Carl Haugen

The Greedy Exhaustive Dual Binary Swap
Methodology for Fuel Loading Optimization in

PWR Reactors Using the poropy Reactor
MASSACHUSETTS INSTIT TE
Optimization Tool OF TECHNOLOGY
by
2014
Carl C. Haugen OCT 2 9
B.Sc., University of Waterloo (2011) LIBRARIES
Submitted to the Department of Nuclear Science and Engineering
in partial fulfillment of the requirements for the degree of
Masters of Science in Nuclear Science and Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 2014
Massachusetts Institute of Technology 2014. All rights reserved.
Signature redacted
Author... ....................
-------J- - - -
.
Department o Nuclear Science and Engineering

S~~t b 5 214
Certified by ..... Signature redacted 'p ""e'

..................
Kord S. Smith
KEPCO Professor of the Practice of Nuclear Science and Engineering
Signature redacted Thesis Supervisor
Certified by....... ..........................
Benoit Forget
Ass 4Vte Professor of Nuclear Science and Engineering
Signature redacted Thesis Supervisor

Accepted by ..... ....................
Mujid S. Kazimi
TEPCO Professor of Nuclear Engineering
Chair, Department Committee on Graduate Students
The Greedy Exhaustive Dual Binary Swap Methodology for
Fuel Loading Optimization in PWR Reactors Using the
poropy Reactor Optimization Tool
by
Carl C. Haugen
Submitted to the Department of Nuclear Science and Engineering

on September 4, 2014, in partial fulfillment of the
requirements for the degree of
Masters of Science in Nuclear Science and Engineering
Abstract
This thesis presents the development and analysis of a deterministic optimization
scheme termed Greedy Exhaustive Dual Binary Swap for the optimization of nuclear
reactor core loading patterns. The goal of this optimization scheme is to emulate
the approach taken by an engineer when manually optimizing a reactor core loading
pattern. This is to determine if this approach is able to locate high quality patterns
that, due to their location in the core loading solution space, are consistently missed
by standard stochastic optimization methods such as those in the genetic algorithm
class, or those in the simulated annealing class. This optimization study is carried out
using the poropy tool to handle the reactor physics model. Initially, optimizations
are carried out using beginning of cycle eigenvalue as a surrogate for core excess
reactivity and thus cycle length. The deterministic Dual Binary Swap is found to
locate acceptable patterns less reliably than stochastic methods, but those that are
located are of higher quality. Optimizations of the full depletion problem result in
the deterministic Dual Binary Swap optimizer locating patterns that are of higher
quality than those found by the stochastic Simulated Annealing, with comparable
frequency. The Dual Binary Swap optimizer is, however, found to be very dependent
on the starting core configuration, and can not reliably find a high quality pattern
from any given starting configuration.
Thesis Supervisor: Kord S. Smith

Title: Korea Electric Power Company (KEPCO)
Professor of the Practice of Nuclear Science and Engineering
Thesis Supervisor: Benoit Forget

Title: Associate Professor of Nuclear Science and Engineering
3
ACKNOWLEDGMENTS
Partial support for this research was provided by the Natural Sciences Engineering
and Research Council of Canada, the Manson Benedict fellowship, and the Bishop
fellowship. Without this assistance, this work would not have been possible.
I would like to thank my co-supervisors Kord Smith and Benoit Forget. Working
with both of them and having had the privilege to benefit from their immense knowl-
edge of and experience in the field of nuclear engineering has made for an extremely
educational and enjoyable Masters experience. I greatly appreciate the patience and
the kindness they have both shown me throughout my time at MIT, and look forward
to continuing to have the privilege to work with them throughout my PhD.
I would also like to thank former graduate student Jeremy Roberts for his help and
guidance in introducing me to the poropy tool and the basics of nuclear engineering
in general and reactor core loading problems in specific. Without his assistance, my
transition to being a graduate student at MIT would not have been nearly as smooth
or as productive.
5
n I II I I
CONTENTS
1 INTRODUCTION
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.2 Greedy Exhaustive Dual Binary Swap . . . . . . . . . . . . . . . . . . . . 23
1.3 poropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.3.1 poropy Loading Pattern Graphics . . . . . . . . . . . . . . . . . . 29
1.4 Model ............................................. 30
1.5 Objectives / Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2 BEGINNING OF CYCLE ANALYSIS
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .....35
.
2.2 Beginning of Cycle Genetic Algorithm Analysis . . . . . . . . .. . . . . 36
2.3 Beginning of Cycle Single Binary Swap Analysis . . . . . . . . . . . . . . 38
2.4 Beginning of Cycle Triplet Swap Analysis . . . . . . . . . . . . . . . . . . 40
2.5 Beginning of Cycle Dual Binary Swap Analysis . . . . . . . . . . . . . . 41
2.6 Restricting Location of Fresh Fuel Assemblies . . . . . . . . . . . . . . . 42
2.7 Beginning of Cycle Simulated Annealing Analysis . . . . . . . . . . . .. 45
3 IMPLEMENTATION OF DEPLETION
3.1 M otivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.2.1 Effect of Timestep Method . . . . . . . . . . . . . . . . . . . . . . 66
3.2.2 Time Cost of Depletion . . . . . . . . . . . . . . . . . . . . . . . . 70
3.2.3 Heuristics in the GEDBS Method . . . . . . . . . . . . . . . . . . 71
3.3 Future Depletion Work .. . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4 DUAL BINARY SWAP WITH DEPLETION

4.1 Introduction . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2 Exhaustive Dual Binary Swap Analysis . . . . . . . . . . . . . . . . . . . 76
4.3 Greedy Dual Binary Swap Analysis . . . . . . . . . . . . . . . . . . . . . 91
5 SIMULATED ANNEALING WITH DEPLETION

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 95
5.2 Simulated Annealing Analysis . . . . . . . . . . . . . . . . . . . . . . . . 95
5.3 Exhaustive Dual Binary Swap of Simulated Annealing Patterns . . . . . 102
6 CONCLUSIONS AND FUTURE WORK

6.1 Conclusions ......................................... 121
6.2 Future Work......................................... 124
1 Appendix 127
A APPENDIX
A.1 Beginning of Cycle Analysis............................... 129
7
A.2 Implementation of Depletion . . . . . . . . . . . . . . . . . . . . . . . . . 131
A.3 Dual Binary Swap with Depletion . . . . . . . . . . . . . . . . . . . . . . 139
BIBLIOGRAPHY
8
LIST OF FIGURES
Figure 1.1 Algorithm of the Simulated Annealing Methodology . . . . .
.
Figure 1.2 Algorithm of the Genetic Algorithm Methodology . . . . . . .
.
Figure 1.4 Algorithm of the Greedy Dual Binary Swap Methodology . .
.
Figure 1.5 Algorithm of the Exhaustive Dual Binary Swap Methodology
Figure 1.3 Order that the Dual Binary Swap Algorithms Search Through
an Input Quarter Core . . . . . . . . . . . . . . . . . . . . . . .
.
Figure 1.6 Single Binary Swap . . . . . . . . . . . . . . . . . . . . . . . . .
.
Figure 1.7 Triplet (Chain Shuffle) Swap . . . . . . . . . . . . . . . . . . . .
.
Figure 1.8 Dual Binary Swap . . . . . . . . . . . . . . . . . . . . . . . . . .
.
Figure 1.9 Structure of Assembly Data in poropy Generated Figures . . .
.
Figure 1.10 Burnup of Fuel Bundles in Yamamoto's Benchmark . . . . . .
.
Figure i. ii Reactivity of Fuel Bundles in Yamamoto's Benchmark . . . . .
.
Figure 1.12 Power Peaking in Yamamoto's Benchmark . . . . . . . . . . .
.
Figure 2.1 Structure of Assembly Data in Loading Patterns . . . . . . . .
.
Figure 2.2 Best Beginning of Cycle Core Loading Pattern Obtained by Dual
Binary Swap for an Imposed Power Peaking Constraint of 1.250
Figure 2.8 Best Beginning of Cycle Core Loading Pattern Obtained by Sim-
ulated Annealing for an Imposed Power Peaking Constraint of
1.250................................................................
1.275 -.......................................................
1.300. . - ...................................................
1.325 -.....-.-.................. .................................
1.350 . . . . ........................................
9
1.375 -----....--.- -------- -------...................................-- 59
-
Figure 3.1 Beginning of Cycle Reactivity of Fuel Bundles in a Loading Pat-
tern with a Mid-Cycle Power Peak . . . . . . . . . . . . . . . . 62
.
Figure 3.2 Beginning of Cycle Power Peaking in a Loading Pattern with a
Mid-Cycle Power Peak . . . . . . . . . . . . . . . . . . . . . . . 63
.
Figure 3.3 Mid Cycle Reactivity of Fuel Bundles in a Loading Pattern with
a Mid-Cycle Power Peak . . . . . . . . . . .... . . . . .. . . 64
.
Figure 3.4 Mid Cycle Power Peaking in a Loading Pattern with a Mid-
Cycle Power Peak . . . . . . . . . . . . . . . . . . . . . . . . . . 65
.
Figure 3.5 Beginning of Cycle Reactivity of Fuel Bundles in High Quality
Reactor Loading Pattern . . . . . . . . . . . . . . . . . . . . . . 67
.
Figure 3.6 Beginning of Cycle Power Peaking in High Quality Reactor
Loading Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
.
Figure 4.1 Structure of Assembly Data in Loading Patterns . . . . . . . . 76
.
Figure 4.2 Initial Configuration as Specified by Yamamoto . . . . . . . . 77
.
Figure 4.3 Initial Configuration of First Randomized Core . . . . . . . . . 78
.
Figure 4.4 Initial Configuration of Second Randomized Core . . . . . . . 79
.
Figure 4.5 Initial Configuration of Third Randomized Core . . . . . . . . 8o
.
Figure 4.6 Initial Configuration of Fourth Randomized Core . . . . . . . 81
.
Figure 4.7 Initial Configuration of Fifth Randomized Core . . . . . . . . 82
.
Figure 4.8 Core Loading Pattern Obtained by Dual Binary Swap with End
of Cycle Depletion 13.425 MWd/kg and Maximum Cycle Power
Peaking of 1.349 . .. . . . . . . . . . . . . . . . . . . . . . . . . 85
.
Peaking of 1.374 . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
.
Peaking of 1.372 . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
.
Peaking of 1.399 . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
.
Peaking of 1.400 . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
.
Peaking of 1.396 . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
.
Figure 5.1 Structure of Assembly Data in Loading Patterns .. . .. .. . 102

.
Figure 5.2 Core Loading Pattern Obtained by Simulated Annealing with

End of Cycle Depletion 13.414 MWd/kg and Maximum Cycle
Power Peaking of 1.385 . . . . . . . .. . .. . . . . . .. . . . 103
.

Power Peaking of 1.384 . . . . . . . . . . . . . . . . . . . . . . . 104
.
10
Power Peaking of 1.404 . . . . . . . . . . . . . . . . . . . . . . . 105
.
Power Peaking of 1.398 . . . . . . . . . . . . . . . . . . . . . . . 1o6
.
Power Peaking of 1.398 . . . . . . . . . . . . . . . . . . . . . . . 107
.
Power Peaking of 1.409 . . . . . . . . . . . . . . . . . . . . . . . 1o8
.
Power Peaking of 1.395 - - - - - - -- .. - - - - - - --.-.-.-.-... 109
Power Peaking of 1.394 .. - - - - - - - - - . --.. -.. - -.. --.-.-.-. 110
Figure 5.10 Core Loading Pattern Obtained by Dual Binary Swap of a Con-
verged Simulated Annealing Pattern with End of Cycle Deple-
tion 13.534 MWd/kg and Maximum Cycle Power Peaking of
1.374 ....................................... 112
1.381 ......... ................................... 113
1.399 .... ----.. --.................................................... 114
1.400 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 115
1.399 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 116
1.409 ..................................... 117
1.398 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
.
11
1.399.................-...-....-................................... 119
Figure A.1 Reactivity of Fuel Bundles in Example High Quality Reactor
Loading Pattern at Depletion Step 2 . . . . . . . . . . . . . . . 131
.
Figure A.2 Power Peaking in Example High Quality Reactor Loading Pat-
tern at Depletion Step 2 . . . . . . . . . . . . . . . . . . . . . . 132
.
Figure A.3 Reactivity of Fuel Bundles in Example High Quality Reactor
Loading Pattern at Depletion Step 5 .. . . . . . . . . . . . -. 133
. tern at Depletion Step 5 . . . . . . . . . . . . . . . . . .. . 134
.
Figure A. 5 Reactivity of Fuel Bundles in Example High Quality Reactor

.
.
Figure A. 7 Reactivity of Fuel Bundles in Example High Quality Reactor
.
.
Figure A. 9 BOC Power Peaking of Initial Configuration as Specified by Ya-
mam oto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
.
Figure A.1o BOC Assembly kinf of Initial Configuration as Specified by Ya-
m am oto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
.
Figure A.11 BOC Power Peaking of Initial Configuration of First Random-
ized C ore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
.
Figure A.12 BOC Assembly kinf of Initial Configuration of First Random-
ized Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
.
Figure A.13 BOC Power Peaking of Initial Configuration of Second Ran-
domized Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Figure A.14 BOC Assembly kinf of Initial Configuration of Second Random-
ized Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
.
Figure A.15 BOC Power Peaking of Initial Configuration of Third Random-

ized Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
.
Figure A.16 BOC Assembly kinf of Initial Configuration of Third Random-

ized C ore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
.
Figure A.17 BOC Power Peaking of Initial Configuration of Fourth Random-

ized Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
.
Figure A.18 BOC Assembly kinf of Initial Configuration of Fourth Random-

ized Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
.
Figure A.19 BOC Power Peaking of Initial Configuration of Fifth Random-

ized C ore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
.
Figure A.2o BOC Assembly kinf of Initial Configuration of Fifth Random-

ized C ore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
.
12
LIST OF TABLES
Table 2.1 Results of Beginning of Cycle Analysis of the Genetic Algorithm

Optimization Scheme, Peaking of 1.5. - - - - - - - - - - - - - - 37
Table 2.2 Improvements of the Eigenvalue of the Optimized Core Load-
ing Pattern of the Genetic Algorithm Methodology from iooo
Generations after Applying Greedy Single Swap Optimization 39
ing Pattern of the Genetic Algorithm Methodology from iooo
Generations after Applying Exhaustive Single Swap Optimiza-
tion ......................................... 39
ing Pattern of the Genetic Algorithm Methodology from 4000
Generations after Applying Exhaustive Single Swap Optimiza-
tion ......................................... 40
Table 2.5 Results of Beginning of Cycle Exhaustive Dual Binary Swap
Optimization of Genetic Algorithm Starting Patterns . . . . . . 42
Table 2.6 Results of Beginning of Cycle Greedy Dual Binary Swap Opti-
mization of Genetic Algorithm Starting Patterns . . . . . . . . . 42
Table 2.7 Optimizations of the Yamamoto Benchmark Using the Dual Bi-
nary Swap Methodology . . . . . . . . . . . . . . . . . . . . . . 43
Table 2.8 Success Rates of the Dual Binary Swap Methodologies when
Placing Fresh Fuel Adjacent to the Reflector is Penalized the
most Heavily in the Objective Function . . . . . . . . . . . . . . 45
Table 2.9 Success Rates of the Dual Binary Swap Methodologies when
Power Peaking Above the Constraint is Penalized the most Heav-
ily in the Objective Function . . . . . . . . . . . . . . . .. . . . 45
Table 2.10 Success Rates of the Simulated Annealing Methodology with
Slow Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Table 2.11 Difference of Quality of Best Solutions Produced by the Dual
Binary Swap and Simulated Annealing Methods . . . . . . . . 47
Table 3.1 Eigenvalue and Peaking Results for a High Quality Loading
Pattern Throughout the Cycle . . . . . . . . . . . . . . . . . . . 66
Table 3.2 Predicted Depletion of Loading Pattern with Constant Deple-
tion Steps for High Quality Loading Pattern . . . . . . . . . . . 69
Table 3.3 Predicted Depletion of Loading Pattern with Constant Deple-
tion Steps for Yamamoto Starting Point Loading Pattern . . . . 70
Table 3.4 Scaling of Computational Time With Number of Depletion Points
Computed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Table 4.1 Results of Full Depletion Optimization with the Exhaustive Dual
Binary Swap, Peaking of 1.275 . . . . . . .. . . . . . . . . - 79
Table 4.2 Number of Evaluations Carried out for Full Depletion Opti-
mization with the Exhaustive Dual Binary Swap, Peaking of
1.275 ..... ............ .................................. 8o
13
. . . . . . 81 Binary Swap, Peaking of 1.300 . . . . . . . . . . -
. . . . . . . . . . . . . . 82 Binary Swap, Peaking of 1.315 . . . . .
Binary Swap, Peaking of 1.325 . . . . . . . . . . .. . . . . . . 83
Binary Swap, Peaking of 1.350 . . . . . . . . . . . . . . . . . . . 83
Binary Swap, Peaking of 1.375 . . . . . . . . . . . . . . . . . .. 83
mization with the Exhaustive Dual Binary Swap, Peaking of
1.400 -.......................................................... .... 84
Table 4.10 Results of Full Depletion Optimization with the Greedy Dual
Binary Swap, Peaking of 1.275 . . . . . . - - - -. . . . . . . . 91
mization with the Greedy Dual Binary Swap, Peaking of 1.275 91
Binary Swap, Peaking of 1.315 . . . . . . . . . -. . . . . . . . 92
Binary Swap, Peaking of 1.325 . . . . . . . . . . . . . . . . . - 93
Binary Swap, Peaking of 1.350 . . . . . . . . . - .. . . . . . - 93
Binary Swap, Peaking of 1.375 . . . . . . . . . . . ... . . . - 93
Table 5.1 Results of Full Depletion Optimization with Simulated Anneal-
ing, Only Single Binary Swaps of Assemblies, Peaking of 1.275,
Cooling Parameter 0.995, 27,563 Patterns Evaluated per Run
(Optimized Patterns Identical to Power Peaking Constraints Be-
tween 1.300 and 1.325) . . . . . . . . . .. . . .. . . . . . . . 96
(Optimized Patterns Identical to Power Peaking Constraints of
1.375 and 1.400) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Cooling Parameter 0.9995, 64,458 Patterns Evaluated per Run 97
Cooling Parameter 0.9995, 64,458 Patterns Evaluated per Run 97
14
ing, Only Dual Binary Swaps of Assemblies, Peaking of 1.275,
(Patterns Identical to Power Peaking Constraints from 1.300 to
1.400) .............................................................. .... 99
Cooling Parameter o.998, 16,103 Patterns Evaluated per Run
1.400) . ............ .................................. .... 99
1.400) .............. .................................. .. 100
Cooling Parameter 0.9995, 64,458 Patterns Evaluated per Run .oo
Cooling Parameter 0.9995, 64,458 Patterns Evaluated per Run . 1
Table 5.10 Results of Deterministic Dual Binary Swap Optimization on
Converged Simulated Annealing Results . . . . . . . . . . . . . 111
Table 5.11 Number of Passes Required to Converge Deterministic Dual
Binary Swap Optimization on Converged Simulated Annealing
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Table 6.1 Best Loading Patterns Found in the Full Depletion Study by all
Optimization Methods, Power Peaking Constraint 1.350 . . . . 123
Table A.1 Complete Results of Beginning of Cycle Exhaustive Dual Binary
Swap Optimization of Genetic Algorithm Starting Patterns . . 129
Table A.2 Complete Results of Beginning of Cycle Greedy Dual Binary
Swap Optimization of Genetic Algorithm Starting Patterns . . 129
Table A.3 Optimizations of the Randomized Yamamoto Benchmark Using
the Dual Binary Swap Methodology . . . . . . . . . . . . . . . . 130
15
1 INTRODUCTION
1.1 BACKGROUND
Nuclear fuel management in reactors is a very important component of the opera-
tional process, as proper management allows for significant reduction of the opera-
tional cost incurred by utilities. However, while utilities and reactor operators may
wish to reduce costs to the absolute minimum possible, doing so would result in op-
erating the reactor in an unsafe state, as well as ignoring many legislative constraints.
Fuel management can be described as being of one of two forms; in-core and out-of-
core fuel management. Out-of-core fuel management involves decisions about what
types of fuel assemblies to order (i.e. number, type, and location of burnable poison,
enrichment, etc), as well as which previously burned assemblies should be re-inserted
into the reactor core [1]. It also is concerned with the interaction of the nuclear power
plant with the electric grid system to which it is connected. In-core fuel management
(the focus of this work) involves the placement of available fuel bundles (potentially
with different burnups, enrichments, and burnable poisons) at specific locations in
the reactor core to best satisfy potentially conflicting constraints [2]. These include,
for instance, maximizing excess reactivity in the core to improve cycle length, keeping
assembly power peaking below a certain threshold, and reducing moderator temper-
ature coefficient. If power peaking in a reactor is too high, fuel melt, especially in
accident situations, becomes significantly more difficult to prevent. Conversely, low
core power peaking tends to be a feature of loading patterns that have little excess
reactivity, and thus short operational life. This is due to the fact that excess reactivity
present in a core is correlated with the amount of time criticality can be maintained
before a refuelling shutdown is required. Considering that a nuclear reactor that is
not operating is extremely costly for utilities, there is a strong incentive encouraging
17
utilities to push power peaking as high as technical constraints will allow to maximize
profits. For example, using the Mean Real Time Dispatch price data for the state of
Texas in 2013 from The Electric Reliability Council of Texas (ERCOT) suggests that
the mean electricity price for a utility operating there was $31.66 / MWh over 2013
[3]. This pricing data shows that for a plant with a power output of 1 GW electric, a
loading pattern that resulted in even 24 hours less of operation in 2013 would have
cost a Texas utility approximately $760,00o in lost revenue.
The difficulty of this problem, however, is that there are many unique possible pat-
terns for a core loading problem [4]. A typical PWR, with 193 fuel assemblies and
loaded with 64 identical fresh assemblies, will have approximately 10267 unique load-
ing patterns [5]. Even if taken to be quarter core symmetric, the number of unique
loading patterns is still on the order of 1060. It is thus completely infeasible to explore
the entire solution space.
When nuclear energy was first being introduced around the world, and computa-
tional models were of limited accuracy, loading patterns had to be developed by hand.
These special patterns were typically chosen to make extensive use of symmetry and
to be extremely conservative from a power peaking perspective. The class of pattern
typically used at this time involved putting all of the fresh fuel assemblies on the edge
of the reactor, adjacent to the reflector. Since the fresh fuel bundles tend to be the most
reactive, this configuration results in a large fraction of neutrons leaking out of the
core. This is very helpful in keeping the power peaking in the core low, but also tends
to result in a short reactor cycle length.
As reactor control methods and computational technology have advanced, these
hand-designed loading patterns have stopped being used. Modern computational
models can accurately predict what the power in every assembly will be at any given
point during the operational cycle of a reactor. The lower uncertainty in assembly pow-
ers allows for less conservative safety margins when restricting the power peaking of
candidate loading patterns. Thus, patterns with higher cycle lengths that would pre-
viously have been excluded due to too large power peaking are now viable. Typically
18
patterns that are now used are described as being of the "ring of fire" class. These types
of patterns place heavily burned fuel on the edge of the reactor, with fresh fuel just in-
side. This tends to lead to a power peaking that is higher, while not being intolerable,
and a significantly better cycle length.
While the entire list of possible solutions will never be fully explored, computa-
tional optimization schemes are often used to explore as many solutions as possible
and (in principle) arrive at a very good solution. There are two broad classes of opti-
mization scheme: deterministic and stochastic. Deterministic methods search through
a space in a methodical manner to arrive at a solution. Deterministic methods are
sometimes relatively easy to implement, but for some problems can either get trapped
in local minimum or waste a lot of time examining bad solutions. Stochastic methods,
by contrast, use sampling of random numbers to explore parts of a solution space in
an intelligent manner. Stochastic methods tend to be preferred for many optimization
problems, including core loading, as they are (in principle) able to escape from local
minima and typically require less computational time to arrive at a high quality so-
lution than deterministic methods. One difficulty that is sometimes encountered with
stochastic methods is that they tend to have parameters that need to be tuned for every
problem that they are applied to, and if these parameters are not at optimal values,
these methods will perform poorly [6].
Two of the most common stochastic methods used in core loading optimization are
Genetic Algorithms (GA) [7], [6], [5], [8] and Simulated Annealing (SA) [6],[5], [9],
[io], [ii], [12], although there are other, less frequently used stochastic methods, such
as the Great Deluge Algorithm, Record to Record Travel, and the Population-Based
Incremental Learning Algorithm [5].
SA methods take inspiration from the process of annealing in the field of metallurgy.
They involve creating a guess solution for the problem, and making an attempted
perturbation. At the beginning of the simulation the system is said to be at a high
"temperature", and any perturbation is accepted. As the simulation progresses, the
temperature is lowered, and perturbations that make the solution worse are accepted
19
with lower and lower probability. As the end of the simulation is reached, only pertur-
bations that make the solution better are accepted. In principle, if the system is cooled
slowly enough, the solution should converge to the global best minimum [13], [14],
[15]. This algorithm is shown visually in Figure 1.1. In the problem of core loading
optimization, by far the most common system perturbation involves swapping the
location of two assemblies within the reactor (a binary swap - see Figure 1.6). SA as
implemented in the poropy tool primarily uses swapping of one pair of fuel assem-
blies at a time as its perturbation (a single binary swaps). The SA optimizer in the
poropy tool is capable of using the swapping of two or three pairs of assemblies (dual
or triple binary swaps) as its perturbation, although this is done less frequently in this
work. Unless otherwise explicitly specified, all SA analyses in this work were carried
out using only single binary swaps.
GA methods, by contrast, take inspiration from the field of evolutionary biology. A
set of guess solutions to the problem are generated, known as the system's "popula-
tion". Each member of the population is assigned a "fitness" based on its quality as a
solution to the problem. This population represents a "generation". Two solutions in
the population are then chosen as "parents", and have their characteristic features (in
the core loading case, this would be where fuel assemblies are positioned) mixed to-
gether, along with a certain amount of randomness, to create a "child". The likelihood
that any two solutions will be chosen as parents increases the higher their fitness
is. This process is repeated until enough children have been created to fill the next
generation [14]. This algorithm is shown visually in Figure 1.2.
The shortcoming of these and other stochastic algorithms is that they often have dif-
ficulty finding the best patterns for core loading. This is due to the fact that even the
movement of only a couple of assemblies has the potential to severely negatively im-
pact the cycle length or power peaking of high quality patterns. Since most stochastic
methods operate by searching around high quality basins on the solution landscape,
these solutions are often missed.
20
Guess a Starting
Pattern and set a
System "Temperature"
p U
Make a Random Does this Change Accept the Change
Perturbation to the Make the Solution and Update the
System Better or Worse? Better
I U
Solution I
Worse
Accept the Change

with a Probability Have we Best Solution is the
Based Temperature converged? Global Minimum
Yes
MEMEM
No
Decrease System
Temperature
MEMO"
Figure i.i. Algorithm of the Simulated Annealing Methodology

