●●　オーバードクター・ポスドク問題　●●Part2

●●　オーバードクタ ..

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タイトル: Study on Advanced In-Core Fuel Management for Pressurized Water Reactors Using Loading Pattern Optimization Methods
その他のタイトル: 装荷パターン最適化手法を用いたPWR炉心燃料管理の高度化に関する研究
著者: Yamamoto, Akio
著者名の別形: 山本, 章夫
発行日: 23-Mar-1998
出版者: Kyoto University
記述: 本文データは平成22年度国立国会図書館の学位論文(博士)のデジタル化実施により作成された画像ファイルを基にpdf変換したものである
学位授与大学: Kyoto University (京都大学)
学位の種類: 新制・課程博士
取得分野: 博士(エネルギー科学)
報告番号: 甲第7440号
学位記番号: 博第3号
請求記号: 新制/エネ/1
研究科・専攻: 京都大学大学院エネルギー科学研究科エネルギー社会・環境科学専攻
論文調査委員: (主査)教授神田啓治, 教授吉川榮和, 教授代谷誠治
学位授与の要件: 学位規則第4条第1項該当
DOI: 10.11501/3135597
URI: URLﾘﾝｸ(hdl.handle.net)
出現コレクション: 博士(エネルギー科学)

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STUDY ON ADVANCED IN-CORE FUEL MANAGEMENT
FOR PRESSURIZED WATER REACTORS
USING LOADING PATTERN OPTIMIZATION METHODS
AKIO YAMAMOTO
Submitted for the Degree
of
Doctor of Energy Science
of
KYOTO UNIVERSITY

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CONTENTS
CHAPTER 1. INTRODUCTION
1.1 Background ---------------------------------------------------------------------------- 1-1
1.2 Basics of in-core fuel management for PWR -------------------------------- 1-2
1.2.1 Description of a PWR core from a viewpoint
of in-core fuel management -------------------------------------------- 1-2
1.2.2 Design process of a fuel loading pattern --------------------------- 1-4
1.2.3 Impact of a loading pattern on the core characteristics ------- 1-4
1.2.4 In-core and ex-core fuel managements ----------------------------- 1-6
1.3 Descriptions of the loading pattern optimization problem ------------- 1-7
1. 3.1 Features --------------------------------------------------------------------- 1-7
1.3.2 Objectives and Constraints -------------------------------------------- 1-8
1.3.3 Traditional Approach ---------------------------------------------------- 1-12
1. 3.4 Advanced Approach ------------------------------------------------------ 1-16
1.4 Purpose of this thesis -------------------------------------------------------------- 1-17
1.5 Contents of this thesis ------------------------------------------------------------ 1-19
REFERENCES FOR CHAPTER 1 ------------------------------------------------- 1-22

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CHAPTER 2. A QUANTITATIVE COMPARISON OF LOADING PATIERN
OPTIMIZATION METHODS FOR IN-CORE FUEL MANAGEMENT
OFPWR
2.1 Introduction -------------------------------------------------------------------------- 2-1
2. 2 Optimization Methods ------------------------------------------------------------ 2-3
2. 2.1 Simulated Annealing Method ---------------------------------------- 2-3
2.2.2 Direct Search Method --------------------------------------------------- 2-5
2.2.3 Binary Exchange Method ---------------------------------------------- 2-5
2.2.4 Genetic Algorithms Method ------------------------------------------- 2-6
2.2.5 Hybrid Search Method-------------------------------------------------- 2-8
2. 3 Calculations -------------------------------------------------------------------------- 2-9
2.3.1 Benchmark Problem ---------------------------------------------------- 2-9
2.3.2 Optimization Calculations -------------------------------------------- 2-10
2.3.3 Results and Discussion ------------------------------------------------ 2-12
2. 4 Conclusions -------------------------------------------------------------------------- 2-15
REFERENCES FOR CHAPTER 2 ------------------------------------------------ 2-17

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CHAPTER 3. LOADING PATTERN OPTIMIZATION USING HYBRID GENETIC
ALGORITHMS
3.1 Introduction ------------------------------------------------------------------------- 3-1
3.2 Optimization Method ------------------------------------------------------------ 3-2
3.2.1 Genetic Algorithms ----------------------------------------------------- 3-2
3.2.2 Application of Genetic Algorithms to
Loading pattern Optimization -------------------------------------- 3-3
3.2.3 Development of the GALLOP Code ------------------------------- 3-4
3. 3 Calculations ------------------------------------------------------------------------ 3-5
3.3.1 Single Cycle Optimization Benchmark -------------------------- 3-5
3. 3. 2 Results and Discussion ----------------------------------------------- 3-7
3. 4 Conclusions ------------------------------------------------------------------------- 3-8
REFERENCES FOR CHAPTER 3 ------------------------------------------------ 3-10

