes a solar energy system with the same basic type of liquid storage and flat plate collectors described in section

3.3. Differences in the dollar values of the equations reflect primarily differences in the assumed size of the solar energy system and the associated com- plexity of the building application. In the case of the solar service hot water

system for the retail stores, the cost functions describe a small, off-the-shelf

system in the size range of 3 m^ to 8 m^ (about 30 ft^ to 85 ft^) of collector

area, a system requiring little specialized design or engineering. For the

office building’s solar service hot water system, the cost functions apply to a system in the size range of 8 m^ to 50 m^ (about 85 ft^ to 540 ft^) of collector

area, with some site-specific design and engineering required. The cost functions for the combined solar space heating/hot water system for both types of

buildings are based on systems with collector arrays over about 50 m^ (over

about 540 ft^) which are assumed to entail considerably more custom design

and engineering than the service hot water only systems. These higher assumed

costs are reflected mainly in the fixed cost components of the cost estimating

equations for the combined systems. A further source of the estimated difference between the costs of solar service hot water systems and of combined space

heating/hot water systems is the year’s dollars in which they are denominated.

The equations for solar service hot water are based on end-of-year 1978 dollars,

and the equations for the combined systems, on mld-1980 dollars.

4.3.2 Reviewing the Cost Estimates

Having assigned estimated values to system acquisition costs, let us now

consider (1) the significance of the values assigned, (2) their probable accuracy, and (3) the selection of alternative values for performing sensitivity

analysis

The values assigned to fixed and variable system acquisition costs are critical

in two ways to the results of an economic optimization analysis: First, their

total comprises the major item of costs against which present value savings are compared in determining the cost effectiveness of a solar energy system.

Second, the unit variable cost determines the system size that will either

maximize net savings or minimize net losses from solar. (Fixed costs, being

by definition independent of system size, do not determine the economically

optimal size of a system, apart from possibly driving the optimal size to zero.)

73
Table 4.3 Solar Energy System Cost Equations^

Fixed and Variable Acquisitions

Type of System Existing Buildings

Retrofit Cost

New Construction

Initial Purchase Cost

Combined Heating and

Hot Water Systems, 1980$^

Materials $40,824 + $270.51/m2 . Area (m2)

($40,824 + $25.13/ft2 . Area (ft2))

$37,045 + $240.80/m2 . Area (m2)

($37,045 + $22.37/ft2 . Area (ft2))

Labor $28,951 + $110.12/m2 . Area (m2)

($28,951 + $10.23/ft2 . Area (m2))

$24,532 + $35.85/m2 . ^rea (m2)

($24,532 + 3.33/ft2 . Area (ft2))

Service Hot Water Systems

Office, 1978$

Materials $6,060 + $190.85/m2 . Area (m2)

($6,060 + $17.73/ft2 . Area (ft2))

$5,759 + $166.52/m2 . Area (m2)

($5,759 + $15.47/(ft2 . Area (ft2))

Labor $4,253 + $68.89/m2 • Area (m2)

($4,253 + $6.40/ft2 . Area (ft2))

$3,644 + $8.07/m2 . Area (m2)

($3,644 + $.75/ft2 » Area (ft2))

Retail Store, 1978$

Materials $1,600 + $179.44/m2 . Area (m2)

($1,600 + $16.87/ft2 . Area (ft2))

$1,500 + $179.44/m2 . Area (m2)

($1,500 + $16.67/ft2 . Area (ft2))

Labor $450 + $29.92/m2 . Area (m2)

($450 + $2.78/ft2 . Area (ft2))

$310 + $23.90/m2 . Area (m2)

($310 + $2.22/ft2 . Area (ft2))

^ These cost equations were developed by Honeywell, Inc., under contract to NBS. The cost

functions for the service hot water system for the retail stores were assumed to apply to an off-the-shelf system in the size range of about 3 to 8 m^ (about 30 to 85 ft^) of collector

area. The cost functions for the service hot water system for the office building reflect the

assumption of higher design and engineering requirements for systems in the size range of about

8 m2 to 50 m2 (about 85 to 540 ft2) of collector area. The cost functions for combined systems were assumed to apply to systems larger than 50 m2 (540 ft2).

^ The costs for combined space heating/hot water systems were adjusted from end-of-year 1978

prices to mid-year 1980 prices based on the change in the Producer Price Index from end of

1978 to mid-1980.

74
Table 4.4 Average Labor Adjustment Factors^

City Factor

Boston, MA 1.04

Washington, DC .99

Nashville, TN .73

Charleston, SC .57

Apalachicola, FL

(nearest city for which

data was available was

Orlando)

.77

Omaha, NE .93

Madison, WI .92

Bismarck, ND

(nearest city for which

data was available was Jamestown, ND)

.66

Miami, FL 00 .

Fort Worth - Dallas, TX .83

Phoenix, AZ 1.00

Los Angeles , CA 1.14

Seattle, WA 1.08

^ These adjustment factors indicate average variation among 21 labor

items widely required for the installation of solar energy systems

in commercial buildings.

Source : Dodge Manual for Building Construction Pricing and Scheduling , Dodge Building Cost Services (New York: McGraw-Hill, 1977)

[56].

75
How accurate are the acquisition cost estimating equations for the intended

applications? Empirical validation of the equations is difficult because

experience with solar energy in the selected types of buildings is quite

limited and the existing data base is small. Furthermore, the available data

applies chiefly to government-funded demonstration projects, which may differ

in costs from projects that are privately funded, as well as from projects

that come after the demonstration phase and benefit from lowered "learning

costs". At best, the available data provide only a rough empirical basis for

verifying the cost equations.

With this qualification in mind, let us compare the estimated cost functions of

table 4.3 against cost functions developed from sample project data drawn from

the Solar Demonstration Program [57]. The demonstration project data that was examined identifies total project cost and collector size for 262 nonresidential solar projects for many different kinds of applications. Of these 262

projects, only a few closely matched the systems/buildings treated by this

study. A reasonably close fit, however, was obtained by selecting a sample of

demonstration project data for combined solar space heating/hot water systems

and solar space heating only systems for office buildings. The sample consisted

of 43 projects for low-, mid-, and high-rise office buildings, ranging in size

from about 30 m^ to about 850 m^ (about 320 ft^ to about 9,150 ft^) of collector area, with a mean size of about 200 (about 2,150 ft^), (The data base

contained an insufficient number of service hot water only systems for office

buildings, or systems of any type for retail stores like those treated here, to

allow direct comparisons with each of the cost equations in table 4.3.)

Least squares regression analysis of the size and cost data for the 43

demonstration projects yielded the following equation for materials and labor

costs combined:

$35,764 + $448.22/m2 • area (m^) nj .. /

r\\ _ „ = total purchase and installation costs (40)

($35,764 + $41.64/ft2 . area (ft2))

Separate functions for material and labor are derivable from equation 40 by

attributing the same proportions of labor and materials costs to total costs as reflected in the equations in table 4.1:

$21,516 + $390.10/m2 • area (m2)

= materials costs

($21,516 + $36.24/ft2 . area (ft2))

$14,248 + $58,13/m^ • area (m^)

„ - = labor costs

($14,248 + $5.40/ft2 * area (ft2))

(41)

(42)

Figure 4.3 plots and compares the NBS cost function for the combined space

heating/hot water system for new construction, from table 4.1, with the

regression-fitted cost function for the 43 demonstration projects. The NBS cost

function (solid line) indicates higher fixed costs than the regression-fitted

cost function (dashed line), and, hence, produces a larger cost estimate than

the latter for smaller projects. The regression-fitted cost function, however,

indicates higher variable cost per unit of collector area, causing it to predict

higher costs for larger projects. The NBS cost function, with its lower variable

76
cost, will result in a larger optimized size than the regression-fitted curve. For a given location, the two cost functions will produce approximately the same estimate of acquisition costs for a system with about 150 m^ (1,600 ft^)

of collectors — the point of intersection of the two functions shown in Figure

4.3.

Although the remaining demonstration projects were less descriptive of the

systems/buildings treated here than those in the above sample, they were also

evaluated for comparison. The application of least-squares regression analysis

to a sample of demonstration projects comprising 63 solar service hot water

systems for the diversity of nonresidential buildings produced estimates of

fixed and variable costs lower than those based on the sample of 43 combined

space heatlng/hot water and space heating only systems for office buildings: a fixed cost of $27,378 versus $35,764, and a variable cost of $347.15/m2 versus $448.22/m2 ($32.25/ft2 versus $41.64/ft^), respectively.

Applying least squares regression analysis to a sample of 104 combined solar

space heating/hot water demonstration projects for the diversity of nonresidential buildings — again a loose comparison with the designated systems —

produced a significantly higher estimate of fixed cost than the office building

data sample ($42,420 versus $35,764) and an estimate of variable cost just

slightly higher $460.71/m2 versus $448.22/ra2 or ($42.80/ft^ versus $41.64/ft^).

Performing the regression analysis for solar service hot water systems only for

a sample of new versus existing buildings produced little difference in cost

estimates between the new and the existing buildings. However, the same analysis for combined solar space heating/hot water systems resulted in a substantial difference in fixed costs between the new and the existing building applications. Based on 23 demonstration retrofit projects, the regression—fitted

cost equation showed $58,224 of fixed costs and $470.29/m2 ($43.69/ft2) of

variable cost; for 30 new demonstration projects, the regression-fitted cost

equation showed $24,323 in fixed costs and $464.69/m2 ($43.17/ft2) in variable

costs. This finding supports the distinction made in table 4.3 between the

cost estimates for new versus existing building applications.

To provide further perspective on recent solar acquisition costs, table 4.5

shows a grouped frequency distribution, by total system cost per m^ (ft^) of

collector area, of the total of 262 solar demonstration projects installed on a wide diversity of new and existing nonresidential buildings during the past

several years. This table reveals a wide range of solar acquisition costs. It

shows an average system cost of $1,076 or less per m^ ($100 or less per ft^) of

collector for 83 percent of the nonresidential demonstration systems.

Hence, although it is difficult to estimate solar acquisition costs for the

systems under study with a high degree of confidence, estimates in the range of

those shown in table 4.3 appear reasonable. However, in light of the uncertainty attached to the estimated values, the case study results were tested for

sensitivity to lower and upper bound cost estimates equal to one-half and one- and-one-half, respectively, of the base-case estimates of table 4.3. Furthermore, for each of the case studies, the system acquisition costs necessary for

solar to break even over the life cycle were calculated.

77
Figure

4.3

Comparison

of

Two

Cost

Functions:

The

NBS-Estimated

Cost

Function

fora

Combined

Space

Heating/Hot

Water

System

for

New

Construction

Versus

the

Regression-fitted

Cost

Function

for

43

Sample

Demonstration

Projects

}soo uouismbov ujaisAs |B101 $ coo3Pncoo

!5

<uCu

di u

03o

W)

C!

ca

0)

03o

03a

CO T3

03d

T-l • JQ ^o 0

CJ • uo mvO. d \D oH CSJ U <o- ud + d

<4-1 ro

4J lO

CO 0k Oo vOo>

CO

0k § 0

03

0) . •H

03

78 ^

The

regression-fitted

cost

function

for

43

combined

space

heating/hot

water

and

space

heating

only

systems

for

new

and

existing

low-,

medium-,

and

highrise

office

demonstration

projects

(i.e.,

$35,764 +

$448.22 •

m2).
4.4 ENERGY PRICES

To estimate life-cycle energy savings, current and projected prices of energy

were needed. The Energy Information Administration (EIA) of DoE provided a single source of reported and forecasted prices for the three types of energy

considered in the case studies — natural gas, distillate oil, and electricity.

The "base-year" prices used in the model were regional market prices. ^ Future

prices were Incorporated into the model through the use of compound annual

rates of change, derived by comparing base-year prices with EIA's projected

benchmark prices for 1985, 1990, and 1995, and adjusting for an estimated rate

of general price inflation.

The estimated annual rates of projected price escalation are uniform within

each of the three time intervals: from the base-year to mid-1985, from mid1985 to raid-1990, and from mid-1990 to mld-1995 and beyond. The energy prices

selected for use apply specifically to commercial buildings as distinct from

the prices that would apply to residential and industrial uses.

EIA's price projections include ranges of possible future prices for each type

of energy, based on different sets of assumptions about demand and supply conditions. For example, figure 4.4 illustrates the range of projected prices for

oil as given in EIA's 1979 Annual Report to Congress [38]. The table insert to

the graph indicates the nature of the demand/supply assumptions underlying each

curve.

The selection of a future price scenario for the case studies was guided by DoE.

This selection shifted during the course of the study from a "medium price" case early in the study to a "high price" case later in the study.

^ The EIA reports projected current market prices on a national basis in

Short-Term Energy Outlook [58]. (Current regional price estimates, developed

from the national data, are published periodically in the Federal Register

[59] as part of the DoE Life Cycle Cost Rule.)

2 The benchmark regional energy prices projected by EIA's Mid-Term Energy

Forecasting System (MEFS), an integrated computer model of the domestic energy

system with explicit representation at the regional level of the projected

supply and demand for energy sources, the costs of petroleum refining, electricity generation, and energy transportation, and the price elasticity of

energy demand. These benchmark prices, given without inflation, are reported

in an EIA Analysis Report entitled, Mid-Term Energy Supply and and Demand,

1985-1995 [60]. The benchmark prices and the derived real escalation rates

are published periodically in the Federal Register [59] as part of the Life

Cycle Cost Rule, and in the Life- Cycle Costing Manual for the Federal Energy

Management Program [61]. For use in this study, the derived real escalation

rates were converted to nominal rates by incorporating a 6 percent per annum

assumed inflation rate.

79
Figure

4.4

World

Oil

Prices:

Recent

History

and

Alternative

Future

Projections,

1960-1995^

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to

80

U.S.

Department

of

Energy,

Energy

Information

Administration,

Annual

Report

to

Congress,

1979,

Vol.

DoE/EIA-0173(79)

[38],
Table

4.5

Acquisition

Costs

for

262

Nonresidentlal

Solar

Demonstration

Projects,

By

Type

of

System®

COB

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81
Case studies were performed for three different sets of energy price data,

reflecting ongoing price updates, revisions in demand and supply assumptions,

and the above-mentioned change in the selection of a price scenario, all of

which occurred during the course of the study. For the analysis of service hot

water systems, energy prices and escalation rates as of December 1978 for the

"medium price scenario" for the commercial sector were used. (They were paired

with solar acquisition costs also in 1978 dollars for the solar hot water case

studies.) These 1978 energy prices and escalation rates are shown in table

4.6. Since these data do not fully reflect the large actual increases in oil

prices during 1979, the cost-effectiveness results for service hot water only

systems (given in section 5.1) are likely low, particularly for the cases involving fuel oil. (The calculated break-even energy prices given in section

5.1, however, allow a comparison with current energy prices and projections.)

The case studies for the combined space heating/hot water systems were performed

using DoE’s early-1980 revised price projections and the "high price scenario."

These 1980 prices and escalation rates are shown in table 4.7.

A comparison of the two sets of data in tables 4.6 and 4.7, approximately one year apart in time, reveals a substantial upward revision in the base-year data,

but, in most cases, comparable or lower projected long-term rates of escalation.

In mid-1980, EIA again revised the base-year energy prices and projected energy

price escalation rates. The effects on the case study results were assessed

and are reported in the sensitivity analysis of section 5.3. The energy price

data used in the sensitivity analysis are given in table 4.8.

The effect on life-cycle energy savings of these several data revisions is

highlighted in table 4.9. Part A of table 4.9 shows for each of the three types

of energy and for each region of the country, the total present value of 1 GJ

(1 million Btu) purchased or saved each year over 20 years, based on the prices

in table 4.6. Part B of table 4.9 shows the same information, based on the

prices in table 4.8. Within either portion of the table, the substantial variation among regions and among energy types is indicative of a wide variation in

solar economic feasibility dependent on the location and the availability of

alternative fuel sources. The most striking difference, however, is between

parts A and B, particularly for oil, whose 20 year present value more than

doubled when based on the later price estimates. Thus, taking a long-run view

of energy savings through the life-cycle costing format does not preclude

substantial changes in estimated results, even in the very short run, due to

rapid, unanticipated movements in price.

4.5 FINANCIAL VARIABLES—DISCOUNT , BORROWING, AND INFLATION RATES

4.5.1 Discount Rate and Inflation

The discount rate used by a corporate firm should reflect the rate of return

required by the ultimate investors - stock and bond holders - for use of their

money. The before-tax rate of return required by Investors generally has at

1 The EIA energy price projections have since undergone further revision. [59].

82
Table 4.6 1978 Energy Prices and Projected Rates of Change^

(Base-Case Data for Service Hot Water System Analysis of Section 5.1)

Part A. Commercial Energy Prices in 1978 (1978 $/Sales Unit)

Fuel Type and DoE Regions^

Unit Purchased 1 2 3 4 5 6 7 8 9 10

Electricity (kWH) 0.050 0.061 0.043 0.036 0.041 0.034 0.037 0.029 0.038 0.018

Natural Gas (ft^) 0.004 0.003 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002

Distillate (gal) 0.429 0.430 0.413 0.406 0.410 0.404 0.409 0.417 0.412 0.412

Electricity (mJ) 0.014 0.017 0.012 0.010 0.014 0.009 0.010 0.008 0.011 0.005

Natural Gas (m^) 0.131 0.097 0.082 0.061 0.068 0.056 0.054 0.054 0.069 0.086

Distillate (L) 0.113 0.114 0.109 0.107 0.108 0.107 0.108 0.108 0.109 0.109

Part B. Projected Rates of Change (Including Inflation Rate of 6% Per Annum)

c

Electricity

1978-1985 5.4 3.8 6.0 8.0 6.2 9.4 7.4 8.0 6.6 9.6

1985-1990 7.2 8.1 6.6 7,0 6.8 7.8 7.0 7.2 6.8 7.3

1990-1998 6.7 6.7 6.3 6.6 7.1 6.5 6.6 6.8 7.0 7.1

Natural Gas

1978-1985 6.7 9.7 10.8 12.8 12.6 13.3 14,0 16.1 13.6 13.8

1985-1990 10.0 10.1 9.4 8.3 9.8 12.2 11.1 10.3 9.5 9.3

1990-1988 8.5 9.7 10.0 12.7 9.6 10.7 10.9 9.5 6.5 8.4

Distillate Fuel Oil

1978-1985 9.1 9.3 10.1 10.4 9.3 10.0 9.0 9.2 8.1 8.1

1985-1990 10.4 10.3 10.3 10.3 10.5 10.4 10.7 10.5 10.8 10.8

1990-1998 10.7 10.7 10.7 10.6 10.8 10.7 10,9 10.7 8.0 7.7

^ Prices and rates were developed from prices for the period 1977 to 1995, estimated and projected by

the Energy Information Administration (EIA) using its Mid-terra Energy Forecasting Systems (MEFS),

medium-price case, prices per million Btu.

^ DoE Regions consist of the following states: DoE 1: ME, NH, VT, MA, CT, RI; DoE 2: NY, NJ, PR,

VI; DoE 3: PA, MD, WV, VA, DC, DE; DoE 4: KY, TN, NC, SC, MS, AL, GA, FL, CZ; DoE 5: MN, WI, MI,

IL, IN, OH; DoE 6: TX, NM, OK, AR, LA; DoE 7: KS, MO, lA, NE; DoE 8: MT, ND, SD, WY, UT, CO; DoE

9: CA, NV, AZ, HI, TT, AS, GU; DoE 10: WA, OR, ID, AK.

^ A 6% Inflation rate, based on a sample of econometric forecasts of Inflation for the 1978-90 period

ranging from 5% to 7%, was added to the real rates of change derived from EIA energy prices.

83
Table 4.7 Early 1980 Energy Prices and Projected Rates of Change

(Base-Case Data for Combined Space Heating/Hot Water System Analysis of Section 5.2)

Part A. Commercial Energy Prices in 1980 (1980 $/Sales Unit)

Fuel Type and Unit Purchased

DoE Regions'’

1 2 3 4 5 6 7 8 9 10 Average

Electricity (kWH) 0.063 0.076 0.055 0.045 0.051 0.042 0.045 0.036 0.062 0.024 0.060 Natural Gas (ft^) 0.006 0.005 0.005 0.004 0.004 0.003 0.003 0.003 0.004 0.004 0.004 Distillate (gal) 0.863 0.854 0.854 0.821 0.821 0.763 0.804 0.788 0.821 0.804 0.829

Electricity (mj) 0.018 0.021 0.015 0.013 0.014 0.012 0.013 0.010 0.017 0.006 0.014 Natural Gas (m^) 0.212 0.177 0.177 0.141 0.141 0.106 0.106 0.106 0.141 0.141 0.141 Distillate (L) 0.228 0.226 0.226 0.217 0.217 0.202 0.212 0.208 0.217 0.212 0.219

Part B. Projected Rates of Change (Including Inflation Rate of 6% Per Annum)'’

Electricity

1980-1985 8.66 5.37 5.29 6.86 6.64 11.72 5.87 8.89 7.97 6.65 7.29 1985-1990 4.64 7.45 5.90 6.75 6.47 7.04 5.88 4.75 5.95 7.51 6.37 1990-1995-2000 5.52 5.17 6.42 6.67 6.56 6.18 5.32 6.46 5.16 7.16 6.14

Natural Gas

1980-1985 1.35 3.74 3.86 6.17 8.32 11.84 9.92 18.11 12.42 13.29 8.80 1985-1990 9.63 10.37 10.14 8.17 7.61 10.30 11.22 8.36 13.19 7.29 8.75 1990-1995-2000 8.07 8.39 8.15 9.12 8.87 8.64 8.33 7.60 5.42 6.47 8.13

Distillate Fuel Oil

1980-1985 7.40 7.91 8.07 8.93 7.89 10.08 8.17 8.75 6.92 7.36 8.17 1985-1990 9.59 9.55 9.56 9.59 9.75 9.66 9.74 9.79 9.85 9.85 9.60 1990-1995-2000 8.35 8.32 8.31 8.33 8.43 8.37 8.47 8.43 8.49 8.49 8.39

® Prices and rates are for the period beginning 1980, as projected by EIA using a short-term forecasting model for

the 1980 prices and MEFS, hlgh-prlce case, to project future prices. These prices were adopted by DoE for Federal

solar and energy conservation analyses performed during the period January to October of 1980. The real prices and rates upon which those were based are reported in the Federal Register , January 23, 1980 [59].

DoE Regions consist of the following states: DoE 1: ME, NH, VT, MA, CT, RI; DoE 2: NY, NJ, PR, VI; DoE 3: PA, MD, WV, VA, DC, DE; DoE 4: KY, TN, NC, SC, MS, AL, GA, FL, CZ; DoE 5: MN, WI, MI, IL, IN, OH; DoE 6: TX, NM,

OK, AR, LA; DoE 7: KS, MO, lA, NE; DoE 8: MT, ND, SD, WY, UT, CO; DoE 9: CA, NV, AZ, HI, TT, AS, GU; DoE 10: WA, OR, ID, AK.

^ A 6% inflation rate, based on a sample of econometric forecasts of inflation for the 1978-90 period ranging from

5% to 7%, was added to the real rates of change derived from EIA energy prices.

84
Table 4.8 Mid-1980 Energy Prices and Projected Rates of Change'

(Energy Prices used In the Sensitivity Analysis of Combined Space Heatlng/Hot Water Systems of Section 5.3)

Part A. Commercial Energy Prices in 1980 (1980 $/Sales Unit)

Fuel Type and Unit Purchased

DoE Regions^ US

1 2 3 4 5 6 7 8 9 10 Average

Electricity (kWH) 0.089 0.082 0.063 0.050 0.057 0.060 0.061 0.060 0.073 0.024 0.062 Natural Gas (ft^) 0.004 0.004 0.004 0.003 0.003 0.003 0.003 0.003 0.004 0.004 0.003 Distillate (gal) 0.963 0.972 0.978 0.979 0.940 0.963 0.933 0.943 0.908 0.908 0.958

Electricity (mj) 0.025 0.023 0.018 0.014 0.016 0.017 0.017 0.017 0.020 0.007 0.017 Natural Gas (m^) 0.141 0.141 0.141 0.106 0.106 0.106 0.106 0.106 0.141 0.141 0.106 Distillate (L) 0.254 0.257 0.258 0.259 0.248 0.254 0.246 0.249 0.240 0.240 0.253

Part B. Projected Rates of Change (Including Inflation Rate of 6% Per Annum) ^

Electricity

Mid-1980-1985 5.99 5.98 5.99 5.99 5.98 5.99 5.99 5.98 5.99 5.98 5.98 Mid-1985-1990 5.80 5.32 6.94 7.77 7.13 7.72 5.33 3.22 6.46 10.21 6.7 Mid-1990-1995-2000 2.18 5.53 5.61 6.51 5.85 5.70 6.00 3.14 3.66 7.20 5.3

Natural Gas

Mid-1980-1985 7.88 7.88 7.86 7.83 7.84 7.84 7.83 7.86 7.88 7.88 7.86 Mid-1985-1990 10.11 9.39 9.82 8.99 9.34 11.58 10.10 10.47 7.76 10.77 9.70 Mld-1990-1995-2000 8.29 8.05 7.91 7.25 7.18 8.83 7.46 8.64 6.31 2.21 7.47

Distillate Fuel Oil

Mid-1980-1985 9.58 9.59 9.58 9.58 9.59 9.58 9.60 9.58 9.58 9.58 9.59 Mid-198-1990 9.08 9.05 9.06 9.05 9.17 9.11 9.19 9.08 9.28 9.28 9.12 Mld-1990-1995-2000 10.32 10.34 10.31 10.30 10.45 10.37 10.47 10.43 10.53 10.53 10.34

^ Prices and rates are for the period beginning mid-1980, as projected by EIA using a short-term forecasting model for mld-1980 prices and MEFS, hlgh-prlce case, to project future prices. These prices were adopted by DoE for Federal solar

and energy conservation analyses performed after September 1980, until such time that the data are further revised. The

real prices and rates upon which these were based are reported in the Federal Register , October 27, 1980 [59].