N'
N.
1
Generate a Children are all now Determine Fitness of
"Population" of Parents for the Next Generation. Best Solution is the
Candidate Solutions Generation No Have we Converged? Global Minimum
Yes
Yes
U U
Determine the
Select Two "Parents" Add Child to Bank for
"Fitness" of Each
from the Population Next "Generation".
Solution
Based on Fitness No Is Bank Full?
Mix the Traits of the

Two Parents to
Produce a "Child"
==||
Introduce Random
Elements with Some
Probability
1
Figure 1.2. Algorithm of the Genetic Algorithm Methodology
AIw
Thus, while software is used to analyse the quality of potential patterns, stochastic
methods are not used to generate the loading patterns that are used by commercial
reactors. Instead, loading patterns are examined by a team of engineers that use their
physical intuition to generate a sample of candidate patterns. These specific patterns
are then analysed computationally to determine which is best suited to the needs of
the utility.
The goal of this work is to develop a deterministic method that emulates the ap-
proach an engineer might take to solving this problem. This deterministic method will
be referred to as the Greedy Exhaustive Dual Binary Swap (GEDBS) methodology.
1.2 GREEDY EXHAUSTIVE DUAL BINARY SWAP
The deterministic Greedy Exhaustive Dual Binary Swap (GEDBS) method is designed
to emulate the process that would be taken by a person experimenting with different
loading patterns to determine which is ideal. To that end, the algorithm performs
dual binary swaps. This involves swapping the position of two assemblies within the
reactor, typically with the goal of either increasing the cycle length of the pattern, or
reducing the power peaking. The unfortunate side effect of swapping assemblies with
the goal of improving the pattern's ability to satisfy either of these constraints is that
it tends to harm the pattern's ability to satisfy the other constraint. Thus a second pair
of assemblies have their positions swapped with the goal of improving the pattern's
ability to satisfy the second constraint. When a good pair of swaps is performed, the
net effect is to improve the pattern with respect to one constraint, while the other is
either also improved, or stays unchanged. There are two subsets of DBS method that
will be examined - one that emphasizes the greedy component of the algorithm over
the exhaustive component of the algorithm and will be referred to as the Greedy Dual
Binary Swap method (GDBS, see Figure 1.4), and one that emphasizes the exhaustive
component over the greedy component and will be referred to as the Exhaustive Dual
Binary Swap method (EDBS, see Figure 1.5).
23
Start with a Single Are there More No Accept the Starting
Candidate Solution Possible Swaps of
Pattem as the Best
Primary Assembly
Result
Pairs?
Yes
rNo
Go Through Assemblies
Swap the Locations of Are there More
Left to Right then Top
the First Valid Pair of Possible Swaps of
to Bottom in Quarter
Assemblies Secondary Assembly
Core Pairs?
'
Yes
~No
I I
Swap the Locations of - Evaluate the New

Yes Accept this Pattern
the Second Valid Pair C Pattern
h as the New Starting
Is the Pattern Better
of Assemblies than the Start Point? Point
I I
In
Figure 14. Algorithm of the Greedy Dual Binary Swap Methodology
- w
Start with a Single Accept the Best of Did any of the Swaps No
Yes Accept the Starting
Candidate Solution these Patterns as the ,C= Result in a Pattern
New Starting Point that was Better than Pattern as the Best
the Startine Point? Result
~No
Go Through Assemblies
Swap the Locations of Yes Are there More
Left to Right then Top
the First Valid Pair of Possible Swaps of
to Bottom in Quarter
Assemblies Primary Assembly
Core
Pairs?
No
Yes
Swap the Locations of Evaluate the New

the Second Valid Pair Pattern
Are there More
of Assemblies
Secondary Swaps?
Figure 1.5. Algorithm of the Exhaustive Dual Binary Swap Methodology

In both cases, in a given iteration, the algorithm systematically evaluates the quality
of all possible patterns that result from switching the location of all possible pairs of
assemblies within the reactor (as in Figure 1.8), determining if the candidate pattern
resulting from the swap will be of higher quality than the starting loading pattern.
When evaluating potential patterns, the dual binary swap algorithm searches left to
right, then top to bottom in the quarter core, as shown in Figure 1.3. It is worth
-1 2 3 4
-5 6 7 8
- 9 10 11
-12 131
-14
Figure i-3. Order that the Dual Binary Swap Algorithms Search Through an Input Quarter
Core
noting that the assembly in the far upper left does not have its position changed
by any optimization algorithm, as it is typically the most heavily burned assembly
within a pattern, and has no symmetric partners within the core. Additionally, since
the assemblies on the left edge are the mirror duplicates of the assemblies on the top
edge, their positions are not explicitly changed.
In the greedy implementation of the GEDBS method, when a swap is found that
would improve the reactor pattern, the swap is made, and a new iteration is begun
with this new pattern as the starting point. In the exhaustive implementation of the
DBS method, in a given iteration, all possible dual binary swaps are checked, and
26
T- -I
HI-- I
- -
m.
- j.mmmm. ~ -
I I
-
U-
U-
a
S
- a
-
I
-
i 1 I
Figure 1.6. Single Binary Swap Figure 1.7. Triplet (Chain Shuffle) Swap
the swap that improves the reactor the most is made. The new pattern is then used
at the starting point for the next iteration. In both cases, the procedure is continued
until an iteration is reached in which no dual binary swap exists that would improve
the loading pattern. Prior to the implementation of full dual binary swaps in this
greedy algorithm, single binary swaps and triplet (chain shuffle) swaps were also
tested. These three types of perturbations are shown in Figures 1.6 through 1.8.
- U- U - U - U
-
LIm1 I
r .Di
Figure 1.8. Dual Binary Swap
27
1.3 poropy
In this study, the quality of a core loading design will be evaluated with the teaching
tool known as Physics Of Reactor Optimization in Python (poropy). This is a tool that
was developed in collaboration between three students as part of a course taught at
MIT in fuel cycle analysis [16].
poropy is written as a Python code that calls routines implemented in FORTRAN
to evaluate the physics of a 2 dimensional quarter-core symmetric array of fuel assem-
blies for a PWR. Each assembly is treated as a single node in the problem, and has
its burnup, enrichment, and poison content specified to generate group cross-sections
based on a standard Westinghouse 17x17 PWR fuel assembly. Flare is the physics
model used for this analysis [16].
poropy assumes a boron concentration of 900 ppm, an average moderator (water
in the assembly that is not flowing) temperature of 560 K, an average coolant (water
in the assembly that is flowing) temperature of 58o K, and a fuel temperature of 900
K. Two-group cross-sections are generated using polynomial fits to CASMO- 4 data
output from a range of burnup and enrichment values [16]. It should be noted that
the CASMO- 4 cross-sections were generated only for a single operating power. This
results in any operating power that is specified for a given reactor in the poropy
framework does not impact the implicit Xenon and Samarium levels and thus the
cross-sections used.
poropy also includes various optimizers for taking a given array of assemblies and
generating a high quality loading pattern. Currently those optimizers use the stochas-
tic methods simulated annealing and genetic algorithms, as well as a pair of determin-
istic methods based off of dual binary swaps.
28
U
1.3.1 poropy Loading Pattern Graphics
In this thesis, figures produced by the poropy GUI will be presented (for example,
Figures 1.10, 1.11, and 1.12 in this chapter). These figures will show a quarter of a reac-
tor core (poropy assumes a quarter core symmetric configuration), with blue squares
representing reflector locations, and squares ranging from green to red representing
individual fuel assemblies.
For a burnup figure, green indicates fresh assemblies, while red indicates heavily
burned assemblies. For a reactivity figure, green indicates assemblies with low kinf
while red indicates assemblies with high kinf. For a power peaking figure, green indi-
cates assemblies below the average assembly power in the reactor, while red indicates
above the average assembly power.
Each assembly in these figures has 6 pieces of data associated with it. The first is
simply a number that indicates where in the data structure that assembly is stored
during computation. The second is what burnable poison, if any, is present in the fuel
assembly. The third is the enrichment of the fuel assembly. The fourth is the burnup in
the assembly, in MWd/kg heavy metal. The fifth is the power in the assembly relative
to the average assembly power in the reactor. And finally, the sixth is the kif of that
fuel assembly. The structure of the data in these figures is shown in Figure 1.9
Numerical Identifier for

the Assembly
Bumable Poison in the

Assembly
Assembly Enrichment
Assembly Burnup
Assembly Power Compared

to Average Assembly Power
k., of the Assembly
Figure 1.9. Structure of Assembly Data in poropy Generated Figures
29
1.4 MODEL
The model being used in this analysis is the benchmark discussed and used by Ya-
mamoto in his examination of core loading pattern optimization techniques. This is a
Westinghouse type 900 MW electric, 3 loop PWR loaded with 17x17 fuel assemblies,
corresponding to 157 fuel assemblies in the core. The core is assumed to be quarter
core symmetric throughout the analysis. The fuel assemblies used in the core are all
at 4.10% enrichment. The fuel assemblies are either fresh or have been through one
or two depletion cycles. This results in assembly burnups that range from o.o to 34.7
MWd/kg heavy metal , with the initial configuration specified by Yamamoto shown
in Figure 1.10. Finally, some of the fuel bundles in the Yamamoto benchmark contain
Gadolinium as a burnable poison, while others do not [6].
The initial configuration specified by Yamamoto is an example of a loading pattern
in which the most reactive fuel (fresh with no Gd) is placed on the periphery of the
reactor (see Figure 1.11) , as discussed briefly in the introduction, resulting in a very
flat power distribution, as seen in Figure 1.12.
In the later parts of this study, fresh fuel assemblies will be prohibited from being
placed face adjacent to the reflector in order to minimize leakage and maximize cycle
length. Power peaking will be minimized within these constraints. It should be noted,
however, that since poropy assumes a constant boron concentration of 900 ppm, the
depletion points computed do not correspond to what would be the true end of cycle
in a more rigorous model. Fuel assemblies are each assumed to contain 50o kg of
heavy metal fuel.
This is a more tractable benchmark problem to work with than a commercial 4
loop PWR with approximately 200 fuel assemblies. The reduction in the size of the
solution space greatly reduces the time for optimization methods to locate a high
quality solution.
30
U'
34 10 11 R
GAD Nole None
W 4.1000 4.1000 4.1000
1 0.0000 12.6000 12.6000
1.0956 12218 1 "999 . 1.0128 0.6552
1.1092 1.0867 1 ' - 1.1521 1.1521
(ALl GAD GAD GAD None

4 01_ 4,1000 4.1000 _ 1000
..100.
00000 . 1R4 .04 0,0000
1.0504 07v.9
1.0158 1 358 10158
1T 31- -, 29 7RR
C4A GAD GAD None
4 4.1000 .1000 4.1000
1 0.0000 '1 0.0000 10.2000
I.MS6. 1.0504 11.0461 0.8060
kt10158 -1.0158 1.1743
T' s 30 9 1 R R
- GAD None None
41000 4.1000 4 1000
- _IB.9DQ30 11.3000 3000. 19.9000 U
12218 ,, 1.1731 11059 1.1308 C'S)7
1.0 67 - 1.99 .0 15 11641 1
34 - 35 12 2 R R
GAD - AD GAD None
4-1000 '-4.1000 4.1000 4.1000
0.0000 6- 0.0000 113000 0.0000
1.1799 L1.1059 1.1539 0.8834
1.0158 10153 1.1427 2846
27 8 4 R R R R
GAD None None
4.1000 4.1000 4.1000
-0.0000 11.3000 0.0000
- 1.0461 1.1308 0.8834
1.51.0158 1 1641 2846
1Q 28 6 3 R R R R R
Nonse GAD None None
4.1000 4 1000 4.1000 4,1000
12.6000 0.0000 10.2000 0,0000
1.0128 0.90013 0.8069 0.7 507
1 1521 1.0158 1.1743 1 2846
11 Q R R R R R R R
None None
4 1000 4.1000
12.6000 0.0000
0,6552 0.5919
1.1521 1.2 6
P k R R R R R R R.
Figure z.1o. Burnup of Fuel Bundles in Yamamoto's Benchmark
1.5 OBJECTIVES / GOALS
It is very important to note that the goal of this project is not to replace the current
stochastic methods of optimization. A deterministic method such as the Greedy Ex-
haustive Dual Binary Swap will never be as efficient as a well implemented stochastic
method, since it needs to search across all of the solution space it is able to access, as
opposed to searching preferentially in regions of good solutions. Instead, the aim of
this project is to determine whether the nature of the solution space for core loading
optimization is such that there exist patterns of very high quality that are "isolated".
31
R
1331 2d t 24 29 - 2 - R
R R
R R R
R R R R
R | R | R R R
R R R R R R R
RR R | R R R R R R
Figure i.vt. Reactivity of Fuel Bundles in Yamamoto's Benchmark
In this context, a pattern is isolated if even small perturbations result in the pattern
becoming significantly worse. If the Dual Binary Swap algorithm succeeds in finding
"silver bullet" patterns that are of higher quality than those that can be found by the
stochastic methods, this is strong support for the hypothesis that the core loading
solution space contains these isolated high quality solutions.
If the GEDBS methodology is shown to reliably and easily find these high quality
solutions, it might suggest potential heuristics that could be adopted into current
stochastic methods to improve their efficiency.
There are multiple stages involved in the proposed project.
32
R
r1 0 R R
None None
4.1000 4.1000
0.0000
I 2.v00
1.2845
R R R R R RI R R R
Figure 1.12. Power Peaking in Yamamoto's Benchmark
Initially, this work will compare the efficacy of the deterministic Dual Binary Swap
algorithms with the more traditional stochastic methods for beginning of cycle analy-
sis. This involves both looking at the quality of the solutions obtained by these meth-
ods, as well as the reliability with which they are able to satisfy the constraints im-
posed upon them.
After an analysis of the quality of the patterns obtained from a beginning of cycle
perspective, this work will aim to compare how effectively the methodologies are able
to produce patterns that satisfy the constraints throughout the entire operational cycle
of the reactor. This is a much more difficult optimization problem to perform, but is
much more indicative of which patterns are good for real applications.
33
This project also aims to look at the efficiency of these different methods in arriving
at their solutions in terms of computational time. One expects the stochastic methods
to be more efficient on this front than the GEDBS, but it would be informative to
determine by how much the deterministic methods fall behind.
34
2 BEGINNING OF CYCLE ANALYSIS
2.1 INTRODUCTION
The first component of the analysis in this project involved comparing the efficacy of
different core loading pattern optimization methods at the beginning of the reactor
cycle life. The reactor cycle length is typically heavily correlated with the amount of
excess reactivity present in a core. Unfortunately, excess reactivity is not something
that can be directly measured. Instead, the eigenvalue of the reactor core can be used
as a surrogate for excess reactivity. This gives a first order approximation of which
loading patterns are preferable, based on the amount that the core eigenvalue exceeds
one. As such, analyses were carried out on the two stochastic methods of interest
(Genetic Algorithms and Simulated Annealing) and the deterministic method of the
Dual Binary Swap to see which methods fared better at finding good loading patterns,
both in terms of consistency and quality.
In the analysis, the benchmark of whether an optimization has successfully found an
acceptable pattern is if the best pattern obtained in the simulation manages to satisfy
the power peaking constraint of the objective function. To represent this, a pattern
with high power peaking is much more heavily penalized than a pattern with a low
eigenvalue. In core loading design, the analyst will often treat the power peaking
across assemblies as a hard constraint. If the power peaking imposed for a given
run is satisfied, the quality of patterns within this set of "good" candidates will be
determined based on which pattern has the highest eigenvalue. The structure of the
objective function is shown below.
objective = kfactor * (kreactor - kbaseline) - Pfactor * max(0, Preactor - Pbaseline) (2.1)
35
It is worth noting that for the deterministic beginning of cycle analysis, before im-
plementing the full dual binary swap, simpler methods were implemented and briefly
tested that will be referred to as the single binary swap and the triplet swap. The for-
mer of these methods involved simply switching the position of just two assemblies
within the reactor core configuration before evaluating the new pattern. The latter
technique involves swapping the position of three assemblies within the core. It is
worth noting that these sets of assembly swaps are a small subset of the swaps that
are explored in the dual binary swap method, and are not missed in that algorithm.
2.2 BEGINNING OF CYCLE GENETIC ALGORITHM ANALYSIS
Due to the fact that the genetic algorithm methodology was already included within
the poropy framework, this was the first optimization method tested. For this first
analysis, an assembly power peaking constraint of 1.5 was chosen. Thus, if the power
level in any assembly is higher than 1.5 times the average assembly power level across
the whole core, that loading pattern is heavily penalized. The value of 1.5 was chosen
because the solution space for this problem was believed to have many patterns that
satisfied this constraint. The major differentiation between the core loading patterns
examined then arises from the eigenvalues.
To this end, in the initial analysis, 30 different runs (where each run uses a dif-
ferent random seed in the Genetic Algorithm) were carried out with the Yamamoto
benchmark, with a variety of different genetic algorithm parameters. The number of
generations was varied from 100 to 4000, the number of candidates in a population
was varied from 100 to 150, and the number replaced (the genetic algorithm will al-
ways keep a certain number of best candidates from the previous generation as part
of the next generation - the candidates that are replaced are replaced with children
of the previous generation) in each generation was varied from 50 to 100. All of the
rms succeeded in meeting the imposed power peaking constraints. The results are
summarized below in Table 2.1.
36
Table 2.1. Results of Beginning of Cycle Analysis of the Genetic Algorithm Optimization
Scheme, Peaking of 1.5
Number of Size of Number Mean Power Mean Minimum Maximum
Generations Population Replaced Peaking Eigenvalue Eigenvalue Eigenvalue
100 100 50 1.4734 1.05687 1.05388 1.06040
200 100 50 1.4869 1.06242 1.05898 1.o6586

400 100 50 1.4824 1.07039 1.06529 1.07310
1000 100 50 1.4918 1.07651 1.07136 1.08083
1000 150 50 1.4816 1.07634 1.07470 1.08086
1000 150 75 1.4952 1.08081 1.07721 1.08353
1000 150 100 1.4961 1.08137 1.07749 1.08547