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CHAPTER 40 INSIGHT: AN INTEGRATED SCOPING ANALYSIS TOOL FOR
IN-CORE FUEL MANAGEMENT OF PWR
401 Introduction ------------------------------------------------------------------------ 4-1
402 Software Environment for Developing INSIGHT ---------------------- 4-2
4 0 3 INSIGHT Methodology --------------------------------------------------------- 4-3
40301 System Overview ------------------------------------------------------- 4-3
40302 Loading Pattern Optimization Module(GALLOP) ----------- 4-5
40303 Interactive Loading Pattern
Design Module (PATMAK~R) -------------------------------------- 4-7
403.4 Multicycle Analysis Module (MCA) ------------------------------- 4-8
40 30 5 Integrated Database --------------------------------------------------- 4-9
4.4 Applications ------------------------------------------------------------------------ 4-10
4.401 Single Cycle Loading Pattern Optimization ------------------- 4-10
4.402 Multicycle Loading Pattern Optimization ---------------------- 4-12
4 0 5 Conclusions ----------------------------------------------------------------------- 4-14
REFERENCES FOR CHAPTER 4 ----------------------------------------------- 4-16

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CHAPTER 50 COMPARISON BETWEEN EQUILIBRIUM CYCLE AND
SUCCESSIVE MULTICYCLE OPTIMIZATION METHODS FOR
IN-CORE FUEL MANAGEMENT OF PRESSURIZED WATER
REACTORS
5.1 Introduction ------------------------------------------------------------------------- 5-1
50 2 Optimization Methods ----------------------------------------------------------- 5-4
5.201 Equilibrium Cycle Optimization Method ------------------------ 5-4
5.2.2 Successive Multicycle Optimization Method ------------------- 5-6
50 3 Calculations ------------------------------------------------------------------------- 5-7
5.3.1 Definitions of Benchmark Problem -------------------------------- 5-7
5.3.2 Optimization Calculations ------------------------------------------- 5-9
5_3.3 Results And Discussions ---------------------------------------------- 5-11
5.4 Conclusions -------------------------------------------------------------------------- 5-15
REFERENCES FOR CHAPTER 5 ------------------------------------------------ 5-18

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CHAPTER 6. CONCLUSIONS AND A FUTURE VIEW----------------------- 6-1
ACKNOWLEDGEMENTS

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CHAPTER 1
INTRODUCTION
1.1 Background
Commercial nuclear reactors haveJ\increasing the i~portance in the role of
power generation during past few decades. Actually, the nuclear power accounts for
approximately 30% of the Japanese electric power supply, today. Since there is little
energy resources in Japan, the nuclear power is considered to be one of the stable
and indispensable energy sources.
The cost of nuclear power remains lower compared with that of other sources,
such as the fossil or hydropower. However, due to improvements in the technology of
utilizing the fossil power, especially in an advanced combined cycle (ACC)
technology using liquid natural gas (LNG), the superiority of nuclear in the cost of
power generation is becoming smaller. To keep the nuclear power competitive,
reduction of the power generation cost is desirable.
The cost of nuclear power mainly consists of the nuclear fuel cost, the capital
cost, the maintenance cost and the replacement power cost during the inspection
period. So there are several ways to reduce the cost of nuclear power. In this thesis,
improvements of in-core fuel management methods, which can reduce the nuclear
fuel cost, will be discussed. Note that the nuclear fuel cost amounts to
approximately 20% in the whole nuclear power cost as shown in Table 1-1 (l) (
2
). Since
the capital cost, which corresponds to the cost of plant construction, is fixed,
improvement of the nuclear fuel cost is considered to be one of the important point
to reduce the cost of nuclear power generation.

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The nuclear fuel cost essentially depends on reactor types, specifications of fuel
1-1
assemblies, the in-core fuel management and reactor operating methods. One of the
major tasks of in-core fuel management is to determine a fuel loading pattern. The
fuel loading pattern is an arrangement of fuel assemblies in a reactor core, and its
design is based on a quite complex combinatorial optimization as will be discussed
later in this chapter. Though the loading pattern optimization is one of the primary
factors to improve the fuel cycle cost, it was difficult to fmd practical solutions
because of its stiff and complicated nature. Therefore, engineers, who are
responsible to the in-core fuel management, optimize the loading pattern in every
cycle using their state-of-art techniques. Since the optimizations by engineers are
mainly based on a trial-and-error approach, a practical limit exists on their
optimization capabilities.
Recently, high performance computers, such as engineering workstations
(EWS) or personal computers (PCs), are being widely used and the calculation cost
is rapidly decreasing. Thanks to these powerful computers, practical and robust
optimization theories are being developed and being applied for many industrial
problems<3
).

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For these backgrounds, this thesis treats optimization problems of the fuel
loading pattern and its applications to . the practical in-core fuel management,
aiming to reduce the nuclear fuel cost. The target reactor type in this thesis is a
pressurized water reactor (PWR), which shares around a half of the commercial
reactors in Japan.

