\AN OPTIMAL WITHDRAWAL POLICY FOR

SPENT NUCLEAR FUEL FROM ON-SITE STORAGE,

by

David Wesley Swindle, Jr.

A Thesis submitted to the graduate Faculty of the

Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE

in

Nuclear Science and Engineering

APPROVED:

H. A. Kursted Co-Chairman

a J. A. Nachlas, Co-Chairman G. H. acer

August 1977

Blacksburg, Virginia ib SEs Voss 97"? S94ny C.K ' ACKNOWLEDGEMENTS

The assistance of Dr. H. A. Kurstedt, Chairman, Nuclear Science and Engineering, and Dr. J. A. Nachlas, Industrial Engineering and

Operations Research, in selecting and developing the technical aspects of this paper is gratefully acknowledged. A special thanks is due the author's wife Carolyn, without whom the motivation to complete this work nor the typing of this work would have been possible.

Li TABLE OF CONTENTS

ACKNOWLEDGEMENTS»

TABLE OF CONTENTS

LIST OF TABLES* »*

LIST OF FIGURES e s e e ° e e » e . e e * * e e e a *

INTRODUCTIONs * © * © * © © © © © © © e© © © ee ee

A. Background and Motivation * * * + ** * *« se «+s

B. The Problem and the Objective * *+ + + * * * « + »

The Approach» e e e e e e ° e ° * e e e e ° ° e ° C.

Results ee © e® ® © © e e© 8@» © @® © © e@ &#© ® © #© &© 8 @ D.

DESCRIPTION OF THE SPENT+FUEL STORAGE PROBLEM «+ «+ « » 11

A. An Overview of the Nuclear Fuel Cycle * * © * » » 11

Current Uncertainties Facing the Nuclear Fuel Cycle in the United States* * © * *+ © © © © « « « 16

Reprocessing of Nuclear Fuel in the United States 19

Examination of the Role of Nuclear Energy in Meeting America's Energy Needs» » + © * + ¢ «© « « 23

DERIVATION OF THE SPENT FUEL WITHDRAWAL MODEL « « « « 27

A. Characteristics of the Spent=Fuel Withdrawal Problem- 27

B. The Dynamic Programming Formulation «+ * + «+ « » . 29

Cc. The Hitchcock Problem Formulation * * + + * «© « « 35

D. The Linear Programming Formulations * * +¢ +© + « «» 40

THE EXAMINATION AND EVALUATION OF THE COMPONENTS OF THE SPENT~FUEL WITHDRAWAL PROBLEM «+ + + « « « «© «© © « « « 46

A. Characteristics of Nuclear Reactors in Regard to Nuclear Fuel + «© « «© 2» «© © «© «© © «© © © » © «© « 46

Lii TABLE OF CONTENTS

(Continued)

B. Spent-Fuel Supply and Demand Projections* * * * + °«

C. The Measure of Effectiveness - Profitability

e e per Assembly e P e e e ° e ° ° e a e e e . e e 63

D. Storage Costs * * * * * * © © © © © © © © # © # # » 81

APPLICATION OF THE SPENT=FUEL*WITHDRAWAL MODEL* »° 85

A. Model and Data Summary’ *- * * * * * * 2 2 e+ * © © & 85

B. Implementation of the Model and Procedural Summary: 88

RESULTS © © ee ee tee we ee he he eh we te te 100

A. Optimistic Reprocessing Scenarioe* * * * * * * * & » 100

B. Realistic Reprocessing Scenario * * * * * * * © » » 106

C. Pessimistic Reprocessing Scenario a 107

CONCLUS IONS e e ° e e e e e . ° e ° e e e ® e e e e e ° 118

SUMMARY AND RECOMMENDATIONS 120

BIBLIOGRAPHY* * * * * * © © © © © © © © © © © © ¢ # « ¢ 122

10. APPENDIX ° _ @ @ @ © e@© © #@ © © © © © © © @© © #@ © © @© #© @& 2@ 125

11. VITA e e e e e e e . e e e e e e e e ® e e e e e e . ° 244

iv LIST OF TABLES

Table Title Page

2.1 U.S. Electric Power Statistics 1947-1974- + >» 24

2.2 Installed Nuclear Capacity* + * + * «© © © s « « 26

3.1. Unit Measures of Profitability for the Linear

Programming Problem eo © ss» © © © © © @ e@© #© © 8 @ 42

4.1 LWR Fuel and Discharge Data + * * * * + © # « « 47

4.2 Spent-Fuel Discharge Characteristics» + * « « »« 50

4.3 Average Composition of Plutonium Available for Recycle- 51

4.4 Installed Nuclear Capacity + + + + + + + e+ © ee eos 54

4.5 Discharge Quantities of Spent-—Fuel Per Gigawatt (electric) ee © e# © e © e© # @ e © © @ ee © @ @© @ @ 55

4.6 Spent-Fuel Supply Projections by Reactor Mix: »- 56

4.7 Reprocessing Plant Capacity Schedule* + * + «= » 59

4.8 Reprocessing Capability - Optimistic Scenario * 60

4.9 Reprocessing Capability - Realistic Scenarios + 61

4.10 Reprocessing Capability - Pessimistic Scenarios 62

4,11 Uranium Price Projections + + * * © «© «© #© « « « 73

4.12 Separative Work Cost Projection * * * * * * « « 75

4.13 Uranium Conversion Costs Forecast * * * * « « « > 76

4.14. Plutonium Value for Uranium Feed and Separative Work Equivalents* *© «© « «© © © «© © «© » « © © « « 83

4.15 On-Site Storage Costs Per Assembly~Year + + « » 84

5.1 Profitability Per Assembly - Westinghouse PWR > 89

5.2 Profitability Per Assembly B & W PWRe « «= « e 90

5.3 Profitability Per Assembly - Combustion Engineering 91

Cy 5.4 Profitability Per Assembly GE BWR/6 e e ° ® . e e 92 LIST OF FIGURES

Figure— Title

2.1 The Light Water Reactor Fuel Cycle + * * * * * * © &

4.1 Spent-Fuel Assembly Demand Rate - Westinghouse PWR >

4.2 Spent-Fuel Assembly Demand Rate - Babcock and Wilcox

PWR. ° e e ° e e e a s e s e e e e e e e e e * e * s 65

4.3 Spent-Fuel Assembly Demand Rate —- Combustion Engineering PWRe = * * © © * © © #© © #© © © #© © # © @ 66

4.4 Spent-Fuel Assembly Demand Rate - General Electric

BWR/6-° s ° e e e e e e e ° . e e ° e ° e e . ° ° ° ° 67

5.1 Profitability Per Assembly - lst Discharge

Westinghouse PWR * 2 e «© e@ e # @ 7. © © © 8 © @ @ @© @ | 93

5.2 Profitability Per Assembly ~ lst Discharge General Electric BWR/6 * e©= ee © © ee e@ # © @ ee ee 8 © # @ e@ © 28 94

5.3 Profitability Per Assembly - Ist Discharge No Plutonium Value; Westinghouse PWRe * * * * © » © @ » 97

5.4 Profitability Per Assembly - lst Discharge No Plutonium Value; General Electric BWR/6+ * + «© + » » 98

6.1 Optimal Selection Rule; Base Optimistic Reprocessing Scenario; Westinghouse PWR Base, +20Z SWU, +20% Storage, -20% Uranium, -20% SWU, -~20% Storage Costs + « © © © « «© «© © « © © «© # 8 © »® 101

6.2 Optimal Selection Rule; Base Optimistic Reprocessing Scenario; Westinghouse PWR +207 Uranium Coste 7 © @ e@ e © © © © 8® #© #© © e 2&© # # 102

6.3 Optimal Selection Rule; Base Optimistic Reprocessing Scenario; Westinghouse PWR No Plutonium Value; Base Cost+ *© «© © © «© «6 « « « « » 104

6.4 Optimal Selection Rule; Base Optimistic Reprocessing Scenario; Westinghouse PWR No Plutonium Value; +204 Uranium Cost+ + + « « + « « 105

6.5 Optimal Selection Rule; Base Realistic Reprocessing Scenario-Westinghouse PWR No Plutonium Value; Base Costs « «+ *© + « « «© © « « « 108

vi Figure . Title Page

6.6 Optimal Selection Rule; Base Realistic Reprocessing Scenario —- Westinghouse PWR No Plutonium Value; +20% Uranium Costs* * * + + * * © © #109

6.7 Optimal Selection Rule; Base Realistic Reprocessing Scenario —- Westinghouse PWR Base, +20% SWU, +20% Storage, ~20% Uranium, -—20% SWU, ~20% Storage Cost * * * * * « « e © © © © © © © © we hl elhlcelUC UL

6.8 Optimal Selection Rule; Base Realistic Reprocessing Scenario —- Westinghouse PWR +207 Uranium Cost * * * * © © # # # # e e e © © © © # s& ¢ 11l

6.9 Optimal Selection Rule; Base Pessimistic Reprocessing Scenario; Westinghouse PWR No Plutonium Value; Base Costs* * * * * * * * © * © © *© #113

6.10 Optimal Selection Rule; Base Pessimistic Reprocessing Scenario; Westinghouse PWR No Plutonium Value; +20% Uranium Cost * * * * * * ° © © «114

6.11 Optimal Selection Rule; Base Pessimistic Reprocessing Scenario; Westinghouse PWR Base, +20% Uranium, +20% SWU, +202Z Storage, -20% Uranium, -20% SWU, -20% Storage Costs* * * * * * * * 115

. 6.12 Optimal Selection Rule; +30% Pessimistic. Reprocessing Scneario; Westinghouse PWR +2074 Uranium Cost ee 8© © © © © © © © © © 8 ©» &© &© © © @ 116

6.13 Optimal Selection Rule; +30% Pessimistic Reprocessing Scneario; Westinghouse PWR +20% SWU Cost ° * * * © © © © © © © © © © © © © © © © © © [U7

vii 1. INTRODUCTION

A. Background and Motivation

In order for an industrialized country like the United States to

continue to grow economically, abundant energy at a reasonable cost

must be available. Otherwise, as was recently evidenced by the Arab

Oil Embargo, there will be a decrease in economic growth and an in-

crease in inflation. As a result of these problems, which could con-

ceivably reduce the economic prosperity of this nation, the United

States government has stressed energy self-reliance and conservation.

To obtain this self-reliance, the energy needs of the nation must be

satisfied from domestic resources. In particular, of the domestic

sources of energy available, only two fuels, coal and uranium, are

abundant in the sense of providing a low-cost, high-energy resource.!3

Upon examining the primary uses of these two fuels, coal and uranium

are more suited for the production of electricity than for any other

purpose./3 ‘However at present, when considering environmental,

economic, and societal points of view, the energy obtained from uranium

appears more acceptable for the generation of electricity than coal-

burning and other technologies .?2

To specifically note the electrical energy picture in the United

States, generation of electricity is predicted to be the fastest grow-

ing area of energy use.13 Examining past history, approximately 13 per

cent of the fuel utilized in the United States in 1947 was for the production of electricity. By 1970, this figure had increased to 25 per cent. By the year 2000, it is predicted that between 40 and 50 per cent of the fuel consumed in the U.S. will be for the production of

electricity./3 An important fact to consider is that as demand for

electrical energy increases, there will be an associated demand in the

resources necessary to produce this electricity. It is evident that

oil and gas will become less important as supplies diminish and prices

increase. Coal will take on a much greater responsibility. However, coal will not be able to do the job alone. At least through the year

2000 and probably well beyond that time, the nuclear option has to be a

Major contributor to the U.S. energy mix if it expects to have an

adequate available electrical power supply.”

Several studies examined the current and projected role of nuclear

power, and concluded that nuclear power must grow from its 1976 share

of 6.9% of total U.S. generating capacity to almost 50% by the year

2000.2 -.A study, conducted by Arthur D. Little, Inc. in February 1977,

forecast a fourfold increase in installed nuclear generating capacity

between 1975 and 1985.2 This represents the nuclear role as 21% of

the U.S. total generating capacity in 1985. Independent as well as

government studies indicate that nuclear power must increase through

this fourfold range if it is to be a major contributor of needed

electrical energy in the year 1985 and beyond.

Noting the above and many other similar projections, it seems

that the nuclear industry.is expanding and growing quickly. But

providing nuclear energy depends upon the fuel cycle, and there are

several problems to be resolved. Specifically, at the front end of the

cycle, there is a concern about the possibility of a uranium supply

shortage. At the enrichment stage, the Nuclear Fuel Assurance Act, which would have brought private capital into uranium enrichment, was

disapproved by Congress. However, at the back end of the cycle, the

situation is much worse. Despite the fact that proven technologies

exist for managing spent nuclear fuel,?® the political debates and

indecisions have lead to an executive order?’ stipulating that spent—

fuel be stored and that a moritorium on reprocessing be enforced until

the economic, political, and environmental implications are investigated

further. As a result of this interim prohibition on reprocessing,

recent governmental policy decisions dictate that strategies for

managing spent nuclear fuel be developed. In particular, a key problem

to be resolved includes the determination of optimal inventory with-

drawal policies for spent nuclear fuel from on-site storage pools to

respond to the re-initiation of reprocessing.

B. The Problem and Objective

The most serious problems within the nuclear industry today occur

at the back end of the fuel cycle. The present standstill of repro-

cessing in the U.S., and the uncertainties surrounding adequate

reprocessing capacity in the future, can be attributed at least in part

to the lack of an appropriate government regulatory policy declaration.

Despite the fact that the current Presidential administration has

declared a moratorium on reprocessing and plutonium recycle, analysts,

although pessimistic, are convinced that reprocessing will be available

in the future. 40 As a result, the examination of any phase of the nuclear fuel cycle should take this likelihood into account. A review of the literature has lead to the support of the con-

clusions reached in a 1976 report by the Nuclear Regulatory Commission.

In this report, the Commission expressed the situation quite clearly:

"Tt has been assumed in the past that the uranium and plutonium in spent fuel would be recovered and recycled. Therefore, detailed analyses of the technology of spent fuel disposal (disposition) are not to be found in the literature."!5, 36

Every year, each of the commercial nuclear reactors operating in

the U.S. replaces between 25 and 40 tonnes of spent—fuel in the form of

60 to 200 fuel assemblies.!4* Current reactor designs utilize the majority of the U-235 in the fuel augmented by the burnup of approxi-

mately 0.2% of the U-238.2% The burnup of the U-238 results in the

production of plutonium, a valuable fuel that can be recovered

through reprocessing. Unfortunately, these power systems usually

extract less than one per cent of the energy theoretically available

from uranium fuel. At best, the energy recovered is only about two per

cent. !* As a result of these low energy yields along with the

depletion of fissile fuel material, accumulation of fission products,

and other unfavorable irradiation effects, the fuel must be replaced

periodically. This replacement of nuclear fuel creates a stockpile of

spent-fuel at the reactor site. The direction the spent-fuel follows

at this point is in limbo and awaits resolution.

In order to close the back end of the fuel cycle, questions with

regard to the disposition of irradiated spent-fuel must be answered.

-Fuel discharged from light water reactors contains significant quantities of fissile material in the form of uranium and plutonium isotopes. These fissile quantities are equivalent to about 50% of the original amount loaded into the reactor.29 As a result, energy system forecasters in the past, well aware of the value of the residual uranium and plutonium, have assumed that this valuable fuel would be recycled into reactors.2+ However, as has been previously noted, no recycling has occurred on a commercial scale in the U.S. since the shutdownof Nuclear Fuel Service's West Valley Plant. The continued delay in the expected re-initiation of reprocessing has forced reactor operators to store spent-fuel beyond the normal "cooling-of £" period in which short-lived radioactive fission products decay to manageable levels. This unexpected long term storage at on-site pools has lead to the accumulation of spent~fuel in the form of numerous assemblies of varying ages. Also the lack of reprocessing capability (capacity and availability) in recent years has resulted in present and foreseeable spent-fuel storage capacity shortages. Therefore, it is imperative that strategies for managing spent nuclear fuel be developed, particu- larly if the questions, primarily societal and economic, surrounding nuclear power are to be answered and justified in an appropriate manner.

The objective of this effort is to address a key problem that must be resolved in determining strategies for managing spent nuclear fuel.

Specifically, the objective is to determine an optimal inventory with- drawal policy (or policies) for stored spent nuclear fuel from on-site facilities. The policies identified will indicate the optimal response to the re-initiation of reprocessing. A model that addresses the problem is developed and presented along with the associated detailed derivation. Results from analyzing and solving the spent-fuel-with- drawal model are also presented and the resulting specific policies are indentified.

This effort is being examined from the utility perspective as opposed to that of a fuel reprocessor. This allows the maximum economic gain from fuel recycle to be examined from the reactor operator's point of view. This in effect will contribute to the minimization of the nuclear fuel cycle costs.