2000 100 50 1.4954 1.07904 1.07465 1.08218
4000 150 100 1.4978 1.08332 1.07790 1.08691
The maximum, minimum, and mean values referred to in Table 2.1 arise from the
fact that for each set of parameters, multiple optimizations were carried out with
different seeds for the random number generator. Maximum, minimum, and mean
refer to within these different runs for the same parameter values.
Table 2.1 demonstrates that the eigenvalue of the optimized patterns from the Ge-
netic Algorithm is affected to a significant degree by the random seed used in that
optimization. In all cases, the difference between the maximum and minimum eigen-
values is on the order of hundreds of pcm. Additionally, the eigenvalues of the opti-
mized patterns change dramatically as number of generations, size of population, and
number of candidate solutions replaced changes. This suggests that the effectiveness
of a genetic algorithm optimization run is highly dependent on the parameters used
for the optimization. There is also the potential issue of the efficacy of whichever cross-
over operator is being used in the optimization, which could further limit how well
any given implementation of a Genetic Algorithm can perform. This was one of the
motivations that led to the implementation of Simulated Annealing within the poropy
framework - a stochastic optimization method that allows a reduction in the number
of parameters that need to be intelligently chosen by the user.
37
2.3 BEGINNING OF CYCLE SINGLE BINARY SWAP ANALYSIS
The Single Binary Swap deterministic methodology was implemented in the begin-
ning of cycle analysis to determine a baseline effectiveness for a very simple deter-
ministic methodology. As with the Dual Binary Swap methodologies, two versions of
the Single Binary Swap algorithm were implemented. One version is the exhaustive
implementation, in which all possible swaps are examined, the one that improves the
core the most is accepted, and the search process is begun anew. The second version is
the greedy implementation. In the greedy implementation, as soon as a swap is found
that improves the loading pattern under consideration, it is accepted and the search
for a better pattern is restarted.
When starting from the initial configuration of the Yamamoto core, with initial eigen-
value of 1.02515 and initial power peaking of 1.2218, the exhaustive implementation
of the Single Binary Swap method was able to improve the eigenvalue of the core con-
figuration to 1.07590 by increasing the power peaking of the core to 1.4993. Since, as
with the Dual Binary Swap methodology, the optimizer never accepts any swaps that
decrease the objective function, it is to be expected that the power peaking constraint
of the optimization was satisfied, as the core configuration started with power peaking
well below 1.5. Any patterns with peaking above that threshold would likely not see
a large enough gain in eigenvalue to result in the pattern being considered improved
by the optimizer.
The greedy implementation of the Single Binary Swap was unable to find as good
a loading pattern as the exhaustive from the above mentioned starting configuration,
converging to a solution with an eigenvalue of I.o6656 with a power peaking of 1.4987.
It is worth noting that the single binary swap algorithm was unable to find pattern
solutions that were better than the average for any of the GA runs that were 1ooo
generations or more. In many cases, the result of the Single Binary Swap were signifi-
cantly worse than the best pattern produced by the Genetic Algorithm. This suggests
38
that a more thorough swapping scheme is needed than a Single Binary Swap to be
able to perform at par with stochastic methods.
An additional test was carried to determine if using the Single Binary Swap on the
result of a converged / completed genetic algorithm optimization would result in a
better core loading pattern. The results are shown in Tables 2.2 through 2.4.
Table 2.2. Improvements of the Eigenvalue of the Optimized Core Loading Pattern of the Ge-
netic Algorithm Methodology from looo Generations after Applying Greedy Single
Swap Optimization
Genetic Algorithm Eigenvalue After Eigenvalue
Test Eigenvalue After Greedy Single Difference
Number looo Generations Swap Optimization (pcm)
1 1.07749 1.07798 49
2 1.07821 1.08326 505
3 1.08433 1.08484 51
4 1.08547 1.08633 50
Table 2.3. Improvements of the Eigenvalue of the Optimized Core Loading Pattern of the
Genetic Algorithm Methodology from loo Generations after Applying Exhaustive
Single Swap Optimization
Test Eigenvalue After Exhaustive Single Difference
Number 1ooo Generations Swap Optimization (pcm)
1 1.07749 1.07798 49
2 1.07821 1.08293 472
3 1.08433 1.08484 51
4 1.08547 1.08634 87
It can be quickly seen from this that even a simple Single Swap optimization can
be used to improve the results of a Genetic Algorithm optimization, so long as the
loading pattern in question is not too deep in a local minimum.
39
Table 2.4. Improvements of the Eigenvalue of the Optimized Core Loading Pattern of the
Genetic Algorithm Methodology from 4ooo Generations after Applying Exhaustive
Single Swap Optimization
Test Eigenvalue After Exhaustive Single Difference
Number 4ooo Generations Swap Optimization (pcm)
1 1.07790 1.07790 0
2 1.08359 1.08359 0
3 1.08488 1.08488 0
4 1.08691 1.08691 0
2.4 BEGINNING OF CYCLE TRIPLET SWAP ANALYSIS
As with the the Single Binary Swap, an exhaustive and a greedy version of the Triplet
Swap (also referred to as a "chain shuffle", where the position of three assemblies
within the core are exchanged) optimization were implemented. In the former, all
possible swaps of the positions of three assemblies were examined before accepting
the pattern that most improved the objective function. In the later, a new loading
pattern was accepted as soon as one was found that improved upon the previous
pattern.
value of 1.02515 and initial power peaking of 1.2218, the exhaustive implementation of
the Triplet Swap method was able to improve the eigenvalue of the core configuration
to 1.08543 by increasing the power peaking of the core to 1.4973.
The greedy implementation of the Triplet Swap was unable to find as good a loading
pattern as the exhaustive from the above mentioned starting configuration, converging
to a solution with an eigenvalue of 1.07421 with a power peaking of 1.4989.
It can be seen here that the exhaustive version of the Triplet Swap was able to find a
core loading configuration of almost the same quality (differing by 4 pcm) as the best
genetic algorithm run performed in Section 2.2. The greedy implementation of this
algorithm was only able to match the averages or minimums of those runs with mooo
or more generations.
40
When a test was run to determine if using the Triplet Swap on the result of a
converged / completed Genetic Algorithm optimization (after 4000 iterations) would
result in a better core loading pattern, it was found that, for both exhaustive and
greedy implementations, 3 out of 4 runs were unable to find any swaps that would
improve the pattern, and 1 out of 4 runs was able only to find a single swap that
improved the pattern by 2 pcm.
2.5 BEGINNING OF CYCLE DUAL BINARY SWAP ANALYSIS
As with the Single Binary Swap and the Triplet Swap methodologies, the Dual Binary
Swap was implemented in both an exhaustive and a greedy form.
value of 1.02515 and initial power peaking of 1.2218, the exhaustive implementation of
the Dual Binary Swap method was able to improve the eigenvalue of the core config-
uration to i.08771 by increasing the power peaking of the core to 1.4999. The greedy
implementation of the Dual Binary Swap was unable to find as good a loading pat-
tern as the exhaustive from the above mentioned starting configuration, converging to
a solution with an eigenvalue of 1.08629 with a power peaking of 1.4998.
These results suggest that the Dual Binary Swap methodology may be able to find
patterns that are missed by stochastic methods. In both the exhaustive and greedy
implementations, the Dual Binary Swap performs better than the best runs from the
Genetic Algorithm optimizer by 224 and 82 pcm, respectively.
A number of tests were also run using the results from Genetic Algorithm optimiza-
tions as the starting points for a Dual Binary Swap optimization. A summary of the
average results are presented in Tables 2.5 and 2.6, with a more detailed outline of
individual results given in the appendix.
41
Table 2.5. Results of Beginning of Cycle Exhaustive Dual Binary Swap Optimization of Ge-
netic Algorithm Starting Patterns
Number of Size of Number Mean Post GA Min Post GA Max Post GA
Generations Population Replaced Eigenvalue Eigenvalue Eigenvalue
400, 100 50 1.07039 1.06529 1.07310
1000 120 8o 1.08111 1.07696 1.08499
4000 150 100 1.08332 1.07790 1.08691
Number of Size of Number Mean Post DBS Min Post DBS Max Post DBS
400 100 50 1.08654 1.08549 1.08751
1000 120 8o 1.08593 1.08543 1.08677
4000 150 100 1.08411 1.07790 1.08697
Table 2.6. Results of Beginning of Cycle Greedy Dual Binary Swap Optimization of Genetic
Algorithm Starting Patterns
Number of Size of Number Mean Post GA Min Post GA Max Post GA
4000 150 100 1.08332 1.07790 1.o8691
Number of Size of Number Mean Post DBS Min Post DBS Max Post DBS
4000 150 100 1.08399 1.07790 1.08697
2.6 RESTRICTING LOCATION OF FRESH FUEL ASSEMBLIES
After an analysis of the patterns that were being produced by the various optimization
methods mentioned above, it was decided to add in a heuristic to prevent fresh fuel
assemblies from being placed adjacent to the reflector. In practice, most modern core
loading patterns tend to place partially burned fuel adjacent to the reflector to limit
leakage and increase cycle length / core average excess reactivity. The goal of this
was primarily to ensure that the optimization process did not get stuck in an area of
phase space that is known to be sub-optimal (having fresh fuel bundles face adjacent
to the reflector), and be unable to transition to a better pattern because doing so
would necessitate increasing the core power peaking above the imposed threshold.
The structure of the objective function is shown below.
42
objective = kfactor * (kreactor - kbaseline)
- Pfactor * max(0, Preactor - Pbaseline)
- FreshAdjacentReflectorfactor * FreshAdjacentReflector,,number (2.2)
It is worth noting that due to the nature of the implementation of the Genetic Al-
gorithm optimizer, it was not feasible to impose this constraint of requiring that fresh
fuel assemblies not be placed adjacent to the reflector in this methodology.
The first set of Dual Binary Swap optimizations was carried out starting from the
input configuration of the Yamamoto benchmark as specified in Reference [6]. The
results for the optimizations are shown in Table 2.7. All optimized patterns in the
table had no fresh fuel bundles adjacent to the reflector.
Table 2.7. Optimizations of the Yamamoto Benchmark Using the Dual Binary Swap Method-
ology
Power Peaking Exhaustive Exhaustive Greedy Greedy
Constraint Eigenvalue Power Peaking Eigenvalue Power Peaking
1.6 1.08933 1.5983 1.09324 1.5990
1.5 1.08838 1.4958 1.08327 1.4998
1.4 1.07568 1.3991 1.07460 1.3998
A more extensive table demonstrating the results of the two optimization methods
from random starting configurations (the configurations were randomized by running
the Genetic Algorithm optimizer on the specified Yamamoto input for a small num-
ber of generations, with no high quality solutions being retained from generation to
generation) of the Yamamoto benchmark core is presented in the appendix.
In this analysis, there is no clear preferred scheme between the exhaustive and
greedy methodologies. In some runs, the exhaustive method produced core loading
patterns with better eigenvalues / more excess reactivity. In other runs, the greedy
method produced better patterns. This is something to note, however, as in many
problems algorithms that are more exhaustive outperform algorithms that are more
43
greedy due to their exploring more of the solution space. In this particular case, the
greedy implementation of the algorithm went down a different path than the exhaus-
tive implementation of the algorithm. This ultimately resulted in a higher quality
pattern that could not be reached directly by the exhaustive algorithm.
The optimization problems that the Dual Binary Swap method has been applied
to thus far have a relatively high power peaking constraint, which is easily satisfied.
However, a further set of optimizations was carried out with a lower power peaking
constraint being imposed. Here, the metric for the success of the two Dual Binary
Swap methodologies is, to first order, not how high the eigenvalue obtained is, but
whether the optimizer succeed in satisfying both the power peaking constraint, and
the constraint that a fresh / unburned fuel assembly can not be placed adjacent to the
reflector. Below, in Tables 2.8 and 2.9, the rate of a given Dual Binary Swap method
finding a pattern that satisfies both the power peaking constraint, and the constraint
that a fresh fuel bundle can not be adjacent to a reflector, is recorded. Table 2.8 records
the ability of the Dual Binary Swap algorithms to find solutions that satisfy the con-
straints when having fresh fuel bundles adjacent to the reflector is penalized severely,
having a power peaking value above the threshold is penalized heavily, and having
a low eigenvalue is penalized lightly. Table 2.9 records the ability of the Dual Binary
Swap algorithms to find solutions that satisfy the constraints when having a power
peaking value above the threshold is penalized severely, having fresh fuel bundles ad-
jacent to the reflector is penalized heavily, and having a low eigenvalue is penalized
lightly. In practice, this distinction affects whether the algorithms tend to make swaps
that move fresh fuel away from the reflector or swaps that reduce the power peaking
preferentially.
It is interesting to note that the exhaustive version of the algorithm demonstrated a
much higher rate than the greedy version of finding a pattern that satisfied the hard
constraints when placing fresh fuel assemblies adjacent to the reflector was penalized
much more heavily in the objective function. However, the data seems to suggest that
the greedy version of the algorithm might be slightly more robust than the exhaustive
44
Table 2.8. Success Rates of the Dual Binary Swap Methodologies when Placing Fresh Fuel
Adjacent to the Reflector is Penalized the most Heavily in the Objective Function
Power Peaking Exhaustive DBS Greedy DBS
Constraint Success Rate (%) Success Rate (%)
1.375 100.0 100.0
1.350 100.0 50.0
1.325 83.3 16.7
1.300 50.0 0.0
1.275 33.3 0.0
1.250 16.7 0.0
Table 2.9. Success Rates of the Dual Binary Swap Methodologies when Power Peaking Above
the Constraint is Penalized the most Heavily in the Objective Function
Power Peaking Exhaustive DBS Greedy DBS
Constraint Success Rate (%) Success Rate (%)
1.375 100.0 66.7
1.350 50.0 66.7
1.325 0.0 33.3
1.300 0.0 16.7
1.275 0.0 0.0
1.250 0.0 0.0
algorithm when having power peaking above the imposed maximum was penalized
more heavily.
Due to the much better performance at low power peaking values, the exhaustive
implementation of the Dual Binary Swap with a heavy penalty in the objective func-
tion for placing fresh fuel adjacent to the reflector can be seen to provide the best
representation of the Dual Binary Swap method's ability to find difficult patterns.
2.7 BEGINNING OF CYCLE SIMULATED ANNEALING ANALYSIS
The limitations of the Genetic Algorithm optimization method have already been dis-
cussed earlier in this work. First and foremost is the large number of user defined
parameters that can have a very significant affect on the ability of the scheme to arrive
45
at a high quality solution. Additionally, from a practical standpoint, the GA optimizer
as implemented in the poropy tool can not be easily altered to include changes to the
objective function such as penalizing the placement of fresh fuel assemblies adjacent
to the reflector.
By contrast, the Simulated Annealing method of stochastic optimization has fewer
parameters that need to be intelligently set by the user of the method. Since it had
not previously been included in the poropy framework, it could also very easily be
implemented in such a way as to allow for penalizing fresh fuel assemblies being
placed adjacent to the reflector.
To this end, the Simulated Annealing algorithm was tested to determine its ability
to find loading patterns that satisfied both an imposed power peaking constraint, and
a constraint that fresh fuel assemblies not be placed face adjacent to the reflector.
While initial tests were carried out with the "temperature" of the simulation de-
creasing by between io% and o.1% each iteration (across 10 to 20 orders of magnitude),
these cooling speeds were found to be too quick for the simulated annealing algorithm
to satisfy the power peaking constraint in all but a very small fraction of simulations.
Thus, to better represent the patterns that the Simulated Annealing stochastic method
could possibly find, the benchmark simulations were cooled at a rate of o.oi% reduc-
tion in temperature each iteration (defined as a cooling rate of 0.9999), across 25 orders
of magnitude in temperature.
The results of this Simulated Annealing benchmark is shown in Table 2.10.
Table 2.10. Success Rates of the Simulated Annealing Methodology with Slow Cooling
Power Peaking Simulated Annealing Mean Successful Max Successful
Constraint Success Rate (%) Eigenvalue Eigenvalue
1.375 100.0 1.07186 1.07317
1.350 100.0 1.07106 1.07286
1.325 100.0 1.06632 1.06788
1.300 100.0 i.o6666 1.06787
1.275 100.0 1.05949 1.06097

1.250 16.7 1.05330 1.05330
46
It can be seen that, given sufficiently slow cooling, the Simulating Annealing opti-
mization methodology is able to very consistently satisfy harshly imposed constraints,
much more so than the exhaustive dual binary swap method, as can be seen by com-
paring Tables 2.8 and 2.10. However, those dual binary swap optimizations that suc-
ceed tend to produce very high quality loading patterns, better than the best found
by the Simulated Annealing method, as shown in Table 2.11. The loading patterns are
shown in Figures 2.2 through 2.13 (recalling that the data presented in these graphics
is structured as shown in Figure 2.1).
Table 2.ii. Difference of Quality of Best Solutions Produced by the Dual Binary Swap and
Simulated Annealing Methods
Power Peaking Dual Binary Swap Simulated Annealing Difference
Constraint Max Eigenvalue Max Eigenvalue (pcm)
1.375 1.07948 1.07317 631
1.350 1.07767 1.07286 481
1.325 1.07656 1.06788 868
1.300 1.06825 1.06787 38
1.275 1.06381 1.06097 284
1.250 1.05436 1.05330 1o6

the Assembly
Burnable Poison In the

Assembly
Assembly Enrichment
Assembly Burnup

k In of the Assembly
Figure 2.1. Structure of Assembly Data in Loading Patterns
These results suggest that, for a beginning of cycle analysis, the deterministic dual
binary swap method is able to find high quality loading patterns that stochastic meth-
47
34 12 .03 20 R
GAD CAD
4.1000 4.1000
0t0000 0.0000'
1 0090
.0 1.0867
15
GAD GAD None
4.1000 4.100: 4.1000
0.0000 0.0000
1.1232 1.2093
1.0158 1.2846
28 8 10 R R
GAD None
4.1000 4.1000
0.0000 12.6000
1.1310 0.8273
1.0159 1.1
3-M 27 1 R R
GAD GAD None
,iJ 4.1000 4.1000 4.1000
0.0000 0.0000 0.0000
1.1063 1.1294 1.2122
1.0158 1.0158 1.2846
1130 5 R R
None GAD None
41000 41000 4.1000
12,4000 0.0000 0.0000
1I24M" 1.1530 1.2212 _
1.1521 10158 1.2945
12
GAD
13
DAD
32
GAD
2
None
4i-~ R RI R R
I
4.1000 4.1000 4.1000

11.3000 00000 0.0000
12448 S1.2-413 10807 1.1625
1.1427 1.1092 10158 126846
33 0 6 R R R R R
'
GAD None None

4.1000 4,1000 4.1000
0.0000 0.0000 10.2000
1.0090 1.2072 0.7"1
1.0158 1 2846 1.1743
R R R
R RI R R
Figure 2.2. Best Beginning of Cycle Core Loading Pattern Obtained by Dual Binary Swap for
an Imposed Power Peaking Constraint of 1.250
ods miss. The deterministic method is, however, unable to find patterns that satisfy
imposed constraints as consistently as stochastic methods.
48
34 7 14 8 4 9 R
GAD None GAD None None None
4.1000 4.1000 4.1000 4.1000 4.1000 4.1000
0 0000 10.2000 18.4000 11.3000 0.0000 11.3000
1 0'77 12742 1.2649 1.2683 1 2424 0.S534
1.0158 1.1743 1.0958 1.1641
~1 1.2846 1.1641
34 t0 33 17 18 32 2 R
GAD None GAD GAD GAD GAD None
4.1000 4.1000 4.1000 4.0000 9.17)00 4.000 4 1000
0.0000 12.6000 0.0000 ra0 1.01000 0.0000 00000
1.0577 1.1748 Sl141b tow9~ 1.143 1.1474 .2099
10158 1.1521 1.0'58 1.0917 1.0408 1cP8 1.2846
7 30 19 13 9 0 ft R
None GAD 5a GAD None
4.1000 4.1000 41000 4.1000 4.1000
102000 0.0000 16.8000 0.0000 0 0000
1.2742 1.1593 1.190 1.2219 1 1724 1.2748
1.1743 1.015a 1.0908 1.1092 1.OS0 1.2846
14 12 11 5 6 28e R R
GAD GAD None GAD None GAD
4.1000 4.1000 4.1000 4.1000 4.1000 4.1000
18.4000 11 3000 12.6000 18.A000 102000 0.0000
1.2649 1.2645 1.2719 12413 1.2369 07872
1.0958 1.1427 1.1521 1.0958 1.1743 1.0158
ill
8
None
4,
GAD
f 1a None
R R R
-
4 1000 41000 4 1000

11.3000 0 0000 ON 0.0000
1,2683 1.0570 ,1.11% 1.2111
1.1641 1.0156 -1;Q41 1-2846
3, 5 3 R R R
GAD Nene None
-- 4 00 4.1000 4.000
0.0000 0 0000 0 0000
0.98,08 1.2722 1.2160
1.0158 1.2846 1.2846
-
4 35 R R R R R
None GAD
4.1000 4.1000
1.2424 0.7161
1.2846 1.0158
R R
R R R R R
Figure 2-3. Best Beginning of Cycle Core Loading Pattern Obtained by Dual Binary Swap for
49
34 9 14 8 11 4 R
GAD None None None None
.11000 4.1000 4.'1000 4.1000 4.1000 4 1000
0.0000 11.3000 18.4000 11.3000 12.6000 0.0000
0.90B0 1.2502 1.2770 1.2967 1,2676 1,2344
.0158 1.1641 1.0958 1.1641 1.1521 1.2846
34 33 6 15 32 31 5 R
GAD GAD None GAD GAD GAD None
a :00 4.1000 4.1000 4.1000 4 1000 4.1000 4.1000
0 0000 0.0000 10.2000 0.0000 0.0000 0.0000
0 9088 0.9753 1.2991 1.1278 1.1589 1.1845
10158 1.0158 1.1743 1.0958 o015 1.0158 1 RAA
9
None
4.1000
11.3000
1.2502
30
GAD
4.1000
0.0000
1.1060 ,
-
19
j 12
GAL
4.1000
11.3000
1.2932
28
GAD
4.1000
0.0000
1.1629
0
None
4.1000
00000
1.2968
R R
1.1641 10158 11427 .0159 12846
16 13 10 18 R R
.
d-An r.An None Crln Ie ,-
8 35 27
None GAD GAD None
4.1000 4.1000 4 1000 40 0 41000 4.1000
11 3000 0 0000 0.0009 18.9000 00000 19.S00
1.2967 1.0961 1.1312 1.2975 1.2097 d.512
1 1641 1 0 1.0158 1107 1.2846 X1_0
,1 F1
None None None
4.1000 4.1000 4.1000
12.6000 0.0000 0. 0000
1.2676 1 2993 1.1972
1.15 21 1.2846 1.2846
4 29
None GAD
4 1000 4.1000
0.0000 0 0000
1 2344 0.6979
12846 1.0158
I
R R
50
34 11 9 17 1S R
CAD None None GAD None
-l
4.1000 4.1000 4.1000 4A00 4.1000
0.0000 12.6000 11.3000 18:900,1 0.0000
0.9501 1.2614 1.3082 1.2931 1 2784
1.0158 1.1521 1.16411
33
1,0017
| 1.0958
7
I 1.2846
12 31 R
GAD GAD GAD None GAD
4.1000 1.1000 4.1000 4. 1000 4 1000
00000 0 0000 11.3000 10.2000 0.0000
09501 1 .. 29 1.3172 1.3019 0.7947
1 0198 1.0867 1.0908 1 Gi'58 .1427 1 1743 1.0158
1 6 a 16 14 R R
None None None GAO GAD Nc.,
4.1000 4.1000 4.1000 4.1000 4.1000 0
12.6000 10.2000 11.3000 18.9000 184000 M
1.2614 1.3166 1.3129 1.1951 1.2918 1u99
1.1521 1.1743 1.1641 1.0917 1.0958 1 '846
9 29 27 5 35 R R
None GAD GAD None GAD
4.1000 4.1000 4.1000 4.1000 4.1000
11.3000 0.0000 0.0000 0.0000 00000
1.3082 11852 1.1678 1.3206 0.7705
11641 1.0158 1.0158 1.2846 10105
+ . 4
10 28 1 30 R R
None GAD None GAD
4.1000 4 1000 4 1000 4.1000
12.000 0.0000 00000 0.0000
1l 1.3145 1.1970 1.2799 0.7773
1.1521 1.0158 1.2846 1.0158
1s 13
GAD GAD
2
None
32
GAD
R I R I R
49 41900 4.1000 4 1000
16900 0.0000 0.0000
to-m 13249 1.3182 0.7687
1.t092 1.2846 1.0158
4 H
None None
4.1000 4.1000
0.0000 0.0000
1.2784 1.2499
1.2846 1.2846
R N
51
34 10 8 13 16 4 R
GAD None None GAD GAD None
4 1000 4.1000 41000 4.1000 4.1000 4 1000
3 003C 12.6000 1 1.3000 16,8000 18.9000 00000
0.0347 1.1920 1.3490 1.3196 1.2723 1.2540
1 0159 1.1521 1.1641 1.1092 1.0917 1.2846
33 9 15 14 6 32 R
GAD CAD None GAD GAD None GAD
4. 1000 4.1000 4.1000 4.1000 4.1000 41000
0.0000 11.3000 18.4000 18.4000 10 2000 0 0000
08347 0 9041 1.2677 1.2939 1.2826 1.2781 0 7912
1.0158 1.0' ; 11641 1.0958 1.0958 1.1743 1.0158
10 30 7 12 17 0 R
None GAD None GAD CAD None
4.1000 4.1000 4.1000 4.1000 4 1000
12.6000 0.0000 10.2000 11.3000 0.0000
1.1920 1.0573 1.3061 1.3396 1.3060
1.1521 1.0158 1.1743 1.1427 12846
8 18 28 21, R R ]
None GAD GAD None
4.1000 4.1000 4,1000 4.1000
11.3000 19:600' 0.0000 0.0000 0.0000
1 3490 t.teb5 1.1437 12116 1.3499
1.1641 7.0908 1.0158 1.0158 1.2846
13 2- 1 29 R R
GAD None None GAD
4.1000 4.1000 4 1000 4.1000
16.800 12:6000 0.0000 0.0000
13194 1,348 1 3446 0.8025
1.10 1.1521 12846 1.0 158
-7 1 -
R R R R
"
4 3 R R R R
None None
4 1000 4.1000
0.0000 0.0000
1.2540 1.2264
1 2846 1.2846
R R
R R R R R R R R R
Figure i.6. Best Beginning of Cycle Core Loading Pattern Obtained by Dual Binary Swap for
52
34 10 11 13 1 R
GAD None None GAD None
4.1000 4.1000 4 1000 4.1000 4.1000
00000 12 6000 12.6000 16.8000 00000
1 1506 1.3738 1.3649 1.3363 1 2637
1.010S 1.1521 1.1521 1.1092 1 2846
8 9 I 3 33
None None None CAD
4.1000: 4.1000 4.1000 374 +ItOi 4.1000 4 1000
19.501 11.3000 11.3000 0.0000 0 0000
1.180 1.2723 1.3670 41911 1.3608 0.7663
1.1641 1.1641 1.09 1 2846 1.0158
34 32 12 7 31 29
CAD R R
GAD GAD None CAD GAD
4.1000 4.1000 4.1000 4.1000 4.1000 41000
00000 0.0000 11 3000 10.2000 0 003) 0 0000
11506 1.1837 1,3641 13725 10nt,9 0.77-0
10158 1.0158 1.1427 11743 1.01 98 1 0158
10 6 2 30 R R
None None None CAD
:98 4.1000 4,1000 1 4.1000 4.1000
12.6000 10.2000 0.0000 0.0000
.J
1.3738 1.3719 12148 0. 0678
1.1521 1.1743 1.0958 1.2846 1.0108
11 35 R R
None GAD None
4,1000
12.6000
4 1000
0.0000
-1 4.1000
0.0000
1.3649 1,2192 1 3748
1.1521 1 0158 1.2846
-3 0 27
CMD None CAD
4.1000 4.1000
0.0000 0.0000
1.3390 0.7973
1.1092 1 2444 1 0158
1302
None None
4.1000 4.1000
00000 0.0000
1.2637 1.2173
1 2846 1.2846
N
I
4 R R R R R R RIR
Figure 2.7. Best Beginning of Cycle Core Loading Pattern Obtained by Dual
Binary Swap for
53
27 In 13 14 7 R
GAD None GAD GAD None
4.1000 4.1000 4.1000 4.1000 4.1000
0.0000 12.6000 16.8000 184000 10.2000
0.9730 1.2496 1.2188 1.1107 0.5852
1.G158 1.1521 11092 1.0958 1.1743
1 17 30 5 R
GAD None
4.1000 4.1000
0.0000 0.0000
1.1717 1.1986
1:0917 1.0' 58 1.2846
35S 18
1 8 R R
GAD None
4 1000 " 4180 4.1000
0.0000 19u0G 11.3000
1.0616 4:1785 1.2304
1 018 1.0908 1.1641
27 34 i5 11 4 9
GAD GAD GA. None GAD None None
4,1000 4.1000 4.1000 4.1000 4.1000 4.1000
0.0000 0.0000 12.4000 0.0000 00000 11.3000
09730 0.9906 1.2401 1.2000 1. 2486 0.5901
-
1.0158 1.0158 1,0958 1.1521 105T 1,2846 1.1641