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1.2 Basics of in-core fuel management for PWR
1.2.1 Description of a PWR core from a viewpoint of in-core fuel management
APWR core can be considered as an array of fuel assemblies from a viewpoint of
neutronics design. Figure 1-1 shows a PWR core of Westinghouse three-loop type
1-2
that contains 157 fuel assemblies. A fuel assembly is consists of fue l pins, which are
arranged in a square grid of 17 by 17 as shown in Fig. 1-2. The dimension of the fuel
assembly is approximately 21cmx21cmx400 em (depth x width x height). Note that
the fuel assembly also includes guide thimbles for control rods (rod cluster control,
RCC) and instrumentation thimble for measurement of in-core power distribution.
A thermal output of the Westinghouse three-loop type PWR is 2650MW, so the
fuel assembly generates about 17MW when the power density of the fuel assembly
is equal to an average value in the reactor. Note that, since the assembly power
density depends on its reactivity and the fuel loading pattern as will be discussed in
Sec. 1.2.4, the actual power density is different in each assembly. Here, the
reactivity of the fuel assembly is mainly depends on its cumulative thermal output;
the burnup.
In accordance with the current Japanese regulations, the periodical inspection
is performed almost once a year. During this inspection, about one-third of burnt
fuel assemblies are discharged from the core as spent fuel assemblies and the fresh
fuel assemblies are loaded into the core instead of the discharged fuels. This kind of
refueling strategy is called as a multi-batch loading. The concept of the multi batch
loading is shown in Fig. 1-3 for reference.

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Since the 235U enrichment of fresh fuel assemblies (feed enrichment) is fixed in
Japan, the number of fresh fuel assemblies is determined according to the expected
operating length of next cycle. The discharged fuel assemblies are usually selected
according to their burnup; well-burnt fuels are discharged with a higher priority.
As mentioned above, the neutronics characteristics of the fuel assemblies
depend on not only the specification of fuel assemblies such as the 235U enrichment
or the content of burnable poisons, but also their burnup. Therefore, various fuel
assemblies with different neutronics characteristics are loaded into the core under
1-3
the multi-batch refueling strategy as shown in Fig. 1-4.

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1.2.2 Design process of a fuel loading pattern
The outline of the in-core fuel management for PWR is shown in Fig. 1-5. When
the numbers of fresh fuel assemblies and burnt fuel assemblies are fixed, a core
designer can define a fuel loading pattern. Since the nuclear characteristics of
individual burnt fuel assemblies are different, number of possible patterns for the
fuel arrangement in the core accounts for enormous one. For example, Table 1-2
shows a typical enumeration number of fuel arrangements in the Westinghouse
type four-loop reactor, which contains 193 fuel assemblies.
During the loading pattern design process, the designer often assumes octant or
quarter core symmetry shown in Fig.1-1 and several empirical rules (heuristics)
about the loading pattern to reduce the possible combination number.
Unfortunately, the enumeration number is still too large to calculate all possible
loading patterns.
The safety concern is a most important point for .the characteristics of the fuel
loading pattern. In order to satisfy the safety criteria, various constraints should be
considered during the loading pattern design process as will be discussed in Sec.
1.3.2. Since the economic aspect is also important in the practical application,
various objectives should be taken into account to maximize the fuel utilization
efficiency as will be discussed in Sec. 1.3.2.

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1.2.3 Impact of a loading pattern on the core characteristics
In the boiling water reactor (BWR), the in-core power distribution can be
adjusted by the control rods, which are inserted into the gap between assemblies
from the reactor bottom to top. Though PWR also has rod cluster controls (RCCs),
1-4
they are generally used to shutdown the reactor; RCCs are not used to control the
in-core power distribution.
Consequently, the power distribution in the PWR core essentially depends on
the fuel loading pattern itself. Since the power distribution is a key parameter for
the reactor core safety and the economics, the loading pattern is considered to be the
most important point in the core design.
For reference, Fig. 1-6 shows three different PWR cores that consist of the same
fuel inventory; only the fuel loading pattern is different among these cores. The (a)
core violates the current safety limit for the radial peaking factor, which should be
less than 1.480. Since the radial peaking factor affects to the maximum pellet
temperature and so on, the integrity of the fuel pin in the (a) core cannot be
guaranteed. The (b) core satisfies the safety criteria on the peaking factor, so the (b)
core is acceptable. The (c) core also satisfies the safety criteria on the peaking factor.
Moreover, the discharge burnup of the (c) core is much higher than that of the (b)
core.

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The discharge burnup is a cumulative thermal output of spent fuel assemblies,
hence a higher discharge burnup indicates an effective usage of nuclear fuel. For
example, a 5% difference in the discharge burnup is estimated to result in hundreds
millions ofYens in the fuel cost per a cycle.
It should be noted that, to simplify the discussion, only the radial peaking factor
is considered as the constraint in the above explanation. However, in the practical
loading pattern design, various constraints such as limitations on the maximum
fuel assembly burnup, the moderator temperature coefficient, the reactor shutdown
margin and so on, must be considered. Hence the situation of the actual loading
pattern design is much more complicated.
The observation described above reveals the importance of optimization on the
1-5
fuel loading pattern. Namely, s1nce the fuel loading pattern greatly affects the
safety and the economics of reactor cores, its optimization becomes quite important
from the industrial point of view<
4
>.