DoE Regions consist of the following states: DoE 1: ME, NH, VT, MA, CT, RI; DoE 2: NY, NJ, PR, VI; DoE 3: PA, MD,

WV, VA, DC, DE; DoE 4: KY, TN, NC, SC, MS, AL, GA, FL, CZ; DoE 5: MN, WI, MI, IL, IN, OH; DoE 6; TX, NM, OK, AR, LA; DoE 7: KS, MO, lA, NE; DoE 8: MT, ND, SD, WY, UT, CO; DoE 9: CA, NV, AZ, HI, TT, AS, GU; DoE 10: WA, OR, ID, AK.

^ The 6% inflation rate, based on a sample of econometric forecasts of inflation for the 1978-90 period ranging from 5% to

7%, was added to the real rates of change derived from EIA energy prices.

85
Table 4.9 Total Present Value of 1 Million Btu (1 GJ)

of Energy Purchased Each Year Over 20 Years

Part A. Present Value Based on 1978 Prices and Projected Escalation^

Electricity Natural Gas Oil

DoE Region (PV$/GJ) (PV$/MBtu) (PV$/GJ) (PV$/MBtu) (PV$/GJ) (PV$/MBtu)

1 162.70 154.21 49.21 46.64 48.03 45.52

2 198.44 188.08 43.35 41.09 48.43 45.90

3 147.38 139.69 38.58 36.57 48.52 45.99

4 136.43 129.31 32.35 30.66 48.39 45.86

5 144.04 136.52 35.20 33.36 46.33 43.91

6 140.14 132.83 32.07 30.40 46.77 44.92

7 135.63 128.55 31.68 30.03 45.82 43.48

8 109.87 104.14 33.94 32.17 46.97 44.52

9 134.40 127.39 35.53 33.68 42.65 40.42

10 75.49 71.50 45.67 43.29 42.49 40.27

Part B. Present Value Based on Mid-1980 Prices and Projected Escalation^

Electricity Natural Gas Oil

DoE Region (PV$/GJ) (PV$/MBtu) (PV$/GJ) (PV$/MBtu) (PV$/GJ) (PV$/MBtu)

1 283.74 268.95 59.05 55.97 107.92 102.29

2 269.61 255.55 52.80 50.05 109.00 103.32

3 215.60 204.36 48.40 45.88 109.55 103.84

4 177.08 167.85 39.53 37.47 109.66 103.94

5 198.69 188.33 43.69 41.41 106.01 100.48

6 211.26 200.25 46.59 44.16 108.12 102.48

7 201.57 191.06 42.93 40.69 105.37 99.88

8 180.89 171.46 49.05 46.49 105.99 100.46

9 238.66 226.22 48.97 46.42 102.84 97.48

10 92.05 87.25 51.30 48.63 102.84 97.48

^ Calculated from prices and rates of change given in table 4,6.

b Calculated from prices and rates of change given in table 4,8
least four components : 1) compensation for postponing consumption now until a later time in order to purchase debt or equity shares in the firm; 2) compensation for risk; 3) compensation for the effects of Inflation over the lifetime

of the investment; and 4) compensation for the rate at which income is taxed.

The required rate of return is influenced by the ratio of debt-to-equity

financing of the firm, a factor which influences its level of risk. In actual

practice, the discount rate varies widely among firms and investments.

The approach to selecting a discount rate for the case studies was based on the

capital Investment theory published by Modigliani and Miller in 1958 [62], and

modified by these authors in 1963 [63]. Modigliani and Miller demonstrated

that any project which raises the market value of the firm’s debt and equity

shares is worth undertaking. In order to Increase the market value, the return

on the project must be higher than the.-weighted cost of capital to the firm.

The return on debt and equity shares of the firm thus becomes the true cost of

capital to the firm, i.e., it is the appropriate discount rate for that firm to use in making Investment decisions. Modigliani and Miller show that this rate

of return on debt and equity to the firm’s stockholders would be the same for

all firms in the same risk class in a perfectly functioning world with no corporate income taxes. Since current Income tax laws allow the firm to deduct

interest payments from taxable income, the cost of debt financing tends to be

lower than the cost of equity financing; thus the discount rate used by a firm

with a high debt-equity ratio tends to be lower than the discount rate used by

a firm in the same risk class with a lower debt-equity ratio, other conditions

remaining the same. At some point, of course, increased debt can be expected

to increase the riskiness of the firm, and its risk class will change.

To select a discount rate according to this approach, historical trends in

after-tax rates of return to Investors over recent decades were investigated.

Data published by Holland and Myers [64] describing real rates of return over

all grades of nonfinancial common stocks and long-term corporate bonds were

averaged over the different periods shown in table 4.10. The after-tax rate of

7 percent (rounded from 7.06 percent) covering the 1947-75 period was selected

as representative of long-term returns. To this rate was added an estimate of

long-term inflation of six percent per annum, the average of several econometric

forecasts of inflation for the 1978-90 period.! resulting rate of 13 percent is Intended to reflect an after-tax, after-inflation rate of return for a firm with an average debt-equity ratio in an average risk class. The nominal

! At the time this analysis was made, the annual rate of inflation in the U.S. was considerably higher than 6 percent. However, a rate of 6 percent is

intended to represent expectations of average long-term conditions, rather

than what may be a short-texmi condition of double digit inflation.

87
Table 4.10 Cost of Capital — Average Returns on Debt

and Equity in Nonfinancial Corporations^

Period

Nominal Returns

(precent)

Real Rates

(adjusted by CPI)

(percent)

1971-1975 5.50 -1.30

1947-1975 10.41 7.06

1946-1975 9.85 6.00

1975 28.60 21.60

1974 -19.30 -30.80

^ These averages are based on after-tax rates of return developed by Holland

and Myers [64] from profit rate data published by Ibbotson and Sinquefield

[65]. The average rate of return, R^, to all bond and stockholders of

nonfinancial corporations considered as a group is the following:

where

fMV.CD), — ^ ^ ) + Rt<R) —

)

MV. MV.

R^CD) = after-tax rate of return earned in year t on a portfolio of

all the net outstanding debt of nonfinancial corporations.

It includes interest receipts and capital gains or losses.

MV^-CD) = market value of that debt at the start of year t (includes

net interest paid).

MV^ = total market value of all nonfinancial corporate securities

at the start of year t.

R^-(E) = after-tax rate of return earned in year t on all equity

shares of nonfinancial corporations. It includes dividends

and capital gains.

MV^(E) = market value of equity at the start of the year t (includes

net dividend payments).

Holland and Myers determined real returns for each period by subtracting

percentage changes in the consumer price index. Market value of debt,

MV^(D), was found by capitalizing net Interest paid using the Moody’s Baa

interest rate; market value of equity, MVj-(E), was found by capitalizing

dividends using the dividend yield in the Standard and Poor’s Composite

Stock Index. The combined rate of return, R^. , thus represents the average

over all grades of nonfinancial corporate stocks and bonds, or the rate of

return on a medium grade asset, or the expected rate of return on Investment

for a firm in an average risk class.

88
13 percent rate, after-taxes, is equivalent to a nominal before-tax rate of

approximately 26 percent. It is equivalent to a real after-tax rate of

approximately 7 percent.^

4.5.2 Financing Terms

For the purpose of the case studies, the latter approach — that Is, Including

financing terms in the analysis — is taken. The effect of Including financing

terms and assuming deferred payment will generally improve the cost effectiveness of the investment if the after-tax cost of borrowing is less than the

borrower’s after-tax opportunity cost, as reflected in the discount rate, and

will lessen the cost effectiveness if the after-tax cost of borrowing is higher.

Consistent with the prevailing lending practice, more favorable terms were assumed for new construction projects than for retrofit applications. Specifically, solar projects for new buildings were assumed to be financed as part of

a 20-year mortgage at an interest rate of 9-1/4 percent. Solar projects for

existing buildings were assumed financed with a five-year uncollateralized

commercial bank loan at an interest rate of 10-1/2 percent. The assumed downpayment for the combined solar space heatlng/hot water system was 25 percent;

for the solar hot water systems, no initial downpayment was assumed. Since

these loan rates were lower than the 13 percent discount rate used in the case

studies, the financial assumptions improved somewhat the economic feasibility

results

In the benefit-cost analyses of investment decisions by large organizations that

raise sizable amounts of debt and equity funds, project investment costs are generally evaluated as lump-sum initial cash outlays without regard to

financing. 2 The assumption is that projects meeting the rate of return criterion (stated in terms of the weighted cost of capital) will be funded out of the

firm’s available supply of after-tax debt and equity funds, without distinction

in the financing terms for individual projects of a given risk class. But for

the smaller investor without stock and bond issue, financing often is specifically tied to individual projects, and, therefore, affects their comparative

economic feasibility. In this case, it is appropriate to include financing

terms in the benefit-cost analysis.

1 The relationship between a nominal rate of return (l.e., one that includes

inflation) and a real rate (i.e., one that excludes inflation) may be

expressed mathematically as R = r + I + rl, where R = the nominal rate, r = the real rate, and I = the inflation rate. Applying this equation to the

assumed real after-tax rate of return of 7 percent and the assumed inflation

rate of 6 percent, one obtains a nominal rate of return of 13.42 percent

(i.e., .07 + .06 + (.07)(.06) = .1342). For the purpose of this analysis,

the nominal rate is rounded to 13 percent.

2 Note that the issue of including financing terms in benefit-cost analyses is

different from the Issue of the impact of cash flow scheduling on project

feasibility.

89
Financing terras are, of course, variable araong commercial investors and across

time. Keyed to the prime lending rate, a long-term mortgage loan rate can usually be expected to be roughly l-to-2 percent lower than the prime rate, and

an uncollateralized commercial bank loan, roughly l-to-3 percent higher than

the prime rate. Recently, most loan rates have been substantially higher than

than those assumed here. The borrowing rates assumed for the base-case analysis are lower than the actual market rates prevailing in 1981, and were selected

to be representative of rates which may prevail in the longer term. ^ The

Impacts of alternative assumptions about the borrowing rate and downpayment on solar economic feasibility are examined in section 5.3.

4.6 TAXES AND INCENTIVES

Taxes may affect the economics of a solar investment through the routine

operation of income tax laws, property taxes, and sales taxes, as well as through the use of taxes as mechanisms for providing special financial subsidies (incentives) for solar investment. Because tax law is the principal way

through which solar incentives for businesses are currently provided, it is

convenient to discuss assumptions regarding taxes and financial incentives

jointly. Not all financial subsidies or incentives for investing are delivered

through the tax system. For example, grants are not and preferential loan

terms, if available, would not be. The primary solar incentives available

generally to solar commercial Investors, however, are related to taxes; they

are income tax credits and deductions, accelerated depreciation allowances, and

property and sales tax exemptions.

To determine the after-tax value of those energy-related expenses that are deductible from taxable income, the 46 percent maximum Federal corporate tax

rate in effect at the time the case studies were performed was combined with

the applicable state corporate income tax rate, and the combined rate was used

as the marginal income tax rate in the economic evaluation model [67]. The

calculation of the combined Federal/state income tax rates are based on the

assumption that State income taxes are an allowed tax deduction at the Federal

level, but Federal taxes are not deductible at the state level, i.e.,

t = tg + (1 - tg)tf, (43)

where

t = combined State and Federal corporate income tax rate,

tg = state corporate Income tax rate (see table 4.11 for rates), and

tf = Federal corporate income tax rate (46 percent assumed).

The following expenses were assumed deductible from taxable income: fuel costs,

other operating and maintenance costs, capital depreciation on energy-related

^ The Solar Energy and Energy Conservation Bank was authorized by Title V of

the Energy Security Act, the Solar Energy and Energy Conservation Act of

1980 to assist in the financing of residential and commercial solar energy

systems; however, no special provisions for financing were available at the

time of this study [66].

90
Table 4.11 Rates of Sales Tax, Property Tax, and State Income Tax by Clty^

City Sales Tax Rate (%)^

Effective Property

Tax Rate (%)

State Income

Tax Rate (%)

Phoenix 5.0 (labor only) 0 (Full Exemption) 10.5

Los Angeles 6.0 (materials only) 1.25 6.0

Miami 4.0 (materials only) 2.60 5.0

Apalachicola 4.0 (materials only) 1.91 5.0

Boston 5.0 (materials only) 0 (10 year Solar Exemption)^

12.6 (After 10 years)^

9.5

Omaha 4.5 (materials only) 3.5 4.4

Bismarck 3.0 (materials only) 0 (5 year Exemption)

1.5 (after 5 years)

6.0

Charleston 4.0 (materials only) 1.5 6.0

Fort Worth 0 (Full Exemption) 0 (Full Exemption) 4.0

Nashville 4.5 0 (Full Exemption)® 6.0

Seattle 5.4 0 (Full Exemption) 0.46

Washington, D.C. 5.0 (materials only) 1.83 9.9

Madison 4.0 (materials only) 2.65^ 7.9

^ Rates are based on a telephone survey of local taxing districts in early 1980, and are subject to change,

^ Where the sales tax is applicable only to labor or only to materials, it is noted in

parentheses,

^ This exemption has now been extended to 20 years [68] , but the extension is not

reflected in the case studies for Boston.

This rate of 12.6 percent, provided by the City of Boston Assessments Office, appears

too high compared with the rate for Boston of 4.94 percent given by the "Taxable

Property Values and Assessment Sales Price Ratios," 1977 Census of Governments [69],

The higher rate, provided directly by the Assessments Office, is used in the Boston

case studies. However, the choice of rates for the Boston analysis makes little

difference in the results because of the 10-year exemption of property taxes on solar

energy systems and the discounting of those property taxes that are assumed to occur

after the 10-year period.

® This provision has been recently revised and an expiration date of 1988 is now given

[70]. This limitation on the exemption is not reflected in the Nashville case studies

^ The State of Wisconsin has since enacted a law [71] to exempt active solar energy

systems from the property tax, a change which is not reflected in the Madison case

studies.

91
equipment, interest payments on the loan principal outstanding, and property

and sales taxes. The combined income tax rate was used to calculate tax

adjusted values of the above expenses.

Sales tax rates, property tax rates, and state Income tax rates specific to the

13 local taxing districts covered by the study were compiled. These are shown

in table 4.11.

The effective property tax rates shown in table 4.11 were derived from quoted

"nominal” property tax rates by multiplying the nominal rates by the quoted

property assessment levels, stated as a percentage of market value. ^ To

estimate the solar property values to which the effective property tax rates were applied, it was assumed that the solar energy system would increase the

value of the building in the year of purchase (the base year of the life-cycle

cost evaluation) by an amount equal to the system acquisition cost. A decay

factor was applied to the solar property values to cause a decline in value

over the 20-year study period, leaving a residual market value in the 20th

year of 10 percent of the original acquisition costs.

As may be seen in table 4.11, the sales tax and/or the property tax are exempted

in part or in full in some of the cities. These exemptions constitute regional

investment incentives by reducing the effective cost of solar energy in the

locations where they apply.

Depreciation of solar capital costs was based on an allowable system tax life

of 15 years and on a depreciation base of approximately 96 percent of the original system acquisition cost. This depreciation base was derived by assuming

a 10 percent nominal net salvage value (including inflation and net of disposal

costs) at the end of 15 years, and adjusting the nominal net value to a real

dollar basis to reflect the assumed annual inflation rate of 6 percent. To

compute depreciation according to the straight-line method, the net salvage

value was subtracted initially to obtain the depreciation base; whereas, using

a declining balance depreciation method, the initial depreciation base Included

net salvage value, but depreciation ceased when the net salvage value was reached. Solar energy systems applied in new construction were depreciated by

the 150 percent declining balance method applicable at the time the case studies were performed. (According to conventional practice, a switchover to

the straight-line method was provided for at the point that the straight-line

method yielded a higher yearly depreciation allowance than the declining balance method.) Systems retrofitted to existing buildings were depreciated by

the straight-line method.

^ Nominal tax rates and assessment levels were obtained from each of the local

property tax authorities through telephone interview.

2 These assumptions comply with the rules for depreciation specified in section

167 of the Internal Revenue Code of 1954 [29], and were based in part on the

advice of IRS staff concerning component depreciation and allowable tax lives

for commercial building heat systems.

92
Several of the cities selected for study were in states which provided economic

incentives for commercial use of solar energy through the allowance of shortened depreciation periods. States providing this type of incentive were

Arizona, with a 36-month amortization period for solar equipment [72]; Texas,

with a 60-month period [73]; and Massachusetts and Wisconsin, with immediate

deduction of depreciable costs from gross Income in the tax year in which they occur [74, 75]. (These state provisions were in effect as of raid-1980 and are reflected in the analyses; subsequent state depreciation legislation is not

taken into account.)

The difference between the small undepreciated system cost and the somewhat

higher actual residual value assumed to remain at the end of 20 years is subject to capital gains tax. To evaluate the capital gains tax, a combined Federal and state capital gains tax rate was used, based on a Federal corporate

capital gains rate of 28 percent and the state income tax rates shown in table

4.11. (The capital gains tax rates for states vary, but are widely set at the

ordinary income tax rate.) The combined rate was derived as follows:

Teg = tg + .28 (1 - tg), (44)

where

= combined Federal and state capital gains tax rate, and

tg = state ordinary income tax rate for corporations.

Income tax credits are allowed by the Federal government and by several state

governments to purchasers of solar energy systems. By providing a direct reduction in income taxes by the amount of the allowable credit, the Income tax

credit is a much more potent incentive than an income tax deduction of equal

percentage value.

Under the National Energy Tax Act of 1978 [76], purchasers of solar energy

systems were allowed a one-time reduction in their taxes equal to 10 percent of

the purchase and installation costs of solar energy, energy conservation, or wind energy systems acquired after September 30, 1978, for heating and/or

cooling commercial buildings. Qualified investment for the 10 percent energy

tax credit was defined in section 48 (1)(4) of the Act. The energy tax credit

was subsequently raised to 15 percent, the effect of which is examined in the

sensitivity analysis of section 5.3. Solar and wind energy property considered

under existing law to be a structural component of a building does not qualify

additionally for the regular 10 percent investment tax credit. However, property qualifying both as regular investment credit property under existing laws

and as solar and wind energy property (e.g., solar industrial process heat

equipment) would generally be eligible for both tax credits, for a total of 25

percent of the system acquisition cost.

Several of the cities were in states which allowed tax credits against state

tax liability. The allowances in California was a 55 percent tax credit for

system costs less than $12,000, up to a $3,000 maximum (less Federal tax

credits), and a 25 percent tax credit (less Federal tax credits) for system

costs of $12,000 and above. The tax credit was in lieu of a state depreciation

93
allowance [77]. North Dakota allowed a tax credit of 5 percent of system

costs for two years [78].^

It should be noted that a tax credit at the state level is not effectively

equivalent to a Federal tax credit of the same stated percentage. Because

state taxes are an allowable deduction from taxable income at the Federal

level, a credit against the state tax liability is effectively reduced by a percentage equal to the Federal tax rate. For example, given a one-time tax

credit of 10 percent of system cost at the state level, a system cost of

$5,000, and a Federal corporate Income tax rate of 46 percent, the tax savings

would equal $270 (.10 x $5,000 x (1 - .46) = $270), as compared with a tax

savings of $500 from a Federal tax credit of 10 percent.

In order to reduce the number of variables in the economic evaluation model

without sacrificing analytical detail or accuracy, the parameters for two types

of state tax incentives — income tax credit and rapid depreciation writeoff —

were converted to their equivalents in terras of an annual Federal tax credit.

To convert a state tax credit to an equivalent annual Federal tax credit, the

following equation can be used:

Sc(l - tf) = Fc (45)

whe re

S(, = state tax credit (a percent of system cost),

tp = Federal corporate income tax rate, and

F^ = Federal tax credit (a percent of system cost).

To convert an annual depreciation deduction allowable only at the state level

to an equivalent annual Federal tax credit, the following equation was used:

F^ = C t (1 - t) (46)

e L® ^

C

where

F(, = Federal tax credit (a percent of system cost annually),

tg = state corporate income tax rate,

tp = Federal corporate income tax rate,

C = acquisition costs of the solar energy system, and

L = allowable depreciation writeoff period for computing state income

tax.

^ Arizona subsequently enacted a solar tax credit—not reflected in the case studies for Phoenix—equal to 35 percent of defined eligible costs of a solar

energy device for commercial or industrial purposes through 1983, to be

decreased each year thereafter. This tax credit was in lieu of the rapid

depreciation writeoff [79].

94
4.7 OTHER DATA AND ASSUMPTIONS

4.7.1 Operation, Maintenance, and Repair Costs

There is little historical data available upon which to base estimates of

long-run operation, maintenance, and repair costs for solar energy systems.

Often these are somewhat arbitrarily assumed to be equal to one percent of

the initial contract cost, plus Inflation, an assumption adopted for the case studies. The estimate was assumed to cover insurance premiums less reimbursables, uninsured damage to collectors or other components of the solar system,

periodic cleaning of collector surfaces, routine replacement of parts, and

electricity required to operate the system. Electrical energy costs to run the

pumps, fans, and other motors of the active solar energy system were in this

study lumped with nonfuel operation, maintenance, and repair costs, and were

assumed to escalate at the rate of general price inflation.

4.7.2 Solar Energy System Life and Salvage Value

For the case studies, it was assumed that the solar energy system, whether

retrofitted to an existing building or incorporated into new construction,

would have an economic life of 20 years. It was further assumed that building

ownership would not change over that period.

A nominal resale value of 10 percent of the initial system cost was assumed to

remain at the end of the 20-year period. This was considered to be a conservative estimate, since the copper tubing would likely have a significant salvage

value. Furthermore, it is possible that some of the system components would

have remaining service life at the end of 20 years. However, there is little

or no data upon which to base the estimate.

4.8 SUMMARY OF ECONOMIC PARAMETERS COMMON TO ALL CITIES

Table 4.12 summarizes the economic parameters that are uniform for all of the case studies. The table gives the values for those parameters used both for

the service hot water analyses and for the combined space heating/hot water

analyses. The data in table 4.12 are paired with the city-specific data in

the case studies which follow.

95
Table 4.12 Economic Parameters Common to All Cities: Base Case

Economic Parameter

Solar Energy System Materials Cost

Discount Rate (Nominal)

General Rate of Inflation

Loan Interest Rate

New Buildings

Existing Buildings

Down Payment

Service Hot Water Systems

Combined Space Heating/Hot Water Systems

Years Financed

New Buildings

Existing Buildings

Depreciation Method

New Buildings

Existing Buildings

Depreciation Period

Recurring Cost Rate

Resale Value (Nominal)

Federal Tax Credit

Economic Life

Assigned Value

See Materials Component,

table 4.3

13%

6%

9.25%

10.5%

0

25%

20 Years

5 Years

150% Declining Balance

Straight Line

15 Years

1% of Initial System

Cost plus annual

inflation

10% of Initial System

Cost

10% of Initial System

Cost

20 Years

96
5. CASE STUDY RESULTS

This part of the report presents the results obtained from the case studies of

solar economic feasibility. The case studies were based on the buildings,

building systems, solar energy systems, and geographical locations described in

section 3, and the parametric values designated in section 4, The case study

evaluations were performed using a computer program (listed in appendix B)

representing the economic model described in section 2. Note that the results

for the solar service hot water system were based on 1978 estimates of base-year

and future energy prices which understate the increase in oil prices that

occurred between 1978 and 1980, and the combined heating/hot water systems are based on 1980 price data.

First, the case studies for the service hot water systems are given, A summary

overview is provided with comparisons among the case studies. This overview

is followed by detailed results for selected cases.

Second, the case studies for the combined space heating/service water heating

systems are given. Results are presented for selected case studies under initial assumptions, revised assumptions, and alternative sensitivity assumptions.

Thus , a more comprehensive and current evaluation is provided for the combined

systems than for the service hot water systems.

5.1 CASE STUDY RESULTS FOR SOLAR SERVICE HOT WATER HEATING SYSTEMS UNDER

BASE-CASE ASSUMPTIONS

5.1.1 Summary Overview and Comparisons

Tables 5.1 and 5.2 summarize the base-case results of the economic analyses of

solar service hot water systems for the new office building and new retail

store, respectively, for each of the 13 city case studies. Only the net present

value savings are shown; the payback period is not shown because the estimated

net savings were less than zero for all of the cities, based on 1978 data.