10 19 R R R
13 56 6 1 R
GAD GAD None None
4.1000 4.1000 4.1000 4.1000
TO000 0 0000 10.2000 0.0000
1.2188 1.1769 1.2418 12290
1.102 1.0158 1.1743 1 2846
0 R
- RA None
4.1000
18A00 0.0000
.
1,1107 1 2298
1.0958 1.2846
'
7 12 R R R I R. 1 R R R
None GAD
4 1000 41000
10.2000 11.3000
0.5852 0.5237
1.1743 1.1427
R R R R R R R
Figure 2.8. Best Beginning of Cycle Core Loading Pattern Obtained by Simulated Annealing
for an Imposed Power Peaking Constraint of 1.250
54
34 33 10 R
GAD GAD tNpne
4.1000 4 1000 4.1000
00000 0.0000 12.6000
1.0920 1.1129 1.2740
10!58 10158 1.1521
32 9 15 4 R
- None GAG None G
41000 4.1000 V= 4 1000
0.0000 11.3000 0.0000
10538 1.2551 1.1661
1.0158 1.1641 1.O95 1284 6
None GAD
11 29 19 1 R
GAD None GAD None GAD~
.
4.1000 4.1000 4.1000 4.1000 4 1000 4 1000 a.ro000
0.0000 0.0000 0.0000 12A000 0.0000 10.2000 19.0000 I
1 2476 0.9905 1.0336 1.2037 1.1526 1.2335 07W
1.2946 1.0158 1.0158 1.1521 1.0158 1.1743 1OrN
34 12 8 , 35 0 r R
GAO GAD None JAD GAD None
4 1000 4.1000 4.1000 4.1b00 4.1000 4.1000
00000 11.3000 11.3000 I 0.0000 0.0000
1.0920 1.1880 1.2692 I 20 11687 12079
1.0158 1.1427 1.1641 ,011 1,0158 12846
GAD
4.1000
0.0000
33 27
GAD
4.1000
0.0000
28
GAD
4.1000
0.0000
'-
'f~
I None
4.1000
0.0000
AR
- H
R R R
1.1129 1.1058 1.1563 12743

1.0158 1 0158 10156 1;0901 1 28.16
r S I I
10 13 6
None GAD None None
4.1000 41000 4.1000 4 1000
12.6000 11.000 10.2000 00000
1.2740 U9527 1,2544 1.2647
1.1521 1.1092 1,1743 1.2846
14 17
i _ __ __ __r
R
rG i None GAo
4:1000 4 1000 4,1000
18.4000- 0.0000 16.9000
1.1813 0.7516
1. 2846 10917
R R Rl R
R R R R Rl R R R
RI
55
27 10 8 14' 9 5
GAD None None None None
4.1000 4.1000 4,1000 4.1000 4.1000
0.0000 12*000 a 11.3000 11.3000 0.0000
0.9444 1.2816 11875 1w~ 1.2981 1.2603
I
1.01S8 1.1521 1.1641 1.1641 1.2846
29 35
T --~ --13--- - I i 3
27 6 15
GAD GAD None GAO GAD None
4. 1000 4.1000 41000 4.1000 tiomo 44 10d0 4.1000
0.0000 00000 10.2000 0.0000 16.8000 16A000 0.0000
09:44 1.0225 1.2797 1.1569 1,2392 1.2173 1.2013
01 10159 1.1743 .0158 1.1092 1.0959 1-M46
10 11 19 33 7 18 R
None None GAD None GAD
1.28 a 223 . - 1 1398 - 1.2992

1 )-le
.
1.1521 1.1521 1.0908 1.1743 1.0908
8 20 12 16 R R
None GAD GAD GAD OAD None
4.1000 4.4000 41000 4A 000 ;000
11.3000 S000 19:5000 11,3000 11W0 0 0000
1 2875 1.0860 f.7231" 1.2444 1.4H 1 1232
1.1641 1.0158 1.1427 12846
14 32 30 R R R
GAD GAD GAD
~ A@
4.1000 4,1000 4.1000
18.4090 0.0000 0.0000 9,100
0$7 1.0135 1.0893 tB047'
1.08 1.0150 1.0158
9
F
28
1
4
1
1
I
None GAD None None
4.1000 4.1000 4.1000 4.1000
11.3000 0.0000 0.0000 0.0000
1,2981 1.0184 1.3000 1.1837
1.1641 1.0158 1.2846 1,2846
None GAD
1.2 603 0 7;85

1.2846 1.0158
R R R
I I R
8 8R
56
34 7 R
GAD None
4.1000 4.1000
0.0000 10.2000
1.1343 1.3153
1.0158 1.1743
31 9 13 4 R
GAD None GAD None
4.1000 4.1000 4.1000 4 1000
0.0000 11.3000 16.800 0.0000
1.0492 1,3205 ix133 1.1693
10158 1.1641 .10g2 1.2846
29 11 12 R R
GAD None GAD
4 1000 4,1000 4.1000
00000 12.6000 113000
1.0708 1.3245 1.2489
1.0158 1.1521 1.1427
8 6
1
35
1 0 R R
GAO None None GAD None
-
4.1000 4.1000 4.1000 4.1000 4.1000

11.3000 10.2000 0.0000 0.0000
1.2483 12822 1.1150 ' 1.2300
1:0
-
1.1641 1.1743 1 0158 1.2846

34 27 28 10 R R R
GAD GAD GAD None
4.1000 4.1000 4.1000 4.1000
0.0000 0.0000 0.0000 12.6000
1.1343 1.1172 1 1561 1.1490
1.0158 1.0158 1.0158 1.1521
- 1
7 14 5 32
None GAD. None GAD
4,1000 4.1000 4.1000
10.2000 0.0000 00000
1.3153 1.3196 0.7816
1 1743 I 0958 i2646
1.0158
R R R R R R R R R
Figure z.ii. Best Beginning of Cycle Core Loading Pattern Obtained by Simulated Annealing
57
;3 32 13 14 3 R
CAD GAD AD None
41 000 4.1000 41000 41000
0 0000 0 0000 19.0000~ 18.4000 0.0000
0.7275 09252 1.3074 1.3004
1.0158 1.0158 1i0906 1:0917 1.0958 1 2646
33 28 2 35 10 6 R
GAD GAD None GAD None None
4 1000 4 1000 4 1000 4.1000 4.1000 4.1000
0 0000 0.0000 0 0000 0.0000 12.6000 10.2000
0 7270 0 8463 1.3132 1 148 1.3417 1.3393
1.0158 1.01S8 1 281( 1 015 1.1521 1 1743
32 30 27 14 15 0 R
GAD GAD GAD .GAD GAD None
4.1000 41000 4.'000 4.1t0 4.1000 41000
0 0000 0.0000 0.0000 I19A00 1A00 0 0000
09320 0 7022 1.0596 1.137 1.280 1.3434
1 0158 1.0'58 1_018 1.090 110958 12946
19 31 7 11 29 R
GAEL GAD None None GAD
4 1000 4.1000 4.1000 4.1000
19.Q0000 0 0000 10.2000 12.6000 0.0000
1;os31 1.0262 1.3499 1.2547 0.7886
-
1.00 1.0158 1.1743 1.1521 1.0158
19 - 8 12 -+,_ 5 R R R
MD None GAD _-_ + None
4'G01 4.1000 4:1000 4.1000
1 11.3000 113000 0.0000
1 3401 1.3412 1.2157
10917 1.1641 1.1427 '' 1.846
14 13 9
GAD None None
A.1000 4100 4.1000 4.1000
18A000 16:8000 11.3000 0.0000
13074 1.3230 1.2634 1.2123
1.09$8 1.1092 1.1641 1 2846
None None
4.1000 4.1000
0.0000 0.0000
1.3004 1.2660
1.2846 1.2846
R R R R R R
R R R
58
32 31 12 29 10 4 R
GAD GAD GAD GAD None None
4.1000 4.1000 4.1000 4.1000 4.1000 4.1000
0.0000 0.0000 11.3000 0.0000 12.6000 0.0000
0.7197 0.9519 1.2632 1 2488 1.3613 1 3329
1-0158 1.0158 1.1427 1.0158 1.1521 1.2846
32 27 - 20 16 11 17 5 R
GAD GAD GAO GAO None GAD None
4.1000 4 1000 1000 4.1000: 4.1000 4.1000 4.1000
0.0000 0.0000 18:9000 12.6000 18.9000 0.0000
0.7197 0.8208 193 1.2487 1.3733 1.3G31. 1.3071
101,8 1 0158 1.084 1.0917 1.1521 1.0917 1.2846
31 i8 9 0 13 0 R R
GAD 0A None None GAD None
4.1000 4.1000 4.1000 41000 4.1000 41000
0.0000 19.0000 11.3000 11.3000 16.8000 j 00000
0.9519 1.0862 1.2912 1.3631 1.2941 1 3717
1,0158 1.0908 1.1641 1.1641 1.1092 1.2846
12 7 14 15 34 35 R R
GAD None 9_ GAD GAD GAD
4.1000 4.1000
OA- 4.1000 4.1000
,
11.3000 102000 0.0000 0.0000

1 2632 1.3669 1230 1.7647 0.9742 0. 7054
1.1427 1.1743 1.0958 1.0958 1.0158 1.0158
29 6 28 3 R R
GAD None GAD None
4.1000 4.1000 4 1000 4.1000
0.0000 10.2000 0.0000 0.0000
1.2488 1.3482 1.1978 1.0076
1.0158 1.1743 1.0158 1 2846
to 30 R
None GAD
4.1000 4.1000
12.6000 0.0000
1.3613 1,1396
1.1521 1.0158
4 33
1
None GAD
4.1000 4.1000
0.0000 0.0000
1.3329 0.7694
1. 2846 1.01 58
t i i a t- t
59
3 IMPLEMENTATION OF DEPLETION
3.1 MOTIVATION
Good loading patterns can be determined, to first order, by calculating the eigenvalue
of the reactor core at the beginning of cycle. From the reactor eigenvalue, one can
infer the excess reactivity that will be held down by the addition of soluble boron.
This is directly correlated with the amount of time that a given reactor core can be
maintained at criticality before it needs to be shut down. Additionally, if the power
peaking is above the threshold at the beginning of the cycle, the core loading design
is not acceptable.
However, basing a core loading pattern's perceived quality on its beginning of cycle
parameters can miss some important effects. Firstly, while the cycle length of a reac-
tor is directly related to the excess reactivity present at start-up, this is not a linear
relationship. Especially when comparing two reactors of extremely similar eigenvalue,
without following the reactor core through its full cycle, it is impossible to know if
the reactor with the slightly higher eigenvalue has a slightly higher or slightly lower
cycle length. Secondly, due to the presence of burnable poisons, a configuration that
has a relatively flat power distribution at the beginning of cycle may see a mid cycle
peak as those poisons are consumed, such as can be observed in the loading pattern
shown in Figures 3.1 through 3.4. A pattern with this feature is unacceptable due to
safety concerns, just as a reactor with high power peaking at the beginning of cycle is
unacceptable.
As such, the goal of this segment of the project was to implement a simple deple-
tion model within the poropy framework in order to be able to better evaluate the
quality of a core loading pattern. By following the reactor and, in small timesteps,
increasing the burnup in each fuel assembly, we can see at what point the eigenvalue
61
R
O.T622
L05 .18 I 7;IS~ 10 I Lb0 I 0.86 0269S
0.9896
35 30 $ 7 31 32 24 $ 1 -9 R
R R R R R R R R R
Figure 3.1. Beginning of Cycle Reactivity of Fuel Bundles in a Loading Pattern with a Mid-
Cycle Power Peak
of the reactor falls below 1, corresponding to the point where criticality is unable to
be maintained. This is a very simplified depletion model, and would not be sufficient
to obtain useful results in any commercial applications. However, for the purposes of
this optimization study, and for the purposes of the already simplified poropy model,
it is sufficient. This will allow optimization algorithms to determine the quality of a
loading pattern more accurately since power distribution throughout the entire cycle
will be known, as well as exact cycle life. Thus, patterns will not be accepted based on
beginning of cycle parameters that in a real core would be unacceptable.
62
R
18
I GAO GAO
4! 1000 4.1000
j M0 194000
R R R R R R R R R
Figure 3.2. Beginning of Cycle Power Peaking in a Loading Pattern with a Mid-Cycle Power
Peak
Depletion is fairly easy to implement conceptually. It simply involves figuring out
how the content of fuel and other elements in each assembly in the reactor changes
during the reactor cycle [17],[18]. From this, one can determine how the spacial cross-
sections within the reactor change over time, and thus where the reactions are taking
place. Since the physics model generates cross-section data from input parameters, all
that needs to be done is capture this information by changing the data for each fuel
assembly. This involves increasing the burnup in each assembly at each iteration. The
average assembly power is calculated by dividing the total reactor thermal power by
the number of assemblies within the reactor, then multiplying the average assembly
63
R
37.1000 34.0000
0.3640 0.1404
0.9562 0.9770
17 R
R R R R R R R R R
Figure 3.3. Mid Cycle Reactivity of Fuel Bundles in a Loading Pattern with a Mid-Cycle Power
Peak
power by the power peaking factor in that assembly. The power in an assembly is then
multiplied by the timestep of interest (in days) and divided by the mass of fuel in the
assembly. This gives the amount the burnup in the assembly must be increased (in
the normal burnup units of MWd/kg heavy metal [19]) due to the depletion step. The
smaller the depletion steps, the more accurate this burnup increment will be.
64
39 11 5 32 3- 35 R
GAD GAD GAD None GAD GAD
4.1000 4.1000 4.1000 4.1000 4.1000 4 1000
43.0600 10.4100 11.7800 30.0400 7.1000
2.1205 2.7639
340007
2.5177 0.7099 0.640
0.9188 1.1382 1.1435 1.0085 0.9562 0.9 /77
11 7 9 34 0 17 R
GAD GAD None None None GAD
4.1000 4.1000 4.1000 4.1000 4.1000 4.1 10C
10.4100 11.5000 13.8200 34,0100 5.0300 20.8000
2.7639 2.7832 2.4891 0.6380 0.4297 0.1460
1.1382 1.1432 1.1411 0.9801 1.2240 1.0761
5 25 6 27 20 R
GAD None GAD GAD GAD
4.1000 4.1000 4.1000
4.1000 4 1000
11.7800 24.5900 11.6200
24.7700 21.7700
2.5177 2.1683 1.9508 - I 0 5659 0.2785
1.1435 1.0501 1.1433 1.0446 1.06B3
23 8 2G 35 R
None GAD lft0Oni3
None GAD GAD
4.1000 4.1000 4.1000 1000
23.7900
Clow 4 a 10 n
11.2900 zi-voo 19.7500 s=00
'
1,8071 1.7757 0400
0.7712 0.422
1.0564 1,1426 1.08%5 1,0494
1 4 J
1s 2 31 R
None None None
4,1000 4.1000 4.1000
20.3500 7.9300 27.0600
0.7145 0.5649 0.2 304
1.0845 1.1958 1 0309
32 29 2 33
None None. GAD GAD None
4.1000 4.1Qq 4.1000 1.1000 4.1000
30.0400 10;410 26.0400 5.1000 30.9400
0.7099 0.84$7 0.5676 0.4088 0.2194
1.0085 1.1761 1.0348 10674 1.0019
37 1 24 28 R R R R R
GAD None GAD None
41000 4.1000 4.1000 4.1000
37.1000 6.6600 23.5100 26.0000
0.3640 0,4985 0.2955 01667
0.9562 12080 1.0544 1.0390
35 16 R R r R R R R
GAD GAD
4.1000 4.1000
34.0000 21.2200
0.1404 0.1626
0.9770 1.0727
N. R R R R R R R R
Figure 3.4. Mid Cycle Power Peaking in a Loading Pattern with a Mid-Cycle Power Peak
3.2 RESULTS
Depletion was successfully implemented in the poropy framework, and can currently
be used for the Simulated Annealing and the Greedy Exhaustive Dual Binary Swap
(both in the greedy and the exhaustive implementation).
It was observed that core loading configurations do exist, for this Yamamoto bench-
mark, that do not have fresh fuel bundles adjacent to the reflector and still have power
peaking throughout the cycle that is less than 1.4.
65
An example of this high calibre of solution is a loading pattern configuration gener-
ated by the Exhaustive Dual Binary Swap algorithm. The depletion tool predicts that
the reactor will become subcritical upon reaching reactor burnup of 13.50 MWd/kg,
with the maximum power peaking at any point during the cycle being 1.374 (with the
imposed constraint being 1.375). The progression of reactor eigenvalue and maximum
power peaking can be seen in Table 3.1. The Beginning of Cycle reactivity and power
peaking can be seen in Figures 3.5 and 3.6.
Table 3.1. Eigenvalue and Peaking Results for a High Quality Loading Pattern Throughout
the Cycle
Depletion Core Burnup Max Power
Step (MWd/kg) Eigenvalue Peaking
B.O.C 0.0 1.05794 1.3645
1 0.064 1.04863 1.3528
2 0.191 1.04407 1.3700
3 0.445 1.04300 1.3629
4 0.954 1.04078 1.3495

5 1.972 1.03629 1.3247
6 3.617 1.02989 1.3082
7 5.262 1.02550 1.3082

8 6.907 1.02309 1.3068
9 8.552 1.02115 1.3281

10 10.197 1.01705 1.3612
11 11.842 1.00852 1.3732
12 13.487 0.99671 1.3742
Further figures for this loading pattern at depletion points 2, 5, 9, and 12 are in-
cluded in the appendix.
3.2.1 Effect of Times tep Method
There are typically two conflicting goals when carrying out a computational analysis,
including the depletion analysis performed here. These goals are speed and accuracy.
On the one hand, in order to compare the quality of different loading patterns, it is
66
R
31 1
CAD I4onel 2s R
GAD GAD GAD I Ij GAD None
4"No ~~ 1.1000a 4.1000 .4.10M,0 Nei .lr= 41tfV
41A 400
34.7000
O.SI36 0.3946
0.9722
R R R R R R R
Figure 3-5. Beginning of Cycle Reactivity of Fuel Bundles in High Quality Reactor Loading
Pattern
necessary to be as accurate as possible in determining the amount of time that a given
configuration can be kept critical. However, allocating more computational resources
to each pattern reduces the number of possible patterns that can be explored.
In order to try and balance these conflicting goals, it was decided that each pattern
should go through approximately io - 12 depletion steps. Initially, a constant depletion
timestep was examined, the results of which are shown in Table 3.2 for the high quality
loading pattern generated by the exhaustive dual binary swap methodology.
Note that if the algorithm predicts that the core configuration will become subcrit-
ical on the next depletion step, it adjusts the timestep for the last point to get
the
67
R
R R R R R R R R R
Figure 3.6. Beginning of Cycle Power Peaking in High Quality Reactor Loading Pattern
eigenvalue as close to 1 as possible. As an illustrative example, if the depletion step
is 1.0 MWd/kg, and the algorithm predicts that the next depletion step will bring
the reactor eigenvalue to o.98, it might reduce the depletion step for that point to 0.3
MWd/kg so that the eigenvalue at the end of the depletion is instead in the o.9998 to
1.0002 range.
However, one of the major difficulties in depleting a wide variety of different core
loading patterns is that they have potentially very large differences in their depletion
times. Contrasting the results for the constant depletion step above with those for
depletion of the initial configuration of the Yamamoto benchmark (a configuration
with a much lower cycle length), as in Table 3.3.
68
Table 3.2. Predicted Depletion of Loading Pattern with Constant Depletion Steps for High
Quality Loading Pattern
Depletion Number of Predicted EOC Core
Step (MWd/kg) Depletion Steps Burnup (MWd/kg)
6.361 2 10.425
3.817 4 15.267
1.272 11 13.066
0.636 21 13.043
0.382 35 13.035
0.191 69 13.033
0.127 103 13.033
0.064 205 13.033
0.025 513 13.033
0.013 1025 13.033
It can be clearly seen that since the cores being examined can have vastly different
cycle lengths, a constant depletion step that works very well for one core may either
give a completely meaningless result or take unacceptably long for another core.
Additionally, as can be seen from Table 3.1, there is a significant drop in the eigen-
value of the core at the very beginning of depletion, which in some cases might result
in significant error if that detail is missed.
To combat that difficulty, the depletion scheme that was ultimately chosen was one
of a varying depletion step increment. A certain number of depletion steps are car-
ried out initially with an increasing increment, as seen in Table 3.1. These depletion
steps are: 0.064 MWd/kg, 0.127 MWd/kg, 0.254 MWd/kg, 0.509 MWd/kg, and I.o18
MWd/kg. At the end of this cycle, the algorithm guesses what core burnup will cor-
respond to the end of cycle life, based on a linear extrapolation from the reactivity
drop from 0.509 MWd/kg to 1.018 MWd/kg. It then divides this predicted depletion
increment by a certain number of extra depletion steps (set to 5 in the analysis carried
out later in this work) and uses this as a new constant depletion step, continuing until
true subcriticality of the core is reached. In the case of the high quailty loading pattern
of Table 3.1, this ended up taking 7 extra steps.
69
Table 3.3. Predicted Depletion of Loading Pattern with Constant Depletion Steps for Ya-
mamoto Starting Point Loading Pattern
Depletion Number of Predicted EOC Core
Step (MWd/kg) Depletion Steps Burnup (MWd/kg)
6.361 1 6.361
3.817 2 4.482
1.272 2 2.498
0.636 9 5.476
0.382 15 5.550
0.191 30 5.587
0.127 44 5.589
0.064 89 5.602
0.025 221 5.607
0.013 441 5.609
The end result of this was to yield a reasonably accurate depletion time for a variety
of pattern qualities in an acceptable number of steps (approximately 10 - 12). For the
starting configuration of the Yamamoto benchmark core, the varying timestep scheme
yielded a reactor burnup of 5.49 MWd/kg in 12 depletion steps. For the high quality
loading pattern from the dual binary swap algorithm, this yielded a reactor burnup
of 13.50 MWd/kg in 12 depletion steps.
3.2.2 Time Cost of Depletion
One additional parameter when doing a depletion analysis is the scaling of the com-
putation time with the number of depletion points one is examining, and, of particular
interest, how much time is taken relative to a simple beginning of cycle analysis. The
more expensive a depletion analysis is to carry out, the more motivation there is to
introduce heuristics that avoid depleting reactor cores if they are likely to be of low
quality or not of interest for some other reason. The results of this analysis are sum-
marized in Table 3.4.
It can be easily seen that depleting a reactor core can be exorbitantly expensive
computationally relative to a single beginning of cycle calculation. As the number
70
Table 3.4. Scaling of Computational Time With Number of Depletion Points Computed
Depletion Time Time Rel. Time Rel.
Points (s) to B.O.C. to 1 Dep. Point
B.O.C 4.8161e-05 1.0
-
1 0.0031979 67.0 1.0
2 0.0047441 100.0 1.50
5 0.0095131 200.0 3.00