C. The Approach

The spent-fuel-withdrawal problem involves the interaction of the spent~fuel generation, time and capacity dependent reprocessing demand, and the marketability of valuable products available from spent-fuel.

The amount of spent-fuel generated is dependent upon the type of reactor utilizing the nuclear fuel and the associated mode of oper- ation. In the United States, the pressurized water reactor (PWR) and boiling water reactor (BWR) serve as the primary nuclear power systems commercially available today. As of January 1, 1977 the installed nuclear capacity of 41,887 megawatts-~electric (MWe) was divided between

24 BWRs and 35 PWRs. Each of these reactors have capacities varying from 48 to 1180 MWe.3° These reactors, produced by four major vendors of nuclear steam supply systems (Nsss) in the United States today, discharge spent-fuel in varying quantities and of various compositions.

Therefore, the characteristics of the spent nuclear fuel are important in analyzing the economic disposition of spent-fuel. Demand for spent-fuel can be examined from the standpoint of reprocessing availability and capacity. Probable dates for the re- initiation of reprocessing are projected on the basis of recent independent 2° and government!? studies.

The application of the spent-fuel-withdrawal model is done on a per-reactor basis. The examination includes systems from each of the four major NSSS suppliers. Specifically, three PWR systems and one BWR system serve as representative models of the light water reactor (LWR) industry. Since the analysis is conducted on a per-reactor basis, all parameters (supplies, demands, and associated costs) are based upon the unit-reactor basis. This is appropriate because the recommended policy is for a single reactor system. However, the determined policies are applicable globally to all existing and planned nuclear plants.

The spent-fuel-withdrawal problem is a time~dependent decision problem that is influenced by the uncertainties in the growth of nuclear power capacity, spent-fuel supply and demand, and the avail- ability of spent-fuel reprocessing. The determination of an appro- priate decision policy depends primarily upon the following:

i. the. time at which a particular decision has to be made,

2. the determination of an appropriate measure of effectiveness

that will lead to an interpretable solution, and

3. the determination of spent-fuel characteristics that allow for

the choice of an appropriate measure of effectiveness.

It is necessary that the decision as to the selection of an inven- tory withdrawal policy be based upon comparisons between the value of spent-fuel currently in inventory and that continually being generated.

The key to determining an appropriate policy is therefore to define and determine the economic gain realizable in spent-fuel and examine this gain dynamically. For this problem, an appropriate mathematical model is developed. However, the key to successful optimization of this model lies in the interaction of a computer oriented data set and the economic characteristics of spent~fuel. The data set must be designed to reflect the time-varying nature of the nuclear fuel cycle.

'- The primary purpose of this study is:

1. the identification and understanding of the key elements that

define the spent-fuel inventory withdrawal problen,

2. the development of an appropriate optimization model describ-

ing the dynamic behavior of spent-~fuel,

3. the implementation of an appropriate computer-oriented code

incorporating the developed mathematical model and associated

economic measures of effectiveness, and

4. the determination of appropriate cost-effective strategies

yielding definable withdrawal policies based upon the time-

dependent behavior of nuclear power systems and their gener-

ation of spent-fuel.

The thesis comprises eight chapters. Following the introductory remarks, additional material on the nuclear fuel cycle and uncer- tainties facing the nuclear industry today are found in Chapter 2.

Also included in the second chapter are examinations of the status of reprocessing in the U.S. and the role that nuclear energy has to play in meeting America's energy needs during the remainder of this centruy. Chapter 3 presents the derivation of the spent-fuel-withdrawal model from three perspectives. First, modeling is secured through a dynamic programming approach. Second, the dynamic model is transformed into a

Hitchcock problem designed to be solved as a minimum cost flow problem; and finally, modeling is completed through the derivation of a linear programming representation of the Hitchcock formulation. An economic measure of effectiveness is evaluated in Chapter 4 based upon potential economic gain that is realized by the recycle of fissile isotopes.

Along with the determination of an appropriate measure of effectiveness considering both uranium and plutonium recycle and with uranium only recycle, are spent-fuel supply and demand projections per Gigawatt

(electric) of installed nuclear capacity. Chapter 5 addresses the application and solution of the lLinear-transportation-modeled-spent—fuel- withdrawal model. Optimization results are presented in Chapter 6 and are organized according to the reprocessing scenarios examined. Con- clusions follow in Chapter 7 with the summary and recommendations discussed in Chapter 8. Additional results and a listing of the computer-oriented data-generating code are included in the appendix.

D. Results

A detailed discussion of the results is found in Chapter 6. Also, complete tabular results for all analyses are included in Appendix A.

These results present definitive policies that serve as foundations for management decisions upon the re-initiation of reprocessing. The principal results of this study are presented below:

1. Given recycle of both uranium and plutonium, and an associated 10

organized market for each, the analyses conducted indicate

that a last-in-first-out (LIFO) policy yields the maximum

economic benefit. This policy is shown to be optimal based

entirely upon the value associated with recovered uranium and

plutonium.

In the absence of marketability for plutonium (i.e. uranium

recycle only), the profitability of spent-fuel is minimized

and no discernable policy is determined that uniquely opti- mizes the recoverable value in spent-fuel over time. However

the solutions appear to lean towards a modified LIFO policy.

The results of this effort indicate that if the issues

surrounding plutonium recycle and reprocessing are not settled

as soon as possible, the valuable isotopes remaining in spent~-

fuel become economically unattractive to recover. This is

particularly true as storage and handling costs increase with

time awaiting resolution of the controversy, and as the probability for assembly failures increase while held in

storage, thus increasing storage costs. 2. DESCRIPTION OF THE SPENT-FUEL STORAGE PROBLEM

A. An Overview of the Nuclear Fuel Cycle

The nuclear fuel cycle is a sequence of operations and facilities which include all of the processes necessary for the utilization of nuclear fuel. This is separated into three categories that consist of the front end stages, irradiation stage, and the tail or back end stage. 3! This sequence of operations perform the functions necessary to process the nuclear fuel from the uranium ore through the reactor, to eventual disposal.

The front end of the nuclear fuel cycle consists of those processes necessary to fabricate fuel assemblies suitable for use in nuclear power systems. The steps of the front end of the nuclear fuel cycle shown.in Figure 2.1, and consists of mining and milling of natural uranium, conversion of uranium mill product to uranium hexaflouride

(UFg), enrichment of uranium, and the fabrication of fuel assemblies.

The mining and milling operations occur primarily in the western United

States. This is due to the fact that the majority of the domestic uranium ore is found in the sedimentary sandstone and mudstone deposits of the Colorado Plateau, the Wyoming Basins, and the Gulf Coastal Plain of Texas.23 Once mined, uranium ore is. processed in a mill which uses a number of conventional mechanical and chemical processes to separate the uranium from the host rock and other minerals. The uranium is recovered as yellowcake, a uranium salt containing between 70 and 80 per cent U30g.3!}

11 12

The Light Water Reactor Nuclear Fuel Cycle

oo EES > Sh URANIUM MINES CONVERSION ENRICHING CONVERSION and MILLS TO UF¢ y TO FUEL

: / RECOVERED URANIUM TAILS / URANIUM STOCKPILE J preterene ne eee i f

! T=ost afoot ae ( PLUTONIUM REACTOR ‘Oe \ eeee, REPROCESSING FUEL STORAGE a ' WASTE STORAGE ~«------”

Figure 2.1

13

The U30g concentrate extracted from the ore is then converted to

the compound uranium hexafluoride (UF¢) - The process utilized is the

fluoride volatility process which involves three steps and concludes

with a high-temperature fluoridation step. 3! This conversion from

yellowcake is necessary in preparation for the enriching process.

Following conversion to UF 6» the uranium must be enriched in its concentration of the fissile isotope U-235 in order to be effectively

used in existing light water reactors. This is usually done by means

of the gaseous diffusion process, a process which is based upon the

difference in rates at which gases of different molecular weights |

diffuse through a porous barrier. This process separates the natural

uranium feed stream in the form of UF, into two output streams:

enriched product and depleted uranium tails. The required product

enrichment for current generation reactors varies between 2.5 and 4 per

cent uranium-235. The remainder of the enriched uranium fuel consists

almost entirely of the fertile isotope, uranium~238.

The final step in the front end of the fuel cycle is fabrication.

The enriched uranium produced in the enrichment plant is converted to

pellets of uranium dioxide (U0,). The assembly of these pellets into a

fixed structure, when completed, yields a fuel assembly ready for

irradiation in the reactor.

The second stage, irradiation of nuclear fuel, is the step in

which energy is obtained from the fuel through the fission process.

Each assembly in the reactor is burned at a specific power that may be

the result of a complex decision process aimed at an overall economic

optimization to allow for a total burnup.! Irradiation of the fuel 14 occurs over a 3~ to 4-year period once the nuclear plant is operating at designed power levels. During this time, each kilogram of fuel will produce approximately 2000 million BTU resulting in the generation of

200,000 kw-hr (electric).3!

The remaining series of stages shown in Figure 2.1, temporary spent- fuel storage, reprocessing of. spent-fuel, and waste storage, constitutes the tail or back end of the fuel cycle. This segment consists of those processes and activities required to dispose of the radioactive wastes produced and to reclaim the remaining uranium and/or plutonium found in the discharged fuel.

Storage of spent-fuel is the first activity related to the back end of the fuel cycle after the irradiation in the reactor. In addition to uranium and plutonium, spent-fuel contains a variety of fission products. Most of these fission products are radioactive and constitute the principal sources of heat and radiation. During the storage period, the radioactivity and heat output decrease approximately according to the law pee, where t. is the time after removal from the reactor.2° Minimal storage time, on the order of 120 to 180 days, is necessary to allow for cooling of the spent-fuel assembly and the decay of short-lived fission products. The purpose of this cooling period is primarily for safety reasons to allow the volatile isotope iodine-131, with a half-life of 8.14 days, to decay to manageable levels.?

Following storage for cooling, the spent-fuel is shipped to a reprocessing facility to reclaim the residual uranium and plutoniun, and to concentrate the radioactive fission products. In general, the reprocessing of spent-fuel consists of three steps: head-end treatment, 15

solvent extraction, and product purification. During the head-end

treatment, fuel elements are chopped into short sections and the fuel

material is dissolved out by nitric acid. The uranium and plutonium

are extracted from the fuel-acid solution using an organic solvent

such as tributyl phosphate (TBP) dissolved in a kerosine~like hydro-

carbon. + Through countercurrent mixing of organic and acid aqueous

‘solutions in vertical columns or centrifugal contactors, the substances

more soluble in one solution than in the other can be efficiently

separated.

Upon discharge from the reprocessing facility, radioactive wastes

have to be solidified into an insoluble, non-leachable form. The

solidified waste is packaged and placed in permanent repositories.

This long term solution is designed to isolate wastes in geologic

formations whose stability has been demonstrated over hundreds of

thousands of years. 3!

Note however that today, the once operable reprocessing and waste

storage segments of the nuclear fuel cycle are inoperable. This

portion of the fuel cycle is an area of primarily political and

economic, not technological uncertainties.38 As a result of this

inoperability of reprocessing and waste disposal, the fuel cycle has

not been closed.

The conclusion reached by this overview of the nuclear fuel cycle

is that uranium, the primary circulatory constituent of the fuel cycle,

can be limited by the capability to process it at both the "front" and

"back" ends. Therefore, if the energy from nuclear fission is to 16 contribute to the nation's energy needs, it is imperative that the fuel cycle be closed.

B. Current Uncertainties Facing the Nuclear

Fuel Cycle in the United States

Since the nuclear fuel cycle is a dynamic system, there is no constant relationship between the amount of uranium mined and the amount of uranium charged into nuclear power plants.3! In particular, the requirements at the front end of the fuel cycle for uranium and enrichment are affected by the resources available and the recovery efficiency of uranium and plutonium from the tail end.

At present, the nuclear industry is faced with continuing concerns for proliferation and waste disposal while at the same time having to cope with problems of an underdeveloped fuel cycle.°> According to a recent study by the Energy Research and Development Administration

(ERDA),2° the dilemma challenging the nuclear industry at the front end of the nuclear fuel cycle relates primarily to the availability of uranium resources and enrichment capacity to support the light water reactor (LWR) industry. The conclusions reached by this study indicate that in most phases of the fuel cycle for light water reactors, there exist present and/or foreseen constraints on the growth of the fission energy option. Specifically, there are insufficient known or projected supplies of uranium to support the growth of the LWR industry beyond about 1990-1995. Also established was that there currently exists insufficient enrichment capacity to support the projected growth of the industry beyond 1983. 17

The expected cumulative demand for uranium in the form of U30, ‘in the United States is expected to increase from the 1976 demand of

27,900 short tons to. over 2,194,800 short tons by the year 2000 if no recycle occurs. This figure can be reduced to 1,740,000 short tons in the year 2000 if plutonium and uranium recycle is allowed. These figures represent needed uranium resource if the diffusion plants operate at a 0.3% tails assay.°! As of August 1976, total uranium resources available at less than $35 per pound recovered were determined to be 1,275,000 short tons in the form of 130g. Of this amount, only 270,000 short tons are proven reserves with the remainder being only potential supplies.!? As can be immediately determined, a difference exists by a factor of two between the estimated cumulative demand for the uranium needed beyond the total consumption thru 1976 and the available resource. Note also that over half of the projected supply is speculative.

Also, it is not clear how adequate enrichment capacity will be supplied to support the nuclear industry beyond 1986. The ERDA diffusion plant complex is currently expanding and uprating through the

Cascade Improvement Program (CIP) and the Cascade Uprating Program

(cup).*! Included in present ERDA expansion plans is the capacity addition of 8.75 million SWU/yr to the Portsmouth Gaseous Diffusion

Plant. In total, upon completion of the CUP, CIP, and the Portsmouth add-on, total U.S. enrichment capacity will be 36.75 million SWU/yr. tt

Projections of annual separative work requirements in the United States exceeds that available, including new capacity, as early as 1985.27 »31 18

The contributing factor to much of the speculation is that processes at

the front end are highly investment-intensive with lead times ranging

from eight to ten years.

The most significant current uncertainties requiring solution are

those. which influence the back end of the fuel cycle. At present,

the back end is at a standstill because of regulatory uncertainties.

The conclusion reached by the Energy Research and Development Adminis-

tration (ERDA) in a recent study is that unless prompt and effective actions are taken for the safeguarding and handling of nuclear fuels,

the use of fission energy will be limited.39 ERDA also points out that

since the objection to closing the fuel cycle is primarily political and societal, there is concern as to whether or not industry, on its own, will be able to carry through with the commercialization of the

“back end". 39

The basic problem facing closure of the fuel cycle is realized as soon as the fuel is discharged from the reactor. As a result of a lack of a definitive decision to settle the controversy surrounding reprocessing and waste storage, utilities are forced to store their discharged fuel in on-site storage pools. This creates a problem due to the accumulation of spent~fuel beyond designed and regulatory capacity. This inability has also led to a delay in the issuance of construction permits for the reprocessing and mixed-oxide fuel fabrication plants that would be needed to initiate plutonium recycle.

On August 21, 1974, the Atomic Energy Commission (AEC) issued for comment.a draft report entitled "Generic Environmental Statement on the Use of Recycle Plutonium in Mixed Oxide Fuel in Light Water Cooled 19

Reactors (GESMO)". The Nuclear Regulatory Commission's (NRC) view and the controversy surrounding the recycle issue has created diffi- culties in closing the LWR fuel cycle in the U.S.?4 The delay in the issuance of the Final-GESMO, which was published in August 1976, has also adversely influenced the licensing of construction permits for reprocessing and mixed-oxide fuel fabrication plants. The resolution of the questions surrounding nixed-oxide fuel utilization are currently awaiting acknowledgement while in the process of a public hearing.@!

Unless intelligent decisions are made now, the breeder reactor program along with millions of dollars spent towards developing fission technology will be in jeopardy.

C. Reprocessing of Nuclear Fuel

In The United States

At present, there is no operating capability in the United States for the processing of spent commercial nuclear fuels. The state of regulatory uncertainty has prevented the nuclear industry from implementing plans to proceed with the required technologically proven programs necessary to close the nuclear fuel cycle. Unlike the breeder reactor, there is no technical necessity that mandates the use of reprocessing with light water reactors. Instead, the benefits of reprocessing the spent-fuel from light water reactors are argued to be three: an increase in the energy available from uranium resources, a reduction of the costs of nuclear power, and an easing of the problem of radioactive waste disposal.*® 20

Interest in the reprocessing of nuclear fuels developed among

' suppliers of nuclear power equipment who felt the need to assure their

customers of a closed fuel cycle. The chemical companies had the

necessary technological skill and background, and the oil companies

hoped to expand their operations into other energy sources. 3

The first plant to be built for reprocessing light water reactor

fuels, was the West Valley, New York plant operated by Nuclear Fuel

Services, Inc. Nuclear Fuel Services (now owned jointly by the Getty

Oil Company and the Skelly Oil Company), designed and constructed a

reprocessing plant with a capacity of 300 tons of spent-fuel] per year.?