For each of the 13 cities listed in column 1 of the two tables, there are two rows of data. The first row, labeled "Base Case" gives the results for the

1978 data and base-case assumptions, including the assumed hot water usage

rates. The second row gives the break-even results, including the calculated

break-even hot water usage rate. The optimal solar fraction given in column 4

is identical in the two rows because the optimal collector area changes at

approximately the same rate as the hot water load changes, thereby yielding a constant optimal solar fraction for a given set of values for technical, economic, and climatological parameters (see section 2.5). The optimal solar collector area (column 5) is higher for the break-even analysis than for the base

case due to the larger hot water load to be met.

Note that the tables show results only for solar as an alternative to electric

resistance heating of service hot water for the new office and new retail store.

The results for the existing buildings and the other fuel sources are not shown

because the break-even collector area for these other cases generally exceeded

the upper size limit on the collector area of about 50 m^ (540 ft^) for office
Table 5.1 Economic Analyses of Solar Hot Water for a New Office Building with an Electric Resistance Backup System; Case Study Results for 13 Cities^

(Based on 1978 Energy Prices and System Costs and on Base-Case Assumptions)

(1) (2) (3) (4) (5) (6) (7)

Optimal PV Net PV Solar

Hot Water Thermal Solar Optimal Energy Energy Savings

Usage Rate Load Fraction Collector Savings minus Solar

City/ Case (gal/min) (107 Btu) (%) Area (ft^) ($) Costs ($)

Phoenix

Base Case .53 4.1 89 167 2,385 - 469

Break-even Case .63 5.2 89 212 3,027 0

Boston

Base Case .50 5.4 81 289 3,416 - 556

Break-even Case .62 6.8 81 359 4,248 0

Bismarck

Base Case .50 5.5 82 223 2,406 - 601

Break-even Case .70 7.8 82 313 3,374 0

Nashville

Base Case .50 4.7 81 213 2,604 - 787

Break-even Case .74 7.0 81 316 3,862 0

Charleston

Base Case .50 4.5 81 180 2,479 -1,174

Break-even Case .89 8.0 81 321 4,422 0

Forth Worth

Base Case .50 3.9 84 161 2,360 -1,221

Break-even Case .90 7.0 84 289 4,230 0

Madison

Base Case .50 5.3 78 227 2,770 -1,364

Break-even Case .93 9.9 78 422 5,157 0

Wash. , D.C.

Base Case .50 4.9 79 216 2,802 -1,446

Break-even Case .94 9.3 79 406 5,267 0

Los Angeles

Base Case .50 3.5 85 143 1,908 -1,470

Break-even Case 1.08 7.5 85 318 4,129 0

Apalachicola

Base Case .50 4.3 82 161 2,422 -1,634

Break-even Case 1.05 9.0 82 338 5,096 0

Miami

Base Case .50 3.9 81 139 2,145 -2,209

Break-even Case 1.37 10.6 81 381 5.883 0

Omaha

Base Case .50 5.2 74 178 2,583 -2,570

Break-even Case 1.46 15.1 74 518 7.520 0

Seattle'^

Base Case .50 5.2 47 135 988 -2,816

Break-even Case tl • 3. • n.a. n.a. n.a. n.a. n.a.

^ Based on the buildings and building systems described in section^ and the data and assumptions

given in section 4, and calculated by the model described in section 2. Results are shown for

solar energy as an alternative to electric resistance heating of service hot water. The com- parison is made on the basis of identical electric systems as backup to the solar energy system

and as the nonsolar alternative to it. The case results for solar as an alternative to distillate oil and natural gas were in all cases less favorable than those shown here for electricity.

The break-even loads for the other cases are depicted in figure 5.1.

^ n.a. means not applicable. (For Seattle, the break-even collector area exceeded significantly the

size limit for which the cost estimating function was assumed to apply.)

Note: For simplicity, the hot water usage rate, the thermal load, and the collector area are given in

English units. Metric equivalents (X) can be found as follows: (X)m3/s = gal/min •0.0379/60.0;

(X)GJ = Btu*1055.87/10^; (X)m2 = ft2.0.0929.

98
1

Table 5.2 Economic Analyses of Solar Hot Water for a New Retail Store with an Electric Resistance Backup System: Case Study Results for 13 Cities®

(Based on 1978 Energy Prices and System Costs and on Base-Case Assumptions)

(1) (2) (3) UT) (5) (6) TtT

Optimal PV Net PV Solar

Hot Water Thermal Solar Opt imal Energy Energy Savings

Usage Rate Load Fraction Collector Savings minus Solar

City/Case (gal/min) (10^ Btu) (%) Area (ft2) ($) Costs ($)

Los Angeles

Base Case .015 .15 91 8 88 -186

Break-even Case .057 .57 91 30 334 0

Phoenix

Base Case .015 .18 88 7 102 -363

Break-even Case .092 1.09 88 45 618 0

Bismarck

Base Case .015 .24 80 9 101 -383

Break-even Case .110 1.78 80 61 753 0

Boston

Base Case .015 .25 77 11 140 -429

Break-even Case .093 1.46 78 70 881 0

Nashville

Base Case .015 .20 76 8 106 -438

Break-even Case .120 1.58 78 69 846 0

Forth Worth

Base Case .015 .17 79 6 96 -500

Break-even Case .140 1.55 82 57 904 0

Charleston

Base Case .015 .19 78 7 102 -520

Break-even Case .150 1.88 78 75 1,013 0

Madison^

Base Case .015 .23 75 9 115 -526

Break-even Case .150 2.22 75 82 o00 0

Wash., D.C.

Base Case .015 .21 74 8 113 -544

Break-even Case .140 2.05 76 78 1,115 0

Apalachicola

Base Case .015 .19 77 6 98 -652

Break-even Case .165 2.04 79 70 1.116 0

Seattle^

Base Case .015 .22 42 5 38 -590

Break-even Case n.a. n.a. n.a. n.a. n.a. n.a. Miami

Base Case .015 .17 81 6 93 -652

Break-even Case .210 2.37 78 78 1,272 0

Omaha

Base Case .015 .22 71 7 107 -730

Break-even Case n.a. n.a. n.a. n.a. n.a. n.a.

® Based on the buildings and building systems described in section 3 and the data and assumptions given

in section 4, and calculated by the model described in section 2. Results are shown for solar energy

as an alternative to electric resistance heating of service hot water. The comparison is made on the

basis of the identical electric systems as backup to the solar energy system and as the nonsolar alternative to it. The case results for solar as an alternative to distillate oil and natural gas were in

all cases less favorable than those shown here for electricity. The break-even loads for the other

cases are depicted in figure 5.1.

^ These break-even results are shown although they exceed somewhat the size limit imposed on the size of

the collector for which the cost estimating function was assumed to apply.

^ n.a. means not applicable. (For Seattle and Omaha, the break-even collector area exceeds significantly

the size limit for which the cost estimating function was assumed to apply.)

Note: For simplicity, the hot water usage rate, the thermal load, and the collector area are given in

English units. Metric equivalents (X) can be found as follows: (X)m^/s = gal/mln *0.0379/60.0;

(X)GJ = Btu*1055.87/10^; (X)m2 = ft2 .0.0929.

99
building systems and about 8 ra^ (85 ft^) for retail store systems for which the

cost estimating functions were assumed to apply. The economic feasibility

results for solar in these other cases were less favorable than those for the

electric resistance alternative in new buildings.

Figure 5.1, part A, summarizes the break-even hot water usage rates for the new

office buildings in each city based on each of the three energy alternatives

and on the 1978 data and base-case assumptions. Part B of the figure shows

the break-even rates for the new retail store based on electricity only. The

other break-even results for the retail store are not shown because the collector size requirements necessary to meet the loads imposed by the break-even

rates generally exceeded the allowable limits.

Of the cases evaluated under the 1978 data and base-case assumptions, the

estimated minimum hot water loads necessary for cost effectiveness exceeded

those designated as representative for the selected buildings, and net present

value losses were estimated (see column 7 of tables 5.1 and 5.2, base-case row

for each city). However, by comparing the estimated hot water usage rates for

the base case with the rates for the break-even case—both in column 2 of

tables 5.1 and 5.2—it can be seen that the solar service hot water system was relatively close to cost effectiveness in some of the cities even based on

the lower 1978 energy prices. With only a slightly larger hot water load than

that assumed for the base case, a break-even outcome was estimated. For example, the solar energy system was estimated to break even relative to the electric resistance heating system in the Boston office building at a hot water

usage rate of 0.62 gal/mln (3.916 x 10“^ m^/s) versus the assumed base case

rate of 0.50 gal/min (3.55 x 10“5 m^/s).

In those cases where the estimated break-even hot water usage rate was close

to the base case rate, cost-effective estimates would likely have resulted if

the building had been slightly larger, if occupancy rate had been larger, or

if the building’s functional use had dictated higher hot water requirements

than those assumed typical of the case study buildings. The estimated breakeven hot water usage rates for solar energy as an alternative to oil would be

lower than those shown if they reflected the higher-than-projected rise in

the oil prices from 1978 to 1980.

New Versus Existing Buildings . The case study evaluations of solar service hot

water systems for both commercial office buildings and retail stores estimated

the new buildings to be much more favorable to the use of solar energy systems

than the existing buildings; in fact, the minimum hot water loads necessary for

system cost effectiveness in the existing retail store generally exceeded the

collector size limits imposed on the extrapolation of the system cost estimating equations of table 4,3. The poorer estimated economic performance of

the systems for the existing buildings reflected three key assumptions: (1)

Higher cost functions were developed to estimate the costs of the systems for

existing buildings. (2) Less favorable depreciation allowances were assumed

for existing building systems (see section 4). (3) Less favorable loan terms were assumed for existing building systems.

100
Figure 5.1 Break-even Results: Minimum Hot Water Usage Rates (Gallons Per Minute) for Solar Energy System Cost Effectiveness®

A. New Office Buildings^

B. New Retail Stores'^

Leqand:

Eitctric

A oil u

1)1 Qas

® Minimum hot water usage rates are based on three alternative fuel types for

the nonsolar alternative energy system with which solar energy is compared.

For usage rates lower than those indicated, solar energy is not estimated to be cost effective; for higher usage rates, net present value savings are estimated.

^ Average hot water usage rates in the representative 3-story office building

occupied five days a week by 300 people was assumed to be .5 gallons per minute (30 gallons per hour or 3.155 x 10”^ m^/s).

^ Average hot water usage rates in the representative 1-story retail store occupied six days per week by 100 people was estimated at .015 gallons per minute (0.9 gallons per hour or 0.095 x 10~^ m^/s). Results are not shown

for oil and gas for retail stores because the collector size necessary to meet the breakeven load exceeded the allowable size limit imposed by the

system cost-estimating equations given in table 4.3.

Note: Metric equivalent hot water usage rates (X) can be found as follows:

(X)m3/s = gal/raln. 0.0379/60.0.

101
Office Buildings Versus Retail Stores . By comparing column 6 of tables 5.1

and 5.2, it further can be seen that estimated total energy savings were much

larger for the office buildings than for the retail stores . However, a comparison between the two tables of columns 2 and 5 also shows show much larger

break-even loads and collector areas for the office buildings than for the

retail stores. Additionally, the collector cost per unit area was assumed

higher for the office buildings than the retail stores. As a result, higher

net dollar losses were estimated (column 7) for the office buildings than

for the retail stores.

City Comparisons . Tables 5.1 and 5.2 array the cities in descending order of

the estimated cost effectiveness of the solar energy systems as compared with

electric systems. For the office building case studies, Phoenix, Boston, and

Bismarck were estimated to be the three most favorable locations for solar

energy.

The favorable outcome for Phoenix, (for solar energy versus electricity) owed

to its high annual solar radiation and relatively high tax incentives. Phoenix

offered no particular locational advantage to the use of solar in terms of

other parameters, such as labor costs to install the systems and the price of

electricity (1978 price of $.038/kWh). A locational disadvantage was the relatively high water pipeline temperatures that effectively lower the energy load

associated with given water usage rates.

Boston's relatively favorable outcome can be attributed primarily to the high

cost of electricity (1978 price of .050/kWh—approximately twice as high as in

most of the rest of the U.S.). Other factors favorable to solar energy were

the relatively high tax incentives in Massachusetts and the relatively large

energy loads resulting from a low water pipeline temperature. A locational

disadvantage was the relatively high cost of labor in Boston.

Bismarck's locational advantages were its relatively low labor costs for solar

installation, its relatively high solar radiation, its low water pipeline temperatures which increased the hot water loads, and its solar energy incentives.

Despite its northerly latitude, its daily radiation values compared favorably

with those of many of the other cities. The major disadvantage for solar energy

versus electricity in Bismarck was the comparatively low cost of electricity.

The least favorable locations were estimated to be Seattle, Omaha, and, perhaps

surprisingly (given its early lead in the use of solar hot water systems),

Miami.! Of the thirteen locations examined, Seattle was found to be the least

favorable to solar energy because of three main factors: substantially lower

electricity prices than in the other locations, relatively high labor costs,

and relatively low radiation values.

! Note, however, the high optimal solar fraction for Miami (81 percent).

Economic feasibility results might be considerably more favorable for a solar

energy system with lower cost/lower efficiency characteristics than that used

for this analysis.

102
Omaha was found to be a relatively unfavorable city for solar energy. Causal

factors were the lack of solar financial incentives, comparatively higher

labor costs, comparatively high water pipeline temperatures, and comparatively

low solar radiation values.

Miami's advantage of high insolation values throughout the year did not offset

the locational drawbacks: a lack of special incentives, high water pipeline

temperatures, and relatively low energy prices.

The break-even hot water usage rates (i.e., the minimum rates necessary for

solar cost effectiveness), shown in the upper map of figure 5.1, illustrate the

variation in solar locational feasibility as a function of the nonsolar fuel

alternative and its price in a given city. For example, with oil as the alternative fuel, Bismarck was estimated to be more favorable for solar cost effectiveness than either Phoenix or Boston, the two most favorable cities for solar

energy with electricity as the alternative. Furthermore, with oil as the alternative fuel, Ft. Worth, Charleston, Apalachicola, and Los Angeles were all more

favorable locations for solar energy than Boston, because the very large price

disparity between Boston and these other cities for electricity did not exist

for oil. As another example, Nashville, a relatively favorable location for

solar energy relative to an electric resistance system was a relatively

unfavorable location with natural gas as the alternative. This is because

natural gas prices were lower than average in Nashville, while electricity

prices were about average.

It can be seen by comparing columns 1 of tables 5.1 and 5.2 that the

solar-favorable ranking of the cities for the retail stores is quite similar to

that for the office buildings, with a striking exception: Los Angeles. The

solar service hot water system in the Los Angeles retail store ranked first

among the retail store locations, both in terms of life-cycle cost effectiveness and in terras of having the smallest break-even load. In contrast, the Los

Angeles location ranked ninth among the office building locations. Los Angeles

was estimated to be a favorable location for the solar service water system in

the retail store because of the higher percentage tax credit in California for

smaller, lower cost systems. ^ For the larger, higher cost systems for the

office buildings, the relatively low electricity costs and high water pipeline

temperatures in Los Angeles prevented the location from being particularly

favorable to solar.

Climatic conditions are often emphasized as the prime factors determining

locational advantages and disadvantages for solar use. However, the locational

variations depicted in tables 5.1 and 5.2 and figure 5.1 reflect the importance

of at least four additional factors that are locationally dependent: (1) energy

prices, (2) labor costs, (3) state and local taxes, and (4) state and local

governmental Incentives.

1 As explained in section 4.6, the effect of the California tax credit in effect

at the time of this analysis was to provide a credit equal to 55 percent of

initial costs for systems costing less than $12,000, up to a $3,000 maximum

tax credit (less Federal tax credits), and a credit of 25 percent for systen#

costing $12,000 or more.

103
5el.2 Detailed Results for Selected Case Studies

For illustrative purposes, two of the case studies are presented in more detail

below. For the Boston and Phoenix case studies city-specific input data and

results of the break-even analysis are summarized in tabular form, and graphs

showing the economic optimization paths are given.

Boston Solar Hot Water Case Studies . Table 5.3 summarizes the city-specific

input data for Boston. The electricity and natural gas prices shown in the

table are approximately 50 percent higher than the corresponding average for

the country. The tax incentives are also relatively high. The water supply

temperature is relatively low. Labor costs are relatively high. To perform

the case studies of solar service hot water for Boston, the city-specific data

in table 5.3 were paired with the set of non-location-specific economic

parameters summarized in table 4.12.

Based on the 1978 data and base-case assumptions, solar energy service hot

water was found to be uneconomical for the selected buildings and systems in

Boston. The break-even analysis estimated that the system would become cost

effective in comparison with electric resistance heating for the new office

building with only a small increase in the hot water usage rate, .62 gal/min

versus the assumed .50 gal/rain (3.916 x 10“^ m^/s vs. 3.55 x 10“^ m^/s). Substantially larger increases in hot water consumption were estimated to be

necessary to achieve solar cost effectiveness in comparison with oil or natural

gas.

The break-even results, based on the 1978 data and base case assumptions, are shown in table 5.4. The first column gives the estimated break-even hot water

usage rates and as the corresponding thermal energy loads. Column 2 gives the

life-cycle energy cost of supplying the break-even hot water load using the

designated conventional fuel. For example, the cost of meeting the break-even

load solely by electricity was estimated at $5,215 in present values dollars.

Column 3 gives the present value cost of meeting the residual break-even hot

water load with a conventional auxiliary energy system when part of the load

is met by solar energy. For example, the table shows that it was estimated to

cost $966 to meet the break-even load with an auxiliary electric resistance

system. Column 4 gives the estimated present value of the energy savings from

having the break-even solar energy system; column 5, the estimated solar fraction; and column 6, the estimated collector area necessary to achieve the

break-even position. For example, it was estimated for the new office building

that a system with 359 ft^ (33 m^) of collector area would provide 81 percent

of the load, with a resulting energy savings of $4,248—just sufficient to

offset the after-tax present value of related investment costs over the life

cycle. Hence, the discounted payback period given in column 7 is equal to 20

years, the assumed life of the system. (For comparison, footnote c to table

5.4 gives the optimization results for the base-case data and assumptions.)

Figure 5.2 depicts the economic optimization paths, derived according to the

model described in section 2.5. In the upper part of the figure the estimated

linear relationship between the optimally sized collector area (measured on the

horizontal axis) and the hot water load (measured on the upper vertical axis)

104
Table 5.3 Economic Parameters Specific to Boston®

Economic Parameter

Labor Cost Adjustment Factor

Current Price Per Unit of Fuel

Electricity

Distillate Fuel Oil

Natural Gas

Sales Tax on Solar Energy System

Materials and/or Labor

Sales Tax on Fuel

or Special Fuel Tax

Fuel Price Escalation Rates, Nominal

Property Tax Rate

and/or Exemption Period

Combined Federal and State Tax Rate

Effective State Tax Credit

(see table 4.11)

Number of Years for Which Credit Applied

Combined Tax Credit

Assigned Value

1.04

$0.050/kWh ($0,014 mJ)

$0. 43/gal ($0.113A)

$3.70/1,000 ft3 (0.131 m3)

5%

Materials Only

0

0

See tables 4.6, DoE Region 1

12 . 6 %

10 yrs.

51.3%

5.1%

1

15.1%

® Summarized from section 4

105
Table

5.4

Break-even

Results:

Boston,

Hot

Water

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106

Note:

Metric

equivalent

(X)

can

be

calculated

as

follows:

(X)

m^/s=

gal/mln

•0.00379/60.0;

(X)J =

Btu»1055.87;

(X)

m2=

ft2.

0.0929.
is shovm for the new office building and the new retail store, based on electricity as the alternative energy source. The optimal solar fraction was estimated to be constant at 79 percent for the retail store and 81 percent for

the new office building. (Note from section 2.5 that the constant relationship between collector area and hot water load is dependent on the cost functions and other assumptions holding for the higher and lower hot water loads

indicated. For the purpose of illustration, these economic optimization

paths are shown extended beyond the collector size limits assumed for the cost

functions.) The lower left end of each of the two lines in the upper part of

the figure designates the break-even hot water loads and the economically efficient collector sizes to meet these break-even loads.

Net savings lines in the lower part of figure 5.2 correspond to the economic

optimization paths. They intersect the horizontal axis at the point of the

break-even load, indicating a zero net savings, and extend from that point

downward to the right, indicating increasing net savings associated with

increasing hot water loads.

Phoenix Solar Hot Water Case Studies . Table 5.5 summarizes the data for

Phoenix. Tax incentives included sales and property tax exemptions and a deduction from taxable income over three years of the acquisition costs of the

solar energy system. Estimated conventional energy costs in Phoenix were moderate relative to those in the other locations. Solar-related labor costs,

on the other hand, were higher than in most of the other locations. Water

pipeline temperatures were also high. Annual solar radiation values exceeded

substantially those in any of the other cities considered.

Table 5.6 gives the estimated break-even results for solar hot water in Phoenix.

The solar energy systems were estimated to be very nearly cost competitive

against electricity for the new office building, based on the 1978 data and

assumptions. The estimated minimum hot water usage rate was .63 gal/rain

(3.98 X 10“5 tn3/s) versus the .50 gal/mln (3.15 x 10“5 m^/s) of the base-case.

Break-even hot water usage rates based on natural gas or oil for the office

building and electricity for the retail store as the conventional fuels were 5 to 7 times the base-case assumed rates.

Figure 5.3 shows the economic optimization paths for Phoenix. In this case,

the projected expansion paths based on electricity, natural gas, and oil as the auxiliary fuels are shown for the new office building. The path based on electricity is shown for the new retail store.

107
TOTAL

LIFE-CYCLE

SAVINGS,

$

ANNUAL

HOT

V^ATER

LOAD,

10^

Btu

ECONOMIC OPTIMIZATION PATHS

F = Fraction Of annual hot water load provided by solar

Figure 5.2 Economic Optimization Paths: Boston

108
Table 5.5 Economic Parameters Specific to Phoenix®

Economic Parameter Assigned Value

Labor Cost Adjustment Factor

Current Price Per Unit of Fuel

1.00

Electricity $0.038/kWh ($0,011 mJ)

Distillate Fuel Oil $0. 412/gal ($0.109i0

Natural Gas $0,194/1,000 ft3 ($0,069 m^)

Sales Tax on Solar Energy System 5%

Materials and/or Labor Labor Only

Sales Tax on Fuel 5%

or Special Fuel Tax 0

Fuel Price Escalation Rates, Nominal See tables 4.6, DoE Region 9

Property Tax Rate

and/or Exemption Period

0

Combined Federal and State Tax Rate 51.7%

Effective State Tax Credit

(see table 4.11)

1.9%

Number of Years for Which Credit Applied 3

Combined Tax Credit 11.9%

^ Summarized from section 4

109
Table

5.6

Break-even

Results:

Phoenix,

Hot

Water

Only'

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CO

vO

CO

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110

Note:

Metric

equivalents

(X)

can

be

calculated

as

follows:

(X)

m^/s=

gal/min»0.

00379/60.0;

(X) J=

Btu»1055.87;

(X)

m2=

ft2.

0.0929.
TOTAL

LIFE-CYCLE

SAVINGS,

$

ANNUAL

HOT

WATER

LOAD,

lO'

Btu

Figure 5.3 Economic Optimization Paths for Phoenix

111

10'’

gal

OF

DISTILLATE

FUELOIL
5.2 CASE STUDY RESULTS FOR SOLAR COMBINED SPACE HEATING AND SERVICE WATER

HEATING SYSTEMS

5.2.1 Results Compared for Two Sets of Assumptions

The economic analyses of the combined space heating/hot water systems were initially performed based on energy price projections of early-1980, given in

table 4.7. A substantial upward revision in base-year energy prices, particularly for oil, was made by DoE in mid-1980, reflecting the faster-than-projected

short-run escalation in oil prices in 1980. Additionally, the Federal energy

investment tax credit was raised from 10 percent to 15 percent. Selected case studies were repeated based on the revised energy price estimates given in

table 4.8, and on the 15 percent Federal investment tax credit. Comparisons of

the two sets of results demonstrates the volatility of solar economic feasibility over time.

Table 5.7 summarizes key measures of economic performance for four cities

(col. 1) and three conventional energy sources (col. 2). The results shown in

columns 3-5 are based on the early-1980 energy price projections (table 4.7)

and the 10 percent Federal investment tax credit. The results shown in columns

6-8 are based on the mid-1980 energy price projections (table 4.8) and the 15

percent Federal investment tax credit. The estimated net losses over the life

cycle were considerably reduced by use of the revised data, and near break-even

outcomes were estimated for the combined solar energy system in Bismarck and

Phoenix as compared with electric resistance heating. (The estimated net losses

for the other case studies were larger than those given in table 5.7).