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1.2.4 In-core and ex-core fuel managements
The multi-batch loading can attain the higher discharge burnup compared to
the single-batch loading<
5
>. However, the multi-batch loading makes the loading
pattern optimization more complex one in two aspects. At first, since every burnt
fuel assembly has different neutronics characteristics, the enumeration number of
possible fuel placements in a core becomes enormous one as mentioned in Sec. 1.2.2.
Secondary, since a fuel assembly stays in-core during several operating cycles,
coupling effects between consecutive cycles should be considered.
For example, in the single-batch loading, all fuel assemblies are discharged at
the end of cycle. This means that each cycle is independent; there is no interference
between successive cycles. Therefore a designer can apply an identical fuel loading
pattern in every cycle, when the cycle lengths are the same among these cycles. On
the other hand, fresh fuel assemblies should be loaded with burnt fuel assemblies in
the multi-batch loading. Since the burn up of fuel assemblies depends on the loading
patterns of previous cycles, the designer should consider the loading patterns of
successive several cycles simultaneously.

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Unfortunately, the loading pattern optimization problem of multiple cycles is
far beyond the ability even for the latest computers. So the current fuel
management is mainly divided into two parts, which are an in-core fuel
management and an ex-core fuel management, by omitting cycle by cycle coupling
effect. The in-core fuel management is mainly responsible for a design of a loading
pattern and a series of follow-up calculations of a core during the reactor operation.
1-6
On the other hand, the ex-core fuel management is mainly responsible for defining
an operating cycle length and selecting the discharge fuels, and so on.
This thesis addresses not only to the loading pattern optimization for a single
cycle that is included in the in-core fuel management, but also to the optimization
for the multiple cycles that is included in both the in-core and the ex-core fuel
managements.

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1.3 Descriptions of the loading pattern optimization problem
1.3.1 Features
Features of the loading pattern optimization problems are summarized as
follows:
(1) It is one of the combinatorial optimization problems since the loading pattern
defines combination of individual fuel and loading position in the core. In the
combinatorial problem, a quality of a solution, which is represented by an
objective value, is discrete among solutions. Therefore, gradient information of
objective values, which can indicate a direction to improve a solution, is very
hard to be obtained exactly. Hence the combinatorial optimization problem is
considered to be more difficult than that of the continuous functions. Note that
the objective value is evaluated by core characteristics using an objective
function in the loading pattern optimization.
(2) AB discussed in Sec.1.2.2, the enumeration number of the loading patterns
reaches really an astronomical one .
(3) The nature of objective values has non-linearity; an objective value cannot be
obtained by the superposition of the other objective values. For example, let's
consider the assembly exchange shown in Fig. 1-7. Perturbation on the power
1-7
distribution caused by the assembly exchanges of (a) cannot be obt ained by
adding the perturbation by the exchange (b) and that by the exchange (c).
Consequently, core calculations must be performed for all candidate patterns.
Therefore, the calculation time becomes incredibly longer.

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(4) Since the nature of objective values has non-linearity, there are many local
optima in the solution space. Figure 1-8 shows the concept of the local optima
and the global optimum. Ordinary optimization methods such as liner
programming cannot escape from local optima, since their search progress
according to the gradient of the objective value in the solution space. Figure 1-9
provides behavior of the objective value on the actual loading patterns, which
will treat at a benchmark calculation in Chap. 3. From Fig. 1-9, a complex
structure of the objective value in the solution space and many local optima can
be observed.
(5) Since the nuclear reactor has a potential of large hazard, the safety concern has
a top priority in the design process. Therefore, various constraints should be
satisfied during the loading pattern design, as will be discussed in the next
section.
From the above reasons, the loading pattern optimization problem is considered to
be quite difficult.

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1.3.2 Objectives and Constraints
In order to maintain the safety of nuclear reactors, various restrictions should
be considered during the loading pattern design as mentioned in the previous
sections. Major restrictions on the neutronics characteristics are as follows:
1-8
(1) Limitation on the radial peaking factor, which is defined by the relative peak
power of a fuel rod in the core. Here the power of a fuel pin is axially integrated.
The limitation on the radial peaking factor is settled to prevent the fuel failure
caused by DNB. Since the radial peaking factor greatly depends on the fuel
loading pattern, this restriction is considered as a primary target during the
loading pattern design.
(2) Limitation on the maximum linear heat generation, which is the heat generation
per a unit length of a fuel rod. This restriction is closely related to the maximum
pellet temperature at transient incidents or loss-of-coolant accidents. In order to
obtain the maximum linear heat generation, an axial power distribution is
necessary. However, since two-dimensional core calculations are usually adopted
for the loading pattern optimization because of the computation time, the
maximum linear heat generation cannot be estimated directly. Fortunately, when
the restriction on the radial peaking factor is satisfied, the restriction on the
maximum linear heat generation is usually satisfied. So this limitation is not
considered in the most calculations of the loading pattern optimization.