There were few power reactors when it opened in 1966, and fully two-

thirds of the fuel reprocessed until 1972 was military fuel. This fuel

was supplied by the Atomic Energy Commission under an agreement that

effectively subsidized the plant. This plant operated intermittently

from 1966 to 1972, when it shut down for modifications to increase its

capacity and efficiency. The plan at the time was that the facility

would be restarted in 1975. The planned expansion called for increas-

ing the capacity to 750 tons of spent-fuel per year. Improvements

specified in the modifications included improvements in the

environmental-—protection features and for the installation of waste

facilities needed to meet new regulatory requirements. 3 However, the

then Atomic Energy Commission decided that the modifications were so

extensive that a new construction permit and operating license would

be required. From last reports, the estimated costs of the modifi-

cations had risen from $15 million to $600 million and Nuclear Fuel 21

Services had withdrawn its application to the Nuclear Regulatory

Commission for permission to reopen the plant.? While West Valley operated, the plant processed a total of 600 tons of spent—fuel, 76 of which 244 tons were commercial LWR fuel. 2°

A second plant located at Morris, Illinois; was constructed by the

General Electric Company, Inc. The company was convinced that relatively small reprocessing plants might be built to serve a group of power reactors within a short shipping radius. As a result, the

Midwest Fuel Recovery Plant was designed and built with a capacity of

300 tons per year. The Morris plant included major departures from the typical Purex-TBP process, with the aim of minimizing the contribution of reprocessing costs to the cost of nuclear power. In the course of testing the plant equipment prior to startup, it was concluded that the problems of handling fine radioactive solids were far greater than anticipated. 3 These problems would preclude the successful operation of the plant, and as a result, General Electric decided to postpone the operation plans pending completion of further studies.39

In 1968, the Allied Chemical Corporation announced plans to build a 1500-ton-per-year fuel reprocessing plant in Barnwell, South Carolina.

Construction of the plant was begun in 1971, and the originally planned facilities are complete.? The design and construction at Barnwell of

"tail end" facilities for the solidification of the waste for shipment to a federal repository and for the conversion of plutonium nitrate to solid plutonium oxide await decisions by the NRC and ERDA on the specifications and destinations of those materials. So far, Allied

General has invested over $250 million in the plant with the additional 22 waste and plutonium handling facilities expected to cost another $250 million.3 At present, Allied General's request for an operating. license is in jeopardy. The President's Policy statement of April 20,

1977 specifically denied government subsidy to support the Barnwell facility. The President also recommended the indefinite post- ponement of commercial reprocessing in the United States.?

A fourth plant, to be built by Exxon Nuclear, is planned to have a nominal reprocessing capacity of 1500-tons-per-year. The proposed construction site has been announced as part of the ERDA acreage at

Oak Ridge, Tennessee.? The company is awaiting final approval of site acquisition and the NRC's satisfaction of company complience with siting and licensing regulations.

Although the past history reflects a great deal of uncertainty | with regard to spent-fuel reprocessing; analysts, although pessimistic, are convinced that reprocessing will eventually be available. A detailed study by Macek?> concluded that the earliest date reprocessing would be available would be 1979. This noted the start-up of the

Barnwell Plant in 1979, followed by Nuclear Fuel Services in 1981, and

Exxon in 1985. A fourth plant of 1500 tonnes per year capacity is projected to begin operation in 1987.

Macek determines that under his scenario of uranium pricing and developing reprocessing capacity, if reprocessing is not initiated prior to mid-1982, the back end of the fuel cycle will forever be a financial liability. 23

D. Examination of the Role of Nuclear Energy

In Meeting America's Energy Needs

Historically, the electric power generating industry has grown at

a rapid rate. In the past few decades, the electric power demand has

grown at an average annual rate of about 7%, resulting in a doubling of load every 10 years.!7 This growth has been related to two basic

trends--a growth in population of approximately 1.3% per year and an

increasing per capita use. Statistics for the United States electric

power industry are given in Table 2.1. As can be observed, in the 27

year span from 1947 to 1974, per capita consumption of electricity has

increased over four and one-half times.

Through its convenience and utility, electricity has become vital

to the function and growth of modern civilization. Although the rates

of growth in the consumption of energy and of electricity have

decreased in the past few years and are expected to be substantially

lower in the future than in the past, the gradual increase in the

fraction of energy devoted to production of electricity seems destined

to continue.!7

Forecasts of the demand for electricity in the future contain

Many uncertainties. As a result, the projections of different indi-

viduals or groups vary widely. Studies surveyed by the Nuclear

Regulatory Commision prior to August 1976!’ indicate that this differ-

ence increases to over 600 Gigawatt (electric) by the year 2000

between the lowest and highest estimates. 24

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In 1975, about 9 per cent of the electricity consumed in the U.S. came from nuclear plants.!? As of January 1977, 59 commercial nuclear power plants with a total generating capacity of 41,887 MWe had been completed and licensed to operate. In addition, 131 plants were under construction or on order. By the end of 1987, the installed nuclear capacity would be 182,418 MWe based upon a ten year lead time necessary to construct and begin operation of a nuclear power facility. S Current projections by the nrc, !? ERDA, and the Edison Electric Institute?!

(EEI) conclude that a low growth projection without breeder reactors appears to be the most likely nuclear capacity growth applicable through the year 2000. The NRC and ERDA forecast specifically projects an installed nuclear capacity of 156,000 MWe in 1985 and 507,000 MWE in the year 2000. The projection by the Edison Electric Institute for the low growth scenario is a few per cent lower. As a result of examining reactor demands and the aforementioned nuclear capacity projections, a compromised nuclear growth projection thru 1997 is presented in Table 2.2 according to the reactor mix, Nuclear generating capacity through 1987 is based upon those plants scheduled for start-up by the end of 1987, given a ten-year lead time as a firm commitment.

The projection beyond 1987 reflects the ERDA, NRC, and EEI low growth projections. This projection shall serve as a basis for this study.

While reports have varied and have been argumentative as to the extent nuclear will contribute to the total electric generating capacity in the future, nuclear energy is still expected to account for more than half of the total electrical generating capacity by the year

2000.13 26

Table 2.2. Installed Nuclear Capacity 35 17» 31

Total Capacity BWR Capacity PWR Capacity Year (GWe) (GWe) (GWe)

1976 41.887 16.033 25.854 1977 50.364 17.921 32.443 1978 55.292 17.921 37.371 1979 63.480 20.629 42.851 1980 74.641 24.950 49.691 1981 92.151 29.295 62.856 1982 107.552 33.240 74.312 1983 125.875 41.344 84.531 1984 145.659 47.060 98.599 1985 161.928 52.948 108.980 1986 176.736 56.453 120.283

1987 182.394 58.781 123.613 1988 195.000 62.983 132.017 1989 204.000 65.983 138.017 1990 213.000 68.983 144.017 1991 225.000 72.983 152.017 1992 246.000 79.983 166.017 1993 274.500 89.483 185.017 1994 313.500 102.483 211.017 1995 345.000 112.983 232.017 1996 381.000 124.983 256.017 1997 409.500 134.483 275.017

3. DERIVATIONOF THE SPENT~FUEL WITHDRAWAL MODEL

A. Characteristics of the Spent-Fuel-Withdrawal Problem

In the background information on the current status of the post-

reactor fuel cycle, it was pointed out that as nuclear power continues

to grow and contributes a larger share to America's energy needs, the

questions with regard to the disposition of irradiated fuel must be

answered. Spent-fuel discharged from the operating reactors will

continue to accumulate at on-site storage facilities. This problem

persists until reprocessing is once again an integral part of the fuel

cycle or until the government makes a decision to manage permanent

waste disposal.

Macek2° demonstrates that spent-fuel should be stored on-site

at the reactor as opposed to off-site at the reprocessing plant or at

a repository. Assuming future reprocessing start-up, the utility will

be faced with the decision of how to relieve large stores of spent-fuel

in the face of a continual spent-fuel supply.

The spent-fuel-withdrawal problem is a planning problem which can

be expanded into a decision model spanning a number of time periods.

Before the model can be constructed, decisions applicable to each

modeling approach must first be made. In particular, the length of

the planning horizon, the maximization or minimization of the objective

function, and the specific assumptions necessary for implementation

are required.

Irradiated fuel contains significant quantities of fissile

' Material; often as much as 504 of the amount originally loaded into the

27 28 reactor.29 Given the relative scarcity of the fissile isotope uranium-

235, in the absence of the breeder reactor program the recycleable uranium in the spent-fuel may constitute a valuable energy resource.

Also, as the need for additional energy sources appear,. the fissile plutonium found in spent~fuel also represents a valuable energy resource.

Activities relating to the management of spent-fuel in the back end of the fuel cycle are primarily controlled by economics. Thus, with the question of whether or not to proceed with reprocessing still unanswered, it is important that the economic gain realized with recycle be firmly understood and established. The greater the economic gain with recycle, the greater the incentive to reprocess. Therefore, for the above reasons, the logical approach is to choose the maxi- mization of the economic gain realized with the recycle of uranium and/or plutonium from the spent-fuel to be the objective.

For modeling purposes, a planning horizon of finite length will be chosen. The time period should be of sufficient length to include important occurances such as the resumption of reprocessing and/or any new developments in reactor technology. Also, since the majority of the commercial nuclear plants in operation and in particular the plants that will serve as reference in the model development, operate on an annual cycle, the planning horizon will be divided into periods of one- year duration. Consequentially, the horizon shall be taken as a twenty- year period extending from 1977 to 1997.

Particular assumptions are necessary for the satisfactory deriva- tion and application of a model. First, it is assumed that a reactor's 29

on-site storage capabilityis not constrained as to the number of

assemblies it can store. This assumes that reactor operators will

expand storage facilities as necessary. Second, upon the resumption

of reprocessing, it is assumed that a market exists to warrant recovery

of the uranium and plutonium from the spent-fuel. This is necessary

so that the potential economic gain can be analyzed. The situation

where no marketability for plutonium exists is being examined later.

Also to be assumed is that demand for reprocessing services in any

given year does not exceed reprocessing capacity.

B. The Dynamic Programming Formulation

The objective in formulating the spent~fuel~inventory-withdrawal

model is to provide a framework within which alternative inventory

depletion strategies can be evaluated and compared. Since the com-

position of spent~fuel changes with time (due to decay of fissile Pu-

241), and since the inventory level of spent-fuel also changes with

time, time dependence is a key attribute of the problem. Therefore,

Dynamic programming”? is an appropriate vehicle for modeling the

problem.

It is first necessary to distinguish the periods in the planning

horizon. This is necessary since the ability to make a decision rests with the developments occuring during a particular time period. For

this purpose, the index set T is defined:

T= {0, 1, 2,...,t,..., tel, (3.1) 30

where t denotes the particular time period and t¢ represents the number

of years in the planning horizon. For purposes of this study, the

number of years in the planning horizon is twenty years.

In order to distinguish and denote the particular characteristics

of each assembly, it is necessary to define an index that can singularly

distinguish each assembly from others as a result of different initial

enrichments, burnups, and other characteristics. Each classification

considers only those assemblies assigned to that class in any one time

period as a function of those discharged. For this purpose, the classi-

fication considers only those assemblies assigned to that class in any

one time period as a function of those discharged. For this purpose,

the classification is defined by the index set J as:

J={1, 2, ...,4, ... =, N(t)}, (3.2) where j denotes the particular classification and N(t) denotes the

number of spent-fuel classes generated in period t.

To be able to distinguish each assembly, it becomes necessary not only to identify by classification, but to also identify by the

particular age of the assembly. The age of an assembly is chosen to

represent the number of time periods in years that a spent-fuel assembly has resided outside of the reactor core in storage. The index set

K= {0, 1, 2, ...,k,..., th, (3.3)

is defined to represent the ages of the stored spent-fuel, where k

denotes the number of time periods residing in storage. Using the

defined index sets, it is now possible to describe the spent-fuel 31 stockpiles.

The quantity of fuel discharged in any one year, is distinguished as the number of fuel assemblies discharged in a particular year t.

In general, the quantity discharged less those withdrawn represents the number of assemblies available for withdrawal. Specifically, this spent-fuel stockpile is represented by:

S54, (t) = the number of spent-fuel assemblies of class j and age k

available for withdrawal at the start of year t.

When k = 0, the number of assemblies available is the number discharged in the given time period t.

Also important as a measure of the spent~fuel stockpile is the inventory level. This quantity is denoted by:

T54,(t) = the number of spent-fuel assemblies of class j and age k

remaining in inventory at the end of year t.

The demand for spent-fuel depends upon the reprocessing capability available as a function of time. This rate is determined from the anticipated growth of the reprocessing capability in any particular time period t. This demand is represented by:

D(t) = the number of spent-fuel assemblies demanded to satisfy

reprocessing capability in year t.

The assemblies furnished from the on-site storage pools to meet the demand for reprocessing denote the decision variables. The assemblies chosen to meet demand vary, since demand is not constant 32

over the horizon or planning period. The decision variables are defined

by:

X54, (t) = the number of spent-fuel assemblies of class j and age k

withdrawn from inventory for reprocessing in year t.

Implicit in these definitions are the assumptions that, at the end of each year, inventory charges are assessed on the material remaining

in inventory and that stockpiled spent-fuel left in inventory increases

in age at the start of each year. Therefore, it is necessary to define

the transition relationships indicating the inventory level from stage

(time period) to stage.

The transitions coupling the stages are: I a ct

o>~ ww = Si, (t) - X4,,(t) ¥oj,k, t (3.4) xy C4.

$4. (t) = Ting (t - 1) ¥oj,k,t (3.5) where ¥ is the nomenclature denoting "for the entire range of".

Implicit in the derivation and definition of the spent-fuel stock- pile quantities is the constraint:

0 < Xj1,(t) < S(t)» ¥ogj,k,t (3.6)

which expresses the fact that the number of assemblies chosen to meet demand is greater than or equal to zero but less than or equal to the available supply in inventory.

Since spent-fuel has a "cooling-off" period in order to allow the highly radioactive fission products to decay to a tolerable activity 33

level, the demand cannot be met from assemblies that have not resided

in storage for at least one time period. This constraint, denoted by:

reflects a minimum on-site storage of at least one time period. There-

fore, the assumption of at least one-year in storage is carried through-

out the entire model derivation and application.

By defining a unit holding cost and unit profit per assembly the

objective function is formulated. The objective is to maximize the

present value of the profits from spent-fuel reprocessing over a

planning horizon of te years. Profits are assumed to represent the

excess expenditures obtained by recycling residual fuel over the costs

of spent-fuel storage and reprocessing. This objective is expressed

as:

te ct Nt) Maximize ) } } aM PfP5q(t)Xyx(t)- Wyk (t)1jK(t)} (3.8) t=0 k=0 j=1

where:

a = (1 + interest rate) = the discount factor,

Hy, (t) = the unit holding cost for a spent-fuel assembly of class

j and age k in year t,

P34 (t) = the unit profit resulting from reprocessing a spent- fuel

assembly of class j and age k in year t, and 34

> . X = {X44,(0) X41), ee 8 4 X51 (t,_)} ¥oj,k

is the optimal solution vector which determines the

optimal withdrawal policy.

The time-dependent material decay costs are reflected in the profit

parameters. The objective equation does not include a term to represent

the value of the spent-fuel inventory remaining at the end of the

horizon. This is because numerous assumptions about the nature of the

spent-fuel-disposition problem at the end of the planning horizon are

plausible. For a specific problem application, the selection of this

boundary assumption is reflected in the values assigned to the para- meters Hy), (tg) and Pi, (tg).

Equations (3.4), (3.5), (3.6), and (3.8) define a dynamic program

that is decomposed first in terms of time and further in terms of spent-

fuel classes. For purposes of continuity, the spent-fuel-inventory- withdrawal model for the determination of an optimal withdrawal policy

is summarized as follows:

tr t N(t) Maximize bo oo ihn oT F{P 31 (t)X 5, (t) - Hay (t) Ts (t)} (3.8)

over the vector

X= {Kj (0), Xy(1), - 2 es Kylt), «+ +s Xex(te)} ¥ Gk subject to

Tjx(t) = S54 (t) - Xjx(t) ¥Voeogj, k, t (3.9) 35

$5 (t) = S44-1 (t-1) - X44-1 (t-1) ¥ oj, k>0O, t>0 (3.10)

O < Xjk(t) < SzK(t) ¥oej,k,t (3.11)

Xjo(t) = 0 ¥oj,t (3.12)

N(t) ; Xi, (t) < D(t). ¥ok,t (3.13) j=l

C. The Hitchcock Problem Formulation

The spent-fuel-inventory-withdrawal model presented in the previous

section may be solved both by static and dynamic optimization techniques.