5.2.2 Break-even Analysis

Tables 5.8 and 5.9 summarize the results of a break-even analysis performed for

initial energy prices, energy price escalation rates, and solar energy systems

costs, holding other assumptions at base-case values. Table 5.8 gives the

estimated break-even prices per unit of the nonsolar fuels. These are the

Initial prices of the fuels necessary to cause solar energy to be minimally

cost effective, assuming that the initial prices will escalate at the rates

forecasted in early 1980. It shows in col. 2, for example, that oil would have

to cost $1.55 per gallon ($0.41 /A) in Bismarck in the base year (as compared

with the assumed price of $0,788 per gallon ($0.21/il) in early 1980) in order

for solar energy to be cost effective, other things being equal. It shows in

col. 3, for example, that the estimated price of oil in early 1980 would have

to escalate 1.82 times faster over the 20 year study period than was projected

in early 1980 in order for solar energy to become cost effective in Bismarck

in comparison with oil.

Table 5.9 indicates in col, 4 the estimated break-even value for solar energy

system costs, given the other assumptions of early 1980, The break-even values

are indicated as a percentage of the base-case system costs calculated according

to the equations in table 4.3. For example, for the solar energy system to

break even relative to an oil-fired system in Bismarck, the optimally sized

solar energy system could cost no more than about half the amount estimated by

table 4.3 cost equations. Relative to electric resistance heating (lower part

112
o

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73

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(V •V M

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•H 3 O rH B N •H 3

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•H 3 3 3 B O

•H E CJ B •3 •H o rH •3 Ph 3 3 3 4H O O 3 .32 3 3 3 3 4J Ph 3 3 CJ •• X X 3 3 H 4-1 •rH 4JO X S 1 1 ^
Table 5,8 Break-even Analysis of Base-Year Fuel Prices and Fuel Escalation

Rates for Selected Cases Based on Early-1980 Assumptions

13 Cities

(1)

Break-even Price

($/Fuel

Sales Unit)^

(2)

Break-even Rate of

Price Escalation

(Base Case Escalation X

Col, 3 Factor)^

(3)

New Office, Oil ($/gal) (Multipliers)

Bismarck 1.55 1.82

Phoenix 2.56 2.43

Charleston 3.13 2.56

Nashville 2.97 2.49

Los Angeles 3.91 2.90

Fort Worth 2.77 2.41

Apalachicola 3.39 2.63

Miami 5.30 3.10

Boston 3.09 2.57

Madison 2.52 2.36

Washington, DC 3.12 2.56

Seattle 3.54 2.78

Omaha 2.76 2.46

Other Buildings,

Bismarck, Oil ($/gal)

New Retail Store 2.20 2.21

Existing Office Building 1.41 1.71

Existing Retail Store 1.90 2.05

Other Fuels

,

Bismarck, New Office

Electric Resistance ($/kWh)

0.059 1.82

Natural Gas ($/l,000 ft3)

8.11 2.06

^ Fuel prices in 1980 that would yield break-even outcomes if escalated over

20 years at the DoE rates forecasted in early 1980.

^ Fuel price escalation rate multipliers that if applied to the DoE rates

forecasted in early 1980 and used in conjunction with the DoE base-year

fuel prices for early 1980 would yield break-even outcomes.

Note; Metric equivalents (X) may be calculated as follows: ($X)/Z = $/galT3,785;

$(X)/mJ = $/kWhO,6; $(X)/m3 = $/ft3T0. 02832

114
Table 5.9 Break-even Analysis of System Cost for Selected Cases Based on Early-1980 Data and Assumptions

qys.i

r'

13 Cities

(1)

Optimal

Solar

Fraction

%

(2)

Optimal

Collector

Area,

ft2

(3)

Break-even

System Cost®

%

(4)

New Office, Oil

System as Solar Alternative

Bismarck 45 4,893 51

Phoenix 78 1,392 32

Charleston 68 1,792 27

Nashville 54 2,777 28

Los Angeles 82 1,203 21

Fort Worth 68 1,866 27

Apalachicola 71 1,534 24

Miami 90 804 16

Boston 47 4,174 28

Madison 46 4,465 33

Washington, DC 54 3,029 27

Seattle 39 3,315 23

Omaha 50 3,785 29

Other Buildings, Bismarck,

Oil System as Solar Alternative

New Retail Store 62 2,886 36

Existing Office Building 35 11,816 56

Existing Retail Store 49 4,696 41

Other Alternative Energy

Systems, Bismarck, New Office

Building

Electric Resistance 45 4,893 61

Natural Gas 45 4,892 31

^ The maximum percentage of fixed and variable system costs, as estimated by

the equations In table 4.3, for which the optimally sized systems would be

cost effective, other factors held constant.

Note: Metric equivalent (X) may be calculated as follows: (X)m2 = ft^ *0.929.

115
of col, 4), the solar energy system could cost as much as 61 percent of the

estimated base-case amount and still break even. (Note that in each of the

above cases, factors other than the value of the parameter specified are held

constant, but the size of the solar energy system is reoptimized for the new

value of the specified parameter).

5.2.3 Effects of Financing, Taxes, Incentives, and Discounting

Figure 5.4 breaks down the changes in estimated solar energy costs and savings

caused by financing, taxes, incentives, and discounting for a selected case study. The illustration is for a new office building in Phoenix with oil as the alternative energy system. The illustration, based on early-1980 data

and assumptions, shows three vertical bars: the first shows adjustments to

the costs of the solar energy system; the second, adjustments to the energy

savings; and the third, net savings resulting from combining adjusted system

costs and adjusted energy savings.

The first vertical bar, labeled "Costs of the Solar Energy System," shows a reduction in the initial contract cost of the system (purchase and installation costs, plus sales taxes) from $76,358 to $63,303 by taking into account

the assumed long-term financing of the investment and by discounting the loan

payments to their present value. (This reduction reflects the fact that the

assumed interest rate on borrowed funds was lower than the firm’s assumed

opportunity cost of capital as expressed by the value of the discount rate.)

Costs are further reduced from $63,303 to $20,560 by deductions from taxable

income for interest pajrments, depreciation, and sales taxes, together with

tax credits at the state and Federal levels, and a credit for salvage value

expected at the end of the system life. Taking into account future nonfuel

operation and maintenance cost (allowing for the associated tax deduction), plus

capital gains taxes, raises long-term solar costs from $20,560 to $24,717, as indicated by the upward pointing arrows next to the shaded portion of the bar.

The second vertical bar, labeled "Energy Savings of the Solar Energy System,"

shows that the dollar fuel savings are effectively reduced from $12,331 to

$5,956 by taking into account that fuel costs are a tax deductible business

expense. Hence, after adjustment, $5,956 of estimated energy savings are paired with $24,717 of estimated solar energy cost, as shown in the third

vertical bar labeled "Net Savings on the Solar Energy System," The outcome is

$18,761 in estimated net losses.

5.3 SENSIVITITY ANALYSIS

5.3.1 Sensitivity of Solar Cost Effectiveness to System Size

To examine the sensitivity of life-cycle costs to the size of the solar energy

system, net dollar savings (losses) were calculated for solar fractions ranging

from 10 to 99 percent, in increments of 10 percent, for the new office building,

with oil as the alternative energy system. This analysis was based on early1980 data and assumptions. Two of the resulting curves, showing net losses,

are plotted in figures 5.5 and 5.6 for Phoenix and Bismarck.

116
3e

(0 3

iH 4J O 3

CO

CO co

Li bO 3 e •H

•H rH

4-) •H C X 3 3 o < o

CO iH

•H •H O O

•3 • 0k 3 X

CO •H33 w O

JC > (X t4

XJ 3 C •H

CJ 3 3 CJ M •H

<4J

K 14-1 CO O 3

X!3 3

IS «

bO • • 3 3

•H 3

CJ 3 3 3 3 3 3 >

T-( •H

4JO

U-l 3 o <4-1 14J

4-1 Cd o3 4-1 CX, 3 e O M O I

in

<U3

bO

•H E

<um

>. (0

(0

(i> « SR ^ ?5 E

0) 0) £ « o

O) I c <u

03 .- — iS 0) o z CO E®M

>«

CO

>.o>

<uc

UJmo

CO®

oac0

•H4-> a1

COm

cd 3C

CO

CO 4J

CO •3000o\1>.

Vj

CO

COo

T3

CO PQ

117

$values

after

adjustments
Phoenix

in in

CO CS4

_L

to

I I I I .1 in in in in

o o> 00 r>-

I in

(O

J—I. -L. -I .1. I L

in in in in

in ^ CO CM

v> ^ <0 ^ 0) z gw

118

Figure

5.5

Solar

Cost

Effectiveness

as

a

Function

of

the

Solar

Fraction:

Phoenix
Bismarck

(/> o

0) 0)^ 2 «»

119

Figure

5.6

Solar

Cost

Effectiveness

as

a

Function

of

Solar

Fraction:

Bismarck
In Phoenix (figure 5.5), estimated net losses, though lowest for a system

meeting 40 to 50 percent of the load, were relatively insensitive to the solar

fraction up to a fraction of about 75 percent. In contrast, estimated net

losses for Bismarck (figure 5.6) Increased rapidly over the whole range as the

solar fraction was raised. (In both cases, the optimization analysis was constrained to nonzero system sizes.)

An examination of similar curves for the other cases showed substantial

variability in the sensitivity of net losses to the solar fraction. In the

cities with high insolation values, for example, Phoenix, Charleston, Miami,

Apalachicola, and Los Angeles, the curves were virtually flat over a wide portion, indicating that for the given data and assumptions the solar fraction

could be varied substantially with little estimated change in net savings or losses. In other cities, the estimated dolar penalty from deviating substantially from the optimized size was considerable.

5.3.2 Sensitivity of Solar Life Cycle Costs to Selected Economic Assumptions:

Phoenix Case Study for New Office Buildings

(Note that the results of this section (5.3.2) are all based on mld-1980 data

and assumptions.)

Down payment . Figure 5.7 illustrates for one case study the sensitivity of the

total life-cycle cost to the size of the down payment. Other things remaining

the same, the lower the downpayment, the lower the life-cycle cost of the solar

energy system, and, hence, the greater its cost effectiveness to the purchaser.

This relationship would be expected only if the discount rate is higher than

the assumed loan rate.

Federal Energy Tax Credit . Figure 5.8 illustrates for a given case study the

relationship between solar life-cycle cost and the size of the Federal energy

tax credit. As would be expected, the life-cycle cost of the solar energy

system declines proportionately with an Increasing tax credit.

Depreciation Period . Figure 5.9 illustrates for a given case study the

relationship between solar life-cycle cost and the length of the depreciation

period. The faster the write-off of system costs, the lower the life-cycle

cost of the system.

System Costs . Figure 5.10 illustrates for a given case study how the total

life-cycle cost of using solar declines with decreasing cost of purchasing and

installing the system.

5.4 THE CHANGING STATUS OF SOLAR ECONOMIC FEASIBILITY

During the course of this study, there were several changes in key data that

dramatically affected the case study results. Furthermore, the outcomes were

found to be sensitive to a number of assumptions that could change in the

future. Table 5.10 compares for several cities the results based on alternative

values for projected energy prices and the size of the Federal tax credit.

Additionally, the impact of a lower percentage downpayment as might result from

120
Present

Value

Life-Cycle

Costs

of

the

Solar

Energy

System,

(VLCC^g)

(thousand

dollars)

Figure 5.7 Life-Cycle costs of the Solar Energy System as a Function of

the Size of the Dovm Payment®

Down Payment as a Percentage of Total Purchase and

Installation Costs

^ Based on mid-1980 data and assumptions, including a 15 percent Federal tax

credit. The results reflect a solar energy system size reoptimized for

each dovm payment percentage.

121
Present

Value

Life-Cycle

Costs

of

the

Solar

Energy

System,

(TLCCga)

(thousand

dollars)

Figure 5.8 Life-Cycle costs of the Solar Energy System as a Function of the Energy Tax Credit^

I I I I I I I 0 5 10 15 20 25 30 35

Energy Tax Credit Percentage

^ Based on mid-1980 data and assumptions, including a 15 percent Federal tax

credit. The results reflect a solar energy system size reoptimized for each

energy tax credit.

122
Figure 5.9 Life-Cycle Costs of the Solar Energy System as a Function of

the Length of the Depreciation Period®

0 5 10 15 20 25

Depreciation Period (Years)

® Based on mid-1980 data and assumptions, including a 15 percent Federal tax

credit. The results reflect a solar energy system size reoptimized for each

j depreciation period.

I 123
Present

Value

Life-Cycle

Costs

of

the

Solar

Energy

System,

(TLCCga)

(thousand

dollars)

Figure 5,10 Life-Cycle Costs of the Solar Energy System as a Function of

the of the Purchase and Installation Cost®

Purchase and Installation Costs of the Solar Energy System as a

Percentage of Base-Case Values

® Based on mid-1980 data and assumptions, including a 15 percent Federal tax

credit. The results reflect a solar energy system size reoptimized for each

purchase and installation cost,

^ Base-case values are those calculated according to the cost-estimating

equations in table 4,3,

124
Table

5.10

The

Changing

Status

of

Solar

Economic

Feasibility

CO

u

(V -H

CJ "O .1-1 0) u

(u

u

>%

X

bo nJ u H

COSu

01

C

H

d)

<0 o

Cu^ 0) O iH O Xi

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125
establishment of the Solar Energy and Energy Conservation Bank is illustrated.

The table shows in some cases a change from net savings to the investor in

response to changes in data and assumptions.

These case studies have demonstrated the economic evaluation methodology and

the estimated cost-effectiveness of solar energy in selected applications. They

also have demonstrated in quantitative terras that the economic feasibility of

solar energy is variable with respect to the nature of the application, time,

and events. Periodic reassessments are necessary for a valid appraisal of its

current status.

126
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132
APPEtroiX A. MATHEMATICAL PROGRAMMING AND STOCHASTIC MODELING: EXPLORATORY

RESEARCH

An exploratory effort was undertaken to assess the feasibility of using

mathematical programming and stochastic modeling as the principal tools for

solar optimization analysis. A mathematical programming approach offers the

potential for optimizing the solar energy system for multiple design parameters

at low computational costs. This capability would obviate the need in the

optimization analysis for the simplifying approach that optimizes the system

only with respect to collector area. A stochastic modeling approach is of

interest because it casts the measures of solar performance in probabilistic

terms, providing critical information for certain types of solar application.

This exploratory investigation of mathematical programming and stochastic

modeling was conducted by the NBS Applied Economics Group as a cooperative

efffort with the Johns Hopkins University and the NBS Center for Applied

Mathematics .

This appendix discusses the importance of multivariate optimization in solar

analysis in light of recent research. It gives a classification framework for

solar design variables, developed by the Johns Hopkins University research

team. The mathematical modeling performed by the same team is described,

together with supporting work performed by the NBS Center for Applied Mathematics. Sample results from the Johns Hopkins Model are compared with results

from a simulation model.

A.l THE IMPORTANCE OF MULTIVARIATE ANALYSIS IN SOLAR DESIGN

In section 2.2, several studies are cited in support of using a single variate

optimization procedure, with collector area as the key design variable. These

studies, however, considered a limited range of building loads. Subsequent

research has suggested that optimal economic results may be significantly sen- sitive to storage volume under certain conditions.

A report of work performed by the Department of Mechanical Engineering of the

University of Toronto for the U.S. Department of Energy, entitled "Solar Space

Heating Using Annual Heat Storage" [1-A] indicates that the minimum life-cycle

cost of a solar energy system is quite sensitive to storage volume if extreme

values of solar fraction or extreme prices for storage are involved. The sen- sitivity occurs for solar fractions greater than 70 percent £r for storage

costs significantly higher than those generally assumed. In addition, this

work finds that, based on typical storage costs, short-term storage in the

range of 10-30 Ibs/ft^ of collector area minimizes solar costs for solar

^ Dr, Arthur E. McGarity and Dr. Charles S. Revelle, of the Department of

Geography and Environmental Engineering, were co-principal investigators of

the research work performed at the Johns Hopkins University for the National

Bureau of Standards [2-A]. Dr. Richard L. Francis, of the Department of

Systems and Industrial Engineering at the University of Florida, and a participant in the NBS "Visiting Mathematical Scientist Program," also

contributed to this effort [3-A].

A-1
fractions below 50 percent, while annual storage minimizes solar costs for

solar fractions greater than 50 percent. Costs are found by the University of

Toronto study to be insensitive to the storage/collector ratio in the 50 percent range of solar fraction.

Figure A. 1 , taken from the University of Toronto study, shows the relationship

between system acquisition costs and the ratio of storage volume to collector

area for systems supplying an energy fraction ranging from 45 to 100 percent.

Over a wide range of storage volume/collector area ratios, the cost curves are steeper the higher the solar fraction. The minimum cost region for solar fractions above 50 percent occurs at a tank volume/collector area ratio or 65-85

Ibs/ft^, This range is characteristic of annual storage systems. The minimum

cost region for smaller solar fractions, of 50 percent and below, occurs at

10-20 Ibs/ft^. ^ This range is characteristic of the short-term storage systems,

suggested by Lof and Tybout [4-A] as well as Klein, Beckman, and Duffie

[5-A], which generally provide for a storage period of from one to three days,

depending on climate factors. In the 45 to 60 percent solar fraction range,

costs appear fairly insensitive to storage/collector area ratios.

Figure A. 12 shows that if the costs of solar storage were three to four times

greater than those assumed in figure A. 11, e.g., Kj = $4. 15/lb^' , the shape of

the solar cost curves would change markedly, with the curves everywhere sensitive to the storage volume/collector area ratio. With the higher solar storage

costs, the minimum cost points are shown to occur in the 10-20 Ibs/ft^ region

for solar fractions of 60 percent and below, and in the 65-85 Ibs/ft^ range for

solar fractions above 80 percent. The minimum cost for systems providing a solar fraction in the range from 60 to 80 percent may occur at either region.

The University of Toronto study indicated that long-term storage systems may be

economically more favorable than short-term storage in the situations indicated.

If long-term storage is used, performance of a system in one month will depend

on performance in previous months.

These findings suggest the need under certain conditions for solar evaluation

methodologies that allow a more flexible and comprehensive economic optimization across multiple design variables (see also [6-A] and [7-A]. Ideally, it

would be unnecessary to assume that relationships among design variables optimized for one collector area, cost structure, and time frame hold for all other

situations.

^ Annual storage systems can be distinguished from short-term storage systems

by the characteristic that annual storage systems reach maximum and minimum

temperatures once a year while short-term systems have temperature cycles

from one to thirty days in length (most typically 1-2 days), depending on the

actual storage-collector sizing configuration and climatological factors.

2 F.C. Hooper ^ al. , Solar Space Heating Systems Using Annual Heat Storage

[1-A].

A-2
Figure A. 1 Sensitivity of Solar Acquisition Cost to Storage Volume/Collector

Area Ratio for Given Solar Fractions and "Typical" Storage Cost

Notation; F refers to solar fraction; Kj refers to the sum of all solar energy

system costs divided by the sum of the surface area of the floor, walls, and

top of the storage tank ($1 , 15/lbs^/^)

^ Solar acquisition costs are described by the function

Csaq = KqV + + K

2 A + K

3

, where

Kq, K2 , and K3 are defined as $2.70 x 10*"^/lb, $35/ft^, and $3,000,

respectively. (See F.C. Hooper, et al., [ 1-A] , figure 2.1, page 15, for

performance data on which the graphs are based.)

Source: F. C. Hooper, et al., Solar Space Heating Systems Using Annual Heat

Storage [1-A, pp. 36 and 38].

A-3
Figure A. 2 Sensitivity of Solar Acquisition Cost to Storage Volume/ Collector

Area Ratio for Given Solar Fractions and High Storage Cost

Notation; F refers to solar fraction; Kj refers to the sum of all solar energy

system costs divided by the sum of the surface area of the floor,

walls, and top of the storage tank ($4. 15/lb^/^)

^ Solar acquisition costs are described by the function

Csaq = KqV + + K

2

A + K

3

, where

Kq, K2 and K

3 are defined as $2.70 x 10~^/lb, $35/ft^, and $3,000,

respectively. See F. C. Hooper ^ al . ,

[ 1-A] , figure 2.1, page 15 for

performance data on which the graphs are based.)

Source: F.C. Hooper et al.. Solar Space Heating Systems Using Annual Heat

Storage [1-A, pp. 36 and 38].

A-4
A. 2 CLASSIFICATION FRAMEWORK FOR SOLAR DESIGN VARIABLES

As a first step towards establishing a more flexible methodological framework,

the Johns Hopkins research team classified solar design variables on the basis

of (1) the degree of influence that each variable has on economic objectives

and on performance requirements, and (2) the nature of the relationships that

each variable bears to other variables.

The first classification designates variables as either "primary" or "secondary;" the second designates them as "independent," "dependent," or "interdependent." For example, collector area is designated a primary design

variable. Interdependent with other variables; and storage insulation thickness

is designated a secondary design variable, independent of other variables at a given site. The overall classification of solar design variables is shown in

table A. 1. Note in the table that collector area, storage volume, and building

load reduction (achieved through energy conservation alterations to the building

envelope) are all designated interdependent variables. The optimal values of

these variables depend on cost and performance of other variables in the group.

These interdependent variables can be "traded off" until the lowest cost combination in conjunction with nonsolar energy use is found. The variables classified by McGarity and Revelle as "primary" and "interdependent" are called "decision variables," since their determinations are the most significant decisions

in the design process. The existence of three decision variables — collector

area, storage volume, and building load reduction — supports the use of a multivariate optimization technique for solving the optimal solar design

problem.

A. 3 MATHEMATICAL PROGRAMMING AS A POTENTIAL TOOL FOR SOLAR ANALYSIS

Mathematical programming is a technique that is particularly suited for finding

numerical values for a number of variables which in combination optimize an objective function subject to the constraints imposed by required resource

relationships. This technique potentially enables one to find the cost minimizing (or net-savings maximizing) combination of several primary Interdependent

solar decision variables, while allowing restrictions to be placed on the values

these variables can take to reflect the operating characteristics of the components of the solar energy system. Mathematical programming is the recommended

approach in multivariate optimization when inequality constraints prohibit, or make it difficult to use, traditional economic optimization methods employing

the Lagrange Multiplier.

If the objective function and the constraints can be expressed as linear

functions of the design variables to be optimized, the equations can be solved

efficiently by linear programming methods, at low computational cost, with

widely available computer programs. Both economic and physical performance

relationships can be built into the mathematical program. Unlike the simulation method, mathematical programming does not require the value of one design

variable to be specified in order to determine the optimal value of another

design variable.

A-5
Table A. 1 Classification of Solar Design and Load Variables^

Classification of Solar Design Variables

Variable

Degree of Influence Relationship to Other Variables

Primary Secondary Independent Dependent Interdependent

Collector Area / /

Storage Volume / /

Storage Insulation Thickness / /

Fluid Flow

Rates / /

Classification of Variables Affecting Load

Variable

Degree of Influence Relationship to Other Variables

Primary Secondary Independent Dependent Interdependent

Building Load

Reduction / /

Rates of

Thermal

Resistance of

Building / /

Design

Temperatures / /

Weather

Variables / /

Thermal Loads / /

^ Classification according to McGarity and Revelle [2-A].

A-6
The TRNSYS (an acronym for "transient simulation program") solution to an optimization problem with 3 design variables of 10 Intervals each, using 15

minute time steps, would require 35 million computer iterations [2-A]. Each of

more than 35,000 time steps would require 1,000 iterations of the simulation.

Fifteen minute, half-hourly, or hourly time steps are generally used with the

simulation methods. With a time step as large as one day, it is likely that

an iterative approach will take too long to converge—if it converges at all

at each time step, because of the sequential nature of the simulation solution

process. In comparison, a mathematical programming solution to an optimization

problem should require just one run for each of 365 dally time steps.

A. 4 THE MCGARITY-REVELLE SOLAR PERFORMANCE MODElI

McGarlty and Revelle Initially perceived the problem of optimizing solar energy

system design within the format of a cost-minimizing linear program and proceeded to investigate the possibilities of this approach. ^ However, due to

nonlinearities in the functional relationships, they were unsuccessful in using

a linear program to solve for minimum cost values of collector area, storage

volume, and performance. McGarlty and Revelle, however, succeeded in developing

a set of simultaneous equations for predicting the thermal performance of a solar energy system with a specified collector area and storage volume. Furthermore, they found the computational cost of their approach lower than that

of alternative methods of predicting solar energy system performance, because

longer time steps could be used without a significant sacrifice in accuracy.

This linear programming of solar performance represents an Important step and an essential component of a mathematical programming model for determining the

economically optimal system design. This section describes the linear program

formulation, solution properties of the program, and test results for two

sample design problems.

It should be stressed that the linear program described here is a model for

predicting solar performance for a system of a given collector area and storage

volume. Since collector area and storage volume are prespecified, this program

does not find the values of system parameters which minimize cost. Instead it

minimizes the weighted sum of all undesirable flows of energy. It can be used

to find solution values for the following nine variables at optimal system

performance, given the specified collector area and storage volume;

T|-_i = the temperature of storage at the end of the period which precedes

period t;

^ This section is based on the unpublished report prepared by Arthur McGarlty

and Charles Revelle, of the Johns Hopkins University, under the sponsorship

of NBS [2-A], supplemented by supporting notes and oral presentations.