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(3) Limitation on the maximum burnup of a fuel assembly, which is cumulative
thermal output per unit heavy metal inventory. Since the corrosion and the
neutron irradiation reduce the strength of fuel clad as increasing burnup, the
limitation on the maximum burnup must be taken into account. The fuel burnup
also much depends on the fuel loading pattern, so the great care should be paid
when the margin to the limit of maximum burn up is small.
(4) Limitation on the moderator temperature coefficient (MTC), which is reactivity
change due to the perturbation on the moderator temperature. To maintain the
inherent safety, the reactor core must have a negative reactivity feedback; the
negative reactivity must be inserted intrinsically when the reactor power
1-9
increases. In Japan, MTC must be negative throughout the power operation. In
general, the longer cycle operation increases the boron concentration at the
beginning of cycle and MTC tends to move toward a positive value. Though MTC
depends on the fuel and burnable poison inventory, it also depends on the fuel
loading pattern. Therefore, MTC should be considered at the loading pattern
design, especially in the longer cycle.
(5) Limitation on the radial power tilt, which represents power imbalance along the
symmetric line shown in Fig. 1-1. Since the radial power tilt increases the radial
peaking factor, it should be reduced as lower as possible. Fortunately, the radial
power tilt becomes significant only for loading of asymmetric fuel pairs in the
core. The radial power tilt should be taken into account in such a case.

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(6) Limitation on the reactor shutdown margin (SDM), which means subcriticality
of the core at the one-rod stuck configuration. Here the one-rod stuck
configuration means all RCCs are fully inserted in the core except one RCC that
has the largest reactivity worth. The shutdown margin guarantees the
subcriticality during the transient of cooling accidents such as the steam line
break. Note that when the core is cooled because of some accidents, the positive
reactivity will be inserted because of the negative MTC of the core. Though the
SDM can be estimated by the two-dimensional core calculation, it requires fullcore
calculation to simulate the one-rod stuck configuration accurately.
Furthermore, many stuck rod configurations should be calculated even if the core
symmetry is considered. Note that the ordinary calculations are performed for
the quarter core geometry using the core symmetry. Since the SDM calculation
requires much computation time, it is difficult to incorporate the SDM
calculation into the loading pattern optimization. Though SDM can be partially
controlled by some heuristic rules on the fuel loading pattern, e.g. the burnt fuel
1-10
should be placed at the stuck rod position, its estimation is still one of the major
issues in the loading pattern optimization problem.
(7) Limitation on other safety parameters, which are necessary to execute accident
analyses, should be taken into account. For example, the uncontrolled RCC
withdrawal, the RCC drop, the RCC ejection, and so on, must be analyzed and
safety parameters should be confirmed. Fortunately, these parameters are
usually adequate when the limitation on the radial peaking factor and SDM are
satisfied. Therefore, these parameters are not usually included in the loading
pattern optimizations as the constraints.

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(8) Other limitations from plant structures must be considered sometimes. For
example, Nuclear Regulatory Committee (NRC) recommended the burnup
limitation at the RCC positions recently. This is a countermeasure for the incomplete
RCC insertion incident observed in some of reactors in USA, as will be
discussed in Chap. 4. Another example is related to the neutron irradiation for
the reactor vessel. Since the ductility of the reactor vessel decreases as
increasing neutron irradiation, a loading pattern that reduces the neutron
irradiation for the vessel is desirable. This is particularly important in the old
reactors whose vessel has much irradiated by neutrons.
Major objectives in the loading pattern optimization are as follows:
(1) Maximization of the cycle length under the fixed feed enrichment and the fixed
number of fresh fuel.
(2) Maximization of the discharge burn up under the fixed feed enrichment and the
fixed number of fresh fuel.
(3) Minimization of the number of fresh fuel under the fixed cycle length and the
1-11
fixed feed enrichment.
(4) Minimization of the enrichments under the fixed cycle length and the flXed
number of fresh fuel.

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Here the objectives mean that these values are not treated as constraints, but
the loading pattern should be designed to maximize or minimize these values from
the economical point of view.
In Japan, the feed enrichment is usually fixed to a unique value, so the
objective (4) is out of consideration. The objectives (1) and (3) have almost the same
context, though the objective (2) has somewhat different meanings on the loading
pattern design for the single cycle. The reason is the trade-off relationship between
the cycle length and the discharge burnup. For example, when the cycle length is
maximized, the discharge burnup tends to become lower in the single cycle
optimization.
This trade-off makes interesting results in the multiple-cycle loading pattern
optimization, as will be discussed in Chap. 5.