7 0.012745 265.0 4.00
10 0.017674 365.0 5.50
15 0.025605 530.0 8.o
40 0.068870 1430.0 21.5
of depletion points increases, this seems to level off to approximately 35x the com-
putational time per reactor evaluation, making heuristics that reduce the number of
patterns that need to be depleted extremely appealing.
The exact reason for this computational time increase is not entirely clear. It is likely
due to some combination of the need to generate new group cross-sections for each
assembly after each depletion point (since the burnups increase each time), and the
inherent speed penalty that is faced because of the use of an interpreted language like
python for the manipulation of the reactor core during the depletion process.
3.2.3 Heuristics in the GEDBS Method
As can be seen from the above analysis, carrying out a depletion study is extremely
expensive computationally. For the dual binary swap algorithms that search through
many patterns that are going to be discarded, it is completely infeasible to deplete
every core that is examined. Carrying out a full exhaustive dual binary binary swap
while only looking at beginning of cycle quality can take on the order of 1 or 2 hours.
With the scaling discussed above, carrying out a full depletion analysis would likely
take multiple days (it is uncertain what fraction of the B.O.C. analysis time comes
from pattern evaluations vs. what fraction comes from Python pattern manipulation).
This would make dual binary swap analyses impractical.
71
As such, heuristics have been introduced into the Dual Binary Swap algorithm to
dramatically reduce the computational time. Since it is known that many of the swaps
that will be tested results in an extremely poor quality core that will be discarded, not
all patterns have a full depletion analysis carried out on them.
All reactor patterns that are examined have a beginning of cycle analysis carried out
on them. The results of this analysis for the best core determined thus far are stored
along with the results of its depletion analysis. When any new pattern is generated, a
beginning of cycle analysis is carried out on it first. If the beginning of cycle objective
function of the new pattern is better than or worse within a certain tolerance than the
beginning of cycle objective function of the best reactor, a full depletion analysis is
performed. If it is not, then the pattern is discarded and a new pattern is generated.
The analyses discussed in the comparison of methods section utilized a threshold of
20% worse.
3.3 FUTURE DEPLETION WORK
There are a number of further improvements that could be made in the future that
would improve the utility of optimization methods within the depletion framework in
poropy.
One method that could potentially improve the accuracy of depletion times within,
for example, the Dual Binary Swap algorithm, would be to utilise a "semi-constant"
timestep for depletion. In this methodology, the computed depletion time for the best
reactor found at any given point in the simulation would be divided by a given num-
ber of points, say 15. This would be taken to be the constant timestep for depletions
until a new best pattern was found. By this methodology, reactors that have low cycle
length (and thus likely to be of low quality and discarded) would not have as much
computational resources spent on their evaluation. Conversely, reactors with a long
cycle length (and thus likely to be of high quality and accepted) would have more
computational resources spent on their evaluation, and hence be of higher accuracy.
72
One potential difficulty of this timestep method would be that it might result in few
depletion steps being carried out on a potentially good pattern in the early parts of
the simulation where quality of a solution is dominated by power peaking, and not
cycle length. This method would likely also not be as effective for stochastic methods
like simulated annealing, since those methods require by necessity the ability to accept
worse patterns.
A second improvement that could be made to the optimizations while including
depletion would be to include heuristics within Simulated Annealing. Currently all
patterns generated in Simulated Annealing are depleted due to the fact that the al-
gorithm needs to be able to accept worse patterns, especially at higher temperatures.
However, at low temperatures, only swaps that improve the pattern are accepted. As
such, one could potentially reduce the fraction of cores undergoing full depletion by
carrying out a beginning of cycle analysis on a given core. Then, the algorithm could
check if, from a beginning of cycle perspective, a core that was a certain amount worse
than the pattern currently being examined would be accepted or rejected. If it would
be accepted, the algorithm would carry out a full depletion analysis to be certain. If
it would be rejected, the likelihood that it would be accepted based on a full deple-
tion analysis would be very low (provided the threshold was tolerant enough), so the
algorithm would move on to attempting another swap.
73
4 DUAL BINARY SWAP WITH DEPLETION
4.1 INTRODUCTION
The last component of this project involved a comparison of the deterministic and
stochastic optimization schemes via a full depletion analysis. Due to the fact that the
depletion time is directly correlated with the economics of a reactor loading pattern,
a full depletion analysis can much more accurately determine which loading patterns
would be of the most use in practical applications than could the simple beginning of
cycle analysis carried out earlier in this work. It is for this reason that a full depletion
analysis is necessary to be able to state with any confidence whether the deterministic
Dual Binary Swap algorithm is capable of finding loading patterns that are of higher
quality than those produced by stochastic methods such as Simulated Annealing.
It is worth noting that only a single cycle depletion was carried out on any given
reactor pattern. In true commercial applications, where a pattern chosen by a team
of engineers is being evaluated as a potential candidate for refuelling, a multi-cycle
depletion analysis is carried out to determine the effects the loading pattern will have
on partially-burned fuel that will be retained for future cycles. This was not, however,
feasible in the analysis being carried out here due both to the limited information
present in the model being used about the end state of the previous cycle, and due
to the significantly increased computational requirements that would be involved in
carrying out a multi-cycle depletion. The information that would be gained by moving
to a multi-cycle depletion model for this optimization problem would not be sufficient
to overcome the increased computational cost.
This section will focus primarily on the deterministic method of the Dual Binary
Swap. It is worth noting that, for the depletion analysis, Single Binary Swaps and
Triplet Swaps (as discussed in the beginning of cycle analysis) were not explicitly ex-
75
amined due to their being subsets of the Dual Binary Swap within the optimizer, and
the fact that the patterns located by the subset with the beginning of cycle analysis
were of markedly lower quality than those found by the full Dual Binary Swap analy-
sis. The objective function used in these optimizations was of the form shown below.
objective = Burnupfactor * (BurnupEOC - Burnupbaseline)
- CycleMaxPfactor * max(O,CycleMaxPreactor- CycleMaxPbaseline)
- FreshAdjacentReflectorfactor * FreshAdjacentRefletornumber (4.1)
4.2 EXHAUSTIVE DUAL BINARY SWAP ANALYSIS
A Dual Binary Swap analysis was carried out from a number of starting patterns
(the original configuration listed in the Yamamoto benchmark, as shown in Figure
4.2 (recalling that the data presented for each assembly is organized as structured in
Figure 4. 1), as well as a number of randomized starting points, as shown in Figures 4.3
through 4.7) using the exhaustive implementation of the dual binary swap algorithm.
Figures coloured according to the beginning of cycle assembly power peaking and
assembly ki7 f are given in the appendix.

the Assembly
Burnable Poison in the

Assembly
Assembly Enrichment
Assembly Bumup

k M of the Assembly
76
- u 34 , 410 11 R
- - GAD None None
4,i J 4.1000 4.1000 4.1000
16 0.0000 12.6000 12.6000
.9S5 1.1799 1.0128 0.6552
1.h ! 1 0158 1.1521 1..521
33' 19 14 31, 5 R
4AD - GAD GAD GAD None
44000 41000 4.1000 4.1000
000000 0.0000
1,0504 0.9004
1.0158 - 1.o15e
f, _29 7 R R
GAD None
4.1000 4.1000
0.00.000 10000
1CODO
-1.0461 0.8060
81,01
1 1.1743
35 30 9 1 R R
GAD None None
4.1000 4.1000 4.1000
00.0000 113000 0.0000
1.1059 1.1308 07507
1 18 1,015 1.1641 1.2845
35 12 . 2 R R R
GAD GAD None
4.1000 4.1000 4 100
0,0000 11.3000 0 0000
1.1OS9 1.1539 0.8834
1 01,18 1.1427 1 2846
i F r r
10 28 6 3
None CAD None None
4.1000 4.1000 4.1000 4,1000
12.6000 0.0000 10.2000 0.0000
1,0128 0.9004 0.8060 0 7507
1.1521 1.01i0 1.1743 1.2846
11 0 k k R R R R R
None None
4.1000 4 1000
12.6000 0.0000
0,6552 05919
1 1521 1 2646
R R R R R R R R
Figure 4.2. Initial Configuration as Specified by Yamamoto
Optimizations were carried out with imposed maximum power peakings ranging
from 1.275 to 1.400. It is worth remarking once more that due to the presence of
burnable poisons keeping the reactivity of many fresh fuel assemblies depressed at
the beginning of the reactor cycle, some patterns that appeared to be of very high
quality in a beginning of cycle analysis have very large mid-cycle power peaking. This
significant reduction in the quantity of perceived high quality patterns in the search
space will likely reduce the frequency at which optimization algorithms are able to
find patterns that satisfy all constraints.
The results of the optimization for a maximum power peaking threshold across
the entire cycle of 1.275 are shown in Tables 4.1 and 4.2. Each run consisting of an
77
35 R
GAD
4,1000
0,0000
1.1160
1.0158
35 6
GAD None None
4.1000 4.1000 4.1000
0 0000 10.2000 -1$:16 1 12.6000
1 0C 1,2848 0.98
-
1.0158 1.1743
H R R
r.Aft, C
11.3000
1.3411
1.1427
i I
4 1 J R
None None GAD /0n
'wnl
{
4.1000 4 1000 4.1000

0 000 0u0000 0.000 ism
16695 1 662 0 7028 d.Sirb
1 2846 1 '84, 1 0<59 'FAT
_ 1 ~' 0 R R R R R R R
Nonre
R R R R R R R R R
Figure 4.3. Initial Configuration of First Randomized Core
optimization from a different starting configuration. It can be seen that such a low
imposed power peaking could not be satisfied by the algorithm.
Similarly, the results of the exhaustive dual binary swap optimization for imposed
power peakings of 1.300, 1.315, 1-325, 1-350, 1.375, and 1.400 are shown below in Tables
4.3 through 4.8. A table of evaluations carried out during the optimization is also
provided for an imposed power peaking constraint of 1.400 in Table 4.9. It can be seen
that due to the deterministic nature of the algorithm and the use of the same starting
configuration for a given numbered run in the tables, changing the imposed power
peaking did not always result in a different final pattern obtained.
78
34 12 33 20 3 4 R
GAD GAD GAD GAD None None
4U* 4.1000 4.1000 4 1000 4.1Q00 4.1000 4.1000
1:95000.
A.
0 0000 11.3000 0.0000 0.0000 0.0000
0.9441 1.1685 .2477
19:00 1.5761 09212
.98W 10159 1,1427 10150 12846 1 0046
19 17 R
GAD GAP GAD None CAD
4.1000 4 101G 4.1000 1 000
0.0000 19.0000 i0000
0.9319 1.3254 t 6315
1.0158 1.0908 12846
34 2 29 R R
GAD None GAD GAD
4.1000 4 1000 +:1yOQ 4.1000 4.1000
0.0000 0 0000 18,4000 0.0000 0.0000
0.9441 1.5773 1.27195 1.0377 0.5296
1.0158 1.2846 1.05 58 1.0158 1.0158
12 30 R
GAD GAD
4.1000 4.1000
11.3000 00000
1.1685 0.3525
1.1427
7 - 1.0158
33 35 0 R R R
GAD GAD None GAID GAD
4.1000 4.1000 4.1000 439Q 4.1000
OQ 0.0000 0.0000 0.0000 CAD- 0.0000
1 2477 1,2934 1 6346 0.6328
1-0158 1.0158 1,2846 1.0158
20 6 9 7
GAD - None None None
4000 4.1000 4.1000 4.1000
195000 . 102000 113000 10.2000
1.5090 14792 1.2459 0.3623
1.0867 1-1743 1.1641 1.1743
3 8 27 1u R R R R
None None GAD None
4.1000 4.1000 4.1000 4.1000
00000 11,3000 0.0000 12.6000
1.5761 12122 0.6707 0.3618
1.284. 1.1641
4 13 R R R R R R R
None-
4.1000 . 41 010
0.0000 10.0000'
09212 OW43.
1.2846 1.D
F S R R R R R R R
Figure 4.4. Initial Configuration of Second Randomized Core
Table 4.1. Results of Full Depletion Optimization with the Exhaustive Dual Binary
Swap,
Peaking of 1.275
Run Computation EOC Burnup Maximum Cycle BOC BOC Power
Number Time (s) (MWd/kg) Power Peaking Eigenvalue Peaking
1 3645 12.586 1.362 1.05184 1.362
2 6071 13.017 1.413 1.06115 1.412
3 4824 13.327 1.422 1.06152 1.419
4 8660 13.856 1.563 1.07993 1.563
5 3699 13.022 1.331 1.05589 1.319
6 4352 13.658 1.419 1.06195 1.418
79
34 8 _. 3 - 33 R
GAD None None y GAD
4,1000 4.1000 4.1000 4.1000
0:0000 11.3000 0.0000 0 0000
1,3226 1,61x7 1.3937 0.3375
1, 0158 1.1641 1.2846 1. _ 1.0158
34 10 1 11 ' R
GAD n AD None i
3 1000 _ 4.1000 - 4.1000 4.1000
np rW ;g00$ 12Z 6000 18,4000 1103000 12.6000
1,344 1.555 3 1.3458 129 7 .6247
S.0563 1.1521 1.0958 1.1,Q7 1.1521
87 9 30 R R
None None None GAD
4.1000 4.1000 461000 4.1000
11.3000 10,2000 11,3000 0.0000
1.6137 1,5590 1,1862 0.7296
1.1641 1,1743 1.1641 1.0159
-52 6 153 R R
GAD None CAD GAD
4.1000 4.1000 4C10 4 1000
0.0000 10.2000 19A0M 0 0000
1.2208 1.3111 1.0670 0.6343
1 0155 1.1743 1.0958 1.0158
20 13 17 16 R R R
GAD. GAD GAO GAa TOAD None
4.10>0s1 43. 000 4.1000 4.1000
ftftd 0 0000 14,4m4 0.0000
1.1279 119114 -- 0.5886
T 10158 1.1092 12846
3 29 '8 4 1 R R R R
None GAD GAD None None
4.1000 41000 41000 4.1000 4.1000
0.0000 0.0000 0.0000 0.0000 00000
1393'" 00134 10201 14026 0.8935
1246 10158 10158 1.284e 12846
iH {7 R R R R R
GAD GAD A e Nne
4.1000 -: !' 4, (.00
19;OIgG 0.000u
WI' 0.6599
1. 1
- 1.0158
33 0 R R R R R R R
GAO None
4.1000 4 1000
R R R R R R R R R
Figure 4.5. Initial Configuration of Third Randomized Core
Table 4.2. Number of Evaluations Carried out for Full Depletion Optimization with the Ex-
haustive Dual Binary Swap, Peaking of 1.275
Run Number of Full Number of BOC Fraction of Patterns
Number Depletion Evaluations Only Evaluations Depleted (%)
1 108,889 3,944,977 2.69
2 149,903 3,226,659 4.44
3 115,606 2,808,132 3.95
4 330,295 3,497,231 8.63
5 109,769 4,391,179 2.44
6 169,933 2,984,621 5.39
80
34 2-1-- 12 R
GAD None GAD
4.1000 4.1000 - 4.1000
0.0000 11.3000 113000
0.9903 12460 0A432
i.D158 ttY41 1.147
31 529
GAD None GAD
4 1000 4 1000 4.1000
I 0.0000 0.0000 O.0000
1.0237 7.4364 0.7693
1.0is8 1 2846 1,0158 jt
28 10 27 .R
GAD None GAD
4.1000 4.1000 4.1000
0 0000 12.6000 0.0000 --
1 0777 1.3314 1 +00,
.
34 33 15 6 9 7 13
GAD GAD GAD None None None GAD
3.1000 4.1000 4.1000 4.1000 4.1000 4 1000 4.1000
0.0000 0.0000 19,0000 10 2000 11.3000 10.2000 16.8000
0.9903 1 0501 1.2012 1.5311 1.S781 1.2049 0.4639
1.0158 1.0 58 1.0908 1 1743 1.1641 1.1743 1.1092
a
None
14 32 3 fI P. R
G/14 GAD None None
4.1000 4 1000 41000
11.3000 0.0000 0.0000 00000
1,2460
1.2382 1.5755 10350
1.1641 1.01 S 1,2846 1.2846
1 35 k R R R
None GAD
4.1000 4.1000
0.0000 0.0000
1.3880 0.6193
J
_
t 1.2846 1.0158
a Y,
0 30 R R R R
None GAD None
41000 4 1000 4.1000
J 0.0000 0.0000 0.0000
1 568 0.7832 0.5994
T, 1 aa6 1 0158 1.2846
12 16 k R R R R R R
GAD W-
4.1000 4.1 0
11.3000 1$9 00
R R R R R R R R R
Figure 4.6. Initial Configuration of Fourth Randomized Core
Table 4.3. Results of Full Depletion Optimization with the Exhaustive Dual Binary Swap,
Peaking of 1.300
1 3276 12.586 1.362 1.05184 1.362
2 3314 1.410 1.06129
13.933 1.408
3 3870 13.327 1.422 1.06152 1.419
4 9681 13.856 1.563 1.07993 1.563
5 2744 13.oo8 1.332 1.05565 1.320
6 3957 13.658 1.419 1.06195 1.418
e, 18 10 R
None GAD None
4.1000 4.1000 4.1000
,
10.2000 19.000 12.6000
1.4692 1.3294 0.3093
1.1743 1.0908 1.1521
33 11 32 31
CAD None CAD GgAD CAD
4 1000 4.1000 4 1000 4.1006 4.1000
0.0000 12.6000 0 0000 19406 (.0000
1.1395 1A4571 1.2348 1.0375 0 2428
1A159 1.1521 10158 1;1>18 1.0158
30 0 29 R
None GAD None GAD
4.1000 -9 4.1000 4.1000 4.1000
10.2000 0,0000 00000 0.0000
1.4692 1.21 27 1.2324 0.6084
1.1743 1.0, 58 1.2846 1.0158
28 4
CAD GAD GAD GAD ~ GAD None
4.1000 4.1000 4.1000 4,100 4.1000 4.1000
190000 11.3000 0 0000 16.6000 0.0000 0.0000
1.3294 1A4046 1 3214 1.3492 ? 08962 0.5415
1.090 1.1427 1.0158 1.1092 1.0158 1.2846
3 R R
None
-1.1000
00000
0.7370
121ihmWa
~i~ 2 I ~ R R R R
I I
0.5464
1.01S8
10 R R R R R R R
None
4.1000
1=.6000
_ 1
R RRj RRR
03093
1.1521 1 &l1i
Figure 4.7. Initial Configuration of Fifth Randomized Core
Peaking of 1.315
1 3052 12.586 1.362 1.05184 1.362
2 3218 13.933 1.410 1.06129 1.408
3 3980 13.327 1.422 1.06152 1.419
4 8507 13.856 1.563 1.07993 1.563
5 2515 12.941 1.360 1.05510 1.359

6 3756 13.658 1.419 1.06195 1.418
82
Peaking of 1.325
1 2835 12.586 1.362 1.05184 1.362
2 2238 13.933 1.410 1.06129 1.408
3 2735 13.327 1.422 1.06152 1.419
4 6975 13.856 1.563 1.07993 1.563

5 3376 12.948 1.328 1.05486 1.305
6 3710 13.658 1.419 1.06195 1.418
Peaking of 1.350
1 1853 12.514 1.367 1.05120 1.366
2 3089 13.933 1.410 1.06129 1.408
3 3445 13.327 1.422 1.06152 1.419
4 9459 13.856 1.563 1.07993 1.563

5 5189 13.425 1.349 1.05418 1.349
6 3582 13.658 1.419 1.06195 1.418
Peaking of 1.375
1 5434 13.487 1.374 1.05794 1.365
2 2371 13.933 1.410 1.06129 1.408
3 2793 13.327 1.422 1.06152 1.419
4 7741 13.856 1.563 1.07993 1.563

5 8645 13.571 1.372 1.05598 1.371
6 3141 13.658 1.419 1.06195 1.418
83
Peaking of 1.400
1 7341 13.590 1.399 1.05632 1.397
2 2270 13.422 1.410 1.o6o81 1.410
3 2665 13.327 1.422 1.06152 1.419
4 78oo 13.856 1.563 1.07993 1.563

5 5138 13.465 1.400 1.05776 1.400
6 3666 13.682 1.417 1.06215 1.417
High 7409 13.823 1.396 1.05891 1.396
Threshold
Table 4.9. Number of Evaluations Carried out for Full Depletion Optimization with the Ex-
haustive Dual Binary Swap, Peaking of 1.400
1 265,839 5,137,345 4.92
2 29,657 3,120,955 0.94
3 39,505 2,884,217 1.35
4 193,949 3,633,577 5.07

5 149,695 3,902,391 3.69
6 125,259 3,704,991 3.27
High 254,584 4,699,324 5.14
Threshold
84
From this data, it can be seen that at imposed power peaking constraints of 1.325 and
less, the exhaustive implementation of this algorithm failed to find loading patterns
that satisfied the imposed power peaking constraint throughout the entire life of the
core. At an imposed power peaking of 1.350, the optimization algorithm succeeded in

one of the six runs in finding a core loading pattern that satisfied the power peaking
constraint. Finally, in the runs with an imposed power peaking of 1.375 and 1.400, two
of the cases succeeded in satisfying the imposed power peaking constraints. These
successful patterns are shown below in Figures 4.8 through 4.12.
27 - -30 7F R
CAD GAD Nonte
-
4,1000 4Q 41000 4.1000
0.0000 - 10.2000
-
0.9157 " 1.1013 1,2119
1.0158 1.0958 - 1
10158 11743 1.0J1
3 33 .9 5 n
1 None GAD None 0A- None
4.1000 4,1000 4,1000 4.9'00 4.1000
0.0000 0.0000 11.3000 16.9000 0 0000
1.3493 1.0066 1.2319 1.2521 1.2866 4
1 2846 1.0158 1.1641 1.1092 12846 - 1007
27 34 16 14 y9 12 F R
GAD GAD GJD: CA GAD GAD
4.1000 4.1000 t-10 4.1000 4,1000
0.0000
0.9157
0.0000
09694
18 4A*l 00000 11.3000
1tQ9b; 2M 1,0151 0.7604
R158 10158 15.058 tom8 147
15 32 11 31 2 -R R
GAD GAD None 1' GAD None
4.1000 4.1000 4,1000 431"0 4.1000 4.1000
18A4000 0.0000 12.6000 193006d 0000C, 0 0000
Q.9972 0.9853 1.1779 1.152Y 1 :3?4 10547
10958 10158 1.1521 1Q0$ 1.015 1.2646
-y GAD None
4.1000 4.1000
1.-2019 L2623
30 28 6 u 10 R R R
GAD GAD None None None
4.1000
0.0000
4 21000 4.1000 4.1000 4.1000
0.0000 10.2000 0,0000 12.6000
1.101.3 1.1325 1.2974 1.3147 0.7589
1.0158 1.0158 1,1743 1.2846 1.1521
7 4 8 R R R R R
None None None
4.1000 4.1000 4,1000
10.2000 0.0000 113000
1.2119 1.2587 0.8770
1.1743 1.2846 1.1641 - ---
17W "I R R R R R
R R R R R R R R R
Figure 4.8. Core Loading Pattern Obtained by Dual Binary Swap with End of Cycle Depletion
13.425 MWd/kg and Maximum Cycle Power Peaking of 1.349
85
29 13 3 R
GAD GAD None
4.1000 4,1000 4.1000
0.0000 16000 0.0000
1.0872 1.2383 1.3645
1.0108 T1092 1.2846
15 7 30 R
GAP None CAD
4:1000 4.1000 1. 1000
18A40g 10.2000 0.0000
1.1506 1.1992 4.8550
1.0958 1.1743 1.0158
0 33 32 8 3:> 9 R
None CAD CAD None CAD None
4.1000 4,1000 4.1000 4.1000 4.1000 4.1000
0.0000 0.0000 0.0000 11.3000 0.0000 11.3000
1 3163 1.0437 1.0024 1.2465 10242 0.6436
1 2046 1 0158 1.0159 1.1641 10158 1.1641
SZ 16 34 4 I R | R
CAD None
4.1000 4.1000
0.0000 0.0000 1
1.2095 1.1428
3 0l [T 1.01 58 1284A6
L ,
6 R
29 2 F.
GAD None None