2 For a discussion of the application of linear programming to economic problems,

see a mathematical economics text such as Fundamental Methods of Mathematical

Economics [8-A].

A-7
T(- = the temperature of storage at the end of period t;

Qt = the energy supplied to storage during period t;

Y(. = the energy that would be lost if energy collection is attempted

during periods with insufficient solar radiation;

X{._i = the temperature of the preheat tank at the end of the period which

precedes period t;

= the temperature of the preheat tank at the end of period t;

Wt = the energy which cannot be collected during periods for which the

storage temperature equals the boiling point of water;

Ft = the auxiliary energy supplied for space heating during period t;

Mt = the auxiliary energy supplied for water heating during period t.

The minimized energy flows in the objective function include (1) energy that

would be lost if energy collection is attempted during periods of insufficient

solar radiation, Yt; (2) energy which cannot be collected during periods for

which the storage temperature equals the boiling point of water (energy in

storage is already equal to storage capacity), W^.; (3) auxiliary conventional

energy supplied for space heating during the period, F^; and (4) auxiliary

conventional energy supplied for water heating during the period, Mj-. The

latter two energy flows are given greater weights in the objective function

because the control actions of the actual heating system which they reflect

have priority in the solar heating system. To establish this priority, the

first two energy flows are given fractional weights in the linear program

objective function.

The objective function is subject to a set of equality constraints and a set of

inequality constraints that describe thermal performance of the solar heating

system of the building. The relationships are expressed for every time period

"t”. Fixed values are used for collector area and storage volume to enable the

constraints to be expressed as linear functions of the nine design variables

listed above and, thus, to enable the system of equations to be solved as a linear program.

A schematic diagram of the solar space and water heating system is shown in

figure A. 3. For each time period, a constraint is required for each component

of the system that is designated by a circled letter in the diagram, l.e., the

collector, storage system, preheat tank, hot water auxiliary system, and the

space heating distribution system. Each constraint represents an energy

balance relationship or a control action required to describe the performance

of the system. The objective function establishes the operational control

strategy of the solar energy system. It minimizes undesirable energy flows,

e.g., it prevents use of auxiliary energy when the solar energy supplied is

sufficient.

A-8
Figure

A.3

Diagram

of

a

Solar

Energy

System

with

Designation

of

Component

Constraints

1

I

1 A-9
In algebraic notation, the linear program is formulated as follows

N

Minimize Z aYt + bWt + Ft + Mt

t=l

(A.l)

subject to the following constraining conditions for t=l, Ni

(collector performance relationship)

F^jAUrAt FpAUxAt _ , ( R . -k-)T^. + = F^A[(n^)Et + ULAte^];

(A.2)

t=l ,. . .

,

N

(space heating load relationship — performance of heat exchanger in

space heating distribution system)

eT^ra^CpAt e^mpCpAt

(- ” o + (_»_£_f_)t^_j + F^ > + SgmgCpAtTg; t=l , N (A.3)

(storage tank energy balance)

U^A„ + epA^Cp U„A„ + epifipC^

[( s s P p _P )^t + pC V]Tj. + [( s 9 P ^-P )At + pc V]T^_j

e_m_C_At e„m„C„At

- ( P P P )x^ - ( P ..g-£_)x,_i - Qt- ^t + Wt =

U^AgAtTg - t=l, N (A.4)

(water heating load — performance of auxiliary water heater)

m„C_,At iTL-C„At

^ 7 )Xt + (—.P )Xfi + > nvC AtTg; t=l , N (A.5)

1 See section A. 6 for component performance formulas used in the development of

the linear program format, together with a list of the mathematical notation.

A-10
(pre-heat tank energy balance)

e_m_C_At e_m_C„ + U_A_ ^ , (_E_P_P_I)T^ - (_P_P_P_)Tj._j + [(. P P . P-......-0)At + pCpVpJXj.

+ [ (

- P P P^ P^)At + pCpVp] - UpApAtTg ~ V^; t-1 N

iTmln < Tt < W; Qt 0? ^t 0; > 0;

t=l , •••> n},

(A.6)

(A.7)

where the variables, T{._i , Tj., 0^, Y^., X^-i , X(- , Wj-, Fj- and are the nine

design variables previously defined; a = the weight of the Y variables and b ®

the weight of the W variables; and all other terras are constant parameters

defined as follows

Energy flows and temperatures during each time period indexed by ”t” —

Et = Solar radiation per unit of area during period t, = Space heating load during period t,

Vj^ = water heating load,

9^ = average outdoor temperature during period t,

Tg = building temperature.

Collector parameters —

A = collector area,

= a dimensionless heat transfer efficiency factor,

Ul = collector heat loss coefficient,

= collector transraittance-absorptance product.

Space heating load parameters —

Ag = mass flow rate in space heat exchanger.

Eg = effectiveness of space heat exchanger.

Storage parameters —

Ag = storage surface area,

Cp = heat capacity of water,

Ug = storage heat loss coefficient

V = storage volume,

p = density of water.

A-11
Pre-heat parameters —

Ap = pre-heat surface area.

Up = pre-heat loss coefficient,

Vp = pre-heat volume.

Intertank heat exchange parameters —

Ap = mass flow rate in intertank heat exchanger,

£p = effectiveness of intertank heat exchanger.

Water heating load parameters —

ra,, = water mass flow rate,

Tg = temperature of hot water,

^w = temperature of cold water,

^min ” temperature of the air surrounding the storage tank, and

"^raax “ the boiling point of water.

Coefficients on the design variables T^-i, Tj-, Oj-, Y^-, ^t* ^t» ^t»

M|-, in equations (A.3) - (A.6) can be summarized as:

FoAUTAt

Cl -

2 (A.8)

FrAUlAI

(A.9)

(fcR)t " ^R^ ®t^5 , . . . , N (A.10)

c^nigC At

(A.ll)

h2 -

2

(A.12)

(A.13)

£-2-2.) At + pCpV (A.14)

P-P-P )At + pCpV (A.15)

A-12
(A.16) S3

S4

e_ni_C„At

P P P

UgAgAt T3 - t“l , •••> N

m, C At

f = V

^H1 ~2

m„C„At

^HR t-1 , • • •

N

e„m_C„At

^P1

= .P.EJL^

2

^P2 2

^P3 2 - PCp’,

^P4

e_m_C_ +

= r P P P

2

PCpV,

fpR = UpApAt Tg - Vy; t=l, N

(A.17)

(A. 18)

(A.19)

(A.20)

(A.21)

(A. 22)

(A.23)

(A. 24)

(A.25)

(A.26)

Using the summary form for coefficients of the design variables, the matrix

form of the linear program is the following:

N

Minimize I aY^ + bW^ + + Mt, (A. 27) — t=l

subject to the following constraining conditions:

For t=l, 2, N

A-13
fci fC2 1 -1 0 0 0

^S1 ^S2 -1 0 "fs3 “^84 1

“fpl “fp2 0 0 fp3 fp4 0

fsi fR2 0 0 0 0 0

0 0 0 0 fRl ^H2 0

0 1 0 0 0 0 0

0 1 0 0 0 0 0

0 0“ "Tt-r = (fCR)t

1 0 Tt =

0 -1 Ot = fpR

0 0 Yt > (^BR^t

0 1 Xt-i > Frr

0 0 Xt > Ymln

0 Wt < Ymax

Ft

Jit _

The system of equations can be solved using the linear program described above

in matrix notation, assuming in addition the following relationships to hold;

QtYt = 0, (A.28)

Ff-Gt = 0, (A.29)

MfcNt = 0, (A.30)

y = Qt “ Zt " Rt “ l*t» if Tj- = Tjjjax (A»31)

^ 0, if

where

Y|-, Gj. and N^- = surplus variables which convert equations (A. 2),

(A. 3), and (A. 5) to equalities,

Zf- = energy delivered from storage to space heating during period t, = energy delivered from storage to pre-heat during period t,

Lt = energy lost from storage during period t; = average storage tank temperature in time period t, and other terms

are as defined previously.

Equations (A.28) - (A. 31) are nonlinear and cannot be entered directly into the

linear program, but they represent important physical properties of the real

operation of the combined system. Control actions in the actual combined solar

auxiliary system are intended to cause these conditions to hold. The control

actions are reflected in the linear program through the operation of the objective function in the solution process of minimizing all undesired energy flows.

Variable Yj- is entered directly into the linear program so the collector

relationship is an equality. G^ and appear only indirectly in the surplus

variable form. The surplus variables, in combination with the conditions

expressed in equations (A.28) - (A.31), represent switching functions in the

A-14
operation of the controls of the actual solar/auxiliary system that turn off

undeslred energy flows not needed to meet the thermal load requirements of the

building. The switching functions represent relationships of the types

X for X > 0

X+ = (A.32)

0 for X 0

for collector performance, heat exchanger performance, and auxiliary hot water

heater performance. These components of the system are labeled S, H, and B In

figure 3. A. Equations (A.28) - (A, 31), which can be called "orthogonality con- ditions," can be interpreted as follows: "Qt^t ~ says that energy will not

be supplied to storage when solar radiation Is insufficient for collection

(either or Y^- = 0 for every time period), i.e., the control system should

shut off the collector system during periods of insufficient radiation; "F^-Gt =

0" says auxiliary energy will not be supplied for space heating at the same

time that solar energy supplied to meet the space heating load exceeds the space

heating load (either F^ or = 0 for every time period), i.e., the controls in

the actual system should shut off auxiliary energy supply for space heating when

it is not required; "M^N^ =0" says auxiliary energy will not be delivered from

the auxiliary water heater to meet water heating requirements if the solar

energy supplied for water heating exceeds the water heating load (either or

= 0 for every time period), i.e., the controls in the actual system should

shut off auxiliary energy supply for water heating when it is not required.

Within the linear program, the objective function dictates that these control

operations reflected in the orthogonality conditions shall occur by minimizing

undesired energy flows. The variables Y^, F|-, and will have zero values if

the energy flows they represent are not required. If energy cannot be collected because radiation is insufficient, then Y^- = 0, i.e., energy collection

will not be attempted. If energy supplied to the building for space heating

from solar collection equals or exceeds the space heating load, then F^ = 0,

i.e., auxiliary energy will not be used for space heating. If the temperature

of water in the preheat tank equals or is greater than the desired water temperature, then Mj- = 0, i.e., no auxiliary energy will be used to heat water

further. The Wj- condition is similar: if storage is not full, then energy

will be delivered to storage, and energy will not be wasted so Wj- = 0, However, if storage is full, energy will be wasted by the amount

Since these orthogonality conditions are not imposed explicitly in the

constraint equations of the linear program, the question arises whether they

will always hold mathematically, as they are assumed to do, in the operation

of actual solar/auxiliary systems. Are there situations under which the

objective function will not cause these conditions to hold mathematically in

every time period? If so, the linear program cannot be used to predict solar

performance under these situations.

A-15
A. 5 SOLUTION PROPERTIES OF THE MCGARITY-REVELLE LINEAR PROGRAMI

The characteristics of solutions to the McGarity-Revelle solar energy linear

program was investigated by Francis [3-A] to establish the conditions under

which the orthogonality conditions described above will automatically hold

without being imposed explicitly in the program.

Francis considered using a more general objective function that includes

economic costs of the energy flows. This more general objective function is

expressed as follows:

N

Z = Z [a^-Ft + btNt + CtMt + + etOt + ft^t + gt^t]. (A.33)

t=l

where Fj-, M(- , Nj;, Q(-> W(-, and Gj- are as defined previously, and aj-, bj., c^-,

d(-, et, ft» and gt appropriate cost coefficients (i.e., the costs of

corresponding energy flows). The surplus variables N^ and Gj. which, like

the other elements in the objective function, are undesirable energy flows,

are entered into this more general objective function, and, like the other

energy flows, are given cost coefficients. Any cost coefficient which is not

of interest, or which may be meaningless for a specific context, can be deleted

by setting it equal to zero.

This formulation retains the desirable properties of the McGarity-Revelle

linear program, and facilitates a rigorous evaluation of the mathematical

properties of solutions to the program.

In the four observations which follow, Francis described the situations under

which the "orthogonality" conditions = 0, and F^G^- = 0) will automatically hold without being imposed explicitly; the conditions under which

they will not hold; and the effect which relative values of space heating

costs, storage costs, and collector costs have upon optimum solutions to the

linear program. (The underlying logic in support of the observations is given

in section A. 7.)

Observation 1

(a) If a(. + dj. < et + g^, (A.34)

then, in any optimum feasible solution to the linear program, at least one of

the variables Qj- and Yj- will be zero. That is, energy will not be supplied to

storage at the same time that energy is lost due to attempted energy collection

when there is Insufficient solar radiation.

(b) If aj- + d(- = ej- + g^, (A.35)

1 This discussion of solution properties of the linear programming model is

based on an unpublished paper prepared by Dr. Francis for the NBS Applied

Economics Group [3-A].

A-16
and if there exists an optimum feasible solution to the linear program, then

there exists an optimum feasible solution for which at least one of the

variables Q^- and is zero.

Note that to have at least one of the variables Ot equal to zero is

equivalent to having = 0. Observation 1 provides conditions for which the

nonlinear mathematical constraint is redundant and thus may be Ignored, The

condition will automatically hold without being imposed and thus disrupting the

linearity of the program,

Observation 2

(a) If at + dj. > et + gt, (A,36)

then, in any optimum feasible solution to the linear program, at least one of

the variables Ft and Gt will be zero. That is, auxiliary energy will not be

supplied for space heating at the same time that solar energy supplied to meet

the space heating load exceeds the space heating load, (Note that the sufficient condition in observation 1 (a) cannot hold simultaneously with the

sufficient condition in observation 2 (a),)

(b) If aj. + dt = + gt» (A,37)

and if there exists an optimum feasible solution to the linear program, then

there exists an optimum feasible solution for which at least one of the

variables and is zero.

Observations 1 and 2 imply that the relative values of a^ + d^ and e^- + g^. have

a significant effect upon the optimal feasible solutions to the linear program.

With knowledge of their relative values, conclusions may be drawn as to when

auxiliary energy supplied for space heating, and energy supplied to storage,

may be zero. Note that a^ and dj. both involve space heating costs, while e^. and g^. Involve storage and collection costs. Observations 1 and 2 thus imply

that the relative values of space heating costs and storage and collection

costs have an important effect upon optimum answers to the linear program.

It is Intuitively obvious that an actual solar heating system would exhibit

such effects, and so their prediction by the model serves to corroborate the

modeling approach. Observations 1 and 2 precisely state this implication.

Observation 3

(a) If at + dt + ft > 0, (A,38)

then in any optimum feasible solution to the linear program, at least one of

the variables Ft, Gt , and Wt will be zero. That is, it will never be the case

that all three of the following events occur simultaneously: auxiliary energy

^ The conclusion of observation 1 was conjectured by McGarity and Revelle, but

without proof or specification of conditions under which the conclusion

would hold.

A-17
is supplied for space heating; the solar energy supplied to meet the space

heating load exceeds the space heating load; solar energy cannot be collected

due to the fact that storage temperature is at the boiling point of water.

(b) If at + dt + ft = 0, (A.39)

and if there exists an optimum feasible solution to the linear program, then

there exists an optimum feasible solution for which at least one of the

variables Ft, Gt, and Wt is zero.

In effect, observation 3 says that if the storage temperature is at the boiling

point of water (i.e., more than enough solar energy is available to keep storage at its maximum temperature), then auxiliary energy will not be needed for

space heating at the same time that solar energy is supplied for space heating

in excess of the space heating demand. This is, again, a reassuring

conclusion.

Observation 4

If 6(. + f^ + gt < 0» (A.40)

then the linear program does not have any optimum feasible solution; the

objective function value may be made arbitrarily small. Observation 4 is a minor observation, but it is possibly valuable in checking for a badly formulated problem. A comparison of et + ft + gt with zero avoids trying to find

an optimum feasible solution when none exists. Further, observation 4 identifies a situation for which Yt and Qt could be positive, and indicates that

such a situation would result from a poorly formulated problem. Table A.

summarizes the four above observations, plus two additional observations.

Justifications for the observations are given in section A. 7.

A.6 MATHEMATICAL RELATIONSHIPS EMPLOYED IN THE SOLAR ENERGY LINEAR PROGRAM

AND A LISTING OF MATHEMATICAL NOTATIONS^

A. 6.1 Component Performance Relationships

Collector Performance ; A difference equation form of the collector performance

equation developed by Hottel in 1942 and modified by deWinter in 1975 is used;

0^ = F^A[(n^)E^ - ULAtCf^. - (A.41)

Space Heating Load ;

Ft = [B^. - egmgCpAtCT^ - Tg)]+, Z,. = - F^ (A.42)

1 The material given in this appendix is taken from unpublished notes provided

to NBS by Arthur McGarity pertaining to the mathematical programming work

which he performed at the Johns Hopkins University, under the sponsorship of

NBS.

A-18
Table A. 2 Summary of Observations on the Solution Properties

of the McGarity-Revelle Linear Program®

Observation

Number

Condition on

Cost Coefficients

Conclusions About

Optimum Solution (O.S.)

(1) ®t + 4t < et + gt Yf or Qi- = 0

(2) a^ + d^ > e^ + g^ Ff or Gf = 0

(3) at- + d|- + f(- > 0 Ff- or Gt or Wt = 0

(4) et + ft + gt < 0 no O.S. exists: Qt, Wt, and Yt

can be made arbitrarily large

(5) e^ + f^ + g^ > 0 Ot or Wt or Yt = 0

(6) aj. + d(- + f|- <0 no O.S. exists: Fj-, G^-, and

Wt can be made arbitrarily large

® In the appendix we also conclude that if

0t + 5t >. 0

and/or at + 6t *" <^t 2. and if ^t 2 ^

and/or at + Bt 2 ^

then it will never be the case that both Qt and Yt are positive in an optimum basic feasible solution. The terms at> Bt» ^t defined,

with reference to equations (A.8) - (A.25), as follows:

“t " (^CR^t “

(^Cl^'^t-l “ (^C2)Tt

3t = (fSR^t ~ - (fs2)Tt + (fsS^Xfl + (fs4)^t

«t - (fBR)t - (fBl)Tt-l - (fB2)Tf

(Here Tt and Xt are the storage and preheat tank temperatures,

respectively, at the end of period t.)

Source: Taken from an unpublished paper prepared by Richard Francis,

NBS Center for Applied Mathematics, for the NBS Applied Economics Group [3-A].

A-19
Intertank Heat Exchange;

Storage Losses:

(A.43)

- Tj)

Storage Tank Energy Balance:

(A. 44)

pCpV(Tt - Tfi) = Ot - Zt - Rt - Lt - Wt (A.45)

Wasted Energy:

^

_ r^t - ^t - ^t - ^t ^t ~ ^max

^ ^ 0 if < T

t max

Water Heating Load:

(A.46)

Ut * - “t

Preheat Thermal Loss

(A. 47)

'’t = - Tb)

Preheat Tank Energy Balance:

(A. 48)

pCpVpCXt - Xfi) = Rt - Ut - Pt

A, 6. 2 Combination of Performance Formulas for Linear Program Format

(A.49)

The following relationships among the average tank temperatures (T^ and

and the tank temperatures at the end of the current and preceding periods (T|-,

, and X^, are assumed:

and

(A.50)

X. = i(X. + ) t 2 ^

Collector Performance:

(A.51)

A new variable Y^. is introduced

Q^. - Y^. + F^A[(ta)E^ - UL^tCT^ - \)], Q^Y^. = 0 (A. 52)

A-20
Applying the assumption above, the following is obtained:

FAAUxAt FMUxAt^ » r, v + ( J -

0, 2^2 - > F^A Kn„)Ej + Ui,Ate,l

QtYt = 0

Space Heating Load:

A new variable Gj. is introduced

or

Ft - Gt - Bt - eBmgCpAt(T(. - Tg) , F^-Gt - 0

+ F^ - Gj. = Bj. + Eg^CpAtTg

(A.53)

(A. 54)

(A.55)

2 2

FtGt = 0

Storage Tank Energy Balance :

Using the assumption concerning tank temperatures and the formulas for

intertank heat exchange and storage losses, the following is obtained:

UA+emC UA+emC [(^-—-P P P)At + pCpvlTt + K-A_s

^ P P P )At - pCpVJVi

er^®„C„4t e m„C„At

-( P ? P-)Xt - (. P p-)Xfi -

Qt

- Fj + Wj = (A.56)

Water Heating Load :

The new variable is introduced:

^t - Nt = ^CpAt[Tg - = 0

or

)x,_, + M, - N, “ m„C„AtT , t t W P S’

Preheat Tank Energy Balance

(A.57)

Using the assumption concerning tank temperatures and the formulas for

intertank heat exchange and preheat tank losses, the following is obtained.

A-21
2

- t 2 t-1 2

+ P^pVpl^t + [ ( y At - pS'^pIX^.j- = UpApAtTg - V^ (A.58)

A. 6. 3 Mathematical Notation

Energy flows and temperatures during each time period indexed by "t"

Et = solar radiation per unit of area during period t

Q(. = solar energy delivered to storage during period t

= energy lost from storage during period t

W{- = energy wasted if storage is full during period t

Z(- = energy delivered from storage for space heating during period t

R|- = energy delivered from storage to pre-heat during period t

F(- = auxiliary energy for space heating during period t

Mj- = auxiliary energy for water heating during period t

P|- = energy lost from preheat during period t

Bj- = space heating load during period t

= water heating load

= energy delivered from preheat to water heating during period t

= average outdoor temperature during period t

T(- = storage temperature at the end of period t

Xj. = preheat temperature at the end of period t

Tg = building temperature

(The diagram of the solar energy system shown in Figure A. 4 is labeled according

to the above notation to indicate the physical location of each parameter.)

Other notation used in the appendix:

(1) Collector parameters

A = collector area F^ = a dimensionless heat transfer efficiency factor

A-22
Figure

A.4

Solar

Water

and

Space

Heating

System

Diagram

Showing

Energy

Flow

and

Temperature

Parameters

O CC 0) QC

A-23

Building

Temp.
)

Ul = collector heat loss coefficient

= collector transraittance-absorptance product

(2) Storage parameters

Ag = storage surface area

Cp = heat capacity of water

Ug = storage heat loss coefficient

V = storage volume

p = density of water

(3) Pre-heat parameters

Ap = preheat surface area

Vp = preheat volume

Up = preheat loss coefficient

(4) Inter tank heat exchange parameters

•mp = mass flow rate in intertank heat exchanger

£p = effectiveness of intertank heat exchanger

(5) Space heating load parameters

•m3

= mass flow rate in space heat exchanger

£3 = effectiveness of space heat exchanger

( 6 ) Water heating load parameters

m^ = water mass flow rate

Tg = temperature of hot water

= temperature of cold water

A. 7 DERIVATIONS OF SOLUTION PROPERTIES OF THE MCGARITY-REVELLE LINEAR

PROGRAMMING MODELS

After temperature variables T*s and X’s, (which have no effect upon the analysis)

are incorporated into the terms on the right side, upper and lower bounds on temperatures (which likewise have no effect) are deleted, and surplus variables

Nt and Lp are introduced to convert inequalities into equations, the McGarityRevelle constraints for time period t appear as follows:

Qt

I

rt II Srt (1)

Ft -Ot+Wt = 3t (2)

1 Taken from an unpublished paper prepared by Dr. Richard Francis, Visiting

Mathematician in the NBS Center for Applied Mathematics, for the NBS Applied

Economics Group [3. A].

A-24
-Mt = Y (3)

Ft 5t (4)

-Nt+Mt = e (5)

where all variables displayed are nonnegative.

Network Interpretation The constraints (1) through (5) above may be depicted

as a network flow problem as follows

1 (gt>

i Yt Nt

This network representation can be useful in providing Insight regarding the

nature of the constraints. Each numbered node 1 through 5 corresponds to the

constraint having the same number, and the condition that the input "flow"

equals the output "flow" is the constraint Itself; e.g., for constraint (1),

at+Yt=0t, where at‘*‘^t Qt total input and output flows respectively

for node 1. The "input equals output" conditions for the other nodes represent

the other corresponding constraints. To have nonnegative "flows" means that

the variables are nonnegative. Unit costs are shown in parentheses above the

variables. To Illustrate the use of the flow interpretations, suppose and

Ot are positive, and let At = min(Y^,0t)^0» IF we decrease the value of Yj.

and Oj- by At, we maintain the input-output condition for node 1. Now the flow

into node 2 from node 1 is reduced by At, so we must increase the flow into

node 2 from node 4 by At to compensate. Likewise we must increase the flow into

node 4 by At, to compensate. Thus we reduce Y^- and Qj- by At, and increase Fjand Gj- by the same amount in order to compensate: in effect, a new feasible

solution has been constructed.

The foregoing illustrates much of the justification for observation 1. Other

observations may be justified similarly. In what follows, algebraic justifications (complementary to these "flow" arguments) are given in some detail.

As a parting speculative remark, it seems possible that this "partial" network

structure of the linear program (L.P.) (temperature variables prevent having a

A-25
complete network structure) might be of use in developing a more efficient

algorithm to solve the L.P.