In general, the choice among the techniques depends upon the character-

istics. of the model in question.

As defined, the dynamic programming formulation results in a very

large problem. For the general problem defined by equation (3.8), the

model has t¢(t¢ + 1)/2 stages.28 For a reduced problem that considers

only one spent-fuel class, the model still has t¢ stages. For either

problem, the number of state variable transformations is te(t +1)/2.

Therefore, it is unlikely that the dynamic program can be solved.

Fortunately, the dynamic-programming formulation structures the relation-

ships in the problem.

Upon examination of the dynamic-programming formulation, several

special features of the model can be observed. First, it should be

pointed out that the state-transition relationships (equations 3.9-3.11)

‘and the objective function (equation 3.8) are linear in terms of the

state variable, S3K(t), and also the decision variable, X5i(t). Note 36

also that the constraint on the decision variable (equation 3.13) is linear. Another useful property is that the state variables at any stage are dependent only upon the initial states (Ss o(t), Tio (t), ¥j, t), and past decision variables. Finally, it can be seen that the solution would depend solely upon the initial states at each stage.

Since the model displays this special linearity of the objective function, constraints, and state transitions, the problem is amenable to transformation to a linear program. The linear-programming approach offers an advantage of allowing the treatment of decision variables as continuous parameters. However, due to the features of the problem addressed, discrete solutions are more advantageous. Examination of the structure of the model and the fact that linearity exists, indicates the dynamic-programming formulation can be transformed into a Hitchcock problem. /8

The Hitchcock problem was originally designed to address a class of transportation or minimum-cost-flow problems. Although the present formulation is a maximization problem as opposed to a minimum-cost-flow problem, the structure indicates that the Hitchcock problem is applic-— able; and the same method of analysis can be used.

In the context of the Hitchcock problem formulation, the material sources correspond to the initial inventory at the start of the plan- ning horizon and to the annual supply of spent-fuel of class j generated in year t. The demands correspond to the available reprocessing capacity in each year and to the final remaining inventory. The age of the spent-fuel, when it is selected for reprocessing, is represented 37 implicity by the relation between the year in which it is reprocessed and the discharge date. The cost coefficients for the Hitchcock model represent the profit from reprocessing less the cumulative holding costs. Thus the problem becomes profit-maximization rather than a cost-minimization problem.

In terms of the dynamic-programming-model parameters, the associ- ated Hitchcock spent-fuel-withdrawal problemis determined by equivalent supply, demand, and costs parameters.

The supply, denoted by a;, is composed of two components. The first transition relationship for the supply is defined by:

az = Siq(t=0), i=l, oe © « 95 N(t=0). ¥ j (3.14)

This relationship denotes the supply of spent-fuel assemblies from the various classifications comprised in the initial discharge. The relationship that transforms the remaining supply of freshly discharged fuel is given by:

aj = Sjo(t), isN(t=0) +1, ...A4 ¥oj,t (3.15)

where

tf A= ) N(t).

t=

This defines the remaining supply as that from each of the different classifications beyond the first discharge that occurred at t=0.

Demand, denoted by b,, is defined in terms of the total unallocated 38

spent-—fuel-reprocessing capacity available for class j spent-—fuel

assemblies in year t. By defining U, (t) as the unallocated spent-fuel-

reprocessing capacity available for class j assemblies in year t, the

Hitchcock demand is formulated by the following transformation:

i U;(n), n= 1, 2, + + - 5 te jel

ba = (3.16)

A te

) aj ~ ) Dn> n=tpti1 i=1 n=1

The term for n=t¢+l represents the number of spent-fuel assemblies residing in inventory at the end of the horizon. This is an artificial demand necessary to satisfy the constraint that the total available supply equals the total demand.

The measure of effectiveness for the Hitchcock problem is the cost coefficient denoted by C in* As mentioned earlier, this cost coefficient represents the profit from reprocessing less the cumulative holding costs from time of discharge and placement into inventory, until with- drawn to meet demand. This cost is represented as:

n-D Cin = Pi-cyn-pD(™ — by Hi-c,g (Dt2) iB (3.17)

where

n-1 |

aD ) N(X), 2=0 39

n B= ) N(2), and 2=0

D is the time period of inventory insertion.

Due to the constraint that an assembly discharged at time t+l

cannot be utilized to meet demand in year t, a computational constraint

is imposed on the cost coefficient. Using the technique known as the

"Big M'' method,?? an arbitrarily large number (M) is chosen for the cost

coefficient to reflect an infeasible solution. For the case of the maximum profit problem, a negative m is chosen to force the cost coefficient for a particular decision element to a value that would reflect an infeasible solution. This constraint is given by:

int | BB (3.18)

where

n B=) N(2). 2=0

by redefining the decision variable X54, (t) as:

Yin = the number of spent fuel assemblies from source i allocated

to demand period n, the corresponding objective function can be written as:

A t ¢tl Maximize » > Ci. Yan (3.19) i=l n=1 40

£ Subject to: ) Yan = ais i=l, ...,A

A Y Yan = bp i=1

A t +1 £ ) a; - ) b, = 0 i=1 n=1

Y > 0 ¥ odi,on

The maximization problem is converted to a minimization problem by

the standard approach of subtracting all cost coefficients, C;,, from

the maximum of the Cj, and replacing the original coefficients with —

these new corresponding computed values.2®8 Given the defined correspon-

dence between the spent-fuel classes, years of generation, and the

sources, along with that between the reprocessing capacities and the

demands, the interpretation is therefore straightforward. Thus, the problem has been converted into a Hitchcock problem that is far more readily analyzed computationally than the dynamic program.

D. The Linear Programming Formulations

The Hitchcock problem was originally designed to address a class of transportation problems or minimum-cost-flow problems. The formu- lation presented in Section C transforms the dynamic programming formulation into a minimum-cost-flow formulation. The purpose of this section is to develop an appropriate representation of the Hitchcock 41 model that is amenable to solution as a linear program.

The basis for this alternative representation results from the work of Bowman,® in which production scheduling was achieved using the transportation method of linear programming. This approach is partic~ ularly amenable to the Hitchcock model since the Hitchcock model is a formulation of the standard linear programming transportation problem.

Bowman proposed the use of the transportation model with the objective of assigning units of productive capacity in such a way that combined production plus storage costs are minimized and sales demands met, within the constraints of available capacity.’ An appropriate representation of the problem can be seen in Table 3.1 which denotes the activity matrix representation of the spent-fuel-withdrawal problem.

Before trying to understand the representation, it is first necessary to. redefine some parameters from the Hitchcock problem and to also define some new elements.

To begin with, note that for the linear programming formulation, the index set J, (equation 3.2) has been reduced to denote a single class of assemblies {N(t) = 1, ¥ t}, and therefore, it can be eliminated entirely for notational purposes. Also, the number of time periods defined by the index set T (equation 3.1) remain the same. The measure of effectiveness for this representation is the cost coefficient defined as C,_, where i = {0, 1, . Ls teh and n = {1, 2, ..., tet. The cost coefficient represents the profit from reprocessing less the cumulative holding costs in the feasible region. Otherwise the infeasible region has cost coefficients given by a negative "big M". 42

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The coefficient is represented by:

c= (3.20)

where

Pin = the unit profit resulting from reprocessing a spent-fuel

assembly in period n that was discharged in period i and,

Hig = the unit holding cost for a spent-fuel assembly in period

& that was discharged in period i.

The supply, denoted by ass and the demand, denoted by bo> are re- defined as: I ~

(3.21) Go N D(t=n), ¥et

Also necessary to complete this model is the definition of the slack inventory term 1, (i) as the number of spent-fuel assemblies dis- charged in period i that are left in inventory at the end of the horizon.

This added term applies the constraint on the model that:

a,= ) Y.+ 1, ¥i | (3.22) 44

where

Yin = the number of spent-fuel assemblies discharged in period i

allocated to meet demand in period n.

An additional constraint imposed is:

CE beat = ) Iga) (3.23) f i=0

and that for supply to equal demand, the condition:

tr t etl } aj - ) by = 0 (3.24) i=0 n=1

must be satisfied.

Summarizing the linear programming formulation by including the

objective function, the problem becomes:

Maximize — y ) C.in in (3.25) i=0 n=1

subject to:

tptl ) Yan tIg(i) = ay ¥ i n=1 45

Yin = dy

ee aj - b n=1 "

te 4, THE EXAMINATION AND EVALUATION OF THE COMPONENTS

OF THE SPENT-FUEL WITHDRAWAL PROBLEM

A. Characteristics of Nuclear Reactors in Regard to Nuclear Fuel

In order to demonstrate the utility of the models developed, it is necessary to first gain an understanding of the characteristics of nuclear fuel. This is a prerequisite for the implementation of the model, since the decisions rest upon the discharge characteristics of the spent-fuel.

At present, there are 59 commercial operating reactors within the

United States ranging in capacity from 48 to 1180 MWe. 3° Over the next decade, as vendors attempt system standardization, the largest plants on-line will have a capacity of up to 1300 MWe. For purposes of this study, the systems to be examined will be standardized designs of the four vendors currently producing nuclear steam supply systems within the United States.

The standardized system of the four vendors, Westinghouse (W),

Babcock and Wilcox (B & W), Combustion Engineering (CE), and General

Electric (GE) ; have nominal ratings of 1150, 1200, 1300, and 1200 MWe each, respectively. As a result of the variations in capacity plus individualized company designs, the nuclear fuel for each system has unique characteristics. This is true for initial and subsequent loadings of fresh fuel, along with the fuel discharged. Listed in

Table 4.1 are discharge and initial fueling data for the four light water reactor systems. The data given in this table will serve as the basis for parameters to’ be. established subsequently.

46 47

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48

During start-up and initial testing of the nuclear steam supply

system, variations in burnup are to be noted between the initial core

and the equilibrium core. These burnups, determined to be an average of

24,000 and 17,000 megawatt-days per metric tonne of uranium (MWD/MTU)

for the initial core of the PWR and BWR systems respectively, yield fuel

at discharge with a uranium assay of 0.85% U-235 and 0.864 U-235 respec-

tively. Similar interpretation can be established for the equilibrium-

core loading data. The difference in design and operating charac-

teristics for the BWR and PWR results in the difference in burnups

established for each system.

At best, nuclear power systems can extract only about 24 of the

energy theoretically available from uranium fuel. As a result of this

low energy yield with respect to what is theoretically available, | nuclear reactor fuel must be replaced periodically. Current generation reactors systems operate on a cycle in which a portion of the core is replaced annually. Each of the PWR systems discharge one-third of

their core annually, while the BWR system discharges one-fourth of its core annually. This difference between the PWR and BWR can once again be attributed to a difference in the basic design and operating characteristics of each system.

As a result of irradiation in the reactor, in which energy is produced by the fission process, a fraction of the uranium originally charged into the core is bred into plutonium and/or converted into fission products. At discharge, over 98% of the original uranium charged into the reactor remains.!? For the purpose of this study, the 49

negligible difference was ignored and the uranium discharged was

calculated based upon the initial charging.

Listed in Table 4.2 are discharge quantities of uranium and

plutonium found in spent-fuel. The calculations for plutonium quantities

are based on an average fissile plutonium discharge of 4.824 kg/MTU and

5.930 kg/MTU for the initial and replacement cores of a BWR; and 5.829

kg/MTU and 6.633 kg/MTU for the initial and replacement cores of a PWR.27

‘The isotopic composition of discharged plutonium varies with fuel

exposure and with repeated recycle of recovered plutonium in LWR's.!7

With successive recycles, the build up of Pu-236, Pu-238, Pu-240, and

Pu-242 increases and a greater amount of fissile plutonium, Pu-239 and

Pu-241, is required to compensate for parasitic capture. This is

illustrated by Deonigi!l? in which it is noted that during the next

decade, the nuclear industry will pass through a transition where most

of the fuel discharged will be at a low, variational exposure, to a

situation in the year 1985 where the discharged fuel will be pre- dominantly at equilibrium exposures. The results of testing fuel to

date show this assumption to be true.!7 Thus, it will be assumed for

this study that the average composition of plutonium in discharged fuel

is that given in Table 4.3. Also assumed is that the composition given

for plutonium in 1975 applies through 1979; that for 1980 applies through

1984; and that for 1985 applies through the remainder of the century.

Also included in Table 4.3 is the percent of the fissile plutonium in

the spent-fuel at discharge. 50

Table 4.2. Spent Fuel Discharge Characteristics

PWR PWR PWR BWR TEE w) | @Bew | (cE) (GE)

Initial Core

Discharge per Assembly (kg U) 412.0. 408.1 . 376.1. 1 66.2

Discharge per Assembly (kg fissile Pu) 2.4015 2.3788 2.1958 0.8017

Equilibrium Core

Discharge Per Assembly 412.0 408.1 376.1 166.2

(kg U) s e . } s

Discharge Per Assembly | 5 7328 | 2.7069 | 2.4987 | 0.9856 (kg fissile Pu)

51

Table 4.3. Average Composition of Plutonium Available for Recycle

Per Cent!’

YEAR | Pu-236 | Pu-238 | Pu-239 | Pu-240 | Pu-241 | Pu-242 | Fissile Pu

1975 | 0.006 1.0 64 . 22 10 3 74

1980 | 0.007 1.5 58 24 11 5 69 1985 | 0.007 1.7 54 25 12 7 66

52

Plutonium-241 is radioactive and undergoes a beta decay to

americium-241 with a half-life of 13.2 years. As a result, there is a

decrease of fissile plutonium content in the spent-fuel with time. This

effect tends to reduce the profitability of the spent-fuel from

reprocessing. The effect of this Pu-241 decay on the fissile plutonium

in the spent-fuel assembly is expressed by:

Pug(t) = Pug (0) {1 ~ r(l-e-At)} (4.1)

where

Pug(t) = quantity of fissile. plutonium (239 + 241) at time t,

Pug(0) = quantity of fissile plutonium (239 + 241) at discharge,

r = the ratio of Pu-241 at: time t=0 to the total fissile

Pu (239 + 241) at time t=0, and

X = the decay constant for Pu-241, A=.05251 year7l,

Equation 4.1 represents the correction for Pu-241 decay in stored spent-

fuel.

Having established the essential characteristics of spent-fuel, it is appropriate to determine the spent-fuel supply and the spent-fuel

demand. These prerequisites for implementation and solution of the model

are represented in the following section.

B. Spent-Fuel Supply and Demand Projections

It is necessary to first examine the supply of spent-fuel globally

in order to determine the reprocessing capacity based on reactor mix.

ERDA has published projections for quantities of spent-fuel generated 53

over the interval 1976-1985.** This projection is based on reactors on-line and those anticipated to be on-line through the end of 1985.

Based upon the projected growth of nuclear generating capacity presented in Table 4.4, a comparison is made to determine an average spent-fuel supply per GWe (1000 MWe). As a result of this comparison through the year 1985, the discharged quantities of spent-fuel in terms of tonnes of uranium and also the number of assemblies per gigawatt

(1000 megawatt) electric were determined. These quantities, given in

Table 4.5, represent the basis for the projected spent-fuel supply by reactor type to be established beyond 1985. By assuming an installed capacity of 1 BWR to 2 PWR's beyond 1987, nuclear growth capacity by reactor mix is established. This projection, coupled with the average discharge quantities of spent-fuel presented in Table 4.5, result in the global spent-fuel supply projections given in Table 4.6. In Table

4.6, tonnes of uranium as well as the number of assemblies of spent-fuel discharged per year are displayed. Calculations of the projected quantities in a given year are based on the installed capacity the previous year, indicating the one year lag in fuel loading and subsequent discharge. Having established the global quantities of spent-fuel discharged per year, it is necessary to establish the spent-fuel discharge per reactor. These quantities are established for the standardized systems examined in Table 4.1 and represent the annual supply available. As discussed earlier, these systems are the repre- sentative models of the reactors to be analyzed. Discharge quantities are 64, 68, 80, 183 assemblies per year for the Westinghouse, Babcock 54

Table 4.4. Installed Nuclear Capacity

Year Total Capacity BWR Capacity PWR Capacity (GWe) (GWe) (GWe)

1976 41.887 16.033 25.854 1977 50.364 17.921 32.443 1978 55.292 17.921 37.371 1979 63.480 20.629 42.851 1980 74.641 24.950 49.691 1981 92.151 29.295 62.856 1982 107.552 33.240 74.312 1983 125.875 41.344 84.531 1984 145.659 47.060 98.599 1985 161.928 52.948 108.980 1986 176.736 56.453 120.283

1987 182.394 58.781 123.613 1988 195.000 62.983 132.017 1989 204.000 65.483 138.017 1990 213.000 68.483 144.017 1991 225.000 72.483 152.017 1992 246.000 79.483 166.017 1993 274.500 89.483 185.017 1994 313.500 102.483 211.017 1995 345.000 112.983 232.017 1996 381.000 124.983 256.017 1997 409.500 134.483 275.017

55

Table 4.5. Discharge Quantities of Spent Fuel

Per Gigawatt (electric)

Reactor Type PWR BWR

Average Discharge (MTU/GWe) 29.002 | 35.329

Average Discharge (Assemblies/GWe) 67 192

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57

and Wilcox, Combustion Engineering, and General Electric systems, res-

pectively.!4

With the spent-fuel supply projections established, it is nec-

essary to determine the reprocessing capability. For the purpose of

this study, the amount of reprocessing available per reactor in any

year is based upon:

“1. the reprocessing capability in the given year,

2. the amount of spent-fuel generated in a given year, and

3. the reactor mix in which the reprocessing capability is

distributed.