4.1000 4.1000 - 4.1000
0.0000 10:2000 0.0000
1.0872 1.3129 1 1,3641
I 0158 1.1743 1b958 12846
13 27 2 i 10 R R
CAD CAD CAD None None
4.1000 4 1000 4. 000 4.1000 4.1000
16.8000 0 0000 0 0000 0.0000 12,6000
1.2383 1.,109 1.0697 12607 0.7683
1.1092 1.058 1.0158 1 2846 1.1521
3 5 12 R R R R R
None None GAD
4 1000 4.1000 4.1000
00000 0.0000 11.3000
1 3645 1.2720 0.7681
1.2846 1.2846 1.1427
Iit R
4 4 I
R R
I ________ 4 ________ .1________ 1
Figure 4.9. Core Loading Pattern Obtained by Dual Binary Swap with End of Cycle Depletion
13.487 MWd/kg and Maximum Cycle Power Peaking of 1.374
While most of the patterns obtained by the optimizer were not impacted by relaxing
the threshold for which patterns would be rejected after a beginning of cycle analy-
sis, starting from the configuration specified in the Yamamoto benchmark model, a
different path was taken when a very loose threshold was used (carrying out a full de-
pletion analysis on any patterns that were up to twice as bad for beginning of cycle as
the current best pattern), a better pattern was obtained. Due to the path taken by this
optimization, 254,584 patterns underwent a full depletion analysis, while 4,699,324
patterns only had a beginning of cycle analysis performed on them. The parameters
of the optimized pattern (shown in Figure 4.13) are:
86
2 29 7 R
- None GAD None
4.1000 4.1000 4.1000
0.0000 0.0000 10.2000
1.3648 1.1018 1.1856
1 2846 1 0158 1.1743
33 19 13 8
GAD CAD GAD GAD GAD None Jcne R
4.1000 4 1006 4,1000 4.1000 4.1000 4 1000 4 04
19.5000 0 0000 t9A1@0a 0.0000 16.8000 11 3000 00000
0.9651 1,0133 1.2079 1.3312 1 262
.
1G 1.0158 1.0158 1.1092 1.1641 1 2246
2 32 9 35 10 R R
None GAD None GAD None
4. 1000 4.1000 4.1000 4.1000 4,T000
0 0000 0.0000 113000 0.0000 02.6000
1 3648 0.9673 1.2555 1.1086 0.040
1.2846 1.0158 1.1641 1.0158 .152.1
31 12 30 4
GAD GAD CAD Nlone
4.1000 4,1000 4. 00) 4. 1000
0.0000 11.3000 0.0000 0.0000
0.9006 1,0718 1847 1.1829
1.0158 1.1427 1.0158 1.2846
34 10 3 R R
GAD None None
4.1000 4.1000 4.1000
0.0000 12.600o 0.0000
4,8A0W
1,086 1.3552 1.3709 -1 A
b9tS9
10158 1.1521 1 2846
29 6 1 16 R8R R R
-
GAD GAD None None GAD

4.1000 4.1000 4.1000 4.1000
0.0000 0.0000 10.2000 0.0000 $10
1.1018 10505 1.1960 1.3273
10158
f
1.0158
I
1.1743 1.2846 1.dlt
y
7 0
None None
4.1000 4.1000
10.2000 0.0000
1.1856 1,1606 b
1.1743 1.2846
I
4 1 1 1 L 1
R R
Figure 4.10. Core Loading Pattern Obtained by Dual Binary Swap with End of Cycle Deple-
tion 13.571 MWd/kg and Maximum Cycle Power Peaking of 1.372
End of Cycle Burnup = 13.823 MWd/kg
Maximum Cycle Power Peaking = 1.396
Beginning of Cycle Eigenvalue = 1.05891
Beginning of Cycle Power Peaking = 1.396
It is worth noting from these results, that the Exhaustive Dual Binary Swap algo-
rithm is not robust enough to find a high quality loading pattern from any starting
point. The initial pattern given to the optimizer strongly influences the results ob-
tained. This is not unexpected, however, as a similar effect was observed in the be-
87
33 13 32 R
GAD GAD GAD
4.1000
0.0000 1A3000-
4 1000
0.0000 i
.
0 96x4 1;1379 09793
1.015 3 1,1092 1,0917
29 8 17 R
GAD None GAD
4. 10U0 4.1000. .1Uil AInnn
I " - " ic
Ucao
- .I- -
.
108
01@.1r 1 284 r
6 12 R R
None GAD
411W@ 4.1000 4.1000
iQboo 10.2000 113000
t 1007w 13540 0.8689
1.1743 1 1d7
35 0 R R
GAD GAD None
4.1000 4.1000 4.1000
0.0000 0.0000 0.0000
1.0552 1.141% 1.1939 1.2917
1.0158 1.4958 1.0158 1.2846
11 20 5 P, R R
GAD GiAD None GAD None
1000
C0000
C 9645
4.IWQ
1.4000
1A83
4.1000
12.6000
1.2742
4 1000
0.0000
12187
4.100
0.0000
122942
':1
10759 1.4958 1.1521 10158 1.2644
13 31 7 19 R
GAD CAd None None GAD
4.1000 4 1000 4.1000 4 1000 43000
16.8000 0 0000 10.2000 00000
1.1379 1.1263 1.2981 1.2889
't9 000
0.9.8
1.1092 1.0158 1.1743 1 7911A
32 2 9
GAD None None
4. 1000 4.1000 4.1000
0.0000 0.0000 11.3000
0.9793 1.1414 0.8377
1.0158 1.2846 1.1641
R R
I I I
I 1A
R R
ginning of cycle analysis, and greedy deterministic algorithms tend to struggle with
problems that have diverse high quality solutions.
88
34 s 8 33 3 R
GAD GAD None GAD None
4.1000 41 00 4.1000 4 1000 4 1000
0.0000 184000 11.3000 0 0000 0.0000
0.9510 1.0489 1.2307 1 2035 1 3975
1.0158 1.0958 1.1641 10153
I2846
3 F R
GAD GAD
4.1000 4 f000
0.0000 t
1.0420 iZfJI
'
1,0159 '1 1.10'i
14 0 12 R
GAD None -CiW j None GAD
4.1000 4.1000 4 i 4.1000 4.1000
0.0000 12.6000 T 4 1I4 0.0000 11.3000
09816 1.2248 1 7t 1 3998 0.7576
1.0156 1.1521 t095 1.2846 1.1427
6 27 35 0 R
None GAD GAD
4,1000 4.1000 4.1000
10.2000 0.0000 0.0000
1.2256 1.0793 0 835 >
1.1743 1.0158 10158
28 9 R
GAD Nine None
41000 4.1000 4 1000
0 0000 0.000" 11.3000
1.1645 1.1971 0.5904
1.0158 1.2846 1.1641
33 31 7 5 T ' R R R R
GAD CAD None None GAy
4.1000 4.1000 4.1000 4,1000
0.0000 0.0000 10.2000 0.0000 r
1.2035 1.1715 1.2625 1 2627
1.0158 1.0158 1.1743 1 2846
3 1 10 20 R R R R R
None None None GAD
4.1000 4.1000 4.1000 am
0.0000 0.0000 12A000 L-
1.3975 1.3634 0.8921
1.2846 1.2846 1.1521
R R R R R R R
R R R R R R R R R
89
1 34 4 R
None GAO None
1E(000
4.1000 4.1000 .1000
0.0000 0.0000 18.4000. 0.0000
1.3596 10451 1.3963
1 2846 1.0159 1'.0958 1.2846
D 33 1l 140 32 8 R
GAD GAD None
4.1000 41000 4,1008
17j; W, 0.0000 0 0000 11 3000
OAM 1.OS03 11542 1.1245 0 9355 0.9312
law 1.0158 1.0908 1.0958 1 01;8 1.1641
1 31 9 6 28 12 R R
None GAD None None GAD GAD
4 1000 4.1000 4.1000 4.1000 4 1000 4.1000
00000 0.0000 11.3000 10.2000 00000 11.3000
13596 0.9932 1.2838 1.2779 0.9475 U.038
12846 1.0158 1.1641 1.1743 1.0159 1;1427
.a
41 J 30 29 R R
GAD GAD None
4.100
0 4.1000 4.1000
i D.ooo0 0.0000 0.0000
1.Y3 gi , 09821 1.1989 1 0839
1.0158 1.01 19 1 284t
34 35 0 R R R
GAD GAD None
41000 4.1000
1.992 4.1000
0.0000 0.0000 0.0000
1.0451 1.0858 1.3769
1.015A 1 0'Se 1.2846
15 27 3 10 R R
GAD GAD None None None
4.;000 41000 4.1000 4 1003 4.1000
18.4000 0.0000 10.2000 0 0000 12.6000
;.15;18 1. t 517 1.2838 1 3918 0.8089
1Al 1.015& 1.1743 1,2846 1.1521
4 2 R
None None
4 1000 4.1000
0.0000 0.0000
1 3963 1.3580
1.2846 1.2846
R R R
R R R R R R R
II
90
4.3 GREEDY DUAL BINARY SWAP ANALYSIS
A Dual Binary Swap analysis was also carried out from a number of starting patterns
(using the same starting points as for the Exhaustive Dual Binary Swap analysis in
the preceding section) using the greedy implementation of the Dual Binary Swap
algorithm.
Optimizations were carried out with the same imposed maximum power peakings
ranging from 1.275 to 1.400 as in the exhaustive study. The results of the optimization
for a maximum power peaking threshold across the entire cycle of 1.275 are shown in
Tables 4.10 and 4.11. It can be seen that such a low imposed power peaking could not
be satisfied by the algorithm.
Table 4.10. Results of Full Depletion Optimization with the Greedy Dual Binary Swap, Peak-
ing of 1.275
1 4307 14.138 1.593 1.07871 1.593
2 2359 13.850 1.479 1.06320 1.479
3 1683 12.703 1.485 1.06482 1.485
4 2272 13.618 1.534 1.07602 1.534
5 1463 14.105 1.449 1.06596 1.449
6 3545 14.603 1.573 1.07192 1.573
Table 4.11. Number of Evaluations Carried out for Full Depletion Optimization with the
Greedy Dual Binary Swap, Peaking of 1.275
1 161,856 1,798,937 4.92
2 54,483 1,937,221 0.94
3 43,267 1,013,689 1.35
4 62,202 1,187,661 5.07
5 38,854 718,716 3.69
6 111,840 1,431,714 3.27
91
Similarly, the results of the greedy dual binary swap optimization for imposed
power peakings of 1.300, 1.315, 1-325, 1.350, 1.375, and 1.400 are shown below in Tables
4.12 through 4.17.
ing of 1.300
1 14.138 1.593 1.07871 1.593
4926
2 2197 13.850 1.479 1.06320 1.479
12.703 1.485 1.06482 1.485
3 1395
4 1991 13.618 1.534 1.07602 1.534
5 1277 14.105 1.449 1.06596 1.449
6 3350 14.603 1.573 1.07192 1.573
ing of 1.315
1 13.106 1.560 1.07439 1-56o
3277
2 2031 13.850 1.479 1.06320 1.479
1478 12.703 1.485 1.06482 1.485

3
2021 13.618 1.07602 1.534
4 1.534
1153 14.105 1.06596 1.449

5 1.449
6 3220 14.603 1.573 1.07192 1.573
The greedy implementation of the Dual Binary Swap was unable to find any pat-
terns that satisfied the imposed power peaking constraints, and even in the cases that
failed, the power peaking values obtained in each converged pattern were in most
cases higher than those obtained with the analogous Exhaustive Dual Binary Swap.
As was seen in the beginning of cycle analysis, the greedy implementation of the
algorithm does not perform as well as the exhaustive in a full depletion analysis.
92
ing of 1.325
1 2629 13.106 1.560 1.07439 1.56o
2 1982 13.850 1.479 1.06320 1.479
3 1345 12.703 1.485 1.06482 1.485
4 1796 13.618 1.534 1.07602 1.534

5 1152 14.105 1.449 1.06596 1.449
6 2706 14.603 1.573 1.07192 1.573
ing of 1.350
1 2966 13.106 1.560 1.07439 1.56o
2 1794 13.850 1.479 1.06320 1.479
3 1091 12.703 1.485 1.06482 1.485
4 1684 13.618 1.534 1.o7602 1.534

5 856 14.105 1.449 1.06596 1.449
6 2625 14.603 1.573 1.07192 1.573
ing of 1.375
1 2796 13.106 1.560 1.07439 1.560
2 1583 13.850 1.479 1.06320 1.479

3 1807 13.738 1.449 1.06734 1.448
4 1379 13.618 1.534 1.07602 1.534

5 1075 13.296 1.410 1.06237 1.410
6 2367 14.603 1.573 1.07192 1.573
93
ing of 1.400
1 2689 13.106 1.560 1.07439 1.560
2 2103 13.313 1.454 1.05961 1.454

3 1868 13.738 1.449 1.06734 1.448
4 1093 13.618 1.534 1.07602 1.534
5 1303 13.299 1.436 1.06446 1.434
6 2010 14.603 1-573 1.07192 1.573
94
5 SIMULATED ANNEALING WITH DEPLETION
5.1 INTRODUCTION
As mentioned in the previous section, the last component of this project involved a
comparison of the deterministic and stochastic optimization schemes via a full deple-
tion analysis. This is due to the fact that a full depletion analysis is necessary to be able
to state with any confidence whether the deterministic Dual Binary Swap algorithm
is capable of finding loading patterns that are of higher quality than those produced
by stochastic methods such as Simulated Annealing.
This section will focus on the stochastic method of Simulated Annealing. As with
the Dual Binary Swap optimization, only a single cycle depletion was carried out on
any given reactor pattern.
5.2 SIMULATED ANNEALING ANALYSIS
The Yamamoto benchmark was optimized by Simulated Annealing using the same
range of imposed power peaking constraints as in Dual Binary Swap optimization in
the previous section (1.275, 1.300, 1.315, 1.325, 1.350, 1.375, and 1.400).
A Simulated Annealing optimization analysis was carried out using purely single
binary assembly swaps. In this optimization, the cooling schedule was set such that
the "temperature" of the system at each state point was 0.995 that of the "temperature"
at the previous state point. The range of the temperatures throughout the optimization
was such that the was a significant period of random movement across the solution
space at the beginning and a significant period of only accepting swaps that improved
the best loading pattern at the end. The loading patterns generated by this optimizer
95
with an imposed power peaking of 1.275 are displayed in Table 5.1. In total, 27,563
patterns were examined and depleted during these optimizations.
Table 5.1. Results of Full Depletion Optimization with Simulated Annealing, Only Single
Binary Swaps of Assemblies, Peaking of 1.275, Cooling Parameter 0.995, 27,563
Patterns Evaluated per Run (Optimized Patterns Identical to Power Peaking Con-
straints Between 1.300 and 1.325)
1 689 12.711 1.586 1.06338 1.586
2 735 12.711 1.586 1.06338 1-586
3 684 13.966 1.519 1.06746 1.519
4 667 13.966 1-519 1.06746 1-519
5 661 13.939 1.535 1.06455 1.535
6 679 13.329 1.463 1.06261 1.462
Changes in the optimized reactor cores due to changes in the imposed power peak-
ing were mostly not present. However, a change in the patterns obtained was observed
when transitioning from a power peaking of 1.325 to 1.350. Whereas imposed power
peakings of 1.275, 1.300, 1.315, and 1.325 resulted in optimized patterns as listed in
Table 5.1, imposed power peakings of 1.350, 1.375, and 1.400 resulted in optimized
patterns as listed in Table 5.2.
Patterns Evaluated per Run (Optimized Patterns Identical to Power Peaking Con-
straints of 1.375 and 1.400)
1 660 13.229 1.454 1.05879 1.454
2 658 13.229 1.454 1.05879 1.454
3 673 13.749 1.566 1.06583 1-566
4 662 13.446 1.576 1.06772 1.576
5 646 12.377 1.600 1.07510 1.600
6 720 13.693 1.440 1.06482 1.437
The results of a Simulated Annealing optimization run with single binary swaps
only and a slower cooling speed of 0.9995 (although a shorter period of random walk
96
of the pattern at the beginning of the simulation, and a shorter period of purely
"greedy" searching at the end of the optimization) are shown below in Table 5.3. A
total of 64,458 patterns were depleted and evaluated.
Patterns Evaluated per Run
1 1669 14.102 1.409 1.06235 1.406
2 1705 14.102 1.409 1.06235 1.406
3 1712 12.890 1.437 1.06578 1.436
4 1725 12.640 1.423 1.06399 1.423
5 1491 12.640 1.424 1.06350 1.423
6 1643 12.929 1.383 1.05124 1.381
From Table 5.3, it can be seen that while none of the patterns come close to satis-
fying the imposed power peaking constraint of 1.275, one of the patterns does have
a maximum cycle power peaking less than the upper threshold in this study of 1.400,
and some of the other converged patterns have maximum cycle peaking very close
to this threshold. This suggests that single binary swap Simulated Annealing with
this cooling schedule is able to find patterns that satisfy an imposed power peaking
constraint of 1.400. The results of this study are listed in Table 5.4.
Patterns Evaluated per Run
1 1762 13.149 1.456 1.06083 1.456
2 1729 13.173 1.443 1.06295 1.441
3 1986 12.466 1.409 1.05361 1.409
4 1795 13.552 1.394 1.06059 1.386
5 1731 13.527 1.395 1.05950 1.395
6 1738 13.544 1.434 1.05961 1.433
97
From these results, we see that the single binary swap Simulated Annealing is able
to locate two patterns that satisfy an imposed power peaking constraint of 1.400. The
other four patterns obtained, however, did not satisfy the imposed power peaking
constraint. It is interesting to note that the path taken by the optimizer in the first two
runs was sufficiently different from the run with an imposed peaking constraint of
1.275 that the patterns obtained had both a worse cycle length and a worse maximum
cycle peaking.
Here, the stochastic Simulated Annealing optimizer struggles to consistently locate
patterns that satisfy the imposed power peaking constraint. It is worth noting, how-
ever, that due to the difficulty in applying heuristics to a methodology that requires
accepting patterns that are worse than the starting pattern, a Simulated Annealing
optimization will examine far fewer patterns in a given computational time than will
the deterministic Dual Binary Swap.
In addition to carrying out a Simulated Annealing analysis purely with single binary
assembly swaps, a Simulated Annealing optimization analysis was carried out using
purely dual binary assembly swaps. Below, in Table 5.5 are summarized the results
for the optimization with all movements of assembly positions that are attempted by
the Simulated Annealing optimizer being dual binary swaps, with a power peaking
of 1.275 acting as the constraint. This is a much larger potential search space than
simulated annealing optimizers that solely carry out single swaps. In this optimization,
the cooling schedule was set such that the "temperature" of the system at each state
point was 0.995 that of the "temperature" at the previous state point. The range of the
temperatures throughout the optimization was such that the was a significant period
of random movement across the solution space at the beginning and a significant
period of only accepting swaps that improved the best loading pattern at the end. In
total, 27,563 patterns were examined and depleted during these optimizations.
Unlike with the deterministic Dual Binary Swap optimization methodologies, how-
ever, there was very little impact due to changing the power peaking threshold on
the optimized pattern obtained (since, for reproducibility, the same random seed was
98
Table 5.5. Results of Full Depletion Optimization with Simulated Annealing, Only Dual Bi-
nary Swaps of Assemblies, Peaking of 1.275, Cooling Parameter 0.995, 27,563 Pat-
terns Evaluated per Run (Patterns Identical to Power Peaking Constraints from 1.300
to 1.400)
Run Computation EOC Burnup Maximum Cycle BOC BOC Pow er
1 727 12.485 1.477 1.05546 1.477
2 704 13.740 1.454 1.06802 1.454
3 705 13.740 1.454 1.06802 1.454
4 688 13.343 1.440 1.05613 1.439
5 675 13.089 1.411 1.05993 1.407
6 665 13.089 1.411 1.05993 1.407
used for a given run number). All of the higher power peaking threshold runs (1.300,
1.315, 1.325, 1.350, 1.375, and 1.400) arrived at the same optimized patterns as listed in
Table 5.5.
It can be quickly seen that the Simulated Annealing algorithm failed to find loading
patterns that satisfy the imposed power peaking constraints. As such, further optimiza-
tions were carried out that used a slower cooling speed (although a shorter period of
random walk of the pattern at the beginning of the simulation, and a shorter period
of purely "greedy" searching at the end of the optimization). These results are shown
below in Tables 5.6 through 5.8. For the cooling schedules of o.998, 0.999, and 0.9995, a
total of 16,103, 32,222, and 64,458 patterns were depleted and evaluated, respectively.
nary Swaps of Assemblies, Peaking of 1.275, Cooling Parameter o.998, 16,103 Pat-
to 1.400)
Run Computation EOC Burnup Maximum Cycle BOC BOC Pow er

1 449 13.423 1.607 1.06781 1.607
2 448 13.109 1.534 1.05967 1.534
3 431 13.749 1.482 1.06281 1.482
4 394 12.893 1.422 1.06658 1.421
5 431 12.893 1.422 1.o6658 1.421

6 461 13.757 1.471 1.06921 1.462
99
Table 5-7. Results of Full Depletion Optimization with Simulated Annealing, Only Dual Bi-
to 1.400)

1 914 12.744 1.540 1.05795 1.540
2 885 13.838 1.522 1.06418 1.522
3 866 13.122 1.497 1.06519 1.497
4 802 13.439 1.443 1.05845 1.443
5 896 13.439 1.443 1.05845 1.443

6 907 12.606 1.438 1.05774 1.438
terns Evaluated per Run
1 1748 13.464 1.452 1.06479 1.452
2 1737 13.464 1.452 1.06479 1.452
3 1814 13.867 1.404 1.o61o6 1.403
4 1665 13.414 1.385 1.06167 1.381
5 1663 13.414 1.385 1.06167 1.381
6 1697 12.614 1.414 1.05689 1.414
If the power peaking constraint for the cooling schedule of 0.9995 is instead set to
1.400, the results obtained are very similar to those listed in Table 5.8, although some
patterns are found that satisfy the imposed power peaking constraint. These results
are listed in Table 5.9.
The optimization of this reactor core with the Simulated Annealing algorithm does
not suggest that a major gain is made by reducing the cooling speed of the simu-
lation, unless the cooling speed is made extremely slow. Even with a much slower
cooling rate, the Simulated Annealing algorithm has difficulty consistently satisfying
the imposed power peaking constraints. This is likely due, in part, to the fact that the
nature of Simulated Annealing's periodic acceptance of patterns that are worse than
the starting pattern hampers the inclusion of heuristics that allow for some patterns
100
terns Evaluated per Run
1 1620 13.464 1.452 1.06479 1.452
2 1563 13.464 1.452 1.06479 1.452
3 1625 13.867 1.404 1.o61o6 1.403
4 1508 13.485 1.398 1.06225 1.389
5 1694 13.485 1.398 1.06225 1.389
6 1965 13.022 1.398 1.05914 1.398
to be passed over without carrying out a full depletion analysis. It should be noted
that the cooling speed of 0.9995 is still quicker than the cooling speed that was found
to function best for the beginning of cycle analysis (o.9999), with the number of per-
ceived high quality patterns present in the search space being significantly smaller in
the depletion analysis, due to the presence of mid-cycle power peaking.
101
5.3 EXHAUSTIVE DUAL BINARY SWAP OF SIMULATED ANNEALING PATTERNS
In order to help determine the ability of the deterministic Dual Binary Swap analysis
to locate patterns that are missed by stochastic methods, during the full depletion
analysis an Exhaustive Dual Binary Swap analysis was carried out on the converged
results of the Simulated Annealing optimizations that succeeded or came very close to
succeeding in satisfying the power peaking constraints. All of these converged results
were obtained from the Simulated Annealing optimizations with a cooling parameter
of 0.9995. These starting configurations are shown in Figures 5.2 through 5.9 (recalling
that the data in the assemblies is structured as shown in Figure 5.1).
Numerical identifier for

the Assembly
Burnable Poison in the

Assembly
Assembly Enrichment
Assembly Bumup

k1 n of the Assembly
102
4 34 33
None GAD
-
GAD None
41000 4.1000 j4
1 4 1000 4.1000
0.0000 0 0000 1 0.0000 0.0000
1.2907 1.1324 - r271 1.1914 1 3723
1.2846 1.0158 1.0158 1.2846
31 7 30
-
GAD None GAD-
4.1000 4.1000 4.1000 F
0.0000 10.2000 0.0000
0.9632 1.3043 1.1435
1.0158 1.1743 1.0159 iF
15 10 11 6 R
None GAD GAD None None None
4 1000 4 1000 4.1000 4.1000 000C 4.1000
0.0000 0.0000 18.4000 12.6000 20000 10.2000
1 2907 1.04 .4 1.0951 1.3447 1 3900 0.7575
1.2846 1.0158 1.1521 1.2846 1.1743
34 il 14 8 27 3S fR
GAD None None GAD GAD
4.1000 4.1000 4.1000 4.1000 4.1000
0.0000 12.6000 1.1.3000 0.0000 0.0000
1.1324 1.2773 1.2734 1.0844 07849
1.0158 1.1521 1.8958 1 1641 1.0158 1.0158
. 9 29 3 R R R
- - Nim GAD None
- 4,1000 4.1000 4.1000
1. 11.3000 00000 0.0000
1.3229 1.1808 1.1344
S 1.1641 1.0158 12846
0.3 13
GAD GAD' GAD None
4 1000 4.1000 4 1000
0.0000 16.000 0.0000 0.0000
S1914 1.2669 1 0891 1.1505
1 0 :4 1.1092 1.0158 1.2846
2 t2 7i R R R R
None None GAD GAD
4 1000 4.1000 4.1000 4.1,000
0.0000 0.0000 11.3000 18.9000
1.3723 1.3544 0.8022 0'513
1.2846 1.2846 1.1427 1.0914
R R R R R R R
R R R R R R R R R
Figure 5.2. Core Loading Pattern Obtained by Simulated Annealing with End of Cycle Deple-
103
10 29 9 R
None GAD None
4.1000 41000 4.100a-
12.6000 0 0000 11.300b
1.2708 1.0838 0.5632
1321 10158 1.641
34 5 R
GAD . None.
816' f
R R
0.9743 0.9 93 1.1559

1.0158 1015& 1.1521 1.1t743 1.1092
27 4 32 3 R R
GAD None GAD
4 100o 4.1000 4.1000
0.0000 0.0000 0.0000
1.0165 1.3795 1.0804
1 0158 1.2846 1.0158
r. ~ 1g 35 2
None
14
GAD
R R
G GAD
4.1000 4.1000 4 1000 4.1000
-18.400O 0.0000 0.0000 18A1000
r' 1:0773 1.1504 13247 0.7260
1.015 1 2946 11.05
su 28 1 R
None N40ne GAD None
4.1000 41000 4.1000 4.1000
1L6000 10.2000 0.0000 0.0000
i 1.2748 1.3809 1.1126 1.2215
1.1521 1.1743 1.0158 1.2846
29 0 12
I
CAD None None GAD
4.1000 4.1000 4.1000 4.1000
0.0000 0.0000 11.3000 113000
1.0838 1.3547 0.6744 0.6050
1.0158 1.2846 ,541 1,U27
9 16 R R R R R R R
None GAO
000 4.1000
11.3000 _
-
R R R R R R R R R
104
34 2 12 33 R
GAD None GAD CAD
4.1000 4.1000 4,1000 4.1000
0.0000 0.0000 11.3000 0.0000
0.9710 1.3971 1.2216 1.1620
1.0158 1.2846 1.1427 1.0158
32 9 10 R
GAD None None
4.1000 4.1000 4.1000
0.0000 11.3000 12000
1.1083 1.2583 1.1322
1.0158 1.1641 1.1521
I 31 28 R
None None CAD GAD
4.1000 4.1000 411000 4 1000
12.6000 10-2000 0 0000 0.0000
1.2338 1.2618 S1347 09218
1.1S21 1.1743 1.0159 1.0158
13 29 8 1 R
GAD GAD None None
4.1000 4.1000 4:1000 4.1000
16.$000 00000 11.3000 0 0000
1.1594 1174 1.3202 1 1099
1.1092 10' 58 1.1641 1.204S
GAD 100
20 R R R
GAG None GA
4.1000 4 1000 4.1000
11.3000 0.0000 0.0000
1.2216 ,.1515 1.2923
1.1427 1.0 158 1.2846
3 6 R R
GADGAD None None
4.1t000 4.1t000 4.1000 4.1000
0.0000 0.0000 0.0000 10.2000
1.1620 1.1875 1.4027 1.0410
1.0156 1.0158 1.2846 1.1743 1.ow-8
4
None
3
None
1 F R R R R
'
4.1000 4.1000 4.1000.