For convenience, the abbreviations F.S. and O.F.S. stand for feasible solution

and optimal feasible solution respectively. The underlying logic needed to

justify the observations is much the same for all the observations. Thus only

the justification for observation 1 is given in substantial detail.

Justification for Observation 1 Given a feasible solution (F.S.) to the L.P.

for which

At = min(Qt,Yt) > 0 (A.60)

construct new variable values 0^=Q^.-At, Y^=Yt-At, F^=F^+At, G^=Gj.+At, leaving

remaining variable values unchanged. Since Qt“Y|-=Qt“Yt=at ,~Ft“0t+Wt-=

-Ft“0t+ and Ft“Gt=Ft-Gj-=6t , the new variable values provide a F.S.

Further, certainly rain(Q{. ,Yj-)=0. Letting z and z' denote the initial and new

objective function values respectively, we have

z-z’=atFt+dtGt+etQt+gtYt ~ (atF^+dtG^+etQ^+gtY^

= -a|-At-d(.At+et.At+gj.At=(et+gt“at“‘^t)^^* (A.61)

(a) Since At>0, if (et+gt~at-dj-)>0 then (et+gt~at“dt)At>0 so z-z’>0, i.e.,

z>z'. Thus the given F.S. could not be an optimal feasible solution (O.F.S.)

Hence for any O.F.S., the minimum of Ot and Y^. must be zero, as otherwise a new

F.S. with smaller objective function value can be constructed.

(b) If (e(.+gj.-aj.-dj-)=0 then z=z*, so if the initial F.S. is an O.F.S., then

the new F.S. is an O.F.S., and one for which the minimum of the new values of

Ot and Yt is zero.

Justification for Observation 2 Given a F.S. to the L.P. for which

At = min(Ft ,Gt)>0, (A. 62)

construct new variable values by decreasing Ft and Gt by At, increasing Ot and

Y^ by At, and leaving other variable values unchanged. It is direct to verify

that the new values provide a F.S., and one for which the minimum of the new

values of Ft and Gt is zero. Further, if z and z' are the initial and new

objective function values respectively, then

z-z'=atFt+dtGt+etQt+gtYt “

[at(Ft-At)+dt(Gt-At)+et(Ot+At)+gt(Yt+At)]

=(at+dt-et“gt)At. (A.63)

(a) Since At>0, if (at+dt-et~gt)>0» then z>z' and so the Initial given F.S.

could not be an O.F.S. Thus for any O.F.S., the minimum of Ft and Gt would

have to be zero.

A-26
(b) If (aj.+dt.-et“gt)’®0 then z=»z' , so if the initial F.S. is an O.F.S. then the

new F.S. is also an O.F.S. , and has the minimum of the new values of F^. and

equal to zero.

Justification for Observation 3 Given a F.S. to the L.P. for which

At = min(Ft ,Gt ,Wt)>0, (A.64)

construct new variable values by decreasing F|., Gj., and Wj. by At, and leaving

other variable values unchanged. It is direct to verify that the new variables

provide a F.S. for which the minimum of the new values of F^, G^-, and Wj- is zero. Further, if z and z' are the initial and new objective function values

respectively, then

z-z’=atFt+d(.Gt+ftWt-at(Ft-At)-dt(Gt-At)-ft(Wt.-At) (A.65)

=(at+dt-+f t)At.

(a) Since A^>0, if (a|-+d|-+f t)>0, then z>z’ and so the initial given F.S.

could not be an O.F.S. Thus for any O.F.S. the minimum of F^, G^, and W^- would

have to be zero.

(b) If (aj-+d^+f |-)=0 then z=z’ , so if the initial F.S. is an O.F.S. then the

new F.S. is also an O.F.S., and one for which the minimum of the new values of

Fj-, Gt, and Wj- are zero.

Justification for Observation 4 Given a F.S. to the L.P., add any positive

term At to Qj-, Wj-, and to obtain new variable values, while leaving other

variable values unchanged. It is direct to verify that the new variables

provide a F.S. Further, if z and z’ denote the initial and new objective

function values respectively, then

z-z’=etQt+ftWt+gtYt-fet(Ot+^t)+ft(Wt+At)+gt(Yt+At)]

(A. 66)

Since At>0, if (e^+f then z>z’ . Further, At can be made arbitrarily

large, implying that z’, the new objective function value, can be made

arbitrarily small, and hence no O.F.S. to the L.P. exists.

Observation 5 (a) If

et+ft+gt>0 (A.67)

then, in any optimum feasible solution to the L.P., at least one of the

variables Q(-, Yj-, and W(- will be zero. That is, it will never be the case that

all three of the following events occur simultaneously: energy is supplied to

storage; energy is wasted because the storage temperature is at the boiling

point of water; energy is lost because energy collection is attempted when

there is insufficient radiation.

A-27
(b) If

6j-+ft+gt=0 (A. 68)

and if there exists an optimum feasible solution to the L.P., then there exists

an optimum feasible solution for which at least one of the variables O^, Y^,

and W{. is zero.

Observation 6 If

a(-+d(-+ft<0 (A. 69)

then the L.P. does not have any optimum feasible solution: the objective

function value may be made arbitrarily small.

Observations 5 and 6 are quite similar to Observations 3 and 4 respectively;

their justifications are also similar, and so they are omitted.

As a final remark, all these observations are related to finding flow-augmenting

paths in network flow optimization problems.

Conclusions from the L.P. Tableau The main conclusion of this section is as follows. If

6t + 5t > 0, (A.70)

and/ or

3t "b <St ^ (A. 71)

and if

6t>0. (A.72)

and/ or

at + 6t 2l (A. 73)

then it will never be the case that both Oj- and are positive in an optimum

basic feasible solution. Also, some conclusions are drawn about what the L.P.

tableau will look like if both Ot and Y^ are basic variables in any basic

feasible solution. Including an optimal basic feasible solution.

Consider the L.P. constraints (1) through (5) for the case when both Ot and Yt

are in the basis. Note that (3) and (5) completely specify Nt and Mt, so that

the constraints of Interest reduce to

/ 0 0 1 0 ^at^

-1 0 -1 1 0 Ot = Wt 1 Bt

^

1 -1 0 0 oy

i'7

A-28
We first observe, if and Yj- are in the basis, that is nonbasic, since

+

That is, Qt, Y^- and W^ are linearly dependent. Thus if Qj- and Yj- are in the

basis, and another variable for time period t is in the basis, it must be

either or

If F{., Qt» and Y^ are in the basis, the basis matrix and its inverse appear as follows

(which verifies linear independence). That part of the tableau which is of

interest appears as follows:

dt ft

Costs Basis BFS Nl Wj,

at Ft «t -1 0

et Qt “(3t+<St) 1 -1

gt Yt -(at+at+^t) 1 -1

Zj-Cj et+St"(at+dt) -(et+ft+gt)

Tableau I

If Gt, Qt» and Yt are in the basis, we have

0

B = 0

-1

1 -1

0

-1

0

0

B

-1

0 -1

-1 0

-1 0

(which verifies linear independence). That part of the tableau which is of

interest appears as follows

at ft

Costs Basis BFS Fjl El

dt Gt «t -1 0

et Qt -3t 1 -1

St Yt -at-8t 1 -1

et+gt“(at+dt) -(et+ft+gt)

A-29
Tableau II

In order for either tableau to specify an optimum basic feasible solution, the

reduced costs (the Zj-Cj) would have to be nonpositive and the basic variables

would have to be nonnegative. Thus, if F^., 0^-, and appear in an optimum

basic feasible solution and Ot are positive, then (from Tableau I)

5t > 0,

3t+5t < 0 (A.74)

at+Bt"*''^t ^ ^ (A.75)

et+gt < at+dt (A.76)

et+ft+gt >. (A.77)

Likewise, if Gj-, Q^, and Y^- appear in an optimum basic feasible solution and Ot

and Yt are positive, then (from Tableau II) 5t ^ 0,

Bt < 0 (A.78)

at+Bt ^ 0 (A.79)

®f'‘St £ at+dt (A. 80)

et+ft+gt 2. (A.81)

Note that conditions (A.76) and (A.77) are identical

respectively, and are in agreement with Observations

Further, either the conditions (A.74) and (A.75), or prove rather stringent, although available data was checking these conditions.

to (A. 80) and (A.81)

2 and 5 respectively.

(A.78) and (A.79), may

Insufficient to permit

When we consider the full L.P. (every time period), if Or and Yf are both

positive in an optimum basic feasible solution, and either F|- or is in the

basis, then the foregoing conclusions again apply. If neither F^ nor G^ is in

the optimal basis (which could conceivably occur when more than one time period

is considered) then the constraints involving Ot and Yt would reduce to

0t“Yt = at

-Ot = Bt

(A.82)

(A.83)

and thus (A. 78) and (A. 79) would still hold if both Ot ^ad Yt are positive.

In conclusion, if both Ot and Yt are positive in an optimum basic feasible

solution, then either (A. 74) and (A. 75), or (A. 78) and (A. 79) must hold. Thus

if (A. 74) or (A.75) is not true, and (A. 78) or (A. 79) is not true, then it will

never be the case that both Ot and Yt are positive in an optimum basic feasible

solution.

A-30
A. 8 TEST RESULTS: THE LINEAR PROGRAM COMPARED WITH TENSYS^

McGarlty and Revelle used the linear program described in section A. 4 to

solve problems in predicting solar heating system performance. In solving

these problems, they encountered no situations for which the orthogonal conditions fail to hold. That is, in actual operation of the linear program in a large number of computer runs, the situations in which violations could occur,

did not arise.

McGarity and Revelle compared the linear programming solution with results

obtained from the simulation program TRNSY [9-A] in the following two problems:

The first problem in performance prediction is for a solar hot water system

for a commercial building in Boulder, Colorado, for one week during the month

of January. The solar energy system is assumed to have a collector area of

65 m^ (700 ft^) and a storage volume of 3,900il (1,030 gal.), and to provide hot

water at a rate of 3,000^ (393 gal.) /day, evenly distributed between the hours

of 7:00 a.m. and 9:00 p.ra. If the temperature in the storage tank falls below

60°C, additional energy is to be added to the water flowing out of the tank to

raise its temperature to 60“C. The temperature of the storage unit at the

beginning of the week is 60° C.

McGarity and ReVelle solved the linear program in time steps of one hour using

the commercially available software linear programming package MPS on an IBM360 computer. They solved the same performance problem with TRNSYS using time

steps of one hour or less.

Solution results from the linear program and from TRNSYS for the temperature

of storage at end of the day (Tt variable) appear in figure A. 5. The two

solutions are almost identical.

The second problem in performance prediction is for a solar space and water

heating system in a single-family residence. The specified collector area

of the system is 50 m^ (538 ft^) and the storage volume is 4005A (1,058 gal.).

The space heating load during each hour and the hourly values of solar radiation and outdoor temperature are supplied on weather computer tapes. The

linear program is solved with time steps of 24 hours using the software MPS

package on an IBM-360 computer. The TRNSYS simulation is used to solve the

same problem in time steps of 15 minutes. The linear programming solution for

temperatures of storage at the end of a weekly period (T^ variable) is compared

with the TRNSYS solution to the same problem in figure A. 6. The solutions are

for the most part quite similar, but there are small differences, possibly due

to the different lengths of time steps used in the two methods — 15 minutes

for TRNSYS and 24 hours for the linear program.

The comparisons demonstrate the accuracy and efficiency of the linear

programming method as a prediction tool for studying solar heating performance

^ This section is based on a report prepared by Arthur McGarity and Charles

Revelle, of the Johns Hopkins University, under sponsorship of NBS [2-A].

A-31
characteristics. Daily temperature and insolation data seem adequate for use with the linear programming technique.

In contrast, dally time steps are not generally adequate for a simulation

solution. The simulation approach depends on sequential calculations as com- pared to the simultaneous solutions of the linear program. With time steps as large as one day, it is quite possible that computer iterations of the simulation will take a long time to converge or will not converge at all at each time

step to give a solution.

A. 9 SUBSEQUENT RESEARCH

The solar mathematical programming approach has since been extended by McGarity,

Revelle, and Cohen [6-A], [7-A]. By employing a hybrid basic descent method,

they were able to generate "optimal area-volume paths" describing the cost

minimizing combinations of collector area and storage volume for providing

different fractions of the heating load.

This multivariate, mathematical programming optimization model, called the

"Finite Difference Direct Calculation Method," produces estimates of thermal

performance using one time step per day with weather and heating load data

averaged over daily intervals. It was developed through a combination of two

other methods: 1) the Direct Calculation Method which solves the nonlinear

differential equation associated with the storage energy balance and 2) the

Simultaneous Solution Method which uses a modified form of the linear program

described in section A. 5 to solve descriptive difference equations for all

time periods being considered.

Preliminary results suggest that the optimal area-volume path is quite

sensitive to solar equipment cost assumptions, and that it is often desirable

to use substantially more storage volume than has been typically suggested in

the literature. These preliminary results support the findings of Hooper et

al. [1-A] discussed in section A. 5. They also support the need for a multivariate optimization approach to solar energy design problems.

A. 10 STOCHASTIC MODELING OF WEATHER VARIABLES^

Weather- related parameters are important determinants of a building's energy

demand and of the performance of a solar energy heating system. These parameters represent energy flows and temperatures that have a random nature. The

intensity of solar radiation, the outdoor temperature, and the windspeed are

all subject to fluctuations independent of other performance factors. Their

variability results in variable performance of the solar energy system.

1 For additional work in this area, see [10-A], [11-A], [12-A].

2 The potential for using stochastic modeling of weather variables in

conjunction with mathematical programming for solar energy analysis was investigated by Arthur McGarity and Charles Revelle of the Johns Hopkins

University under sponsorship of the National Bureau of Standards [2-A],

and expanded by McGarity and Quadir [13-A].

A-32
Temperature

of

Storage

at

the

End

of

the

Day

(Degrees

Centigrade)

Figure A. 5 Comparison of Performance Test Results for a Commercial Solar

Water Heating System in Boulder, Colorado: Linear Program

Versus TRNSYS®

Day • Linear Program

o TRNSYS

^ Linear program and TRNSYS both use hourly time steps in this sample

coiaparison.

Source: Arthur McGarity and Charles Revelle [2-A].

A-33
Figure A. 6 Comparison of Performance Test Results for a Residential Solar

Water and Space Heating System in Madison, Wisconsin; Linear

Program Versus TRNSYS^

• Linear Program

O TRNSYS

^ Linear program uses daily time steps; TRNSYS uses 15-minute time steps.

Source: Arthur McGarity and Charles Revelle [2-A].

A-34
Many of the methods developed for predicting solar performance have used

radiation and temperature data that lead to reasonably accurate "average"

expected performance figures over a several year period. Liu and Jordan

developed general utilizability curves useful in predicting long-term perfor- mance of water heating systems utilizing flat plate collectors [14-A]. They

I used years of monthly meteorological data to construct their curves for predicting long-term average performance. Lof and Tybout used actual data

for a single year [4-A] , while Klein, Beckman, and Duffle constructed monthly

I data for an artificial year by choosing months from an 8-year period which

I reflected average monthly conditions [5- A].

A study of performance during a single year — real or artificial — however,

provides no information about the year-to-year variability of a particular

month's or week's performance. In addition, the data for an artificial year

\ does not allow for serial correlation of weather conditions from one month to

the next, a common feature of hydrologic systems that also depend on the

i weather [15-A].

These are limitations to using the more traditional weather data methods in

predicting solar performance. How important they are depends on the circumstances and objectives of the building owner or system designer. The con7

structed average-year data seem capable of yielding acceptable average long-term

performance predictions. For some building owners, year-to-year variability in

monthly performance of a system may not be important if the "average" or "typical" year provides a reasonable estimate of system performance over a number of

years. Also, in most cases, the serial correlation of weather values from month

to month will likely be small. The serial correlation tends to be small because

i the performance of a system in one month will not be affected greatly by excess

(; energy in storage at the end of the previous month if storage is sized for

'j periods of only a few days — and short-term storage is typical. For other

J building owners, however, it may be Important to know how much variation can be

Ij expected in the amount of auxiliary fuel that will be required with a solar

I energy system. This information is likely to be particularly important to

!! owners of commercial buildings who are given relatively low allocation priority

*! if shortages of fuel oil or natural gas occur.

I To estimate the probability that the predicted performance of a solar energy

Ij system will occur and to estimate the year-to-year variability in monthly

1, performance, stochastic models of meteorological data are needed. A stochastic

! model (a statistical model for treating random phenomena) can be developed by

applying the methods of time series analysis to recorded observations. The

• statistical properties of actual meteorological data collected for a number of

j years can then be used to generate synthetic meteorological data which duplicate

I the statistical properties of the actual data. These synthetic data then

I become the weather parameters used in the solar performance model.

>

I Weather data can be separated into two components. The first component is

i deterministic, representing the long-term characteristics of the weather varj iable, (i.e., radiation, temperature, or windspeed) in a given geographical

; region. The second component is stochastic, or unpredictable, reflecting the

"dynamic behavior" of long-run weather phenomena resulting from random meteoj, rological events. The first can be described by average values (or means) of

, A-35

1
observations of the same time period for different years. The second can be

described by the variance of observations from the average values and by the

correlation among sequential observations ("autocorrelation"). For any time

period, the stochastic component, is given by:

Xi = 7i - Yt, i = 1, 2, . . .

,

n,

where y^^ is the value of the individual observation,

and is the computed deterministic component.

Several stochastic models of solar radiation are reported in the literature.

Goh, Tan, and Brlnkworth describe models for long-term insolation forecasting

using first-order autoregressive equations that pick up autocorrelation in

radiation over two-day periods [16-A, 17-A]. (Autocorrelation over periods

greater than two days has been shown to be insignificant.)

A stochastic model of insolation can be used to generate synthetic data for

any number of years. However, it is of limited value for analyzing the

expected performance of solar heating systems unless synthetic data for the

other driving meteorological forces, such as outdoor temperature can also be

generated. An Important feature of a stochastic model for generating outdoor temperature data is that it must account for the correlation between

dally Insolation and average dally temperatures.

Additional work is necessary to develop a stochastic model for each of the

forces affecting solar heating system performance and building load. Models

are needed that incorporate any correlation which may exist among the different driving forces, such as the day-to-day correlation between solar insolation and outdoor temperature, as well as the cross-correlation between the

insolation and temperature.

Once a complete stochastic model has been developed, it can be used with

solar performance prediction models such as the linear program described above

to derive a probability distribution of the thermal performance of a specified

solar energy system. The linear programming solar performance model described

in section A. 4 is particularly suited to the use of stochastic models of solar

insolation and temperature. Note that the insolation and outdoor temperature

variables all appear on the right side of the constraint equations in the

linear program (section A. 4). Once an initial solution to the linear program

has been obtained for one year of solar energy system operation, results for

different years and different values of temperature and radiation data can be

obtained with little additional computational effort because the matrices of

coefficients from the constraint equations need not be adjusted again either

in the hand calculations or computer operation of the simplex method. One can

find an initial solution with the original data, modify the right side of the

constraint equations, and proceed with the simplex method until a new solution

is found.

In addition, the test results reported in section A. 6 show that daily time

steps are sufficient when the linear programming method is used for performance

A-36
prediction. This greatly facilitates use of either stochastic models or good

average year data. Weather data is available on a dally basis for a large number of cities for several years, and the stochastic models reported in the

literature thus far generate dally data.

In contrast, simulation models require the same amount of computation each time

they are run and generally require runs for a minimum of hourly, preferably 15-

ralnute, time steps. This data is more difficult to obtain from historical

files or to generate by a stochastic model.

APPENDIX A - REFERENCES

1-

A Hooper, F. C., Attwater, C. R. , et al. Solar Space Heating Systems Using

Annual Heat Storage . (Progress Report prepared by the Department of

Mechanical Engineering, University of Toronto for the U.S. Department of

Energy) NTIS COO/2929-5. Washington, D.C. , February 1978.

2-

A McGarity, Arthur E. and Revelle, Charles S. "Mathematical Modeling and

Optimization in the Design and Economic Analysis of Solar Heating

Systems." (Unpublished contractor's report prepared at the Johns

Hopkins University for the National Bureau of Standards), December 1977.

3-

A Francis, Richard L. "Solar Properties of the solar Energy Linear Program

of McGarity and Revelle." (Unpublished paper prepared at the National

Bureau of Standards, Center of Applied Mathematics), January 1978.

4-

A Lof, G.O.G. and Tybout, R. A.. "Solar House Heating." Natural Resources

Journal, Vol. 23, No. 2, April 1970, pp. 268-325.

5-

A Klein, S. A.; Beckman, W. A; and Duffle, J. A. "A Design Procedure for

Solar Heating Systems." Solar Energy , Vol. 18, pp. 111-127.

6-

A McGarity, A. E. "Solar Heating Simulation: Accuracy and Precision of a Model Using Dally Data." SUN II. Proceedings of the 1979 Congress of

the International Solar Energy Society . Atlanta, Georgia, May 1979.

7-

A McGarity, A. E.; Revelle, C. S.; and Cohen, J. L. "Solar Heating System

Performance: Continuous and Finite Difference Solutions for Hourly

and Daily Data." Proceedings of Solar Rising . Meeting of the American

Section of International Solar Energy Society. Philadelphia,

Pennsylvania, May 1981.

8-

A Chiang, Alpha C. Fundamental Methods of Mathematical Economics. 2nd ed.

New York: McGraw-Hill, Inc., 1974.

9-

A TRNSYS - A Transient Simulation Program . Solar Energy Laboratory, Report

38. Madison, Wisconsin: University of Wisconsin, November 1976.

10-A McGarity, A. E. "Optimum Collector Storage Combinations Involving Annual

Cycle Storage." Proceedings of the Solar Energy Storage Options

Workshop , sponsored by the U.S. Department of Energy, San Antonio, Texas,

March 1979.

A-37
11-

A McGarity, A. E. and Revelle, C. S. "Solar Energy Optimization: CostMinimizing Collector-Storage Combinations." Proceedings of the

International Association of Science and Technology for Development

Symposium on Modelling, Planning and Decision in Energy Systems . Montreux, Switzerland, June 1979.

12-

A McGarity, A. E.; Revelle, C. S,; and Cohen, J. L. "Solar Heating Design

with a Performance Requirement." Proceedings of Solar Rising . Meeting

of the American Section of the International Solar Energy Society,

Philadelphia, Pennsylvania, May 1981.

13-

A McGarity, A. E., and Quadlr, I. Z. "Modeling of Renewable Energy Systems:

An Application of Time Series Analysis." Proceedings of the Twelfth

Annual Pittsburgh Modeling and Simulation Conference . Pittsburgh,

Pennsylvania: TJniverslty of Pittsburgh, May 1981.

14-

A Liu, B. Y. H. and Jordan, R. C. "A Rational Procedure for Predicting the

Long-Term Average Performance of Flat-Plate Solar Energy Collectors."

Solar Energy , Vol. 7, No. 2, 1963, pp. 53-71.

15-

A Fiering, M. B. and Jackson, B. B. Synthetic Steam-flows . American

Geographical Union Monograph 1, 1971,

16-

A Goh, T.N. and Tan, K. J. "Stochastic Modeling and Forecasting of Solar

Radiation Data," Solar Energy , Vol. 19, No. 6, 1977, pp, 755-757,

17-

A Brinkworth, B, J. "Autocorrelation and Stochastic Modeling of Insolation

Sequences," Solar Energy, Vol. 18, 1977, pp, 343-347.

A-38
APPENDIX B. COMPUTER PROGRAM USED TO APPLY THE ECONOMIC EVALUATION MODEL TO

SELECTED CASE STUDIES FOR COMBINED SPACE HEATING AND HOT WATER^

The computer program, written In BASIC, Is designed to analyze the technical

and economic performance characteristics of a specified solar energy system

and to search for the size of that system which will minimize life-cycle costs.

Environmental data, such as air temperature, ground temperature, and cloud

cover, pertaining to the solar energy system's technical performance are incorporated in the computer program. The program calculates life-cycle costs with

and without the optimally sized solar energy system, the resulting net savings

or losses in present value dollars, and the discounted payback period if net

savings are positive. If net savings are negative, i.e., if net losses are estimated for the optimally sized system larger than zero, values of key parameters are calculated which will result in an economic equality between the

solar and alternative conventional energy systems. The parameters for which

break-even values are calculated (holding other variables except system size

constant in each case) are (1) the initial fuel price, (2) the future rate of

escalation in fuel price, and (3) the purchase and installation cost of the

solar energy system.

City Data Files . The city specific data used for the case studies are located

in 13 individual files within the program. Each file contains for the

designated city location the following data: (1) monthly average dally radiation on a horizontal surface, (2) cloudiness index, (3) quarterly ground

temperatures, (4) monthly heating load data for each of the four buildings—the new office building, the existing office building, the new retail store, and

the existing retail store, (5) early-1980 fuel prices, (6) fuel price escalation rates, (7) city-specific labor cost adjustment factor, (8) latitude, angle

of collector tilt, and ground reflectance, (9) location-specific tax rates,

including state income tax, sales tax, and property tax rates; fuel tax, if

applicable; and any state income tax credits.