As a result of the first and second statements above, the re-

processing capability in any given year are distributed to process

only the additional new fuel added to the supply each year. This es-

tablishes a reprocessing demand distribution in which per-reactor re-

processing demands can be established. From the third statement

above, the assumed reactor mix is 2 PWR’s to 1 BWR. This in effect

allots reprocessing capacity by two-thirds of the total to PWR's, and one-third of the total to BWR's. This is necessary since both

systems are to be examined on a unit-reactor basis.

Before proceeding further, the next question to be answered re-

lates to the availability of reprocessing capacity during the time

frame examined. As a result of a study by Macek,!7 three scenarios reflecting reprocessing plant start-up are established. They are: 58

1. Optimistic, with. start-up mid-1979,

2. Realistic, with start-up 1981, and

3. Pessimistic, with start-up 1983.

For purposes of plant start-up, it is assumed that the plants op-

erate at one-third and two-thirds capacity in their first and second

years of operation, respectively; and at their rated capacities

thereafter.!? This is in accord with recent government and indepen-

dent studies of reprocessing capability.’ 0 The earliest start-up

date (optimistic scenario) projects start-up of Allied General Nuclear

Services (AGNS) plant at Barnwell in 1978, Nuclear Fuel Services (NFS)

in 1983, Exxon in 1985, and a fourth plant in 1987. The rated capac- “ities of the plants are 1500, 750, 2000, and 2000 metric ton heavy

metal per year (MTHM/yr) respectively. Listed in Table 4.7 is the

reprocessing plant capacity schedule. Assuming the sequence of

reprocessing capacity additions over time remains the same for all

scenarios, the different scenarios to be examined differ only by

the start-up date of the reprocessing activity. Under this assump-

tion, the three reprocessing scenarios to be examined are presented in

Tables 4.8 through 4.10 based upon reactor mix. These projections are

compared to a recent independent study +? and prove to be conservative

by an average difference of 7 per cent.

To establish the reprocessing demand rate, in number of assemblies

per year, for a particular system, the discharge characteristics of

spent~fuel from Table 4.2 are utilized. Using average heavy metal

discharge quantities for each of the systems, the unit reactor 59

Table 4.7. Reprocessing Plant Capacity Schedule

Capacity -— MTHM

Year AGNS NFS Exxon 4th Plant

1977 - ~ = -

1978 - - - -

1979 500 = - -

1980 1000 - ~ -

1981 1500 - ~ - 1982 | 1500 - ~ -

1983 1500 250 ~ -

1984 1500 500 - -

1985 1500 750 750 -

1986 1500 750 1400 -

1987 1500 750 2000 750

1988 1500 750 2000 1400

1989 1500 750 2000 2000

1990 1500 750 2000 2000

1991 1500 750 2000 2000

1997 1500 750 2000 2000

60

Table 4.8. Reprocessing Capability - Optimistic Scenario

(MTU)

Year Total BWR PWR Year Total BWR PWR

1977 0 0 0 1988 5650 1883 3767 1978 0 0 0 1989 6250 2083 4167 1979 500 167 333 1990 6250 2083 4167 1980 1000 333. 667 1991 6250 2083 4167 1981 1500 500 1000 1992 6250 2083 4167 1982 1500 500 1000 1993 6250 2083 4167 1983 1750 583 1167 1994 6250 2083 4167 1984 2000 667 1333 1995 6250 2083 4167 1985 3000 1000 2000 1996 6250 2083 4167 1986 3650 1217 2433 1997 6250 2083 4167 1987 5000 1667 3333

61

Table 4.9. Reprocessing Capability - Realistic Scenario

(MTU)

Year Total BWR PWR Year Total BWR PWR

1977 0 0 0 1988 3650 1217 § 2433

1978 0 O- 0 1989 5000 1667 3333 1979 0 0 0 1990 5650 1883 3767 1980 0 0 0 1991 6250 2083 4167

1981 500 167 333 1992 6250 2083 4167

1982 1000 333 667 1993 6250 2083 4167

1983 1500 500 1000 1994 6250 2083 4167

1984 1500 500 1000 1995 6250 2083 4167

1985 1750 583 1167 1996 6250 2083 4167

1986 2000 667 1333 1997 6250 2083 4167 1987 3060 1000 2000

62

Table 4.10. Reprocessing Capability - Pesimistic Scenario

(MTU)

Year Total BWR PWR Year Total BWR PWR

1977 0 0 0 1988 2000 667 1333

1978 0 0 0 1989 3000 1000 2000

1979 0 0 0 1990 3650 1217 2433

1980 0 0 0 1991 5000 1667 3333

| 1981 0 0 0 1992 5650 1883 3767

1982 0 0 0 1993 6250 2083 — 4167

1983 500 167 333 1994 6250 2083 4167

1984 1000 333 667 1995 6250 2083 4167

1985 1500 500 1000 1996 6250 2083 4167

1986 1500 500 1000: 1997 6250 2083 1987 1750 583 1167

63

discharge demands are established. These demands, in terms of number of assemblies demanded per year, are represented in Figures

4.1 through 4.4 for each of the systems under examination. By plotting the annual discharge, as long as the demand curves remain below this rate, there is always a backlog of spent-fuel. When the yearly rate of demand exceeds the annual discharge, the backlog begins to reduce, with the possibility of consuming the entire backlog until the supply equals demand. For purposes of sensitivity analyses, the projected demand rates for each of the scenarios ex- amined are varied by +152 and +30% from the base projections. These demand profiles are also included in the figures. |

The remaining elements of the spent-fuel withdrawal model to be determined are the cost coefficients. Their derivation is discussed in the next section.

C. The Measure of Effectiveness - Profitability per Assembly

1. Independent Value Components

The choice of an optimal withdrawal policy for spent nuclear fuel rests primarily with the economics of the spent-fuel. Spent-fuel may be considered as a "mineral" which can be "mined" (reprocessed) to recover the unused nuclear fuel.2° A measure of the net value of unreprocessed fuel is given by equation 4.217 where

Net Value = U Value+ Pu Value - Fuel Storage Costs

~ Reprocessing Costs - Waste Disposal Costs (4.2) 64

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zoj0eay Jog pepuewaq seT[Tquessy jo azsquny 68

Of the five components of the value of spent-fuel, only three components have been established with reasonable assurance and validity. Speci- fically, uranium value, plutonium value, and storage costs have been studied» 19 517 ana pricing methods established; whereas, due to the uncertainties surrounding the back end of the fuel cycle, repro- cessing and waste disposal costs are debateable and uncertain.

When considering the question of the reprocessing alternative, it is to be noted that certain costs must be met if a profit is to be | realized. This is particularly true of reprocessing and waste disposal costs. These two components of the value equation (4.1) are costs to be incurred if any measure of profitability is to be realized. There- fore, when considering the establishing of an economic measure of effectiveness, these costs do not have to be included in the profit- ability of spent-fuel for the analyses of this particular problem.

As a result, equation 4.2 is redefined to reflect the profitability per assembly in terms of the time and characteristic dependent costs.

Thus equation 4.2 becomes:

Profitability| _ [U Value Per] 4+ |Pu Value _ |Storage Costsir4 . 3) Per Assembly |. Assembly Per Assembly Per Assembly

When each of these parameters are taken into consideration, the salvage value of the spent~fuel is utilized as the measure of the effective economic gain. Thus, the basis for the cost coefficients of the models derived is that given above by equation 4.3.

2. Uranium Value of Spent-Fuel 69

The uranium recovered in the reprocessing facility can be recycled into the feed steam of the nuclear fuel cycle. Since the discharged fuel from the operating light water reactors under equilibrium conditions contains fissile uranium (U-235) that is higher in concen- tration than that of natural uranium, use of spent-fuel provides a savings in natural uranium feed and separative work.

The potential savings in natural uranium feed and separative work are proportional to the price of the enriched product from the enrich- ment plant. Enrichment expenses in a gaseous diffusion plant are proportional to separative work, whose cost reflects energy consumption needed to produce enriched uranium. Upon analysis of the gaseous diffusion cascade, ?4t this value of the enriched uranium product based upon the unit cost of separative work and the unit cost of natural uranium feed is given by:

D = C.{(2x.-1) gn __“P_ ~ (2x ~1)£n_7*w__+ Ss P 1-xp ~Ky

_*p7*w_Ey ((2x,-1) Qn Ix,“woe (2x_-1) an IKfy}

+ C, *p7*w (4.4) Kp Ky where

D = price of the enriched product, $/kg;

Cs= unit cost of separative work, $/kg;

70

Cy = unit cost of natural uranium feed, $/kg;

x. = assay of uranium-235 in the enriched product weight fraction;

X- = assay of uranium-235 in the feed stream, weight fraction; and

x = assay of uranium-235 in the diffusion plant tails, weight

fraction.

Also included in the original analysis of the diffusion plant cascade

for the enriched product is a unit cost assigned to the diffusion tails.

At present, no value is assigned to the tails and as a result, it is ©

not included in equation 4.4.

Utilizing the discharge assays of the spent nuclear fuel for the

PWR and BWR systems given in Table 4.1, the uranium-value equations

are established for the intial- and equilibrium-core discharges. The

uranium-value equations are determined on the basis of an enrichment

plant tails assay of 0.304 and the product enrichment being the spent-

fuel assay. By substituting this data into equation 4.4, the value of

recovered uranium for the initial core discharge is given by:

1. BWR

D= (0.12397)C_ + (1.36253)Ce- Co (4.5)

2. PWR

D = (0.11456)C_ + (1.33820)C, - C, (4.6)

are for the equilibrium core discharges, the value of recovered uranium

is given by: 71

1. BWR

D= (0.10529)C. + (1.31387)C, = Cc. (4.7)

2. PWR

D= (0.08719)C_ + (1.26521)C. - Cc. (4.8) where C, is added to represent the unit cost of conversion of uranium to UF¢ gas.

These value equations can be interpreted to indicate (using equation 4.5) that one unit of recovered uranium can serve to reduce the front end requirement for natural uranium by 1.36 units and for separative work by 0.124 units. Similar interpretation can be applied to equations 4.6-4.8. Formulas 4.5~-4.8 corrected for losses during reprocessing and fabrication are rewritten as:

D = {(0.12397)C, + (1.36253)C, - C,}{1 - ny- no} (4.9)

D = £(0.11456)¢, + (1.33820)C, - C,H1- n, - no} (4.10)

D= {(0.10529)C. + (1.31387)C, - CHI -1 7 No} (4.11)

D = {(0.08719)C_ + (1.26521)c, - C Hl - ny - no} (4.12) where

ny = loss during reprocessing!’ (=0.005)

7 = loss during fabrication!’ (#0 .003)

A reduction in uranium value is noted between the initial- and 72

equilibrium-core discharges due to higher burnups which result in a

greater utilization of the fissile U-235.

From the formulas derived, it is shown that the values of recovered

uranium from spent-fuel is related to the prices of uranium and sepa- rative work less the conversion costs. The price of uranium is in- fluenced by the uranium market, but eventually the price depends on

the balance: between supply and demand. Projections made by the Energy

Research and Development Administration? indicate that as the cumulative production of U,0g increases, the value of the uranium will also increase. This projection results approximately in an 18% increase in the price of U,0g as the cumulative production increases by one million short tons. Correlating this projection with a historical analysis of the uranium mining industry by Liebermann,? the price of uranium is projected to increase at an average rate of 5.5% per year. This average rate is determined based upon proven reserves cf less than $30 per pound forward cost. A study by Voltin and Draper!! estimated that the rate of uranium price increase is reduced as more reserves are proven and as international trade increases; the rate of 5.52% per year is effective only through the year 1985. Afterwards, as a result of the increased supply, the price is assumed to increase at a reduced rate of 3.9% per year./1 it is forecasted that the present U30g price of $41 per pound?" increases to $63.66 by 1985, and reaches $101.65 by

1997, The forecasts of future uranium prices are summarized in Table

4.11.

In the U.S. today, the only major supplier of enrichment is the

ERDA diffusion plant complex. The diffusion process is energy 73

Table 4.11. Uranium Price Projections

Price Price | Price Price Year Year $/1b U0 $/kg U302 $/1b U302 $/kg U308

1977 41.00 90.39 1988 71.56 157.77

1978 43.32 95.50 1989 74.41 164.04

1979 45.77 100.90 1990 77.37 120.57

1980 48.36 106.60 1991 80.44 177.35

1981 51.09 112.63 1992 83.64 184.40

1982 53.98 119.00 1993 86.97 191.74

1983 57.03 125.73 1994 90.43 197.36

1984 60.25 132.84 1995 94.03 207.29

1985 63.66 140.35 1996 97.77 215.54

1986 66.19 145.93 1997 101.65 224.11 1987 68.83 151.73

74 intensive; and as a result, a sharp increase from $26 per kg of sepa- ative work for fixed-commitment contracts is projected to escalate at

1.4% annually by Voltin and Draper in 1976.!! This is to allow for the increase costs due to the required energy input per SWU. The reason that the escalation is not greater after 1985, when energy costs are expected to rise sharply, is that it is assumed that new technology

(centrifuge, nozzle, and/or laser separation) will make a contribution to lowering costs. . However, upon comparison of this projected rate with recent ERDA projections,» the estimate is changed to an increase of

2.44 annually, to more accurately reflect recent trends. Separative work cost projections for the time frame of this study are summarized in Table 4.12. Also as a result of the work of Voltin and Draper, the conversion costs are forecast to increase by 2% per year from the 1977 price of $4.41 per kg.° Table 4.13 presents the forecast of uranium conversion (to UF ¢ gas) costs.

3. Plutonium Value

The value of the residual plutonium in the spent-fuel is deter- mined by the manner in which plutonium is used. In order to establish a value, it is necessary to distinguish between plutonium price and plutonium value.?° A "use value" is understood to be the sum of money paid in order to use a commodity. A "market price" depends upon the balance between supply and demand. When considering the current stand- still of reprocessing capability in the U.S., the establishment of a firm market price for plutonium is impractical.

However, for the purpose of this study, an artificial "market" is 75

Table 4.12. Separative Work Cost Projection

SWU Cost SWU Cost Year Year $/kg-SWU $/kg-SWU

1977 69.80 1988 90.84 1978 71.50 1989 93.10 1979 73.23 1990 95.36 1980 75.01 1991 97.67 1981 76.83 1992 100.65 1982 78.70 1993 102.48 1983 80.61 1994 104.97 1984 82.57 1995 107.52 1985 84.57 1996 110.13 1986 86.63 1997 112.80 1987 88.73

76

Table 4.13. Uranium Conversion Costs Forecast

Cost Cost Year Year $/kg-U $/kg-U

1977 | 4.41 1988 | 5.59 1978 | 4.50 1989 | 5.70 1979 | 4.59 1990 | 5.82 1980 | 4.68 1991 | 5.93 1981 | 4.77 1992 | 6.05 1982 | 4.87 1993 | 6.17 1983 | 4.97 1994 | 6.30 1984 | 5.06 1995 | 6.42 1985 | 5.17 1996 | 6.55. 1986 | 5.27 1997 | 6.68 1987 | 5.48

77 established for plutonium based upon the use value (limiting value for plutonium in the plutonium market). To define the use value, it is necessary to know the technical and economical circumstances of plutonium utilization.2? Obviously, the plutonium market, when esta- blished in this manner, is linked with uranium and enrichment services since if recycled, fissile plutonium can replace some U-235.

Plutonium generated in light water reactors can be used in several different modes. Among the utilizations established are: in-situ (the plutonium, when formed in the reactor, is consumed before recycle), for recycle in LWR's, for the High Temperature Gas Cooled Reactor based upon plutonium fuel design, and for early fuelings of the fast breeders. 25 Only light water reactors are expected to contribute to the nuclear generating capacity through the end of this century. This is due to the establishment of firm technology and services.to support it.17 Thus, for the time frame of this study, it is assumed that only piutonium use in-situ and in recycle contributes to the establishment of a plutonium market.