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1.3.3 Traditional Approach
Since the loading pattern optimization is highly required from the safety and
economical points of view, many researchers have studied various approaches
during the past few decades. Though an well-organized historical review of the
loading pattern optimization is provided in Ref. (6), some of the major approaches
are described here to make an image about the loading pattern optimization.
The linear programming is a popular optimization method and widely applied
to the industrial purpose. The optimization by linear programming is formulated as
follows:
1-12
where
0 : Objective value that represents the core performance.
all' : Sensitivity parameter of the objective value caused by the assembly exchange
between I and /'.
Xu. :Flags of assembly swap between I and 1'. When the exchange between I and I'
exsists, Xu. becomes 1, otherwise 0.
The XII. can be determined from Eq.(l) under several constraints. Once XII.
has determined, the optimum loading pattern is obtained by exchanging fuel
assemblies according to the flag represented by XII, .
Though the linear programming is simple, it has several difficulties for
applying it to the loading pattern optimization problems as follows:

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(1) The linear programming requires sensitivity parameters for the optimization.
Therefore, enormous sensitivity parameters should be calculated when a problem
treats various constraints. For example, sensitivity coefficients for the radial
peaking factor and the maximum burnup are different, so these coefficients
should be prepared before the optimization.
(2) The linear programming can treat the "linear" problem that allows the
superposition of the solutions. Unfortunately, the loading pattern optimization
problem has non-linearity, so the linearization is required. By applying the
linearization, the obtained "optimum solution" is no more the optimum exactly
because of the approximation error. To minimize this approximation error, a
successive linear programming is often used. However, the since successive
linear programming requires many sensitivity coefficient calculations, the
calculation time tends to become much longer.
(3) The linear programming is based on the gradient projection; the search
1-13
progresses only to improve the objective value. Therefore, the linear
programming cannot accommodate the multi-modality that is essential in the
loading pattern optimization.
The direct search method, which is based on the multiple shuffles of fuel
assemblies, is also studied by several researchers. The direct search method is
applied to a loading pattern optimization problem as follows:

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(a) Assume an initial solution,
(b) Swap fuel assemblies randomly or according to the heuristic rules to generate a
candidate of solution,
(c) Estimate the objective value of the candidate,
(d) If the candidate improves the objective value, the swap is accepted. Otherwise
the swap is rejected and another swap is tried,
(e) Repeats the procedure from (b) to (d) until the improvement of the objective
value converges.
The direct search method does not require the sensitivity coefficient, but the
core characteristics should be estimated for every loading pattern. Since the direct
search method only accepts the improved solutions during the search process, it
cannot treat the multi-modality. Detailed discussion about this point will be
performed in Chap.2.
Since the engineer well optimizes the loading pattern using their state-of-art
knowledge, the artificial intelligence, which performs the search based on the rule
sets, was considered to be efficient for this problem. Examples of the rule set are as
1-14
follows:
(1) High reactivity fuels and low reactivity fuels should be placed side by side in the
interior of the core to reduce the power peaking factor.
(2) A fresh fuel that does not contain burnable poisons is not loaded in the interior of
the core; it should be loaded to the core periphery.

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The artificial intelligence performs quite well under the typical situations that
can be accommodated by its "knowledge". The "knowledge", however, sometimes
disturbs the optimization. For example, the reactivity of a burnt fuel assembly is
low, so it is often loaded besides a high reactivity fuel in the interior of the core to
suppress the power peaking factor. Therefore, a loading pattern optimization code
that is implemented with this knowledge will put the burnt fuel assembly in the
interior of the core. But when the cycle length is too long, the well-burnt fuels
should be placed at the core periphery to satisfy the limitation of the maximum
burnup. Note that the power density of the fuel at the core periphery tends to be
lower because of the neutron leakage from the core.
The above example suggests that the appropriate rule set is highly sensitive to
the miscellaneous analysis conditions such as the cycle length, the inventory of the
fuel assemblies, and so on. To respond almost all situations that include exceptional
cases, enormous rule sets should be implemented. Moreover, new rule sets should be
implemented when the new-type fuel assemblies are loaded. However, the
development and implementation of rule sets are considered to require an extensive
work. Consequently, the artificial intelligence lacks the robustness that indicates
the toughness of the method; the artificial intelligence cannot perform optimization
under unexpected situations.

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1.3.4 Advanced approach
In the previous section, some traditional optimization methods were briefly
reviewed. However, these methods have inherent shortcomings that are difficult to
overcome in practical applications. Therefore, the loading pattern optimization had
remained in the academic area for a long time.
Recently, thanks to the rapid improvement of the computer hardware, emergent
computing methods are gaining their admiration<3
) (7). They provide the selforganized
behaviors from estimations for local situation; it does not utilize the
knowledge or rules that controls an overall system. The emergent computing
methods are based on a completely different approach from most of the traditional
ones. The traditional methods utilize so-called top-down approach that defines the
behavior of system from the top to bottom. On the other hand, ·emergent
computations define their behaviors from the bottom to top.
The relation between the traditional optimization and the emergent
computation corresponds to the relation between the neutron transport calculation
based on the Boltzman equation and that by the Monte Calro method. In the
emergent computation methods, Genetic algorithms and simulated annealing
methods are the popular ones.