0.0000
1,3300
1.2846
0.0000
1.2953
1,2846 _____1
ItZ
R RR r t R tfRt Rl R
Figure 5.4. Core Loading Pattern Obtained by Simulated Annealing with End of Cycle
Deple-
105
1 34 33 4 R
-
None GAD GAD None
4.1000 4.1000 . 4.1000 4.1000 ' Aiv
-
0.0000 0.0000 00000 00000 ( 0
1.3131 1.1483 1. 1969 1.3886 1
-
1.2846 1.0158 1.0158 12846
32 7 30 R
GAD None GAD',
4.1000 4.1000 4.1000
0.0000 10.2000 0.0000

0.9695 1.3089 1.1366
1 010? 1.1743 1.01s8
-
9 0 12 R R
None GAD None None CAD
4.1000 4.1000 4.1000 4 1000 4.1000
0.0000 0.0000 11.3000 0.0000 11.3000
13131 1.0577 1.3532 1.3660 0.7164
1.2846 1.0158 1.1641 1.2846 1.42
34 8 15 07 35 R R
CAD None GAD None GAD GAD
. ?000 4.1000 4.1000 4.1000 4.1000 4.1000
0 000. 11.3000 18A4000 12.6000 0.0000 00000
1 .00 1,3067 1.2473 1.2346 10752 0.7867
11641 1.0958 1.1521 1.0158 1.0158
X17 14 6 29 3 R R
GAO None GAD None
41000 4.1000 4.1000 4.1000 4.1000 :1100w
18.9000 114000 10.2000 0.0000 0.0000 19L6l00
1.2351 1.3065 1.3359 1 1633 1.120 0.4976
1.0917 1.095 1.1743 1.0158 1.2846 1,0906
4 + + 4 1- _
33 13 31 5 R
GAD GAD GAD None
4 1000 4 1000 4.1000
0.0000
1004IE 0.0000 0.0000
1,1969 1f44 1.0832 1 1147
10158 1092 1.0150 11A. -.
4 2 11
None None None
4. 1000 4.1000 4.1000
0.0000 00000 12.6000
1.3886 13641 0.7954
1.2846 1.2846 1.1521
R
I
R R R R R R R R R
1o6
34 2 33 30 4 12 R
GAD None GAD GAD None CAD
4.1000 4.1000 4.1000 1 4.1000 4.1000 4.1000
0.0000
0.9770
0.0000 0 0000 0.0000 00000 11.3000
1 3_50 1068 r 1 1703 13976 0.6187
1.0158 12846 1158 t 1,0153 12646 1.1427
3.1 3; 13 29 R
GAD CAD GAD GAD
4 1000 4.1000 4;3000 41000
0.0000 0.0000 1T8000 0.0000
0.9770 1.0076 1.1476 0.8750
1.0158 1.059 1.1092 1.0158
2 0 7 R
None None None
4.1000 4.1000 4,1000
0.0000 0 0000 10.2000
13550 13931 0.7607
1 084d 1.0400 1 2846 1.1743
33 27 10 R R
.
GAD GAD None

4.1000 4.1000 4.1000
0.0000 0.0000 12000
1.0687 1.1084 0.9657
10158 1.0158 1.1521
9 26 8 5 1A R R R
None GAD None NJcre GAD
4.1000 4.1000 4 1000 41000
0 300
11.3000 0 000( 11.3000 0 0000
1.3127 1.2470 1.2977 12519
1.1641 1,015 1.1641 12846
30 6 R
GAD None
4 1000 4.1000
00000 10.2000
1 1763 0.6303
1.0158 1.1743
4 R R R R R
None None
4.1000 4.1000
0.0000 0.0000
1.3976 1.3573
12846 1.2846
12
GAD GAD
4 1000 4.1000
11 3000 18.4000
I-
R R R R R R R R R
107
28 1 30 4 R
GAD None I. GAD None
4.1000 4.1000 4.1000 4.1000
0.0000 0.0000 0.0000 0.0000
0.9986 1.3932 12103 1.3865
1.2846
'
1.0158 1.2846 1.09158' 1 0158
28 47 19 R
GAD GADGA| GAD None
4.1000 4.1000 4.100 4.1000 4.1000 1091
00000 IMOGO 463MO 0 0000 10.2000
0 9986 699$8 1;19$ 1 0829 1.3509
'
1.0158 1.095B 1,0937 1 0158 1,0908 1 1743

1 33 11 32 0 Za R
None GAD None GAD Non- GAD
4.1000 4.1000 4.1000 4.1000 4 !000 4.100
0.0000 0,0000 12.6000 0.0000 0.0000 19:500
1.3932 09574 1.1769 1.2535 1.4065 48755
1.2846 1.0158 1.1521 1.0100 1 2646
35 3 R R
GAD None
4.1000 4.1000
0.0000 0.0000
1 1173 1.4052
1.0158 1.2846
10 13 R R
.160 11540 10631 1.1978 0.9441

1.095 11427 1 0158 11521 1.1092
30 27 6 R
GAD GAD None None
4.1000 4.1000 4.1000 .x1004 +1100
0.0000 0 0000 10.2000 0 0000 19,0000
12103 1 1798 1.2427 12640 0.5779
1.01S8 10158 1.1743 12846 1.0908
4 2 9 8 R R R R
None None None None
4.1000 4.1000 4.1000 4.1000
0.0000 0.0000 11.3000 11.3000
1.3865 1.3464 0.9086 0.6330
1.2846 1 2.846 1.1641 1.1641
R
R
R
R
R
R
SR
R R R R R R R R R
Io8
11 12 R
None GAD Nlone
4 1000 4.1000 4 1000
12.6000 11.3000 0 0000
1.1819 1.2998 1.3947
1.1521 1.1427 1 2846
8 13 31 R
None GAD GAD GAD
4.1000 4.1o0 4.1000 41000
11.3000 16.8000 00000 0-0000
1.2658 1.1896 1.0828 0 8417
1-1641 1.1092 1.0158 1 0108
17 33 7 9 R
None GAD None None
4.1 000 4.1000 4.1000 4.1000 4.1000
0.0002 18.4000 0.0000 10.2000 11.3000
1.3310 1.1797 1.2266 0.7257
1.2846 1.0917 1.0150 1.1743 1.1641
110 30 6 15 3 l4 F R
None GAD None CA,_ None
.
4.1000 4.1000 4-1000 4.1000
12.6000 0.0000 10,2000 r 0.0000
1 1819 1.0864 1.3018 ' 1.3432
11521 1.0158 11743 1.058 1.2846
34 28 10 R R
GAD CA) None None
4.1000 1000 4.1000 4.1000
0.0000 0 0000 0.0000
1.0090 1.0818 1.2865
1.0158 1 01'8 1 284t'
12 27 1 R
GAD GAD None 'AD GAO
4 1000 4.1000 41000 ?l>
11.3000 0.0000 0.0000
1.2998 1.2052 1.3459
1.1427 1.0158 1.2846
5 R R R R
None Nn
4 '000 4,1000
0 0000 000
1.3947 132 1 . r
12846
R R R R R R
R R R R R R R R R
109
9 16 10 2 R
None GAD None None
4.1000 41000 4.1000 4.1000
11.3000 12,6000 0.0000
1.2588 L_ U4 1.3470 1.3855
1.1641 tea? 1.1521 1.2846
'
F
29 -1 11 13 32 33
GAD GAD None GAD GAD GAD
z 1000 41000 4.1000 4.1000 4.1000 4 1000
00000 12.6000 16.8000 0.0000 00000
0.00,00
0.8923 0.9922 1.2788 1.2173 1.0930 08392

1.0158 1.0158 1.1521 1.1092 10158 10158
R
1 35 6 8
None GAD None None
4.1000 4. 1000 4.1000 4.1000
0.0000 I 0 0000 10.2000 11.3000
1.3811 1.1637 1.2035 0.7120
1.2846 1.0159 1.1743 1.1641
7 1 14 R R
9 27
None GAD None CAD None

4.1000 4.1000 4.1000 4.1000 4.1000
11,3000 0.0000
1.1240
10,2000
1.2744 1I.10V0
1.2658 0.0000-
1.2905
-
1.2588
1.0158 1.1743 1.0917 1.2846
-
1.1641
1f; 34 30 3 12 R
GAD GAD None GAD

4.1000 4.1000 4.1000 4.1000
0.0000 0.0000 0.0000 11.3000
J 1.0524 1.0468 1.2164 0.7206
,(j ti t 0159 1 0158 1.2846 1.1427
0 R R~a R R
10 28
None GAD None
4.1000 4.1000 4.1000
12.6000 0.0000 0.0o00

13470 11125 1.32 56
i
1.1 S21 1.0158 11846
R R R R R
None None
1000 4.1000
-
0.0000 0,00x
1.385S 1.3104
1, 2846 1.2846
R R R R R
R R R R R R
R R
110
The results of the Dual Binary Swap optimization are shown below in Tables 5.10
and 5.11.
Table 5.10. Results of Deterministic Dual Binary Swap Optimization on Converged Simulated
Annealing Results
Simulated Simulated DBS Dual Dual
Annealing Annealing Power Binary Swap Binary S wap
Pattern EOC Burnup Maximum Cycle Peaking EOC Burnup Maximum Cycle
Number (MWd/kg) Power Peaking Constraint (MWd/kg) Power Pe aking
1 13.414 1.385 1.375 13.534 1.374
2 12.929 1.384 1.375 12.925 1.381
3 13.867 1.404 1.400 14.335 1.399

4 13.485 1.398 1.400 13.914 1.400
5 13.022 1.398 1.400 13.071 1.399

6 14.102 1.409 1.400 14.102 1.409
7 13.527 1.395 1.400 13.584 1.398

8 13.552 1.394 1.400 13.626 1.399
Table 5.11. Number of Passes Required to Converge Deterministic Dual Binary Swap Opti-
mization on Converged Simulated Annealing Results
DBS Dual Dual P asses to
Power Binary Swap Binary Swap Converge
Pattern Peaking EOC Burnup Maximum Cycle Dual
Number Constraint (MWd/kg) Power Peaking Bin ary Swap
1 1.375 13-534 11
1.374
2 1.375 12.925 1.381 4
3 1.400 14.335 1.399 16
4 1.400 13.914 1.400 13
5 1.400 13.071 1.399 7

6 1.400 14.102 1.409 1
7 1.400 13.584 1.398 4

8 1.400 13.626 1.399 8
For almost all of the converged Simulated Annealing patterns, the deterministic
Exhaustive Dual Binary Swap optimization succeeded in finding better patterns that
were missed by the stochastic method. It is also worth noting that the pattern with
111
a cycle length of 14.335 is the best pattern that was found that satisfied an imposed
power peaking constraint of 1.400.
These optimized patterns are shown in Figures 5.10 through 5.17.

1 35 - o 31n 5
None GAD GAD None
4.1000 4.1090 . 4.1000 4 1000
0.0000 0.0000 0 - 0.0000 0 0000
1. 3546 1.1116 11463 1.3742
1.2846 1.0158 1A 158 1 2846
29 R
GAD None GAD
4 1000 4.1000 .-1000
0 000 11.3000 0.0000
1.020 e .. - 1 2 39, 1.1127 -
-
1 11111 1.0158
11 0 7 R R
13
None GAD ,1None None None
4 1000 4.1000 4.1000 4 1000 4.1000
00000 16A000 12.6000 0 0000 10.2000
1.3680 0.7512
1.3546 II 1.3281
1.1111
1.2846 1.1521 1.2846.
R R
35 12 10 27 34
GAD GAD None GAD GAD
4.1000 4,1000 4.100O 4.1000 4.1000
0.0000 11.3000 12-6000 0 GO000 0.0000
1.1116 1.2490 1.2549 1.0821 0,7912
10158 1.1427 1.0958 1.1521 U'_7 1.0158
8 30 4 R R
None GAD None
~I 4.1000 4.1000 4 !000 4.1000
1i iooo 11.3000 0 0000 0.0000 18
1.3028 1.1696 .1449 4A f4
1.1641 1.0158 1.284
0 - . -'rIII
31 33 3 k
CAD GAD None

4.1000 4.1000 4.1000
0.0000 0.0000 0.0000
1.1463 1.0702 1.1226
1.0158 1.01 58 1.2846
I I I
5 2 6
None None None
.21000 4.1000 4.1000
0.0000 0,0000 10.2000
1.3742 1 3711 0M218
1.2846 1.2846 1.1743
4- F I
R R
8 4 -8 I I - 1
-
__ _ _ .1 I I __ _ _ J __ _ _ _ __ _J-__ _ 1_ _ _ _-
Figure 5.10. Core Loading Pattern Obtained by Dual Binary Swap of a Converged Simulated
Annealing Pattern with End of Cycle Depletion 13.534 M Wd/kg and Maximum
Cycle Power Peaking of 1.374
112
4 fG 27
R
GAD None GAD None
4.1000 4.1000 4.1000 4.1000
t _ 0.0000 12.6000 0.0000 11.3000
1.0207 1.2761 10841 0.5621
is 1.0158 1,1521 1.0158 1.1641
32 33
I GAD GAD None
4 JO ; 4.1000 4.1000 4.1000
a-
1- 0.M00 0.0000 0.0000
-
1.0387 R
R
1.1657 1.2006
O 1.0158 1.0159 12846
30 39 1'1
L .. .I 6 13
CAD GAD None None GAD
4.1000 4.1000 4.1000 1,g 41000 4.1000
0.0000 0,00K 126000 I0.2000 16,8000
0.9741 ia;' 11593 1.2835 0.7965
1.0158 1.0158 1.1521 1.1743 1.1092
27 4 3T
R R
GAD None GAD ayg+ None
4.1000 41000 4.1000 4,11100 41000
0.0000 0.0000 00000 133 ; 0.0000
1.0207 1.3793 1.0786 1.3311
1.0158 1.2846 M 10158 i 1,2846
35 18 R R
GAD None
4.1000 4.1000
r.
0.0000 0.0000
1.1434 1.3180
1.0158 1.2846
10 7 28 1 R R R R
None None GAD None
4.1000 4,1000 4.1000 4.1000
12.6000 10.2000 0.0000 0.0000
1.2761 1,3803 1 106- 1.2141
1 521 1.1743 10158 1.2846
29 0 8 12 R
GAD None None GAD
4.1000 4.1000 4.1000 4.1000
0.0000 0.0000 t i.3000 11.3000
1 0841 1 3539 0.6018
10158 1.2846 1.1641 1.1427
9 t R R R R R R R
Npae.
-
R R R R R R R R R
Annealing Pattern with End of Cycle Depletion 12.925 MWd/kg and Maximum
113
Rt
33 2
GAD None GAD None
4.1000 4.1000 4.1000 4.1000
0.0000 00000 0.0000 0.0000
1 .9954 1.3999 1.1921 1 1829 1.3497
1 0158 1.2846 1.0958 10158 .2846
9 7 0 Rt
33 18 31 I I
None None
GAD ffAD GAD GAD None
4.1000 4 .1000, 4.1000 4.1000 4.1000
4.1000
0.0000 - 0.0000 113000 10.2000 0.0000
1 1.0901i
L
1.3354 1.3792 1.3457
0.9954 4-a
1.0158 y _{? 1.0159 1.1641 1.1743 1.2846
13 Rt
2 19 17 28
None GD GA GAO GAD GAD
4.1000
,
41000
4.1000 > 4.1000
0.0000 0.0000 16S,000
0.0000 _ 1.059
1.3989 1.1345 1.0733 0.7338
1.2846 0g0 1.01;8 10150 1.1092
29 6 1 R Rt
CAD GAD GAD None None
4 1000 4.ip .1 1003 4.1000 J 1Doc
0.0000 19.501 0.0000 10.2000 0 0000
-
1.0041 4+i4Z 1.1140 1.3421 11907

.0158 - 1.0867 10130 1.1743 12846
Rt R
14 10 34 8 3n A16
GAD None GAD None Nonn cmD
S32 27 5i
S
GAD GAD None None
4 1000 4.1000 41000 4.1000
0001 0 0000 0.0000 12,6000
1.1029 1.0686 1.2608 0.9328
1.413 10158 1.2840 1.1521
a t-
R
4.1000 4'0
0.0000 113000
1 3497 0.8939
11846 1.1427
-
+ I - t
-
R 8
114
34 33 R
GAD GAD GAD None
4.1000 41000 4 0 4.1000 4 1000
0.0000 0 0000 19.500- 0.0000 0.0000
09075 1.0248 ti53 11815 1.3744
0158 1.0158 1 1,01500 1.2846
4 30 y 28 10 R
None CAD 6 -- GAD None
'
4.1000 4,1000 4.1000 4.1000
0.0000 0.0000 0.0000 12.000
1.3855 1.0459 112 11743 1,0857 0.9458
1.2846 1.0158 0O 1.0158 1.1521
-
34 2
'
13 0 0 R
GAD r GAD None None 1
4 1000 41000 4.1000
r' 1
4 1000 i
0.0000 - 16.8000 10.2000 0.0000 18;4 de
0.9675 {{ 1.1943 1 3284 1.3200 0.6755.
1.0158
,
1.1092 1.1743 1 2846
33 8 15 11 35 27 R R
GAD None GAD None GAD GAD
4 1000 4 1000 4.1000 4.1000 4.1000 4.1000
00000 11 3000 18.4000 12.6000 0.0000 0.0000
1.0248 1.2949 1. 40S 1.2537 10661 0.7812 1
10158 1.1641 . 1.1521 1.0158 1,0158
_1 16 9 29 5 12 +< R R
- None GAD None GAD
4.1000 41000 4.1000 4.1000
11.3000 0.0000 0.0000 11,3000
1 1.2968 11371 1 1090 0.5275
10957 1,1641 1.0158 1.2846 1.1427
32 7 3:. R R R R
GAD None GAO None
-
4.1000 4.1000 4.1000 4.1000

0.0000 10.2000 0 0000 00000
1.1815 1,3719 1.0612 1 0583
1.0158 1.1743 1.0158 1.2846
3 1 1 R R R
None None
4.1000 4,1000
-
0.0000 0.0000
1.3744 1,3575
1.2846 1.2846
R R R R R R R
R R R R R R R R R
115
4 35
R
29
GAD None GAD GAD None

4.1000 4.1000 4.1000 4 1000 4:1000
'
4.1000
0.0000 00000 0.0000 0.0000 11.3000

0.0000
0.9842 1.3633 1.0649 1.1707 1 3992 0.6287

1.0158 1.2846 1.0158 .0158 12846 1.1427
N I I #- .- 1
29 t_
32 11 30 1R
GAD f
4,1000 4I000 4.1000
4.1000
0.0000
1.
00000 166000
1.1493
0,000
0.8911
0.9642 o 7T
10024
1.0158 L-ji 0" _. i
1.0!59 1.1092 1.0158
4 -- - 27 -? 0 6 R R
None_ ' GAO None None

41000 , 41000 1 1000 4.1000
0.0000 0.0000 1&000 00000, 10.2000
1.3633 1.0244 1.1397 1 3958 0.7558
1.2846 1.09511 1.0158 1.0917 1284t 1.1743
. - - 1D :8 8 R R
35 't9 -
GAD QW, Nane GAD None
4.1000 4. f 000 4.1000 4.1000
0.0000 ='1 126000 0.0000 11.3000
1,0649 1.1138 0.9746
1 o15Q T.ql ' 1 1.is21 1.0158 1.1641
11 31 7 3 15 R R R
None GAD None None AD
4.1000 4 1000 4.1000 4.1000 4.1000
12.6000 00000 10.2000 0.0000 18.4000
1.2832 1 2407 1.3157 1.258; 0.5935
1.1521 1.0154 1.1743 1.204 1.0958
a a 99
-~ F R a R a R
-
2 34 P R R R
GAD None GAD None
4.1000 4.1000 4,1000 4.1000
0.0000 0.0000 0.0000 11.3D00
1.1707 1.3919 0.8913 0.6223
10158 1.2846 1.0158 1.1641
R R R R R
1
None None
41000 4.1000
00000 0.0000
13992 1.3435
1 2846 1.2846
12
GAD
4 1000 4.1WI
11 3000
0.6287
R R R R R R R
R R
116
I f
28
W ?o 4 R
-
GAD None GAS, None
4,1000 4.'000 4 1000 41000
0.0000 0.0000 184000. 0.0000 0.0000
0 9986 1.3992 MOW 1.2103 1.3865
I018 1.2846 1.0158 1.2846
28 29 7 R
GAD GAD None
41000 4.1000
0 0000
toss 4.1000
0.0000 14,000 10.2000
09986 1.0829 1.206L 1.3509
1.0158 1.0158 d198 1.1743 1.Q807 It i
1 11 32 F R
None None GAD Zoo
4 1000 4.1000 4.106(:
0.0000 12.6000 0.0000
4 .2 1.1769 1.2535
1.1521 1.0158
35 3 R R
GAD GAD None
4.1000 41000 4.1000
0.0000 00000 0.0000
0.9462 1.1173 1.4052
1.0158 1.0158 1 2846
12 .1 a 10 13 R R
GAD GAD GAD None GAD
4.10p0 4.1000 4.?000 4.1000 4.1000
18.400 11.3000 0.0000 12.6000 16.8000
IJ6 1.1540 1.0631 1.1978 0.9441
1.0958 1.1427
h ~~~
1 0118 1.1521 1.1092
-- h1f I 1
30 27 6 5 R
GAD GAD None None
4.1000 4.1000 4.1000 4 1000
0.0000 0.0000 10.2000 0.0000
1.2103 1.1798 1.2427 1.2640
10158 1.0158 1. 1743 1 2846
4 2 9 8 k
None None None None
4 1000 4.1000 4.1000 4.1000
0.0000 0.0000 11.3000 11.3000
1.3865 1.3464 0.9086 0.6330
1.2846 1.2846 1.1641 1.1641
R R R R R R R
117
R
None GAD !;one
4.1000 1000
:.1000 q.O
12.6000 11.3000 1 2000 18.9000
. i.00
1.1919 1 2839 1 39P4 0.5540
tow~1 1.1427 1.1846 1.0 It 7
1.1521
30 8 16 31 32
GAD None GA GAD CAD
4.1000 4.1000 4.1000G 4.1000 4.1000
0.0000 11.3000 18.9m0 0 0000 0.0000
0.9855 1.2685 LIS%5 1.0736 0 8528
1.0158 1.1641 1;917 1.0158 1.0158
* t 05 I - t S 1 T
14 35 7
None GAD GAD None None
4.1000 4,1M 4. 1000 4 1000 4.1000
00000 -3.000 18.4000 0 Gt02. 10.2000 11 3000
1.3650 .108 1.2357 1,2283 0.7297
1.1743 1641
1.2846 1.0628 1.0958 1
i
I 3
_
11 33
None GAD None CAD None 41Q~
4.1000 4.1000 4.1000 4.1000 4.1000
12.6090 0.0000 10.2000 00000
1.1919 1.0961 1.3131 1,3543 _0.5552
1.1743 1.1092 12816 1.0959
1.1521 1.0158
29 34 4 10
GAD GAD None None
3 1000 4.1000 4.1000 4.1000
0.0000 00000 0.0000 12.6000
104S4 1.0673 1,2754 0.7673
--- 1.0154 1.015s 1.2846 1.1521
GAD GAD None