Fissile plutonium is formed in the reactor core as a result of the radiative capture of neutrons in the U-238 nucleus. When formed, the fissile plutonium (Pu-239 and Pu-241) participates in a similar fashion to U-235 in the general fission reactions. The more fissile plutonium that is formed in the reactor, the more important it is in-situ con- sumption. The quantity of fissile plutonium available to participate in the fission process depends upon the degree of core burnup and the neutronic characteristics of the reactor. No value can be assigned to this plutonium since it is a general product of neutron capture in U-238 78 and since it is freely available to be utilized without reprocessing.

Plutonium remaining in the spent-fuel can be chemically separated

(reprocessed) from the uranium and fission products contained in the spent-fuel. It can be utilized as a fuel again by mixing it with natu~ ral, depleted, or enriched uranium to form a mixed-oxide fuel. Thus, the separated plutonium can be reintroduced into the same reactor.

As discussed in section A of this chapter, there will be a tran- sition from fuel discharged at low exposures, to a situation in 1983, where the fuel will be discharged predominantly at equilbrium exposures.

The effect of the recycled plutonium depends on this situation, since the operating characteristic of the recycle reactor will depend on the composition of the plutonium. Referring back to Table 4.3, the numbers presented are useful in determining the average value of plutonium in the time periods when it can be expected that the reprocessing plants will be operational. As mentioned earlier, for this study, this recycle mode is primarily for the purpose of creating an “artificial market" to establish a value for plutonium.

The value of plutonium can be established in several ways. The method most frequently used is the indifference or break-even method proposed by Eselbach.!© The indifference method defines plutonium values as the value yielding the same fuel cycle costs whether the plutonium discharged is sold or recycled. For this method, firm’ numerical values for each fuel cycle cost component for both enriched uranium fuel and plutonium recycle must be known, leaving only the value of plutonium unknown. Obviously this method depends heavily upon a mixed-oxide fuel cycle and the establishment of a well defined 79 plutonium market. Therefore, it seems that this method does not appear useful in the present situation of a non-existent firm plutonium market.

This results from the present uncertainties surrounding reprocessing and recycle.

A generalized equation for plutonium value is derived by Deonigi which establishes relationships with fully enriched uranium, Plutonium-

242 penalty factor, and the differential fabrication cost correction for mixed-oxide fuel above that of uranium fuel.!’ Deonigi shows that

the plutonium value is VPu) = U(ie1.6R) - Pu fabsication penateys S/kgwOK (4.13) where

A = plutonium replacement value (relative worth of Pu-239 to U-235

as a fissile material - gm U-235/em Pu — fissile),

U = the cost of fully enriched uranium (93% wt) at the fabrication

plant - $/kg, and

R = ratio of concentration of Pu-242 to the concentration of

fissile plutonium, (Pu-239 and Pu-241).

For purposes of this study, the value of plutonium is best estab- lished by considering the savings realized in the reduction of require- ments for uranium ore and separative work units that results from re- processing.

In the GESMO study, an estimate of savings in uranium ore and separative work realized by plutonium recycle is evaluated.!” This 80

evaluation establishes equivalent savings based on the present offer-

ings of major fuel suppliers. It is found that one gram of fissile

plutonium in a PWR is equivalent to 0.1906 kgs of separative work plus

0.180 kgs of natural uranium. A similar analysis for BWR recycle is

that 0.2037 kgs of separative work plus 0.1870 kgs of natural uranium

is equivalent to one gram of fissile plutonium. The amount of savings realizable is dependent on the price that must be paid for natural

uranium and separative work during the year that recycle occurs. Since

the actual value of the plutonium at a specific time is dependent upon

the quantity of fissile plutonium available at that time, a correction

is allowed for the decay of Pu-241. Accounting for losses during re-

processing and fabrication, the effective plutonium value is determined

by:

V(Pu) = (Cg * A+ Ce * BY) (1 - ny - ng - 13) (4.14) u rg

G plutonium value, $/gm - fissile;

C_ = unit cost of separative work, $/kg;

Ce = unit cost of natural uranium feed, $/kg;

nL = loss during reprocessing!” (= 0.005);

= loss during fabrication!’ ( 0.003); and

= loss in fissile Pu during reprocessing and fabrication due

to decay of Pu-2412° (= 0.01).

The factors A (equivalent of one gram of fissile plutonium in terms of separative work units), and B (equivalent to one gram of fissile 81

plutonium in terms of natural uranium), which vary with the type of reactor, are summarized in Table 4.14. This equivalency approach serves as the method of determining the value of plutonium. The cor- rection for decay of: plutonium-241 with time is reflected in the com- puter code developed to determine the profit realizable from recycle of the plutonium and uranium. It is appropriate to note that equation

4.14 does not include a fabrication penalty for mixed-oxide fabrication.

This penalty is a cost factor that is incurred if recycle and subse- quently an economic gain is to be realized.

D. Storage Costs

An examination of the storage costs is conducted by Macek.2° From the examination, it is established that the range of the costs of spent- fuel storage extends from $5500/MT-spent-fuel per year for a utility- owned storage facility, to $20,000/MI-spent-fuel per year for a commer- cial facility. Macek establishes a figure of $7000 per tonne of fuel per year as typical cost for facilities where utilities supply some form of long term commitment or front-end money. It is expected that the storage costs will increase at the annual rate of inflation. For the current decade, this rate will be at 8% per year, during the 1980's at 6%/yr, and during the 1990's at 4%/yr. Decrease in the rates for the 1980's and 1990's are attributed to a slowing in the rate of inflation

Based upon the assumptions above, on-site storage costs per assembly- year for each of the reactor systems are summerized in Table 4.15.

Each of the parameters discussed in this section are utilized in 82 developing a data generating code. The implementation of the code, plus the solution technique is discussed in the next chapter. 83

Table 4.14. ‘Plutonium Value for Uranium Feed

and Separative Work Equivalents!7

Type of Reactor Equivalents to 1 gm of Pu (fissile)

. PWR BWR

SWU Equivalent - Kg 0.1906 | 0.180 Natural U Equivalent ~- Kg 0.2037 | 0.1870

84

Table 4.15. On-Site Storage Costs Per Assembly - Year

$/Assembly

PWR PWR PWR BWR Year (W) (B & W) (CE) (GE)

1977 3271 3240 2991 1319 1978 3543 3510 3240 1429 1979 3839 3802 3510 1548 1980 4158 4119 3803 1677 1981 4416 4374 4038 1781 1982 4689 4644 4288 1891 1983 4979 4931 4553 2008 1984 5286 5236 4834 2132 1985 | 5613 5560 5133 2264 1986 5961 5904 5451 2404 1987 6329 6269 5788 2553

1988 6721 6657 6146 2711 1989 7136 7069 6526 2878 1990 7577 7506 6929 3056 1991 7887. 7812 7212 3181 1992 8209 8131 7507 5311 1993 8544 8463 7813 3446 1994 8892 8808 8132 3587 1995 9255 9168 8464 3733 1996 ~—-9633 9542 8809 3886 1997 10026 9931 9169 4044

5. APPLICATIONOF THE SPENT-—FUEL-WITHDRAWAL MODEL

A. Model and Data Summary

The linear models for the spent-fuel-withdrawal problem developed in Sections C and D of Chapter 3 are analyzed from an optimization standpoint. These models are chosen to be applied to the optimization of the inventory withdrawal problem because standard methods exist for their solution. Before discussing these models, it is necessary first to discuss the dynamic programming formulation and the problems to -be encountered in its implementation.

The dynamic program formulated in Section 3-B is, computationally a large problem. For the general problem, considering N(t) classes of spent-fuel for each time period t, the model has te(t,t1)/2 stages.

Considering N(t)=1 for all t in order to reduce the dimensionality of the problem, the program has tr stages. For either problem the number of state variable transformations is tg(tgtl)/2.28 At each stage, the number of state variables exceeds or is equal to the number of decision variables. This adds additional computational difficulty, if not impossibility, in obtaining a dynamic programming solution of the model.

As a result of these difficulties encountered with the dynamic program, the linear approach is determined to be the best.

Upon considering the linear models developed in Chapter 3, a reduction in dimensionality is obtained for the Hitchcock formulation when considering only one spent-fuel class at each discharge period.

Upon reduction in dimensionality, the Hitchcock problem formulation reduces to the Linear Programming Problem or Transportation model

85 86

developed in section 3-D. The model that is applied to the actual

optimization of the inventory model is analyzed from the point of view

of only one class of spent-fuel at a given discharge period. This one

classification represents the average characteristics of the spent-fuel

in the discharge. Having examined actual discharge data for the H. B.

Robinson plant, ® it is found that each assembly can be uniquely classi- fied according to the burnup it has experienced. For this particular

plant, this results in 53 different classifications at each discharge.

Statistically, for each of the assemblies examined from any given dis-

charge period, deviation from mean burnup is at a maximum, only 92%.

Thus, for the purposes of reduction in dimensionality of the problem,

only one classification of assemblies in a discharge period is examined.

It should be noted that differences in burnups experienced by the aver-

age assemblies are reflected in the economic measure of effectiveness

to be determined. This measure is based upon the characteristics of

the spent-fuel from each discharge period.

The specific model to be solved is given as:

Maximize ) y Cia Yin . i=0 n=l

subject to: | (5.1)

tptl 1) )} Yip tigi) =a, ¥ i 87

te 2) )} Yan = Dy ¥on i=0

3) ) aj 7 } by = 0 i= =]

5) Y¥.,. > 0 ¥oi,on

where Y;,, n is the decision variable to be determined to yield optimal results.

Based upon the evaluation and determination of the components of the measure of effectiveness for the problem as presented in Chapter 4, a computer code is developed. This code is used specifically for the purpose of cost data preparation. Included in Appendix B, this code is developed for more use than that needed for this study. However, for this study, the code determines the effective value of the spent-fuel based upon the value of residual uranium and plutonium less cumulative storage costs during the period of examination. Included in this code are corrections for the decay of fissile Pu-241 which reduce the effective plutonium value while in storage. Based upon the developments in Chapter 4, the profitability or measure of effectiveness that serves as cost coefficients (Cc, are determined for each pair of discharge periods i and demand periods n in the feasible region. Summarized 88 in Table 5.1 are the profitability or cost coefficients for spent~ fuel discharge in the first, fifth, tenth, fifteenth, and twentieth year of the horizon. In the base case, this is for a Westinghouse assembly. As is determined from the data, assemblies discharged later in the horizon actually decrease in value. This is due to the rate of increase in storage costs exceeding the cumulative rate of increase in the value of uranium and plutonium in the spent-fuel. Similar data is obtained for the other PWR systems and the BWR system. This data is listed in Tables 5.2-5.4.

B. Implementation of the Model and Procedural Summary

The spent-fuel-inventory~-withdrawal model is implemented on the

Virginia Polytechnic Institute and State University IBM computer

System/370 Model 1585 using a proprietory mathematical. programming system--MPS IIr,26 and by using the Out-of-Kilter- algorithm. 19

The Out-of-Kilter algorithm developed by Fulkerson is a method of solving minimal-cost network flow problems. The method begins with an arbitrary flow, feasible or not, together with an arbitrary pricing vector, and then uses a labeling procedure to adjust an arc of the net- work that fails to satisfy the appropriate optimality properties. This algorithm is particularly applicable to the Hitchcock problem formula- tion. |

MPS III is a modular mathematical programming system consisting of several components. Its BASIC system provides the system control and the standard optimization procedure. This system incorporates a very high-speed matrix inversion method utilizing the preassigned pivot 89

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procedure for sparse matrix inversion. The VARIFORM module prepares a model for solution, solves the model, and prints the answers from the solution. The VARIFORM Module can be instructed to minimize or maximize the objective function, thus yielding a user determined optimization.

Additional features of the MPS III system are described in the liter- ature.26

Initial trial executions are attempted to determine which of the two solution techniques are to be used. The Out-of-Kilter algorithm offers a reduction in the data handling but proved less computationally efficient. An average execution of the Out-of~Kilter code yields optimal results after 43 minutes of CPU time. The MPS III system has the disadvantage of much data handling but a typical execution time averages only 30 seconds. Thus, the withdrawal problem is solved using the MPS III system.

The procedure for obtaining solutions for the spent-fuel-withdrawal problem involves two basic computations. The first is the analysis of the base case data as established in Chapter 4. This involves the calculation of the cost coefficients as given partially in Tables 5.1-

5.4 from the data generating code. Each of the primary cost components, uranium costs, separative work costs, and storage costs, are utilized at their projected values.|

The second phase of the computation involves testing the sensi- tivity on the first discharge's profitability are shown in Figures 5.1 and 5.2 for the Westinghouse and General Electric systems. Note that uranium prices and storage costs prove to be the most sensitive para- 96

meters that determine the profitability of the spent-fuel with time for

this study. When storage costs increase at a rate greater than that

for the value of uranium and plutonium in an assembly, a loss in value

occurs with time. This is particularly true for 120% storage cost and

80% uranium cost projections. When the value of the uranium and

plutonium in the assembly increases at a rate that far exceeds the rate

of increase of the storage costs, a noticeable increase in profit-

ability per assembly over time is noted. This is the case for 120% of

the uranium price, and for 80% of the storage cost projections. Little

effect on the profitability profile is noted for variations in the

price of separative work.

The final step in the analysis of the model is to determine the

effect of prohibiting plutonium recycle so that only uranium recycle

occurs. Depicted in Figures 5.3 and 5.4 are the Westinghouse PWR and

GE BWR profitability profiles for the first discharge under the con- dition of no plutonium market. As can be seen, from the profiles, a notable decrease in the profitability versus time occurs for all cases

except 120% of uranium price and 80% of storage costs. This is attributed to the fact that without plutonium recycle, the value of the

spent-fuel is minimized to a point that the addition of holding costs each year exceeds the rate at which the uranium in the spent-fuel

increases in value. A very slight increase in value with time is noted for the 120% uranium price and 80% storage costs profiles. How- ever, once again, as time in storage increases, the value begins to decrease and a measureable loss is incurred.

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Results of the several situations examined are discussed in the next chapter. In particular, the analysis of the results are indicated by the reprocessing scenarios established in Chapter 4. 6. RESULTS

A. Optimistic Reprocessing Scenario

The optimistic reprocessing scenario assumes the re-initiation of

reprocessing in mid-1979. For this assumption, the spent-fuel-withdrawal

model as developed in Section 3-D, is optimized using the MPS-III module.

The analyses performed are primarily for two distinct cases. The first

situation examined assumes the recycle of both plutonium and uranium

and associated markets for each. . Results for this case are examined

first on the basis of the projected optimistic reprocessing demand

capability for each reactor system. Then the demands established are

varied by +15% and +30% for purposes of testing solution sensitivity

to these changes. The projected optimistic demand discussed in Section

4-B is referred to as the base demand case, while the sensitivity

‘increases reflect their changes appropriately.

As discussed in Chapter 5, each of the principal cost elements of

the measure of profitability of the spent-fuel assembly are varied by

+ 204 to determine the sensitivity of the solution to variations in

these parameters. Results from each of these variations except for a

120% increase in the uranium price indicate that the Last-In-First—Out

(LIFO) policy is an optimal selection rule. This is shown in Figure

6.1. However, when the price of uranium increases from its base by

+20%, a modified LIFO policy is determined as the optimal selection

rule. This is depicted in Figure 6.2. In general, the results for the

optimistic scenario considering both plutonium and uranium recycle

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6.2 are for the Westinghouse PWR, which is representative of the PWR systems. An examination of the GE BWR/6 system also yields the LIFO selection rule. Individual results for the cases examined are included in Appendix A.

To interpret these figures, and those that will appear in the remainder of this chapter and Appendix A, a note of explanation is appropriate. Associated with the rows of the table are the supply of fuel assemblies. The left end of the table lists the year of discharge and right side lists the associated number of assemblies discharged

(supply). Listed at the top of the table is the year of demand and at the bottom of the table, the associated number of assemblies demanded.

Also included in the column section is a column denoted as T.: The term

Ig denotes the slack inventory or the number of spent~fuel assemblies residing in inventory at the end of the horizon. Optimal decisions are given in the rectangular box enclosed in double lines for emphasis.

For example, from Figure 6.1, a partial interpretation is that of the 64 assemblies discharged in 1978, 21 assemblies are used in meeting demand in 1980, with the remaining 43 assemblies left in the storage pool at the end of the horizon. Similar analyses can be performed for each discharge period in the figure.

Upon analysis of the different cases in the absence of a plutonium market, non-unique or non-distinctive policies are found. This can be seen in Figures 6.3 and 6.4. Figure 6.3 is the base case analysis for 104

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‘As can be observed from the analyses of the situations examined, the specific results are that the Last-In-First-Out policy is the optimal choice for selecting spent-fuel from on-site storage when Pu and U recycle occurs. However, no unique or discernable policy exists for the situation in which no plutonium recycle occurs.