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Since the emergent computations utilize only estimations of the local situation,
it is quite robust. For example, consider a problem about the burnt-fuel discussed in
the previous section. The artificial intelligence approach defines the fuel loading
pattern according to the rule sets and the information such as the cycle length or
the fuel inventory. In the emergent computation, a candidate obtains a good
objective value if the maximum burn up does not violate the limitation, otherwise it
obtains a bad value. The search progresses utilizing these objective values.
1-16
Therefore, the emergent computation can easily respond various constraints or
objectives. A detail description of the emergent computation will be provided in
Chap. 2.
Note that since the emergent computations require many trial calculations to
find out the optimum or the near-optimum solution, the computation time tends to
become longer. However, progresses on the computer hardware make such
calculations feasible.
Recently, the emergent computations such as the simulated annealing were
applied to the loading pattern optimization problems and showed good
feasibility<s).(
9).
The development of an efficient optimization method for the loading pattern
using the emergent computation methods is one of the major topics in this thesis
and will be discussed in Chap. 2.

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1.4 Purpose of this thesis
The efficient utilization of nuclear fuel is important to reduce the cost of nuclear
power and to save the limited Uranium resources. As mentioned in the previous
sections, the fuel loading pattern, which means the fuel arrangement in the core,
has a quite large impact on the core performance from viewpoints of safety and
economics. Therefore, there are strong needs for its optimization from an industrial
point of view.
Since the loading pattern optimization is a stiff problem, the traditional
optimization methods cannot cope with it. However, the emergent computations,
which are based on the stochastic process, can make a breakthrough for this
problem. Though the emergent computations are robust and applicable to practical
problems, it takes a much computation time. Therefore, the first purpose of this
1-17
thesis is:
(1) To develop a more efficient optimization method for the fuel loading pattern
Here the optimization method should be robust and applicable for the practical
and industrial-scale problems

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Since the in-core fuel management requires various kind of data, a software
workbench to manage these data is highly required especially in the multicycle
analysis. There are several software tools for the in-core fuel management, though
they were mainly designed to perform optimization for the single cycle. AB
mentioned in Sec. 1.2.4, an accurate analysis and optimization for the multiple
cycles are important to reduce the overall fuel cycle cost. Therefore, second purpose
of this thesis is:
(2) To develop a software tool for the in-core fuel management that has capability to
treat loading pattern optimizations for successive multicycles Furthermore the
software should be user-friendly for practical applications

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Though the true multicycle optimization that is necessary to perform
simultaneous optimizations of the multiple cycles is far beyond the ability of to day's
computers, it is important to evaluate the effect of loading patterns for the whole
multicycle performance . In the loading pattern optimization for a single cycle, the
low-leakage loading pattern was considered to be desirable because it could attain a
longer cycle length than the traditional out-in loading pattern. However, as
mentioned in Sec. 1.3.2, the discharge burnup tends to become lower in the lowleakage
loading pattern. Namely, a trade-off exists between the cycle length and the
1-18
discharge burn up. Therefore, the third purpose of this thesis is:
(3) To examine a desirable loading pattern strategy for the overall fuel cycle cost by
evaluating the trade-off between the low-leakage strategy and the highdischarge
burnup strategy.
These problems will be discussed through Chap.2 to Chap.5.

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1.5 Contents of this thesis
This chapter provides basic concepts of the in-core fuel management for PWRs
and a conceptual description of the loading pattern optimization problem for the
better understanding about this thesis. This chapter also points out the
backgrounds and the purposes of this thesis.
Chapter 2 describes the development and verification of loading pattern
optimization methods<10). AB described in Sec. 1.3, the loading pattern optimization
is a quite difficult problem, and requires novel optimization theories to make a
breakthrough for the practical use. The genetic algorithm is applied to this problem
since it is considered to be quite suitable to a complicated combinatorial
optimization problem. The genetic algorithm is one of the optimization techniques
whose concept is based on the evolution of life. According to the Darwinian theory,
the survival of fittest and the crossover of chromosomes promote the life towards
the adaptive direction. For the loading pattern optimization, a candidate of the
loading pattern is treated as an individual, and the performance of the individual,
which is defined by the core characteristics such as the cycle length and the radial
peaking factor, defines the probability of survival or mating.

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Though the genetic algorithm is powerful for the global search, its local search
capability is rather poor. Therefore, a new hybrid optimization method, which
combines genetic algorithms and the local search method, will be proposed. To verify
the capability of the proposed optimization method, the quantitative comparison
with other optimization methods such as the direct search or the simulated
annealing, is performed and optimization results are compared with each other.
Chapter 3 provides calculated results of a benchmark problem using the hybrid
optimization method that is proposed in Chap. 2. In the combinatorial optimization
problem, a guarantee of the optimum solution can be obtained only through the
exhaustive enumeration calculations. However, since the enumeration number
accounts for an enormous one in the practical optimization problem, it is difficult to
confirm the optimization capability that can find out the optimum solution.
A benchmark problem is set up assuming restrictions about the fuel placement
in the core and simplifications of the fuel inventory. By evaluating all loading
patterns in the benchmark problem, an optimum solution is identified. After that,
the hybrid optimization method is applied to the benchmark problem and its
capability is examined.