4.1000 4.1000 4.1000
0.0000 0.0000
-
11.3000
12889 11852 13077 0.3274
1.1427 1.0158 1.2846 .09960
4 1 1
J i F I
R R R0
118
17 R
None No None
41000 a 4.1000 4 100
12.600 11.3000 0.0000
1.2172 F 1.3612 1 3986
1.1521 1.1641 1.2846
28 30 9 19 31 32 R
GAD GAD None GAD GAD GAD
4.1000 4.1000 4.1000 4.1000 4 1000 4 1000
0.0000 0-0000 11.3000 190000 0.0000 0.0000
0.8729 0.9772 1,2713 1.1747 1.0804 0.8272
1,01%8 1.0158 1.1641 1.0908 1.0158 10158
1 15 35 7 12 R
None GA40; GAD None GAD
4...000 X1i004 4.1000 4.1000 4.1000
0.0000 IBS;0 0.0000 10.2000 11.3000
1.3535 a.1.2222 1.1666 1.1980 0.6843
1.2846 ' - ' 1093$ 1.01S8 1.1743 1.1427
10 33 6 13 ? L R R
NOWi GAD None GAD None --
4.1000 4.1000 4.1000 4-00O 4.1000
12.6000 0.0000
-
10.2000 - AtIdM 0.0000
1,2172 1.1020 1.2973 1,3233 1 1in7 O.3P29
1.1521 1.0158 1.1743 1.1092 ' h4d 1.0917
29 34 0 11
GAD GAD None None
4.1000 4.1000 4.1000 4.1000
-0.0000 O.00OC 0.0000 12.000
- L0538 1.0678 1.2686 0.7521
1.0158 1.0158 1.2846 1.7521
8 27 2 i R
Norne
4.1000
GAD None - CAD,
4.1000 4.1000 418
11.3000 0.0000 0.0000 8il.
1.3612 1.2150 1.3309.61
10524
;15
I
None None
-
4.1000 4.1000 'r

0.0ooo 0.0000 f =
1 3986 1.3168
1.2846 1.2846 _S
R R
IK I
119
6 CONCLUSIONS AND FUTURE WORK
6.1 CONCLUSIONS
In the beginning of cycle study carried out, the deterministic Dual Binary Swap meth-
ods were found to less reliably produce loading patterns that satisfied imposed power
peaking constraints than did the stochastic simulated annealing algorithm. However,
the best pattern found by the dual binary swap method was, for imposed power peak-
ing constraints in the range of 1.25 to 1.35, of consistently higher quality than the best
pattern found by Simulated Annealing.
This beginning of cycle analysis suggests that there are indeed "silver bullet" pat-
terns existing in the solution space of core loading optimization that stochastic meth-
ods miss. The deterministic Dual Binary Swap that explores all possible assembly
swaps and discards the many that do not help improve the pattern is better able to
find these unique solutions than stochastic methods that might miss an important
assembly swap that is required to find these patterns due to their random nature.
The fact that reactor kinf was serving as a surrogate for excess core reactivity and
cycle length, however, necessitated the inclusion of a depletion model, as this is not
a perfect approximation. Additionally, a depletion model is capable of handling the
mid-cycle effects of burnable poisons, which is something that could not be accounted
for in the beginning of cycle analysis.
The difficulty of the depletion model is that it negates some of the computational
speed benefit that was obtained by the use of a FLARE evaluator to determine the
reactor eigenvalue. The computational overhead was such that a depleted reactor core
would take approximately 35x as much CPU time per state point as a simple beginning
of cycle analysis. Heuristics introduced into the Dual Binary Swap algorithm to avoid
depleting patterns that appear as though they will be of much lower quality than the
121
current best pattern help alleviate the cost. Heuristics were not introduced into the
Simulated Annealing algorithm both because the algorithm needs to be capable of
accepting worse solutions, so simply ignoring patterns that are significantly worse at
the beginning of cycle is not as feasible, and because the proportional computational
gain from the heuristics would be lower. The proportional gain from heuristics would
be lower in Simulated Annealing due to the fact that so many more potential assembly
swaps are accepted over the course of the optimization (especially at the beginning of
the optimization where the searching of the solution space is at or close to a random
walk). Any swap that would be accepted, or any swap that sufficiently close from a
beginning of cycle analysis that there is a chance it will be accepted after a depletion
analysis, must be depleted.
The full depletion problem is a much more difficult one than the beginning of cycle
optimization, primarily due to the issue of large mid-cycle power peaking appearing
as burnable poisons are consumed. This can be seen very easily from the Dual Binary
Swap optimization where, while patterns that satisfied the power peaking constraints
were found, those patterns were located less consistently than in the beginning of
cycle analysis and only at higher imposed power peaking thresholds. Additionally,
while the exhaustive implementation of the Dual Binary Swap was able to succeed in
satisfying the constraints at power peaking values of 1.4oo and below, the greedy im-
plementation was not. A summary of the best patterns found in the depletion analysis
by the different optimization methods is shown in Tables 6.1 through 6.3.
Simulated annealing was found to be able to locate these high quality patterns that
satisfied such a low power peaking threshold, but to be unable to do so consistently.
There was no marked difference between the deterministic Dual Binary Swap and the
stochastic Simulated Annealing in this regard, unlike in the beginning of cycle analysis.
This is likely due both the the fact that the more complicated solution space can make
some of these high quality patterns more isolated from other good solutions (which
can often hamper stochastic methods that rely on random elements) and the fact that
the lack of effective heuristics in the Simulated Annealing algorithm limits how low a
122
Table 6.1. Best Loading Patterns Found in the Full Depletion Study by all Optimization Meth-
ods, Power Peaking Constraint 1.350
Exhaustive Dual Exhaustive Dual Greedy Dual Greedy Dual
Binary Swap Binary Swap Binary Swap Binary Swap
Pattern EOC Burnup Maximum Cycle EOC Burnup Maximum Cycle
Number (MWd/kg) Power Peaking (MWd/kg) Power Peaking
1 13.425 1.349 None None
Found Found
Simulated Simulated Exhaustive DBS on Exhaustive DBS on
Annealing Annealing Simulated Annealing Simulated Annealing
None None Not Not
Found Found Applicable Applicable
cooling parameter could be used. An extremely slow cooling parameter would result
in a stochastic algorithm that requires more computational resources than the Greedy
Exhaustive Dual Binary Swap, which is itself too computationally expensive to be of
practical use.
The analysis with depletion thus supports the results observed in the beginning of
cycle analysis. The deterministic Dual Binary Swap optimizer is able to find special
"silver bullet" patterns that standard stochastic methods have difficulty locating. In
particular, the Dual Binary Swap algorithm was successful in locating a pattern that
satisfied an imposed power peaking constraint of 1.350 and 1.375, while Simulated
Annealing was not. Additionally, the best pattern found by the Dual Binary Swap
method from poor starting points (either the starting Yamamoto configuration or ran-
domized starting points) for an imposed power peaking constraint of 1.400 had an
End of Cycle burnup 0.271 MWd/kg higher than the best pattern found by the Sim-
ulated Annealing algorithm. Additionally, the deterministic Dual Binary Swap from
converged Simulated Annealing starting points was able to find a pattern with End
of Cycle burnup 0.783 MWd/kg better than the best pattern found by Simulated An-
nealing.
123
Binary Swap Binary Swap Binary Swap Binary Swap
1 13.571 1.372 None None
2 13.487 1.374 Found Found
1 None None 13.534 1.374
Found Found
The Dual Binary Swap optimizer is, however, dependant on the configuration of
the starting point, and is not sufficiently robust to always arrive at a pattern that
satisfies the constraints. It is also computationally expensive due to the necessity of
examining many poor patterns, and will likely never be practical for use in commercial
applications.
6.2 FUTURE WORK
This work strongly supports the hypothesis that in the optimization problem of nu-
clear reactor core loading there are special patterns that stochastic methods miss. How-
ever, due to the size of the solution space and the practical limitations on the amount
of computational resources that can be allocated to stochastic optimizations, this con-
clusion can not be conclusively proven for real reactor problems. Future work in this
area could be applied to make this work model real-world reactor optimization prob-
lems more closely.
This would necessitate primarily improving upon the poropy reactor model. This
could be done by steps such as depleting at a constant reactor eigenvalue of 1 while
instead changing the reactor boron concentration, by introducing a thermal-hydraulic
124
Binary Swap Binary Swap Binary Swap Binary S wap
Number (MWd/kg) Power Peaking (MWd/kg) Power Pe aking
1 13.823 1.396 None None
2 13.590 1.399 Found Found
3 13.465 1.400

1 13.552 1.394 14.335 1.399
2 13.527 1.395 13.914 1.400
3 13.485 1.398 13.626 1.399
4 13.022 1.398 13.584 1.398
5 13.071 1.399
model and allowing different temperatures across the core, or by using a more com-
plex physics model than the FLARE model. Changing any of these parts of the
model,
however, would dramatically increase the computational cost of optimizing the reactor
loading pattern. Similarly, carrying out a multi-cycle depletion analysis would greatly
increase the fidelity of the optimization problem [20], but would be prohibitively ex-
pensive computationally. This would also require a model with more detail than is
provided in the Yamamoto benchmark. The parallels between this work and realistic
core loading pattern optimization problems would also be increased if either a full 4
loop PWR was optimized or if more heuristics to represent practical constraints were
introduced. Both of these would increase the complexity of the search space to be
more in line with that of a realistic core loading optimization problem.
Finally, one additional step that could be taken to support the results obtained here
would be to introduce a Genetic Algorithm optimizer that is capable of carrying out
an optimization with full depletion. This would allow comparison of the deterministic
125
methodology with both of the most commonly used stochastic optimizers as opposed
to just the simpler Simulated Annealing.
126
Part I
Appendix
127
___
A APPENDIX
A.1 BEGINNING OF CYCLE ANALYSIS
Table A.s. Complete Results of Beginning of Cycle Exhaustive Dual Binary Swap Optimiza-
tion of Genetic Algorithm Starting Patterns
Number of Size of Number Post GA Post GA Post DBS Post DBS
Generations Population Replaced Power Peaking Eigenvalue Power Peaking Eigenvalue
400 100 50 1.4927 1.07039 1.4996 1.08751
400 100 50 1.4641 1.07310 1.4992 1.08749

400 100 50 1.4924 1.06529 1.4993 1.08549
400 100 50 1.4805 1.07278 1.4998 I.o8568

1000 120 8o 1.4956 1.08499 1.4980 1.08569
1000 120 8o 1.4812 1.07779 1.4994 1.08584
1000 120 8o 1.4992 1.08471 1.4964 1.08543
1000 120 8o 1.5000 1.07696 1.4999 1.08677
4000 150 100 1.4994 1.07790 1.4994 1.07790
4000 150 100 1.4948 1.08359 1.4978 1.08580
4000 150 100 1.4996 1.08488 1.4999 1.08576

4000 150 100 1.4972 1.08691 1.4997 1.08697
Table A.2. Complete Results of Beginning of Cycle Greedy Dual Binary Swap Optimization
of Genetic Algorithm Starting Patterns
Number of Size of Number Post GA Post GA Post DBS Post DBS
Generations Population Replaced Power Peaking Eigenvalue Power Peaking Eigenvalue
4000 150 100 1.4994 1.07790 1.4994 1.07790
4000 150 100 1.4948 1.08359 1.4998 1.08581
4000 150 100 1.4996 1.08488 1.4992 1.08526

4000 150 100 1.4972 1.08691 1.4997 1.08697
129
Table A.3. Optimizations of the Randomized Yamamoto Benchmark Using the Dual Binary
Swap Methodology
Power Peaking Exhaustive Exhaustive Greedy Greedy
Constraint Eigenvalue Power Peaking Eigenvalue Power Peaking
1.4 1.07934 1.3996 1.07321 1.3997
1.4 1.08032 1.3998 1.08094 1.3999
1.4 1.07380 1.3993 1.07568 1.3994
1.4 1.07993 1.3988 1.07346 1.3997
1.4 1.08023 1.3997 1.07424 1.4000
1.5 1.08288 1.4997 1.08385 1.4966

1.5 1.08585 1.4995 1.08136 1.5000
1.5 1.08560 1.4995 1.08137 1.4999

1.5 1.08219 1.4992 1.08613 1.4998
1.5 1.08727 1.4991 1.08711 1.4987
1.6 1.09066 1.5999 1.09217 1.5987
1.6 1.08850 1.5999 1.09411 1.5993

1.6 1.09026 1.5993 1.09266 1.5998
1.6 1.09499 1.5994 1.09095 1.5988

1.6 1.09513 1.5993
130
A.2 IMPLEMENTATION OF DEPLETION
. 10.800 SO11 1.06 1.1073 j=*Z76 1 1.01

36 23 3i 2 14 6,30 74 R
C No GAD GAD2
-
R
R a Ra R
Figure A.i. Reactivity of Fuel Bundles in Example High Quality Reactor Loading Pattern at
Depletion Step 2
131
17 t
GAD
4.1000
19.0900
0.4999
1.0901
24 R
None
4.1.000
R R
R R R R
R
12 26 R R R R
- ' GAO None
- - -4.1000
R R R R R R
R
R R R R R
RR R R
Figure A.2. Power Peaking in Example High Quality Reactor Loading Pattern at Depletion
Step 2
132
R
- t.
0A6 +k
'1 A 32
n R
9I 2b
,: 1 R R
R R
R R
I I
Figure A.3. Reactivity of Fuel Bundles in Example High Quality Reactor Loading Pattern at
Depletion Step 5
133
R
R R R R
y'YM : GABo None

00 4,]000
RR RRt R R R R R R
R RR R R R R R R
Step 5
134
CAED CAt) None + A CAD - GAO
41x000 4.1000 x1000 410W0 4.1000 4O5iwD0 4.1000
'.2300 - Va
O.90073
~- A 32.5000
1.t689
93800.
13Z19
26.6600
1.2197
-:L 22.9300
0.9361 0.4587
0.9450 .09907 1.134 1.0301 !1.0$90
37
3926 0 17
31R
R R R
R i R I R R t R t R R
A AR
Figure A. 5 . Reactivity of Fuel Bundles in Example High Quality Reactor Loading Pattern at
Depletion Step 9
135
R K K K K K R K
Step 9
136
38 37 1_ 34 120 S 18 ft
GAD CAD) f None?.4D GAD 1G11!
.*. 1000 4.1000 ' .000 44T00D: 4.1000' - 4.1000
43.1000 42.3400 1.3QAW 37.8400 15.8100 3213400 S~0 25.0900
0.5007
09185
0.7303
0.92 31
1CitSSZ 1.0983 1.?177 1.2255 i4 ~ 0.5060
37 33 + 1I1$ <:j 0,9541 -111W8 0.9888 1I 1.04______

GAD None
4.1000
42.3400
4.1000
35.1012
39
GAD
4.1000.
~ GDNn
4.1000
A30 20
-4;1000.
2.5r R
0.7303 44 9,700 $3:9100 32a160 L,.

0.0463 .1.18 80M
Rt Rt
U.911 jd 1.0255 1-1,-

34 23 32 19 Rt Rt
None None CAD
1.'.000 4.1000 4.1000
GAD_
37.8400 28,2600 34 4400
1.0983 1.1283 I 1.0580
,9541 1.0217 0
09740 0.9792 1.1f61' , 1lAI4
13
GiaD
30.
14.n .
21
Nne
27
CAI)
4-
Eiinm-.
24 R [ Rf
None
f.11?6 0.9289 1.0397 098421 i.75
GAD 66k Wbe None

R Rt R R
4 00 .410 . 4.1000..
ft
18 35 Rt Rt R R Rt Rt
GAD GAD)
4.100
ft ft
Figure A.7. Reactivity of Fuel Bundles in Example High Quality Reactor Loading Pattern at
Depletion Step 12
137
i8 R
GAD
4.1000
25.0900
25 R
None
4.1000
R R
R R
R Rt
-- 17 29 R R R R R
a +. GAD None
4.1000
R R R R R R R
R R R R R R
RR R
Step 12
138
A.3 DUAL BINARY SWAP WITH DEPLETION
A R R R
Figure A.9. BOC Power Peaking of Initial Configuration as Specified by Yamamoto
139
R
0 6250 0.7194 1.1799 1.0531 . _

0.9722 11182 y,867 1.0169 i_ _ i
i 0.9722 1,0158
'
38 37n 33 23 19 AS 32 |R
GAD GAD GAD oGAD
R R
t'.7.1 '. --- " l 0TW

1261 FR V.IM -'"""""
1.0867 1.0564 1..0917 01986 1.015E 1:lAd4
34 1R 25 35 - 12 - 2 - R R R
GAD GAD None GAD G
R R R R
R R R R R
R R R R R | R R
R R R R R R R
R R
Figure A.zo. BOC Assembly k;, f of Initial Configuration as Specified by Yamamoto
140
R
f 32 20
NO A
Nc GAD
. In
R R R R R R R R R
Figure A.1i. BOC Power Peaking of Initial Configuration of First Randomized Core
141
R
R R R R R
R R | R R R R R
R | R | R | R | R R R R R
Core
Figure A.1i2. BOC Assembly kinf of Initial Configuration of First Randomized
142
R
R R
Figure A.1 3 . BOC Power Peaking of Initial Configuration of Second Randomized Core
143
R
Figure A.14. BOC Assembly kinf of Initial Configuration of Second Randomized Core
144
R
18
GAO
?7 Z1 I 2 K
R
CAD None Rn
I 4110 I 4.1000 4.1000 a410OW
33 0 K H R R R R R
GADL None
4.1000 4 1000
0.0000 0.0000
0,3375 0 3558
JAM~g
R R R R R R R R R
Figure A.1 5 . BOC Power Peaking of Initial Configuration of Third Randomized Core
145
R
1.61- 1.54. . t P 0.9860 ta
Q~r= tiL -None GAD -E ~1 ; --F GAD GAAR

~miis -2. 410 4.7000 4. Q00 4.100
R R
R I R R
R R R R
R R R R R
R R R R R R R
R R R R R R R. R R
Figure A.16. BOC Assembly kinf of Initial Configuration of Third Randomized Core
146
R
GAD UAU
4.1000 4.1000
113000 18.9000
04432 0.4304
1.1427 1.0917
R R R R R R R
Figure A.1 7 . BOC Power Peaking of Initial Configuration of Fourth Randomized Core
147
R
19.5000 32.7000 {9Oa04: 0.0000 34,7000 0.0000

0.7330 0.7358 0;9216 1.0237 1.0418 - 0.7693 03150
TM867 0.9860 1.0908 1.0158 0 972 1 1.0158 1.0958
25 11 28 10 27 21 26 R
hone None G AD NOWu GAD | None NO"e
S fO.
4.10 ig--4.1W70 4,i00 4.1W04.100x7
R R
R R R
32.200 I .I)6D5
114@ 12?fl 10941 1.2382
-
71641 U9:5 0.9896 1.0158
24 35 R R R
R R R R R
R R R R | R | R R
R RR R R R R R R
Figure A.28. BOC Assembly ki,,f of Initial Configuration of Fourth Randomized Core
148
R
a1^^. T 1.0564
37 35 3C 14 R
GAD CAD CAO GA
4.1000 4.1000 4.4000 4 000
32.7000 0.0000
0.5504 0.5464
0.9860 1.0158
10 5
None None_
R R R R R R R R { R
Figure A.19. BOC Power Peaking of Initial Configuration of Fifth Randomized Core
149
R
R R
149: 1:3327 1.3705 1.2i27 i1.or7 1.2|> ' 0.6084

tq 1.68 1;0958 1.ot58 1.0169 f 46 1.0158
Fi A 2 27 13 23 28 R R
R R R
R R R R
R R R R R
R R R R R R R
RR R R R R R R R
Figure A.20. BOC Assembly kinf of Initial Configuration of Fifth Randomized Core
150
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20Hub%20Prices&showHTMLView=&mimicKey.
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144-152, 1995-
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153

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The Greedy Exhaustive Dual Binary Swap

Methodology for Fuel Loading Optimization in

Department o Nuclear Science and Engineering

Certified by ..... Signature redacted 'p ""e'

Signature redacted Thesis Supervisor

Fuel Loading Optimization in PWR Reactors Using the

poropy Reactor Optimization Tool

Submitted to the Department of Nuclear Science and Engineering

Thesis Supervisor: Kord S. Smith

Thesis Supervisor: Benoit Forget

to continuing to have the privilege to work with them throughout my PhD.

4 DUAL BINARY SWAP WITH DEPLETION

5 SIMULATED ANNEALING WITH DEPLETION

6 CONCLUSIONS AND FUTURE WORK

Figure 1.1 Algorithm of the Simulated Annealing Methodology . . . . .

Figure 5.1 Structure of Assembly Data in Loading Patterns .. . .. .. . 102

Figure 5.2 Core Loading Pattern Obtained by Simulated Annealing with

Figure 5.3 Core Loading Pattern Obtained by Simulated Annealing with

Figure A. 5 Reactivity of Fuel Bundles in Example High Quality Reactor

Figure A.15 BOC Power Peaking of Initial Configuration of Third Random-

Figure A.16 BOC Assembly kinf of Initial Configuration of Third Random-

Figure A.17 BOC Power Peaking of Initial Configuration of Fourth Random-

Figure A.18 BOC Assembly kinf of Initial Configuration of Fourth Random-

Figure A.19 BOC Power Peaking of Initial Configuration of Fifth Random-

Figure A.2o BOC Assembly kinf of Initial Configuration of Fifth Random-

Table 2.1 Results of Beginning of Cycle Analysis of the Genetic Algorithm

Nuclear fuel management in reactors is a very important component of the opera-

enrichment, etc), as well as which previously burned assemblies should be re-inserted

with different burnups, enrichments, and burnable poisons) at specific locations in

accident situations, becomes significantly more difficult to prevent. Conversely, low

before a refuelling shutdown is required. Considering that a nuclear reactor that is

the entire solution space.

to be extremely conservative from a power peaking perspective. The class of pattern

to result in a short reactor cycle length.

As reactor control methods and computational technology have advanced, these

hand-designed loading patterns have stopped being used. Modern computational

and a significantly better cycle length.

mization scheme: deterministic and stochastic. Deterministic methods search through

a space in a methodical manner to arrive at a solution. Deterministic methods are

by contrast, use sampling of random numbers to explore parts of a solution space in

an intelligent manner. Stochastic methods tend to be preferred for many optimization

these methods will perform poorly [6].

Incremental Learning Algorithm [5].

perturbation. At the beginning of the simulation the system is said to be at a high

"temperature", and any perturbation is accepted. As the simulation progresses, the

out using only single binary swaps.

GA methods, by contrast, take inspiration from the field of evolutionary biology. A

solution to the problem. This population represents a "generation". Two solutions in

generation [14]. This algorithm is shown visually in Figure 1.2.

these solutions are often missed.

Accept the Change

Figure i.i. Algorithm of the Simulated Annealing Methodology

Mix the Traits of the

physical intuition to generate a sample of candidate patterns. These specific patterns

be referred to as the Greedy Exhaustive Dual Binary Swap (GEDBS) methodology.

1.2 GREEDY EXHAUSTIVE DUAL BINARY SWAP

Binary Swap method (EDBS, see Figure 1.5).

Swap the Locations of - Evaluate the New

Swap the Locations of Evaluate the New