Engineering Data and Assumptions . The total combined energy load to be met is

calculated by the program from (1) the space heating load input by the user and

(2) the hot water load generated by the program based on user-specified rates

of hot water usage, operating schedules, and climate data. The portion of the

total load, or of either of its two components, met by a solar energy system of

given design is calculated for a range of system sizes by a subroutine of the

1 The computer programming code listed in this appendix was originally

developed by G. Thomas Sav and modified by Joel Levy. It was designed to

Implement the solar economic evaluation model for combined space heating

and service hot water systems presented in section 2.3. This brief description of the program is condensed from a preliminary draft report prepared

by Joel Levy, A revised edition of the NBS computer code for evaluating

solar energy systems for commercial buildings, and accompanying user's guide,

is in preparation under the name "COMSOL."

2 Energy price data stored in the computer files are those given in table 4.7

of the teKt.

B-1
program based on the LASL solar load ratio method as described in section 4.2

of this report. Monthly insolation values are calculated by the program

according to a procedure equivalent to that given by S.A. Klein in his paper

"Calculation of Monthly Average Insolation on Tilted Surfaces" [35]. The

equipment efficiencies for the conventional and solar energy systems are data

inputs.

Economic Data and Assumptions . Four sets of purchase and installation cost

data for the solar energy system are contained in the program (lines 13610-

13750). There is one for each building type, with each set comprised of fixed

cost and variable cost elements for materials and labor. 1 (Labor costs are adjusted in the program by the city-specific labor cost adjustment factor. )2

Financial parameters include the discount rate, inflation rate, loan rate,

downpajnnent as a proportion of purchase and installation costs, loan life,

Federal income tax rate, depreciation allowance, and capital gains tax rate.^

Both the straight-line and declining balance depreciation methods are programmed, Annually recurring cost is input as a fixed percentage of the initial

system cost, and is increased annually at the rate of general price inflation.

Optimization Algorithm . The solar energy system is optimized with respect to

the size of the collector area that will minimize total life-cycle costs. The

size selection algorithm applies not only when variable costs are linear but

also when these costs are any convex function of collector area,^

1 Cost data for purchasing and installing the solar energy system contained in

this program are those given in table 4.3 of the text,

2 Labor cost adjustment factors stored in the city data files are those given

in table 4,4 of the text,

3 Data contained in the program for these financial parameters are consistent

with that given in section 4,5 of the text.

^ It may be demonstrated as follows that the cost function used to assess

solar energy system performance is convex in collector area: the initial

purchase and installation cost of a solar energy system is assumed linear in

collector area, i.e., can be expressed as K1 + K2 • A, where K1 is total

fixed cost and K2 is the coefficient of the variable cost taken as linear in

collector area. Interest payments, maintenance costs, taxes and tax exemptions are all linear in the applicable cost base. Letting F denote the fraction of the total heating load met by solar energy, the life-cycle cost of

fuel for the conventional system equals (1-F) • (heating load) • K3, where

K3 is a positive constant depending on energy type, prices, tax rates, etc.,

but not varying with collector area. Similarly the heating load does not

depend on collector area and therefore, may be considered a positive constant

for present purposes. It may then be shown as follows that (1-F) is convex

in collector area:

For each month, J, the fraction of the building load in that month met by

solar energy, Z(J), is calculated in SUB2 of the computer program by lines

B-2
Footnote 4 Continued

5890 thru 5990. Line 5890 defines LI =• A • • A, where the term In

brackets Is a positive constant with respect to A.

If LI < 1.2 then effectively, Z(J) » A1 • l(J) * A, and so 1 - Z(J) Is

convex as a function of A for LI > 1.2.

It remains to check the function (1 - Z(J)) at the point A yielding LI 1.2.

For A1 = .318, A2 *1.132, and A3 * .504, (values given In lines 4191 - 4193

of the main program), there Is a limit of 1 - Z(J) as A Increases such that

LI tends to 1.2 as 1 - .318 * 1.2 - .618. For A at LI - 1.2 there Is a limit

of 1 - Z(J) as A Increases such that LI tends to 1.2 as 1 - .318 * 1.2 .618.

For A at LI * 1.2,

1 -Z(J) = 1.132 • e-. 504*1. 2 = .618.

So 1 - Z(J) Is continuous.

I(J)

The factor of the positive term l(j) In the slope of 1 - Z(J) to the left of

I(J)

A at LI = 1.2 Is -.318. The factor of l(j) for the right hand derivative at

at A corresponding to LI = 1.2 Is:

-(1.132)(.504)“1*2 * -504 = -.312,

l.e., at that point the slope of 1 - Z(J) Is greater to the right than It Is

to the left. Thus for A > 1 the function (1 - Z(J)) Is convex.

Taking the sum of positive multiples of the convex functions discussed above

we find that life-cycle cost Is a convex function of A.

This result that total life-cycle cost of the solar energy system Is convex

In the collector area Is exploited to justify the algorithm used to minimize

total life-cycle cost.

B-3
A number of parameters of economic significance independent of collector area

are calculated in the program. Using these parameters, the program calculates

the total life-cycle cost of a solar energy system with a given collector area. The sequence of steps used in the optimization algorithm employs principles of

search techniques to find an extremum of the uniraodel function.

Other Economic Evaluation Measures , The payback computation Involves

considering initial Investment cost, Cq, and anticipated net return in each

subsequent year, Nj j=l ,,,, k, where by definition Nj = Rj “ Cj , and Rj is revenue from the candidate investment in year j and Cj is the cost of operating

the investment in year j.

The simple payback period is the first k such that;

k

E N^j - Co > 0.

j=l

The measure of discounted payback period formulated in the computer program is

designed to overcome two problems: The first problem is that obtaining a positive life-cycle value of an investment does not guarantee that in every year

after the investment net return will be positive. Consequently there exists

the possibility of a sequence of positive and negative values for the partial

sums of discounted returns. The second problem is that the payback measure

often does not adequately account for the use of borrowed funds.

To deal with the latter problem, the discounted payback measure used here does

not treat future loan payments as an initial cost. Purchases made with borrowed

funds are entered as costs at the time the indebtedness is eliminated. If financing terms allow the debt to run to year t, then for a year prior to t to be a candidate for the payback year, it is necessary that the sum of discounted net

returns be equal to or greater than the discounted value of the remaining unpaid

principal on the debt.

Furthermore, if we denote by the unpaid balance at end of year k, by D the

discount rate, and use the simple payback notation above, then the payback year

is constrained to the first year t such that

k

E (1+D)j (1+D)k

for all k-t, t+1, ,,.,L.

If this condition is not met for any year prior to the end of system life, the

program prints "cumulative savings first equals cost beyond system life." This

is a euphemism to state that discounted life cycle savings do not cover cost of

the solar energy system.

If the life-cycle analysis determines that the solar energy system is not cost

effective based on input parameter values, break-even analysis is then performed

on the premise that if other variables are held fixed;

B-4
a. for a sufficiently high fuel price in the base year the net life-cycle

savings will be positive;

b. for a sufficiently high rate of future Increase in fuel prices the net

life-cycle savings will be positive; or

c. for a sufficiently low system cost the net life-cycle savings will be

positive.

First, the program calculates the minimum value of the base-year fuel price

necessary for the system to equal the conventional system in economic performance based on a reoptimization of system size. Next it finds the break-even

escalation rate of fuel in terms of how much faster than predicted will fuel

prices have to rise in order that the solar energy system will just pay for

itself, assuming other cost parameters as Initially specified, but again allowing the system size to be reoptimized. It then calculates the factor by which

the initially assumed system cost must be multiplied in order for the system,

based on a reoptimized size, to be equivalent economically to the conventional

system. The break-even system cost factor is the multiplicative inverse of the

break-even base-year fuel price factor and the fuel price escalation rate. The

optimal collector area and solar fraction for the break-even system costs will

be the same as that calculated for the break-even base-year fuel price and the

fuel price escalation rate.^

The computer program consists of a main part, three subroutines and 13 city

files. The computer program code is listed below. This is followed by a sample

printout of a city file (Boston) in table B-1. Table B-2 shows a sample output

of the program, performed for a new office building in Washington, D.C.

Operating The Program. After the program and data files are loaded into the

computer, the program is accessed. The user than specifies the city for which

the economic efficiency of the solar energy system is to be analyzed, the type

of building for which the analysis is being conducted, and the conventional

fuel that is to yield the energy requirements supplementary to that supplied by

solar energy.

This is done by specifying three lines of instruction;

(1) "30 Files ZNFL#".

For the last two symbols the relevant number of the city is entered. Numbers

identify corresponding cities as follows;

01. Boston, Massachusetts

02. Washington, D.C.

03. Nashville, Tennesseee

^ A variation of this computer program, not included in appendix B was used to

evaluate the case studies for hot water only. That version of the program

additionally provides for the calculation of breakdown hot water loads.

B-5
04. Charleston, West Virginia

05. Miami, Florida

06. Apalachicola, Florida

07. Madison, Wisconsin

08. Bismarck, North Carolina

09. Omaha, Nebraska

10. Fort Worth, Texas

11. Phoenix, Arizona

12. Seattle, Washington

13. Los Angeles, California

The name of the city for which the analysis has been run is the last item

printed on the first line of output information. (Print instruction line 1810).

(2) "124 04 =

This line specifies the type of fuel supplying the energy requirements

supplemental to and complementary to solar energy for the analysis. The

alternatives are:

1. Gas

2. Oil

3. Electricity

The fuel type specified is reported in the program output in line twelve of the

printed output. (Print Instruction is line 2110).

(3) "125 05 = //"

The line selects the building type for which the analysis is conducted. Four

alternatives can be chosen by setting 05 equal to 1 thru 4. The alternatives- are

1. New office building

2. Existing office building

3. New retail building

4. Existing retail building.

The type of building for which the analysis has been run is reported in the

program output on the first printed line just before the name of the city in

which the building is located. (Print instruction line 1810).

B-6
COHPOm PROOIAN LI8TXII6

1 REM***ZMABE2***

10 FILES * 20 PRINT "INPUT CITY PILE NAME:"

25 PRINT "ZNFL#|"

30 INPUT F$(l)

40 FILE|1»F$(1)

50 DIM H(12) ,I (12) ,K(12) ,L(12) ,M(12) ,Z(12) ,E(20) ,N(20) ,P(20)

70 DIM D(500) ,Q(400) ,Y(400)

90 DIM W(3,3) ,A(3)

110 DIM B(500)

111 DIM S(4,12) ,T(12)

120 READ#1,K

122 PRINT "INPUT FUEL TYPE #:"

123 PRINT "1. GAS, 2. OIL, 3. ELECTRICITY."

124 INPUT 04

125 PRINT "INPUT BUILDING TYPE #:"

12r> PRINT "1. NEW OFFICE, 2. EXISTING OFFICE, 3. NEW RETAIL, 4 . EXISTING RETAIL."

127 INPUT 05

150 READ T1,G3,Z1,P8,D,R3,N,R2,L,F1,I,D2,M,U2,S8,T5,V2,V3,V4

155 LET V3=10000

160 T5=.15

151 D2=.25

170 READ 08

190 FOR J=1 TO 12

210 READ M(J) ,D(J)

230 NEXT J 250 FOR J=1 TO K 255 READ A$

260 NEXT J 266 FOR J=K+1 TO 15

267 READ Z$

268 NEXT J 270 READ#1,Y1,Y4,G4,S2,S3,W1,W2,P4,T3,T2,T8,Z6,Z7,K7

280 Y4=Y1+15

290 FOR Jl=l TO 12

310 READ#1,H(J1) ,K(J1)

330 NEXT J1

350 READ#1,G(1) ,G(2) ,G(3) ,G(4)

360 MAT READ»1,S

370 FOR J=1 TO 05

550 READ B$,C$,M4,M5,I4,I5,D6,H9,H8

560 M4=18523

561 M5=11.19

562 14=12266

563 15=1.67

570 NEXT J 590 IF 05=4 THEN 730

610 FOR J1=J+1 TO 4 630 READ Z$,Y$

650 FOR J2=l TO 7 670 READ W9

690 NEXT J2

710 NEXT J1

730 IF 05<=2 THEN 770

750 LET G3=.015

770 LET El=.6

790 IF 04=2 THEN 910

810 IF 04=3 THEN 990

830 LET F$="GAS"

370 LET W7=2

890 GO TO 1090

910 LET F$="OIL"

930 LET B=140000

950 LET W7=l

970 GO TO 1090

990 LET F$="ELECTRICITY"

1010 LET B=3413

1030 LET El=l

1050 LET W7=0

850 LET B=1000000

B-7
1070

1090

1130

1150

1170

1190

1210

1230

1250

1370

1390

1410

1430

1450

1470

1590

1610

1630

1650

1670

1710

1810

1830

1850

1870

1890

1910

1930

1950

1970

1990

2010

2030

2050

2070

2090

2110

2130

2150

2170

2190

2210

2230

2250

2270

2290

2310

2330

2350

2370

2390

2410

2430

2450

2470

2490

2510

2530

2550

2570

2590

2610

2630

2650

2670

2690

2710

2730

2740

2750

2770

2790

2810

2830

COMPUTER PROGRAM LISTING - (CONTINUED)

GO TO 1150

IF B$="EXISTING" THEN 1150

LET El=.75

READ P5

FOR J=1 TO P5

READ N(J)

NEXT J MAT READ# 1,

P9=A(04)

MAT READ#1,W

FOR J=1 TO P5

E(J)=W(04,J)

NEXT J J1=(1/.54)*(.72*T2-.18)

READ#1 , A9

LET I4=I4*A9

LET I5=I5*A9

LET N9=l

LET H1=G3

IFB$="EX1STING" THEN 1750

LET I=.0925

PRINT "THE FOLLOWING ANALYSIS IS FOR A(N) "B$,C$" IN "A$

PRINT

PRINT " LATITUDE, DEGREES "Y1

PRINT " HOT WATER TEMP.,F "T1

PRINT " AVERAGE ANNUAL SUPPLY TEMP.,F "G4

PRINT " GALLON/MINUTE USE "G3

PRINT

PRINT

PRINT " FIXED MATERIAL COST,$ "M4

PRINT " VARIABLE MATERIAL COST/SQ/FT/COLL "M5

PRINT " FIXED LABOR COST,$ "14

PRINT " VARIABLE LABOR COST/SQ/FT/COLL "15

PRINT " SALES TAX ON MATERIAL, % "S2*100

PRINT " SALES TAX ON LABOR, T "S3*1C0

PRINT

PRINT " FUEL TYPE IS "F$

PRINT " CONVENTIONAL EQUIP. EFF. "El*100 "%"

PRINT " BTU CONTENT OF FUEL "B PRINT " CURRENT(197S) PRICE PER UNIT,$ "P9

PRINT " SALES TAX ON FUEL,% "Wl*100

PRINT " SPECIAL FUEL TAX,% "W2*100

PRINT " NUMBER OF FUEL ESCALATION PERIODS "P5

PRINT " ESCALATION RATES (NOMINAL) &CENGTH OF PERIODS" , FOR J=1 TO P5

PRINT " ”E(J) ,N(0)

NEXT J PRINT

PRINT " DISCOUNT RATE (NOMINAL), % "D*100

PRINT " GENERAL RATE OF INFLATION, % "P8*100

IF F1=0 THEN 2470

PRINT " LOAN INTEREST RATE,% "1*100

PRINT " DOWNPAYMENT, % "D2*100

PRINT " YEARS FINANCED "M

IF D6=l THEN 2550

PRINT " DEPRECIATION METHOD DECLINING BALANCE"

PRINT " DECLINING BALANCE RATE,% "R3*100

GO TO 2570

PRINT " DEPRECIATION METHOD STRAIGHT LINE"

PRINT " NUMBER OF DEPRECIATION YEARS "N PRINT " RECURRING COST RATE,% OF CONTRACT COST "R2*100

PRINT " PROPERTY TAX RATE,% OF CONTRACT COST "P4*100

IF T3=l THEN 2670

PRINT " PROPERTY TAX EXEMPT UNTIL YEAR "T3

PRINT " SALVAGE VALUE, % OF CONTRACT COST "S8*100

PRINT

PRINT " COMBINED FEDERAL/STATE TAX RTE,% "T2*100

PRINT " FEDERAL TAX RATE,% "U2*100

PRINT " EFFECTIVE CAPITAL GAINS TAX RATE,% "Jl*100

PRINT " FEDERAL TAX CREDIT, % ."T5*100

PRINT " EFFECTIVE STATE TAX CREDIT, % .."T8*100

PRINT " # OF YEARS STATE CREDIT APPLIES "Z6

IF Z7=0 THEN 2350

PRINT " STATE CREDIT IN LIEU OF DEPRECIATION"

B-8
COMPUTER PROGRAM LISTING - (CONTINUED)

2850 REM****************************************************************;

2870 REM CALCULATE HOT WATER LOADS & SOLAR RADIATION ON TILTED SURFACE

2890 LET Q1=0

2910 LET L2=0

2930 LET Q=. 017453293

2970 FOR J=1 TO 12

2980 U5=INT( (J+2)/3) ^

3150 T (J)=C 3*60* H9*H8* (D(J)/7) *8.34* (Tl-G (U5)

)

3155 L(J)=T(J)+S(05,J)*in“6

3170 LET b(J)=L(J)

3190 LET L2=L(J)+L2

3210 LET Y2=23.45*SIN (Q*360/365* (284+M(J) )

)

3230 LET W3=ATN (3QR(l-(-TAN(Q*Yl) *TAN(Q*Y2) ) *2 ) / (-TAN (Q* Y 1

) *TAN (Q*Y2 ) ) ) /Q

3235 IF W3>=0 THEN 3250

3240 W3=K3+180

3250 IF Y10Y4 THEN 3310

3270 LET Y=90

3290 GO TO 3390

3310 LET Y3=-TAN (Q* (Y1-Y4) ) *TAN (Q*Y2)

3330 LET Y=ATN(SQR(1-Y3''2)/Y3)/Q

3350 IF Y>0 THEN 3390

3370 LET Y=Y+180

3390 09=Q*MIN (Y,W3)

3430 Y5=COS(Q* (Y1-Y4) ) *COS(Q*Y2) *SIN (09) +09*SIN (Q* (Y1-Y4) )*SIN(Q*Y2)

3450 LET YG=Y5/(C0S(Q*Y1) *COS(Q*Y2) *SIN(Q*W3)+Q*W3*SIN(C*Y1) *SIN(Q*Y2)

)

3470 LET Y5=1.39-4.027*K(J)+5.531*K(J)'‘2-3.108*K(J)''3

3490 LET Y(J)=(l-Y5)*Y6+Y5*(l+COS(Q*Y4) )/2+K7*(l-COS(Q*Y4) )/2

3510 LET I (J)=Y(J)*H(J)*D(J)

3530 NEXT J 3550 REM*****************************************************************’

3570 REM ECONOMIC RARAMETEHS INDEPENDENT OF COLLECTOR AREA

3590 LET Pl= ( (1+D) “M-1)/(D* (1+D) "M)

3610 LET P2=(1+P8)/(D-P8)*( (1+D)''L-(1+P8)“L)/(1+D)“L

3630 LET H6=I/12* (1+1/12) ~ (12*M)/( (1+1/12) “ (12*M)-1)

3650 M2= (1+1/12) ~1 2* (1+D) “M- ( l+I/l 2

) “ ( 1 2*M+1 2

)

3670 M2=M2/( (1+D) “M* (l+D-( 1+1/12) “12)

)

3690 M1=T2* (1-D2) * (12*R6*P1+(1-12/I*R6) * (1-1/(1+I/12) “12)*M2)

3710 LET X=L* (1-S8“ (1/L)/(1+P8)

)

3730 PG=(1- ( (L-X) * (1+P8)/(L* (1+D) )

) “ (L+1-T3) ) *L* ( 1+D) / (X+X*P8-P8*L+D*L)

3750 LET P6=P4* (1-T2) * ( (L-X) * (1+P8)/(L* (1+D) )

) “T3*P6

3770 IF D6=l THEN 3850

3771 D(300)=S8/(1+P8) “L 3773 D(301)=l/(1-R3/N)

3774 D(302) =N-N/R3+1

3775 J2=INT(N+2-N/R3)

3777 IF (J2-N/R3*D(300) *D(301) “ (J2-1) ) >=D(302) THEN 3780

3778 J2=J2+1

3779 GO TO 3777

3780 N2=l/(1+D)

3785 Nl= (1-R3/N) *N2

3789 D1=(1-N1“ (J2-1) )/(l-Nl)

3792 U1=(R3/N)*N2*D1

3795 N3=(D(301) “ (1-J2)-D(300) )/(N+l-,J2)

3800 N3=N3* (N2'‘J2) * (1-N2“ (N+1-J2) )/(l-N2)

3810 D1=T2* (D1+N3)

3830 GO TO 3890

3850 LET D1=T2/N* ( (1+D) “N-1)/(D* (1+D) “N)

3870 U1=D1* (1-S8/(1+P8) “L)

3890 LET X2=D2+ (1-U2) *12*R6*P1

3910 REM* ********************************************************* *****<

3930 REM PVFFOR P5 FUEL ESCALATION PERIODS IN L 3950 LET Q2=l

3970 LET P3=0

3990 FOR J=1 TO P5

4010 IF E(J)=D THEN 4070

4030 LET P(J)=Q2* (1+E ( J ) ) / (D-E ( J )

) * ( ( 1+D) “N ( J

) - ( 1 +E ( J )

) “N ( J ) ) / ( 1 +D) “N(J)

4050 GO TO 4090

4070 P(J)=Q2*N(J)

4090 LET P3=P(J)+P3

4110 LET Q2 = Q2* ( (1 + E (J) )/(l+D) ) “N (.1)

B-9
COMPUTER PROGRAM LISTING - (CONTINUED)

4130 NEXT J 4150 LET X1=P3*P9* (1+Wl) * (1+W2)/(E1*B)

4170 REM **************************************

4190 REM SELECT PERFORMANCE PARAMETERS

4101 LET Al=.318

4192 LET A2=l. 132

4193 LET A3=.504

4194 GO TO 4910

4210 ON Z1 GO TO 4250,4710,4830

4230 REM DIFF. TEMP. FOR 2 GLAZ INGS , NONSECTI VE

4250 ON (Tl-1 10) /20+1 GO TO 4290,4390,4490,4590

4270 REM TEMP IlOF

4290 LET A1=.5G0

4310 LET A2=1.153

4330 LET A3=.933

4350 GO TO 4910

4370 REM TEMP 130F

4390 LET Al=.499

4410 LET A2=1.08

4430 LET A3=.729

4450 CO TO 4910

4470 REM TEMP 150F

4 4 9(J 4 510

4 530

4550

4 57 0 4590

4610

4 6 30

4650

4 670

4690

4 710

4730

4750

4 77 0 4790

4310

4330

4850

4870

4890

LET Al=.44

LET A2=.978

LET A3=.514

GO TO 4910

REM TEMP=170F

LET Al=. 343

LET A2=.S66

LET A3=. 365

GO TO 4910

REM DIFF. COLLECTOR TYPES FOR TEMP=110F

REM ONE GLAZING, SELECTIVE

LET a 1=.568

LET A2=l . 165

LET A3=. 945

GO TO 4910

REM TWO GLAZING, NONSELECTIVE:SEE TEMP=110F

REM ONE GLAZING, NONSELECTIVE

LET Al=.503

LET A?=1.081

LET A3=.748

4910 D(150)=X1

5010 D(304)=M4

5030 D(305)=M5

5050 D(314)=I4

5070 D(315)=I5

5760 CALL ZOPTl

34 30 PRINT

84 50 IF E4=l THEN 8830

3470 PIUMT " 8490 PRINT

8 510 PRINT

8530 : MONTH HOR. INSOL

8550 : AV. DAILY

8570 : BTU/IT2

3590 PRINT USING 8530,

36 1 0 PRINT USING 8550,

8 6 30 PRINT USING 8570,

0650 PRINT

8670 : ft H II If # If # It S HII . 0690 FOlt J = I TO 12 8710 PRINT USING 8670,

8730 NEXT J 0750: ANNUAL LOAD

0770: LOAD MET &Y SOLAR

8790 PRINT USING 0750, L2

8810 PRINT USING 8770,51

8830 PRINT

8850 PRINT

8990 PRINT

THERMAL ANALYSTS

CORR.

FACTOR

(R-BAR)

MO. INSOL.

TILT=L+1

BTU/FT2

SPACE HEAT.