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Chapter 4 describes the development and applications of INSIGHT, which is a
software for the practical in-core fuel management0 1). The hybrid optimization
method for the fuel loading pattern is used as a key technology in INSIGHT.
In the practical in-core fuel management, various data should be treated; these
include the power distribution from the core calculation, the fuel loading history, the
fuel burnup, and so on. Therefore, an integrated software, which realize a seamless
management of these data, is highly desired especially in the multicycle analyses.
1-20
INSIGHT includes capabilities of the core calculation, the loading pattern
optimization, the management of database, and so on.
INSIGHT is applied to the practical problem of in-core fuel management, and
its capability is tested.

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Chapter 5 describes the loading pattern optimization considering the coupling
effect in the multiple cycles02>. As mentioned in Sec.l.2.4, the fuel assemblies are
loaded during several cycles. Though the optimization for the multiple cycles is
desirable to reduce the overall fuel cost, it is far beyond the ability of today's
computers, as mentioned before. Therefore, a cycle by cycle successive optimization
is often used in the current in-core fuel management.
In Chap. 5, the equilibrium cycle is treated because it can be considered as the
special case of the multiple cycle. Namely, the optimization of the equilibrium cycle
is considered almost compatible with that of the multiple cycles.
A new optimization method for the equilibrium cycle is proposed s1nce the
optimization is somewhat complicated than that of the single cycle. Using the new
optimization method and the successive multicycle optimization method developed
in Chap. 4, strategy for the multiple cycle optimization is investigated.
Chapter 6 summarizes the results obtained through the course of this study,
and a future view of this study is also provided.

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REFERENCES FOR CHAPTER 1
(1) "The Economics of the Nuclear Fuel Cycle," OECD/NEA, Paris, (1994).
(2) H. W. Graves, Jr., "Nuclear Fuel Management," John Wiley & Sons, Inc., New York,
(1979).
(3) "Novel Optimization Methods and Its Application," Japan Society of Mech. Eng., 930-
59, (1993).
(4) A. Yamamoto, et al., "New or Improved Computational Methods and Advanced
Reactor Design," Journal of the Atomic Energy Society of Japan, 39, 10 (1997), in
Japanese.
(5) M. J. Driscoll, et al., "The Linear Reactivity Model for Nuclear Fuel Management,"
American Nuclear Society, Illinois, (1990).
(6) J. G. Stevens, "A Hybrid Method for In-Core Optimization of Pressurized Water Reactor
Reload Core Design," Dr. Thesis, Purdue Univ., (1995).
(7) J. H. Holland, "Adaptation in Natural and Artificial Systems," MIT Press, London,
(1992).
(8) D. J. Kropaczek and P. J. Turinsky, "In-Core Fuel Management Optimization for
Pressurized Water Reactors Utilizing Simulated Annealing," Nucl. Techno/., 95, 9
(1991).
(9) P. W. Poon and G. T. Parks, "Application of Genetic Algorithms to In-Core Fuel
Management Optimization," Proc. Topical Mtg. Math. Methods and Supercomputing
for Nuclear Applications, Karlsruhe, Germany, 2, 777 (1993).
(10) A. Yamamoto, "A Quantitative Comparison of Loading Pattern
Optimization Methods for In-Core Fuel Management of PWR," J. Nucl. Sci.
Techno/., 34, 339 (1997).

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(11) A. Yamamoto, et al., "INSIGHT:An Integrated Scoping Analysis Tool for InCore
Fuel Management of PWR," J Nucl. Sci. Techno!., 34, 84 7 (1997).
(12) A. Yamamoto and K. Kanda, "Comparison Between Equilibrium Cycle and
Successive Multicycle Optimization Methods for In-Core Fuel Management of
Pressurized Water Reactors," J Nucl. Sci. Techno!., 34, 882 (1997).

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Table 1-1 Breakdown of typical nuclear power generation cost
Table 1-2 Enumeration number of the fuel loading pattern for Westinghouse 4
loop PWR
Note1:The number of fresh fuel assemblies is assumed to be 72, including 48
Gadolinia bearing fuels.
Note2:The calculation time is assumed to be 1 second/pattern.
Fig. 1-1 A fuel arrangement ofPWR core; Westinghouse three loop type
Fig. 1-2 Cross-sectional view of a fuel assembly; 17x17 type
Fig. 1-4 A fuel loading pattern under the multi-batch loading strategy
Fig. 1-5 An outline of the reactor operation and the in-core fuel management
Fig. 1-6 Different PWR cores consisted of same fuel inventory.
Fig.l-7 Effect of non-linearity on the fuel shuffling
Fig.l-8 Concept of local optima and the global optimum
Fig.l-9 Local optima in the actual problem of loading pattern optimization.

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