B. Realistic Reprocessing Scenario

The Realistic reprocessing scenario discussed in Section 4-B assumes the start-up of reprocessing in 1981. The sequence of the addition of reprocessing capacity over time is the same for the Realistic

Scenario as for the Optimistic Scenario. The only difference is the start-up date.

As discussed in the preceeding section, two basic situations are analyzed. The first involves the profitability of spent-fuel with established plutonium and uranium recycle, and the second situation involves only U recycle. Once again for purposes of sensitivity analyses, the Realistic demand rates are varied by +15% and +30%. Also 107

for each execution, the principal value components that determine the profitability of spent-fuel (U prices, SWU costs, and storage costs), are varied from the base measure by +202.

Results from these analyses once again indicate that in the absence of Pu recycle, no definative policy is observable. This is depicted in Figures 6.5 and 6.6 for the Westinghouse PWR. Results are for base costs and 120% uranium price, respectively. Similar results of non- unique or non-distinguishable policies are evident where demand is increased by +15% and +30% over the base, and when examined under the different cost component variations of +202.

When Pu and U recycle occurs and a market exists for each, the

Last-In-First-Out (LIFO) selection rule is dominant. Some modifications are noted as is the case once again with an increase of +204 in the projected uranium prices. Representative results for the Realistic

Scenario indicating a LIFO policy are given in Figures 6.7 and 6.8.

These figures represent the Westinghouse PWR system at all cost vari-~ ations except 120% U prices, and for 120% uranium prices, respectively.

The remainder of the results obtained for all cases examined under the Realistic Scenario are given in Appendix A.

C. Pessimistic Reprocessing Scenario

The Pessimistic reprocessing scenario defined in Section 4-B assumes the availability of reprocessing in 1983. As established with the Realistic scenario, the addition of reprocessing capacity over time is the same for the Pessimistic scenarioas for the Optimistic scenario, 108

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Results from the analyses conducted once again indicate that in the. absence of Pu-recycle, no unique policy is observable. This is depicted in Figures 6.9 and 6.10 for the Westinghouse PWR. Results are for base costs and 120% uranium prices, respectively. Similar results of non-uniqueness are evident when demand increases by +15% and +302 over the base demand, and when examined under the different cost component variations of +202.

When Pu and U recycle occurs and a market exists for each, the

Last-In-First-Out selection rule is again optimal. The only modifi- cations noted are that when demand is increased by 30%, a modified LIFO selection rule is observed when uranium prices are increased by 20% and when separative work prices are decreased by 20%. These representative results for the Pessimistic Scenario are given in Figures 6.11-6.13.

The individualized results for the other systems examined are given in

Appendix A. 113

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3 3 & & 7. CONCLUSIONS

It is concluded that the inventory of spent-fuel at on-site

storage pools can be modeled both by dynamic programming and linear

programming. The formulation of the spent-fuel-inventory-withdrawal model specifically lead to solution-yielding optimal quantities of spent-

fuel from a specific discharge to meet reprocessing demand when avail-

able. The dynamic formulation proves to be computationally intractable but the linear models are readily solved.

From the analyses performed, the conclusion reached by this study

is that a Last-In-First-Out policy is the optimal selection rule for ordering the withdrawal of spent-fuel from on-site storage. This policy prevails with the existence of plutonium and uranium recycle.

However, current issues, including the use of plutonium and the environmental controversy over nuclear power, have resulted in the cessation of the reprocessing activity. Thus, there has been no closure of the nuclear fuel cycle and there is going to be a continuous backlog of spent-fuel. The policy determined will help reactor operators specify a selection rule to withdraw spent-fuel from the on- site storage pools upon the re-initiation of reprocessing.

It is determined in this study that the earliest (optimistic) date for the re-initiation of reprocessing is 1979. Unfortunately, under the current administration policy, even this date appears unlikely.

Without a market for plutonium (i.e. no recycle), the question as to the economic disposition of spent-fuel appears even more difficult to answer in an appropriate manner. Without Pu recycle, the remaining

118 119

value in the spent-fuel is'at a minimum if only U recycle occurs. This

study concludes that in the absence of a Pu recycle, the economic dispo-

ition of spent-fuel is in even greater jeopardy, as no definitive or general policy is established.

An important trend to be noted is that if the negative costs associated with determining a net economic gain from spent-fuel (i.e. storage costs, reprocessing costs, and the fabrication penalties) continue to increase at a rate faster than inflation and/or an asso- ciated increase in the recoverable value for uranium and plutonium, the net value of a spent-fuel assembly decreases with time. This results in an increase in the power costs from nuclear power as long as the questions with regard to the disposition of spent~fuel remains unanswered.

To be concluded from this study is that uranium prices and storage costs are the parameters to which the model and thus the system are most sensitive. Fluctations in the cost of separative work has little influence on the optimal policies. 8. SUMMARY AND RECOMMENDATIONS

The models developed consititute a worthwhile tool for evaluating spent-fuel management stategies. In light of the present controversy over spent nuclear fuel reprocessing, the models developed should be employed to analyze the importance of contemplated policy changes upon the nuclear industry. Numerous plausible reference cases should be identified and analyzed to measure the impact on the industry of policy changes.

It is worth noting that reprocessing of both plutonium and uranium appears to offer significant benefits to the nuclear industry in the form of economic gain, and to the public in the form of resource conservation with a greater availablity of energy. As a consequence of this study, it is recommended that the present and anticipated govern- ment policy on spent nuclear fuel reprocessing be re-examined relative to the results of this analysis.

There are several areas of future study which are of interest for extending this analysis of the spent-fuel-inventory-withdrawal problem.

The first of these areas is to determine the full range of the useful- ness of the models developed. In the analysis presented, three models have been developed ignoring the constraint of pool capacity. It should be noted that an extension of these nodels would be to constrain the storage pool capacity as to the number of assemblies that can be held in storage, and analyze the effect of this constraint on the optimal policy. A second area of interest would be to incorporate probability into the model. This would involve establishing an additional con-

120 421

straint that would reflect the probability of assembly failure which could possibly influence the optimal policy. The study involving this probability is beyond the scope of this thesis and has not been examined.

It is appropriate to note however that present fuel cladding designs and fuel burnup rates are not oriented toward extended storage. Asa result of these physical constraints, severe cost penalties, not represented in this model may develop. For example, the cost penalty associated with a cladding failure may force a strict First-In-First-

Out (FIFO) policy as opposed to the optimal LIFO policy.

Specifically as a result of this study, it is recommended that a

Last-In-First—Out policy be adopted for managing the inventory of spent-fuel in on-site storage. This policy would take effect upon the resumption of the reprocessing activity and the establishment of | plutonium and uranium recycle. 9. BIBLIOGRAPHY

"Alternatives for Managing Wastes From Reactors and Post~Fission Operations in the LWR Cycle," ERDA 76-43, May 1976.

Anderson, Earl V., "Nuclear Energy: A Key Role Despite Problems," Chemical and Engineering News, March 7, 1977.

Bebbington, William P., 'The Reprocessing of Nuclear Fuels," Scientific American, December 1976.

Benedict, Manson, and Thomas H. Pigford, Nuclear Chemical Engi- neering, McGraw-Hill Co., Inc., Toronto, 1957.

"Benefit Analysis of Reprocessing and Recycling Light Water ‘Reactor Fuel,'' ERDA 76-121, December 1976.

Bowman, Edward H., "Production Scheduling by the Transportation Method of Linear Programming," Operations Research, Volume 4, Number 1, February 1956, pp. 100-103.

Buffa, Elwood S., and William H. Taubert, Production-Inventory Systems Planning and Control, Richard D. Irwin, Inc., Homewood, Iil., 1972.

Carolina Power and Light Company, Raliegh, N. C., personal commu- nication ~- Mr. Bill Stocks, March 24, 1977.

“Carter Vs. Plutonium: The Battle is Joined," Nuclear News, Volume 20, Number 7, May 1977.

10. Deonigi, D. E., "The Value of Pu Recycle in Thermal Reactors," Nuclear Technology, Volume 18, Number 80, May 1973.

11. Draper, E. Linn, and M. John Voltin, Jr., "Sensitivity of Total Fuel Cycle Costs to Variations in Enrichment Tails Assay Stategies,' t Transactions of the American Nuclear Society, Volume 22, November 1975.

12. Duderstadt, James J., and Louis J. Hamilton, Nuclear Reactor Analysis, John Wiley and Sons, Inc., New York, 1976.

13. "Environmental Impact of Electrical Power Generation: Nuclear and Fossil," Pennsylvania Department of Education, ERDA-69, 1975.

14. "Environmental Survey of the Nuclear Fuel Cycle," WASH-1250, 1974.

15. "Environmental Survey of the Reprocessing and Waste Management Portions of the LWR Fuel Cycle," NUREG-0116, 1976.

122 123

16.. Eschbach, E. A., "Plutonium Value Analysis," Proceedings of the Third International Conferences on the Peaceful Uses of Atomic Energy, Geneva, 1964, Volume 11, pp. 47-55, United Nations, New York, N. Y., 1965.

17. "Final Generic Environmental Statement on the Use of Recycle Plutonium in Mixed Oxide Fuel in Light Water Cooled Reactors,” NUREG-0002, August 1976.

18. Ford, L. R., and D. R. Fulkerson, Flows in Networks, Princeton University Press, New Jersey 1962.

19. Fulkerson, D. R., "An Out-Of-Kilter Method for Minimal-Cost Flow Problems," Journal of the Society for Industrial and Applied Mathematics, Volume 9, Number 1, 1961.

20. ‘Gaussens, J., and H. Paillot, Study of the Long Term Values and Pricesof Plutonium, CEA-12-2795, 1964.

21. "GESMO Hearing Begins,'’ Nuclear News, Volume 20, Number 1, January 1977.

22. Leskovjan, Larry L., David J. Rose, and Patrick W. Walsh, "Nuclear Power -~- Compared to What?,'' American Scientist, Volume 64, May- June 1976.

23. Lieberman, M. A., "United States Uranium Resources -- An Analysis of Historical Data," Science, Volume 192, April 30, 1976, pp. 431- 436.

24. "LWR Spent Fuel Disposition Capabilities 1976-1985," ERDA-25, May 1976.

25. Macek, Victor, Optimization of Time and Location Dependent Spent Nuclear Fuel Storage Capacity, Unpublished Ph.D.-Dissertation, Virginia Polytechnic Institute and State University, March 1977.

26. Mathematical Programming System -- Extended MPSX, and Generalized Upper Bounding (GUB), Management Science Systems, Inc., Rockville, Md., 1974.

27. Modern Energy Technology, Research and Education Association, New

York, Volume 1, 1975.

28. Nachlas, Joel A., Harold A. Kurstedt, Jr., David W. Swindle, Jr., and K. 0. Korez, "Modeling the Optimal Management of Spent Nuclear Fuel," Eighth Annual Pittsburgh Conference on Modeling and Simu- ‘lation, Pittsburgh, April 1977.

29. Nemhauser, George L., Introduction to Dynamic Programming, John- Wiley and Sons, Inc., 1966. 124

30. "Nuclear Fuel Cycle," ERDA-33, March 1975.

31. "Nuclear Fuels Policy, Report of the Atlantic Council's Nuclear Fuel Policy Working," The Atlantic Council of the United States, 1976.

32. Nuclear News, Volume 19, Number 1, January 1976.

33. Nuclear News, Volume 19, Number 11, September 1976.

34. Nuclear News, Volume 19, Number 15, December 1976.

35. Nuclear News, Volume 20, Number 3, Mid-February 1977.

36. "Reprocessing: How Necessary Is It For The Near Term?," Science, . Volume 196, April 1, 1977.

37. "Reprocessing Linked With World Accords," Nuclear Industry, Volume 23, Number 11, November 1976.

38. Roberts, Richland W., "Technical Reports on the Nuclear Fuel Cycle,’ ' Transactions of: the American Nuclear Society, Volume 24, Number 1, November 1976.

39. Taha, Hamdy A., Operations Research An Introduction, Macmillan Publishing Co., Inc., New York, 1971.

40. Tennessee Valley Authority, Knoxville, Tenn., personal communi- cation -- Mr. Bob Mullens, November 1976.

41. Vanstrum, P. R., and Wm. J. Wilcox, Jr., "Alternative Technologies for Meeting Uranium Enrichment Demands," Paper presented at American Institute of Chemical Engineers' 69th Annual Meeting, December 1-2, 1976. 10. APPENDIX

125 126

Appendix A

This section contains the results of implementing the spent fuel

withdrawal model on the IBM-370 computer system, These tabulated

results reflect the optimal selection rules as established for each of

the reprocessing scenarios examined. The title accompanying each

figure indicates:

1. Specific reprocessing scenario, i.e. Optimistic, Realistic, or

Pessimistic as established in Chapter 4, Section B,

2. the specific reprocessing scenario case examined, i.e. Base

Optimistic reprocessing, +15% Optimistic reprocessing, or

+30% Optimistic reprocessing as established in Chapter 4,

Section B for purposes of sensitivity analyses,

3. the reactor from which the spent assemblies originated, i.e.

Westinghouse PWR, Babcock and Wilcox PWR, Combustion Engineering

PWR, or General Electric BWR/6 as discussed in Chapter 4,

Section A, and

4. the status of the economic parameters that establish a measure

of effectiveness, i.e. Base Costs (all costs at values estab-

lished in Chapter 4, Section C), +20% SWU Costs, +20% Uranium

Prices, and +20% Storage Costs from the base costs established

in Chapter 4, Section C.

Following this page is a listing of all figures in AppendixA indicating

the results for the Optimistic Reprocessing Scenario, the Realistic

Reprocessing Scenario and the Pessimistic Reprocessing Scenario. This 127 section includes listings for each of the four NSSS examined. An explanation of interpreting the figures is given in Chapter 6, Section

A. 128

LIST OF FIGURES -- APPENDIX A

Figure Title

Optimal Selection Rule; Base Optimistic Reprocessing Scenario, Westinghouse PWR Base, +20% SWU, -20% Swu, -20% Uranium, +20% Storage, -~20% Storage Costs* * * * * * © * # «© » 137

Optimal Selection Rule; Base Optimistic Reprocessing Scenario; Westinghouse PWR +20% Uranium Cost * * * © * © * © © « « 138

Optimal Selection Rule; Base Optimistic Reporcessing Scenario; Westinghouse PWR No Plutonium Value; Base Costs* * « «= « 139

Optimal Selction Rule; Base Optimistic Reprocessing Scenario; Westinghouse PWR No Plutonium Value; +20% Uranium Cost ° e 140

Optimal Selection Rule; Base Optimistic Reprocessing Scenario; Westinghouse PWR No Plutonium Value; -20% Uranium Cost °> 141

A-6 Optimal Selection Rule; Base Optimistic Scenario; Westinghouse PWR No Plutonium Value; +20% Storage Cost « 142

Optimal Selection Rule; Base Optimistic Scenario; Westinghouse PWR No Plutonium Value; -20% Storage Cost °* 143

Optimal Selection Rule; Base Optimistic Scenario; Babcock and Wilcox PWR Base Costs* * * * © © © © 8 8 ¢ © « « » 144

Optimal Selection Rule; Base Optimistic Scenario; Combustion Engineering PWR Base Costs o © e e ee «© #© © © e© © #@ @ @ 145

A-10 Optimal Selection Rule; Base Optimistic Scenario; General Electric BWR/6 Base, +204 Uranium Costs* * * * * * « » 146

A-11 Optimal Selection Rule; Base Optimistic Reprocessing Scenario; General Electric BWR/6 No Plutonium Value; Base Costs* * * « ° 147 129

Figure Title

A-12 Optimal Selection Rule; Base Optimistic Reprocessing Scenario; General Electric BWR/6 No Plutonium Value; +20% Uranium Cost * « 148

A-13 Optimal Selection Rule; +15% Optimistic Reprocessing Scenario; Westinghouse PWR Base, +20% SWU, -20Z SWU, +20% Uraniun, -20% Uranium, +20% Storage, -20% Storage Costs* * * ° ° 149

A-14 Optimal Selection Rule; +15% Optimistic Reprocessing Scenario; Westinghouse PWR No Plutonium Value; Base Costs* * « * » « 150

A-15 Optimal Selection Rule; +15% Optimistic Scenario; Westinghouse PWR No Plutonium Value; +20% Uranium Cost - © 151

A-16 Optimal Selection Rule; +15% Optimistic Scenario; Westinghouse PWR No Plutonium Value; -20% Uranium Cost °* © 152

A-17 Optimal Selection Rule; +15% Optimistic Scenario;. Westinghouse PWR No Plutonium Value; +204 Storage Cost °* ¢ 153

A-18 Optimal Selection Rule; +15% Optimistic Scenario; Westinghouse PWR No Plutonium Value; -20Z% Storage Cost ° © 154

A-19 Optimal Selection Rule; +15% Optimistic Scenario; General Electric BWR/6 Base, +204 Uranium Costs* * * * * ss * ¢ 155