+HOT WAT. LOAD

BTU/10“6

If If.llltll #!Hfmilf8.# 8N8.8S88

J,H(J) ,Y(J) ,I (J) ,L(J)s/10“6,Z(J)

im: . ft n # «

8

It It . 8 n 8 8 It

FRACTION

MET BY

SOLAR

8. 8888
COMPUTER PROGRAM LISTING - (CONTINUED)

9010 PRINT " ECONOMIC OPTIMIZATION

9030 PRINT

9050 PRINT

9070 PRINT "OPTIMAL COLLECTOR AREA, SQ.FT. "A

9090 PRINT "OPTIMAL SOLAR FRACTION "F

9110 PRINT "STORAGE VOLUME , GALLONS "A*1.0

9130 PRINT

9150 PRINT "CONTRACT COST, INCL. SALES TAX "C3

9170 PRINT

9190 PRINT "PV CAPITAL COST "C5+S4

9210 PRINT "SALES TAX DEDUCTION "S4

9230 PRINT "TAX CREDIT "T4

9250 PRINT "MORTGAGE INTEREST DEDUCTION "M9

9270 PRINT "RECURRING COST "R

9290 PRINT "PROPERTY TAX "P 9310

9330

9340

9350

9370

9390

9410

9 4 30

94 50

9470

94 90

9510

9550

9500

PRINT "DEPRECIATION DEDUCTION

PRINT "SALVAGE VALUE

PRINT "CAPITAL GAINS TAX

PRINT "ENERGY COST WITH SOLAR

PRINT " NOTE: BEFORE TAX FUEL SAVINGS

PRINT " TAX DEDUCT. LOSSES

PRINT " AFTER TAX FUEL SAVINGS.

PRINT "TOTAL LCC WITH SOLAR

PRINT

PRINT "TOTAL LCC WITHOUT SOLAR

PRINT "TOTAL LCS DUE TO SOLAR

PRINT "PV SYSTEM COST

RESTORE

PRINT

"D3

,-_"S5

"B1

"K2

"K2/(1-F)/(1-T2)*F

"K2*T2*F/(1-F)/(1-T2)

"K2*F/(1-F)

"T

"K2/(l-F)

"T9

"T-K2

9520 IE T9>=0 THEN 9700

9630 PRINT "CUMULATIVE SAVINGS FIRST EQUALS COST:"

9640 PRINT "BEYOND SYSTEM LIFE"

9550 GO TO 9750

9700 PRINT "SYSTEM RECOVERS COST"

9710 CALL ZPBKSl

9730 GO TO 12350

9750 D(151)=X1

9760 D(171)=T9

9770 D(150)=2*D(151)

9780 CALL ZOPTl

9810 IF T9>=0 THEN 9850

9320 D(171)=T9

9830 D(151)=D(150)

9040 GO TO 9770

9850 D(152)=D(150)

9870 D(172)=T9

9890 GO TO 9950

9910 D(151)=D(150)

9930 D(171)=T9

9950 D(175)=D(172)-D(171)

9960 IF ABS(T9)<1 THEN 10090

9970 D(150)= ( (D(172)+1)*D(151)-(D(171)-1)*D(152) )/(D(175)+2)

9990 CALL ZOPTl

10030

10050

10090

10210

10220

10225

10230

10240

10250

10270

10280

10290

10 310

1 0 3 3.0 10350

IF T9<0 THEN 9910

GO TO 9850

PRINT

PRINT "BREAKEVEN lUEL PRICE : " , D ( 1 50 ) /XI *P9

PRINT "OPTIMAL COLLECTOR AREA,SQ. FT. "A PRINT "OPTIMAL SOLAR FRACTION "F PRINT "CONTRACT COST "C3

PRINT

D(200)=D(150)/X1*P3

D(221)=l

D(261)=P3

D(220)=2*D(221)

CALL ZSVEl

IF D(230) >=D(200) THEN 10390

D(221)=D(220)

B-11
5,2 CASE STUDY RESULTS FOR SOLAR COMBINED SPACE HEATING AND SERVICE WATER

HEATING SYSTEMS

5.2.1 Results Compared for Two Sets of Assumptions

The economic analyses of the combined space heating/hot water systems were

Laltlally performed based on energy price projections of early-1980, given in

table 4.7. A substantial upward revision in base-year energy prices, particularly for oil, was made by DoE in mid-1980, reflecting the faster-than-projected

short-run escalation in oil prices in 1980. Additionally, the Federal energy

investment tax credit was raised from 10 percent to 15 percent. Selected case studies were repeated based on the revised energy price estimates given in

table 4.8, and on the 15 percent Federal investment tax credit. Comparisons of

the two sets of results demonstrates the volatility of solar economic feasibility over time.

Table 5,7 summarizes key measures of economic performance for four cities

(col. 1) and three conventional energy sources (col, 2). The results shown in

columns 3-5 are based on the early-1980 energy price projections (table 4,7)

and the 10 percent Federal investment tax credit. The results shown in columns

6-8 are based on the mid-1980 energy price projections (table 4,8) and the 15

percent Federal investment tax credit. The estimated net losses over the life

cycle were considerably reduced by use of the revised data, and near break-even

outcomes were estimated for the combined solar energy system in Bismarck and

Phoenix as compared with electric resistance heating. (The estimated net losses

for the other case studies were larger than those given in table 5.7).

5.2.2 Break-even Analysis

Tables 5.8 and 5.9 summarize the results of a break-even analysis performed for

initial energy prices, energy price escalation rates, and solar energy systems

costs, holding other assumptions at base-case values. Table 5.8 gives the

estimated break-even prices per unit of the nonsolar fuels. These are the

Initial prices of the fuels necessary to cause solar energy to be minimally

cost effective, assuming that the Initial prices will escalate at the rates

forecasted in early 1980. It shows in col. 2, for example, that oil would have

to cost $1.55 per gallon ($0.41/il) in Bismarck in the base year (as compared

with the assumed price of $0,788 per gallon ($0.21/il) in early 1980) in order

for solar energy to be cost effective, other things being equal. It shows in

col, 3, for example, that the estimated price of oil in early 1980 would have

to escalate 1,82 times faster over the 20 year study period than was projected

in early 1980 in order for solar energy to become cost effective in Bismarck

in comparison with oil.

Table 5,9 indicates in col. 4 the estimated break-even value for solar energy

system costs, given the other assumptions of early 1980, The break-even values

are indicated as a percentage of the base-case system costs calculated according

to the equations in table 4.3. For example, for the solar energy system to

break even relative to an oil-fired systeia in Bismarck, the optimally sized

solar energy system could cost no more than about half the amount estimated by

table 4.3 cost equations. Relative to electric resistance heating (lower part

112
COMPUTER PROGRAM LISTING - (CONTINUED)

1 REM***ZOPTl***

5760 B(?.98)»l

5775 B(299)"20000

5777 B(295)«B(299)-B(298)

5780 B ( 300) -B ( 298 )+INT (.45*8(295)

)

5785 B(301)»B(298)+INT(.55*B(295)

5787 IF B(295)<25 THEN 7000

5790 A-B(300)

5800 CALL ZSBEl

5810 B(200)-T

5820 A«=B(301)

5830 CALL ZSBEl

5840 B(201)=T

5900 IF B(200)>B(201) THEN 6100

5910 B(299)»B(301)

5920 GO TO 5777

6100 B(298)*B(300)

6110 GO TO 5777

7000 A=B(298)

7010 CALL ZSBEl

7020 B(250)=T

7050 FOR B9=B(298) TO B(299)

7060 A=B9+1

7070 CALL ZSBEl

7080 IF B(250)<=T THEN 7150

7090 b(250)=T

7100 NEXT B9

7150 A=B9

7160 CALL ZSBEl

7230 T9=K2/(1-F)-T

7250 RETURN

14170 END

1 REM***ZPBKS1***

30 FOR B9=l TO 7 70 D(300+B9)=0

110 NEXT B9

150 D(400)=T9

190 D(399)=L

230 D(400)=D(400)-S5+B1

270 D(301)=P9* (1+Wl) * (1+W2) *L2/(E1*B)

310 Q2=l

350 FOR B9=l TO P5

390 Q2=Q2*( (1+E(B9) )/(l+D) )“N(B9)

430 NEXT B9

470 D(301)=(1-T2)*F*D(301)*Q2 ' 510 IF M<L THEN 710

550 D(302)=-(1-D2)*12*R6*C3/(1+D) ~L

590 D ( 303) =( 1+1/12) “ (12*L)-( 1+1/12) “(12*L-12)

630 D(303)=(D(303)* (1-R6*12/I)+12*R6)/(1+D) “L

670 D(303)=T2*D(303) * (1-D2) *C3

710 GO TO 750

750 D(305)=-(1-T2)*C3*R2*( (1+P8)/(1+D) ) “L

780 IF T3>D(399) THEN 830

790 D(306)=-(1-T2)*P4*C3*S8/(1+D)~L

830 IF N<L THEN 870

870 FOR B9=l TO 7 910 D(400)=D(400)-D(300+B9)

950'NEXT B9

990 D(401)=D(400)

1030 IF M<D(399) THEN 1190

1070 0(401) = ( (1 + 1/12) '' (12*L-1?) * (1-R6*12/I)+R6*12/I)

1110 D(401)=(1-D2)*D(401)/(1+D) "(L-l)

1150 D(401)=D(400)-D(401)+C3

1190 IF D(401)<0 THEN 11000

1230 FOR B9=] TO (L-1)

1270 0(309)=D(399)-1

1310 O(300)=3

1350 IF D(399) >=N (1)+N (2) THEN 1510

1390 Q(300)=2

1430 IF D(399)>= N(l) THEN 1510

1470 Q(300)=l

B-13
COMPUTER PROGRAM LISTING - (CONTINUED)

1510 D(301)=D(301) * (1+D)/(1+E (Q(300) )

)

1550 IF D(399)>M THEN 1750

1590 D(302)=-(l-D2) *12*R6*C3/(1+D) ~D(399)

1630 D( 303)= (1+1/12) “{ 1 2*D { 399) )-( 1+1/12) ~ (1 2*D ( 399) -1 2)

1670 D(303)=(D{303)* (1-R6*12/I)+12*R6)/(1+D) “D{399)

1710 D(303)=T2*D(303) * (1-D2) *C3

1750 IF D(399)>1 THEN 1870

1790 D(400)=D(400)-S4

1830 IF K*13 THEN 2070

1870 IF D(399)>Z6 THEN 2110

1910 D(304)=T8*C3/(1+D) *D(399)

1950 IF D(399)>1 THEN 2110

1990 D(304)=D(304)+T5*C3/(1+D)

2030 GO TO 2110

2070 D(304)=T4

2110 D(30 5)=D(305)*( { 1 + D) / ( 1+P8 ) ) 2150 IF T3>D(399) THEN 2230

2190 D(306)=D(306) * (1+D)/S8'‘ (1/L)

2210 GO TO 2250

2230 0(306)=0

2250 IF D(399)>N THEN 2790

2270 D(307)=M3/(1+S2)+I3/(1+S3)

2290 D(330)=S8/(1+P8) “L 2310 IF D6=l THEN 2630

2350 IF J2>D(399) THEN 2510

2390 D(307)=D(307) * (1-R3/N) “ (J2-1)

2 4 30 D (307) =D (307)* (1-0(330) ) / (N + 1-J2) * ( l+D) (-0(399) ) 2470 GO TO 2670

2510 D(307)=0(307)*R3/N*(1-R3/N) “(0(399)-l)

2550 0(307)=0(307)/(1+0) ~D(399)

2590 GO TO 2570

2630 0 (307) =0(307) * (1-0(330) )/N* (1+0) ^ (-0(399)

)

2670 0(307)=T2*0(307)

2710 IF Z7=0 THEN 2790

2750 0(307)=0(307)/T2*U2

2790 FOR B8=l TO 7 2830 0(400)=0(400)-0(300+B8)

2870 NEXT B8

2910 0(401)=0(4(TO)

2950 IF M<0(399) THEN 3110

2990 O(401)=( (1+I/12)“(12*0(399)-12)*(1-R6*12/I)+R6*12/I)

3030 0(401) = (1-02) *O(401)/(l+O) '*(0(399)-!) 3070 0(401)=0(400)-0(401) *C3

3110 IF 0(401)<0 THEN 11000

3150 PRINT "NET LCS IN YEAR"D ( 3 99 ) " IS : $"D ( 4 0 1

)

3190 NEXT B9

> k x, 10980 PRINT "NET LCS NOW:$"O(401)

10990 GO TO 12350

11000 PRINT "OISCOUNTEO PAYBACK YEAR:"0(399)

12360 RETURN

14170 ENO
COMPUTER PROGRAM LISTING - (CONTINUED)

1 REM***ZSBE1***

5850 LET S1=0

5870 FOR J=1 TO 12

5890 LET L1=A*I (J)/L(J)

5910 IF LKO THEN 5030

5920 IF Ll>100 THEN 5980

5930 IF Ll>=1.2 THEN 5990

5950 LET Z(J)=A1*L1

5970 GO TO 5090

5980 Ll=100

5990 LET Z (J)=1-A2/EXP(A3*L1)

5010 GO TO 5090

5030 PRINT "COLL. AREA* INSOLATION/LOAD<0 FOR MONTH "J

6050 PRINT "CHECK INPUT"

5070 STOP

5090 LET S1=Z (J) *L(J)+S1

5110 NEXT J 5130 LET F=S1/L2

6150 REM SOLAR CONTRACT COST: INCLUDING SALES TAX

5170 M3=(D(304)+D(305)*A)*(1+S2)

6190 I3=(D(314)+D(315)*A)*(1+S3)

6210 REM SALES TAX DEDUCTION

6230 LET S4=T2*{M3/(1+S2)*S2+I3/(1+S3)*S3)

6250 LET S4=S4/(1+D)

6270 LET C3=M3+I3

6290 HEM PV CAPITAL COST NET OF SALES TAX DEDUCT

6310 LET C5=C3*X2-S4

5330 IF F1=0 THEN 5410

5350 REM MORTGAGE INT. DEDUCTION

6370 LET M9=M1*C3

6390 REM TAX CREDIT USING STATE TAX CREDITS NET REM OF FEDERAL INCOME TAX DED

5410 IF T5+T8=0 THEN 6770

5430 IF K013 THEN 6530

6450 IF C3>=12000 THEN 6570

6470 IF C3>=5454.54 THEN 6530

5490 LET T4=.45*C3* (1-T2)

6510 GO TO 6590

5530 LET T4=(3000-.1*C3)*(1-T2)

6550 GO TO 5590

5570 LET T4=. 15* (1-T2) *C3

6590 LET T4=(T4+T5*C3)/(1+D)

6510 GO TO 6770

5630 LET T4=(T5+T8) *C3/(1+D)

6550 IF T8=0 THEN 6770

5670 IF Z6=l THEN 6770

5590 FOR J=2 TO Z6

6710 LET T4=T4+T8*C3/(1+D) "J 5730 NEXT J 6750 REM RECURRING COST

6770 LET R=(l-T2) *R2*C3*P2

5790 REM PROPERTY TAX

6810 LET P=C3*P6

5830 REM DEPRECIATION CALC ON FULL CONTRACT COST LESS SALVAGE AND SALES TAX

6850 LET D3=D1*(M3/(1+S2)+I3/(1+S3)

)

58 55 B3=S8* (1-1/(1+P8) "D * (M3/ ( 1 +S2 ) +I 3/ ( 1+S 3 )

)

5855 B1=J1*B3/ (1+D) ~L 6870 IF Z7=0 THEN 6930

6890 LET D3=D3/T2*U2

6900 B1=.28*B3/(1+D) ~L 5910 REM SALVAGE VALUE

5930 LET S5=S8*C3/(1+D) *L

6950 REM CONVENTIONAL ENERGY COST

6970 K2=(1-T2)*(1-F)*L2*D(150)

6990 REM TOTAL COST

7010 LET T=K2+C5-M9-T4+R+P-D3-S5

7020 T=T+B1

7230 RETURN

8000 END

B-15
COMPUTER PROGRAM LISTING - (CONTINUED)

1 REM***ZSVE1***

10 FOR B9=l TO P5

30 D(250+B9)=D(220)*E(B9)

50 NEXT B9

110 Q2=l

130 D(230)=0

150 FOR B9=l TO P5

170 D(24P)-U+D(250+B9) )/(l+D)

190 IF p<24Q)=l THEN 250

210 D(23P+B9)=Q?i*D(240) * (1-0(240) ~N (B9) )/(l-D (240) ) 230 GO TP 270

250 D(23D+B9)=Q2*N(B9)

270 D(230i«P(230)+D(230+B9)

290 Q2=Q2*D(240)“N(B9)

310 NEXT B9

490 RETURN

500 END

B-16
Table B-1 City Data File for Boston

90 1 100 42.4 47.4 49 .05 0 110 0 0 .126 11 .513

120 .051 1 1 .2

130 505.5 .41

140 738 .426

150 1067.1 .445

IfJO 1355 .438

170 1759 .499

180 1864 .495

190 1860.5 .507

200 1570.1 .48

210 1267.5 .477

220 896.7 .453

230 635.8 . 372

240 442.8 .4 250 41 41 58 54

260 13 5.8 118 95.3 29.5 10.5

270 3.4 5.2 3 . 8 . 6 . 27.2

280 6 3.4 112.7

290 601.9 539 539.3 385.7 330.2

300 177.9 180.4 160 260.6 376

310 462.3 575.1

320 50.3 44 39.8 17.7 6.6

330 1.7 2 .8 4.4 14.4

340 28 44.4

350 147.2 131.2 124.7 84.1 72.3

360 48.5 48.9 46 50.4 80.9

370 99.5 136.8

400 4.12 .963 .089

430 .079 .101 .083

420 .096 .091 . 103

430 .060 .058 .022

500 1.04

B-17
Table B-2 Sanple Program Printout

The Following Output Is for a New Office In Washington

LRT I TUHE j DEGREES 33 .

HOT HATER TEMR.jF 130

BUERRGE RmURL SLPRLY TEMR.jF 70

GRLLOri-'MIhUTE USE 5

FIXED MATERIAL COST? 5

i..'AR I ABLE MATER I AL COST.'-SOxFT.'-COLL FIXED LABOR COST»$

UAFTABLE LAECf: CCGT.'-SO/FT.' C OLL

SALES TAX OM MATERIAL* X SALES TAX OM LABOR:

37045

(3£. 37

£453£

•«' • oo

0

FUEL TYRE IS OIL

CONJEMTIOMAL EQUIR.EFF 75 X BTU COMTEMI OF FLEL 140000

CURREMT(1980) PRICE PER: UMIT-.^: 908

:SALES TAX OM FUEL*'; 5. SPECIAL FUEL TAX*'-. 0

MUMBER OF FUEL ECALATIOM PERIODS 3

ECALATIOM RATEStmMiritfHU SLEMGTH CE PERIODS

. 09S 5

. 093 5

.105 io

DISCOUMT RATE UOIIMFL )

*

''i 13.

GEMERAL PATE OF IMFLATIOM* ''i 6. LOAM IMTEREST RATE*^ 9.35

DOMMRAYMEMT*'. £5

YEAR'S FIMAMCED £0

DEPPECIATIOM METHCiD DECLIMIMG BOLArCE

DECLIMIMG BiRLAMCE RATE*'; 150

MUMBER OF DERRECIATIO^^ 'tEARS 15

RECURRIMG COST RATE*'/; OF COMTR'fiCT COST 1. PROPERTY TAX RATE*'/; CE COMTRAC:T COST 0

SALUAGE UALUEjX CE COMTRACT COST 10.

COMB I MED FEDERAL/-STATE TAX PTE*X 51.7

FEDERAL TAX RATE*'; 46.

EFFECT I UE CAPITAL GAIMS TAX RATE*''; •. . 35.6

FEDERAL TAX CREDIT*'; 15.

EFFECT I UE STATE TAX CREDIT*''; 1.9

# CE 'tears state credit APPLIES 3

STATE CREDIT IM LIEU OF DEPRECIATIOM

THERMAL AMALYSIS

MTH HOP. IMSOL. CORR. MO. IMSOL. SPACE HEAT.

AU. DAILY FACTOR TILT=L+15 +HOT WAT. LOAD

BTU/-FT8 i:R-BAR) ETIJ/FT8 BiTLl/'10t6

1 1 1£6.

6

1 . 73'SE+OO 60731.6 44.7889

LI 1514.7 1 . 470E+00 6£347 .

£

£7. 10£9

1'967. 1 1 . 184E+00 78171.5 17.8899

4 £3SS!. £ 9. 386E-01 6681'?. 6 '?. :39££

C* £709.

6

7. 748E—01 65081.5 15. £119

fc- £781.5 7. 131E-01 5'?4£4.0 13.59££

"7 £450.

5

7.500E-01 56'977. 6 19. '9363

c* cc!'?'3. 6 8 . 6££E—01 61461.1 13. 1363

9 £131.3 1 . 076E+00 68814.5 1£. 6416

10 1688.

9

1 . 375E+00 71'9'94. 1 14.30£5

11 1£90.

0

1 . 661E+00 64880.4 £3. 50£4

1£ 1040.

9

1 . 839E+00 59:350. £ 4£. 70£'5 AMMUAL LOAD £5. 41'SS:36E+07

LOAD MET B;Y SOLAR 15. 345£S£E+07

FRACTIOM

MET BY

SOLAR

3.4711E-01

5.54S4E-01

7. 7’;i72E:-ei 9. 36S6E-01

S. 0048E-01

8.0791E-01

6. 44-97E-01

8. 3039E-01

8. 7564E-01

8.5314E-01

6. £68£E~01

3.5579E-01

B-18
Table B-2 Sample Program Printout (continued)

CCCrCiMIC OPTIMIZPTIONOFTIMfiL COLLECTOR RREfl» SO.FT.

CFTIMPL SOLRR FRACTION .60S5?3

STORAGE UOLUrC» GALLONS 1449*

CCr^TRACT COSTjINCL.SALES tax- 83626. 1

P'.i CAPITAL COST 69329.1

SALES TAX DETLICTIOr( 622.519

TAX CREDIT 14S52.4

NCRTGAGE INTEREST DEDUCTICiN 17083.4

RECURRING COST 4414.08

PROPERTY TAX 0

DEPRECIATION DEDUCTION^ 17297.2

SALUAGE UALUE 725.727

CFPITAL GAINS TAX 137.568

ENERGY COST WITH SOLAR 6580.49

NCiTE! BEFORE TAX FUEL SAUINGS 20752.

TAX DEDUCT. LOSSES 10728.8

AFTER tax: FUEL SAUINGS 10023.2

TOTAL LCC WITH SOLAR 29969.9

TOTAL LCC WITHOUT SOLAR 16603.7

TOTAL LCS DUE TO SOLAR 13366.2

PU SYSTEM COST 23389.5

CUMULATIUE SAUINGS FIRST EQUALS COST:

BE^'OND SYSTOI LIFE

BREAKEUEN FUEL PRICE: 1.93177

OPTIMAL COLLECTOR AREA»SQ. FT. 1392

OPTIMAL SOLAR FRACTION .783427

CONTRACT COST 98809.8

BREAKEUEN FUEL ESCALATION RATE:

1.81593 TIMES PROJECTED RATES

ESCALATION RATE FOR INTERUAL 1 174329

ESCALATION RATE FOR INTERUAL £ 168881

ESCALATION RATE FOR INTERUAL 3 190672

BREAKEiJEN SYSTEM FACTOR: .470035

B-19
NBS-114A (REV. 2-80

U.S. DEPT. OF COMM. 1. PUBLICATION OR 2. Performing Organ. Report No. 3. Publication Date BIBLIOGRAPHIC DATA

SHEET (See instructions)

REPORT NO.

NBSIR 82-2540 July 1982

4. TITLE AND SUBTITLE

Economic Evaluation of Solar Energy Systems in Commercial Buildings: Methodology

and Case Studies

5. AUTHOR(S)

Rosalie T. Ruegg; G. Thomas Sav; Jeanne W. Powell; E. Thomas Pierce

6. PERFORMING ORGANIZATION (If joint or other than MBS. see instructions) 7. Contract/Grant No. national bureau of standards

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

8. Type of Report & Period Covered

Final

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State. ZIP)

U. S. Department of Energy

Office of Solar Heat Technologies

Active Heating and Cooling Division

Washington, D.C. 20585

10. supplementary notes

Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11.

ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bi bliography or literature survey, mention it here)

This report develops a comprehensive economic optimization model for evaluating the

economic feasibility of active solar energy systems to provide service hot water and

combined space heating/service hot water in commercial buildings. The model is

demonstrated in a number of case studies for office buildings and retail stores.

Data and assumptions for use in the model are compiled for the selected case studies.

Using these data, the model is applied to estimate present value net savings (or net

losses) of the solar energy systems over a 20-year life cycle. Break-even values for

hot water loads, solar energy system costs, and current and future energy prices are also calculated to determine the minimum conditions under which the solar energy

systems become cost effective for the selected buildings. Economic optimization paths

which show the optimal solar collector areas and the corresponding present value of

net savings (or net losses) associated with a range of hot water loads are developed

in the case studies. Sensitivity analysis is conducted for key variables. The

relationship between total live-cycle costs and the solar fraction is tested for

selected cities to demonstrate how net savings (net losses) change as the solar

fraction is increased. In its approach, this report is of interest to solar

analysts; in its results, to the solar policy, research, and building communities.

12.

KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolon s) building economics; commercial buildings; economic analysis; energy economics;

life-cycle costing; solar energy

13.

AVAILABILITY

Uni imite(d

I I For Official Distribution. Do Not Release to NTIS

Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402.

Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

205

15. Price

$18.00

USCOMM-DC 6043-P80

1J