A-20 Optimal Selection Rule; +15% Optimistic Scenario; General Electric BWR/6 No Piutonium Value; Base Costs* * * ¢ » » 156

A-21 Optimal Selection Rule; +15% Optimistic Scenario; General Electric: BWR/6 No Plutonium Value; +20% Uranium Cost °¢ » 157

A-22 Optimal Selection Rule; +30% Optimistic Reprocessing Scenario; Westinghouse PWR Base, +20% SWU, -20% SWU, +20Z% Uraniun, ~-20% Uranium, +20% Storage, -20% Storage Costs* + * * * 158 130

Figure Title Page

A-23 Optimal Selection Rule; +30% Optimistic Reprocessing Scenario; Westinghouse PWR No Plutonium Value, Base Costs* * * * * * © © * © © © * * 159

A-24 Optimal Selection Rule; +30% Optimistic Reprocessing Scenario; Westinghouse PWR No Plutonium Value; +20% Uranium, -20% Uranium Costs* + * 160

A-25 Optimal Selection Rule; +304 Optimistic Reprocessing Scenario; Westinghouse PWR No Plutonium Value; +20% Storage Cost * * * * * * * + © © 161

A-26 Optimal Selection Rule; +30% Optimistic Reprocessing Scenario; Westinghouse PWR No Plutonium Value; -20% Storage Cost * * * * * * © © * © 462

A-27 Optimal Selection Rule; +30% Optimistic Reprocessing Scenario; General Electric BWR/6 Base, +20% Uranium Costs* * * * * * © © © © #© © *© *© e ¢ © 163

A-28 Optimal Selection Rule; +30% Optimistic Reprocessing Scenario; General Electric BWR/6 No Plutonium Value; Base Costs* * * * © © © #© © © © © «© © 164

A~29 Optimal Selection Rule; +30% Optimistic Reprocessing Scenario; General Electric BWR/6 No Plutonium Value; +20% Uranium Cost * + * * © * © © + « 165

A-30 Optimal Selection Rule, Base Realistic Reprocessing Scenario; Westinghouse PWR Base, +204 SWU, -~20% SWU, -—20% Uranium, +20% Storage, -20% Storage Costs* * * * © © # © # e e © s © © 8 #® #® # » 166

A-31 Optimal Selection Rule, Base Realistic Reprocessing Scenario; Westinghouse PWR +207 Uranium Cost > © e@ © © © © © # © © © © © © © # #© & #8 167

A~32 Optimal Selection Rule; Base Realistic Reprocessing Scenario; Westinghouse PWR. No Plutonium Value; Base Costs* * * * * * * © s © © # © «© 168

A-33 Optimal Selection Rule; Base Realistic Reprocessing Scenario; Westinghouse PWR No Plutonium Value; +204 Uranium Cost * * * © © © «© © © «© 169 131

Figure Title

A-34 Optimal Selection Rule; Base Realistic Reprocessing Scenario; Westinghouse PWR No Plutonium Value; -20% Uranium Cost * * * * * « » 170

A-35 Optimal Selection Rule; Base Realistic Reprocessing Scenario; Westinghouse PWR No Plutonium Value; +20% Storage Cost * * * * * * 171

— A-36 Optimal Selection Rule; Base Realistic Reprocessing Scenario; Westinghouse PWR No Plutonium Value; -20% Storage Cost * + * * * « » 172

A-37 Optimal Selection Rule; Base Realistic Reprocessing Scenario; Babcock and Wilcox PWR Base Case * © © © © © © © @© © © © © © © © © © &© 8 @ 173

A-38 Optimal Selection Rule; Base Realistic Reprocessing | Scenario; Combustion Engineering PWR

Base Case e e e ° e ° ° e ° s e ° ° ° e e e e e e ® -174

A-39 Optimal Selection Rule; Base Realistic Reprocessing Scenario; General Electric BWR/6 Base, +20% Uranium Costs* * * * © © * * © *« « «© # « 175

A-40 Optimal Selection Rule; Base Realistic Reprocessing Scenario; General Electric BWR/6 No Plutonium Value; Base Costs* * * * * * * *© «© « » 176

A-41 Optimal Selection Rule; Base Realistic Reprocessing Scenario; General Electric BWR/6 No Plutonium Value; +20% Uranium Cost * * * * * « « 177

A-42 Optimal Selection Rule; +15% Realistic Reprocessing Scenario; Westinghouse PWR Base, +20% SWU, -20% SWU, -20% Uranium, -20% Storage, —-204% Storage Costse* * «© © © © © © © © e © # © # « e 178

A~-43 Optimal Selection Rule; +15% Realistic Reprocessing Scenario; Westinghouse PWR: +20% Uranium Cost * «© © © © © © © 8» © © © © 8 8» @© @ 179

A-44 Optimal Selection Rule; +15% Realistic Reprocessing Scenario; Westinghouse PWR . No Plutonium Value; Base Costs* * * * * * * © * « « 180 132

Figure Title

A-45 Optimal Selection Rule; +15% Realistic Reprocessing Scenario; Westinghouse PWR No Plutonium Value; +20% Uranium Cost * * * * * « » 181

A-46 Optimal Selection Rule; +15% Realistic Reprocessing Scenario; Westinghouse PWR No Plutonium Value; -20%Z Uranium Cost * * * * * « » 182

A-47 Optimal Selection Rule; +15% Realistic Reprocessing Scenario; Westinghouse PWR No Plutonium Value; +20% Storage Cost * * * * * @ » 183

A-48 Optimal Selection Rule; +15% Realistic Reprocessing Scenario; Westinghouse PWR No Plutonium Value; -20% Storage Cost * * * * * « =» 184

A-49 Optimal Selection Rule; +15% Realistic Reprocessing — Scenario; General Electric BWR/6 Base, +20% Uranium Costs* * * * * * © * © © * * # » 185

A-50 Optimal Selection Rule; +15% Realistic Reprocessing Scenario; General Electric BWR/6 No Plutonium Value; Base Costs* * * * * * * * © « » 186

A-51 Optimal Selection Rule; +15% Realistic Reprocessing Scenario; General Electric BWR/6 No Plutonium Value; +202 Uranium Cost * * * * * « « 187

A~-52 Optimal Selection Rule; +30% Realistic Reprocessing Scenario; Westinghouse PWR Base, +20% SWU, -20Z SWU, -20% Uranium, +20% Storage, -~2?07% Storage Cost * * «© © # #© 2 e e eo © © ee © @ @« 8 188

A-53 Optimal Selection Rule; +30% Realistic Reprocessing Scenario; Westinghouse PWR +207 Uranium Cost ee e@ «© © © e#© ®@® e@# # @# © 8© 8 09 @© © @ 189

A-54 Optimal Selection Rule; +30% Realistic Reprocessing Scenario; Westinghouse PWR’ No Plutonium Value; Base Costs* * * * * * * « * # » 190

A-55 Optimal Selection Rule; +304 Realistic Reprocessing Scenario; Westinghouse PWR No Plutonium Value; +20% Uranium Cost ° cs et es 191 133

Figure Title

A-56 Optimal Selection Rule; +304 Realistic Reprocessing Scenario; Westinghouse PWR No Plutonium Value; -20% Uranium Cost + **** * + * 192

A-57 Optimal Selection Rule; +30% Realistic Reprocessing Scenario; Westinghouse PWR No Plutonium Value; +20% Storage Cost * * * * * © «= *¢ 193

A-58 Optimal Selection Rule; +30% Realistic Reprocessing Scenario; Westinghouse PWR No Plutonium Value; -20% Storage Cost * * * * * * 2 s 194

A-59 Optimal Selection Rule; +30% Realistic Reprocessing Scenario; General Electric BWR/6 Base, +20% Uranium Costs* * * * * * © © © * ¢ «© «© « 195

A-60 Optimal Selection Rule; +30% Realistic Reprocessing Scenario; General Electric BWR/6 No Plutonium Value; Base Costs* * * * * * * © © © « » 196

A-61 Optimal Selection Rule; +30% Realistic Reprocessing Scenario; General Electric BWR/6 No Plutonium Value; +20% Uranium Cost * * * * *© © « » 197

A~62 Optimal Selection Rule; Base Pessimistic Reprocessing Scenario; Westinghouse PWR Base, +20% SWU, -20% SWU, +20% Uranium, ~20% Uranium, +20% Storage, ~20%Z Storage Costs» * * + * * «© © « « « 198

A~63 Optimal Selection Rule; Base Pessimistic Reprocessing Scenario; Westinghouse PWR No Plutonium Value; Base Costs* * * * © * © © © *© © » 199

A-64 Optimal Selection Rule; Base Pessimistic Reprocessing Scenario; Westinghouse PWR No Plutonium Value; +20% Uranium Cost * * * * * * « 200

A-65 Optimal Selection Rule; Base Pessimistic Reprocessing Scenario; Westinghouse PWR’ No Plutonium Value; -20% Uranium Cost * * * * * * = 201

A-66 Optimal Selection Rule; Base Pessimistic Reprocessing Scenario; Westinghouse PWR No Plutonium Value; +204 Storage Cost * * * * * * ¢ » 202

A-67 Optimal Selection Rule; Base Pessimistic Reprocessing Scenario; Westinghouse PWR No Plutonium Value; -20% Storage Cost * * * * * * « « 203 134

Figure Title

A-68 Optimal Selection Rule; Base Pessimistic Reprocessing Scenario; Babcock and Wilcox PWR Base Costs» eo © © «© «© © © © © © @© © © © © © © © © #© «€ @ 204

A-69 Optimal Selection Rule; Base Pessimistic Reprocessing Scenario; Combustion Engineering PWR

Base Costs? . ° e ° e oe ® e © e e e e ° ® e e * e s e e 205

A-70 Optimal Selection Rule; Base Pessimistic Reprocessing Scenario; General Electric BWR/6 Base, +20% Uranium Costs, No Plutonium Value Base Costs 206

A-71 Optimal Selection Rule; Base Pessimistic Reprocessing Scenario; General Electric BWR/6 No Plutonium Value; +20% Uranium Costs* * * * * © * « » 207

A-72 Optimal Selection Rule; +15% Pessimistic Reprocessing Scenario: Westinghouse PWR Base, +20% SWU, -20% SWU, +20% Uranium, -20% Uraniun, +20% Storage, -20% Storage Costs»: * * * * * * © *« # « « ¢ 208

A-73 Optimal Selection Rule; +15% Pessimistic Reprocessing Scenario; Westinghouse PWR No Plutonium Value; Base Costs* * * * * * *« © ¢ « ¢ « e * 209

A-74 Optimal Selection Rule; +15% Pessimistic Reprocessing Scenario;, Westinghouse PWR No Plutonium Value; +20% Uranium Cost * * * * * © © © » 210

A-75 Optimal Selection Rule; +15% Pessimistic Reprocessing Scenario; Westinghouse PWR No Plutonium Value; -20% Uranium Cost * *+ * * * * + « 211

A-76_ Optimal Selection Rule; +15% Pessimistic Reprocessing Scenario; Westinghouse PWR No Plutonium Value; +20% Storage Cost * * * * * © * * » 212

A-77 Optimal Selection Rule; +15% Pessimistic Reprocessing Scenario; Westinghouse PWR. No Plutonium Value; -20% Storage Cost * * * * * * * © » 213

A-78 Optimal Selection Rule; +15% Pessimistic Reprocessing Scenario; General Electric BWR/6 Base, +204 Uranium Costs* * ** * * * * * * * * © ¢ # 8 214 135

Figure Title

A-79 Optimal Selection Rule; +15% Pessimistic Reprocessing Scenario; General Electric BWR/6 No Plutonium Value; Base Costs* * * * * * © * * *« * « 215

A-80 Optimal Selection Rule; +15% Pessimistic Reprocessing Scenario; General Electric BWR/6 No Plutonium Value; +20% Uranium Cost * * * * * * + » 216

A-81 Optimal Selection Rule; +30% Pessimistic Reprocessing Scenario; Westinghouse PWR Base, +20% SWU, -20% Uranium, +20% Storage, -20% Storage Costs e e ° e e e e e e e ° o e eo . ° e e ° e » e e e 217

A-82 Optimal Selection Rule; +30% Pessimistic Reprocessing Scenario; Westinghouse PWR -207 SWU Cost eo «© © © © # e@© © # © © #8 @© © #© © # #© @ @ ° 218 |

A-83 Optimal Selection Rule; +30% Pessimistic Reprocessing Scenario; Westinghouse PWR +2027 Uranium Cost « ° ee © © © © © ee fe ee ee lt 219

A-84 Optimal Selection Rule; +30% Pessimistic Reprocessing Scenario; Westinghouse PWR No Plutonium Value; Base Costs* * * * * * © * *« © « » 220

A-85 Optimal Selection Rule; +30% Pessimistic Reprocessing Scenario; Westinghouse PWR No Plutonium Value; +20% Uranium Cost * * * * * © © « 221

A-86 Optimal Selection Rule; +30% Pessimistic Reprocessing Scenario; Westinghouse PWR No Plutonium Value; -20% Uranium Cost * * * * * © « « 222

A-87 Optimal Selection Rule; +30% Pessimistic Reprocessing Scenario; Westinghouse PWR No Plutonium Value; +20% Storage Cost * * * * * © © « 223

A-88 Optimal Selection Rule; +30% Pessimistic Reprocessing Scenario; Westinghouse PWR- No Plutonium Value; ~20%Z Storage Cost * * * * * © « « 224

A-89 Optimal Selection Rule; +30% Pessimistic Reprocessing Scenario; General Electric BWR/6 Base, +20% Uranium Costs* * * * * * ss ee fe et es 225 136

Figure Title Page

A-90 Optimal Selection Rule; +304 Pessimistic Reprocessing Scenario; General Electric BWR/6 No Plutonium Value; Base Costs* * * * * © © * © * # * « » 226

A-91 Optimal Selection Rule; +30% Pessimistic Reprocessing Scenario; General Electric BWR/6 No Plutonium Value; +20% Uranium Cost * * * * * *© © © © « 227 137

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The author was born in Nashville, Tennessee on June 20, 1954. He was educated in the public school system of Metropolitan Nashville-

Davidson County and in June 1972 graduated from Maplewood High School.

In September of 1972, the author entered Tennessee Technological

University in Cookeville and graduated summa cum laude in June 1976.

During the time enrolled in Tennessee Technological University, the author gained technical experience in the field of engineering through employment with Nashville Machine Company, Inc. as a draftsman, and

I. €. Thomasson and Associates, Consulting Engineers as an associate engineer in the solid waste department. After receiving a Bachelor of Science degree in Engineering Science in June 1976, the author worked as a research assistant to Dr. Ray Kinslow in the area of hypervelocity impact studies. In the fall of 1976, the author entered

Graduate School at Virginia Polytechnic Institute and State University as an ERDA Fellow and completed requirements for the Master of Science degree in Nuclear Science and Engineering in August 1977. After graduation,the author will begin employment with Union Carbide Corpora-~ tion (Nuclear) in the division of Operations Analysis and Planning.

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David Wesley Swindle, Jr.

244 AN OPTIMAL WITHDRAWAL POLICY FOR

SPENT NUCLEAR FUEL FROM ON-SITE STORAGE

by

David Wesley Swindle, Jr.

(ABSTRACT)

The need to extend light water reactor spent-fuel on-site storage requirements and the future need to relieve resulting stockpiles necessitates the determination of optimal spent~fuel-withdrawal patterns under various end-use scenarios. End-use scenarios include no- economic-return throwaway and uranium recycle with and without plutonium recycle. Results from developing, analyzing, and solving a spent-fuel- withdrawal model are used to recommend specific stategies.

The spent-fuel-withdrawal problem involves the interaction of spent-fuel generation, time and capacity-dependent reprocessing demand, and expected spent-fuel value. Spent-fuel characteristics based upon burnup history and initial composition, are considered along with uranium, separative work, and storage cost projections to realize profitable spent-fuel dispositon. Application of the spent-fuel- withdrawal model is done on a per-reactor basis.

Assumptions inherent in the application of the model developed include, 1). unconstrained on-site storage capacity, 2) realizable uranium and plutonium values, and 3) capacity constrained reprocessing demand. Examining supply, demand, and characteristics of spent-fuel during a twenty~year horizon, the model application is developed through, 1) a dynamic programming approach, 2) a Hitchcock problem to be solved similarly to a minimum-cost~flow problem, and 3) a linear program defineable as a Transportation problem.

In the model analyses, the dynamic programming formulation proved to be computationally infeasible. The analyses of the Hitchcock and linear program problem is done by the use of the Out-Of-Kilter Algorithm and the proprietary mathematical MPS-III system,. respectfully.

Specific results indicate that the economically optimal withdrawal pattern is:

1) for uranium and plutonium recycle, a Last-In-First-—Out pattern,

and

2) for uranium recycle only, no discernable pattern.