Study of Nuclear Reactor Reload Using Different Approaches of
Quantum Inspired Algorithms
Andressa dos Santos Nicolau and Roberto Schirru
Institute of Nuclear Engineering, Federal University of Rio de Janeiro, Ilha do Fundão, Rio de Janeiro, Brazil
Keywords: Nuclear Reactor Reload Optimization Problem, Quantum Delta-Potential Well based Particle Swarm
Optimization Algorithm, Quantum Inspired Algorithm, Quantum PBIL, Nuclear Power Plant.
Abstract: The purpose of this article is to show the performance of different approaches of quantum-inspired
algorithms as optimization tool of Nuclear Reactor Reload of Brazilian Nuclear Power Plant. Nuclear
Reactor Reload is a classical problem in Nuclear Engineering that has been studied for more than 40 years
that focus on the economics and safety of the Nuclear Power Plant. The main goal of this article is to show
the performance of Quantum Delta-Potential Well Based Particle Swarm Optimization Algorithm to solve
the Nuclear Reactor Reload compared with its classical counterpart Particle Swarm Optimization with
Random Keys method. Furthermore, others quantum inspired algorithms are also used to demonstrate the
feasibility of quantum inspired algorithms to solve cycle 7 of Brazilian Nuclear Power Plant Angra 1. The
results show that Quantum Delta-Potential Well Based Particle Swarm Optimization Algorithm found the
best result with less computational effort than its classical counterpart. Besides shows that quantum inspired
algorithm are well situated among the best alternatives for dealing with optimization problems that number
of evaluations is crucial due to the high computational cost of the evaluations, such as Nuclear Reactor
Reload.
1 INTRODUCTION
In the last few years, quantum-inspired algorithms
have been developed and gained attention both in
Physics, Mathematics, Computer Science and others
fields. These algorithms are based on different
theory of quantum mechanics and are created in
order to increase the performance and velocity of
traditional optimizations algorithms of the literature.
Nuclear Reactor Reload Optimization Problem
(NRROP) is a classical problem in Nuclear
Engineering that consists in replacing part of the
nuclear fuel of a Nuclear Power Plant (NPP).
Generally, the remaining elements of previous
cycles that can still be used are rearranged in the
reactor’s core and the remaining positions are filled
up with new fuel elements in order to provide
operation of the NPP at nominal power. However,
this is not a simple process. In the reactor core of
Angra 1 NPP, for example, 10
25
arrangements are
possible, making it impossible to verify all the
arrangements to determine the best one. Moreover,
NRROP presents high-dimensionality, large number
of feasible solutions, disconnected feasible regions
in the search space as well as high computational
cost of the evaluation function and lack of derivative
information. For decades, the NRROP was carried
out by specialists that used their knowledge and
experience to build configurations of the reactor core
to fulfill the requirements of the NPP.
The purpose of this article is to show the
performance of different approaches of quantum-
inspired algorithms as optimization tool of Nuclear
Reactor Reload of Brazilian Nuclear Power Plant.
The algorithm implemented in this study was
Quantum Delta-Potential-Well-based Particle
Swarm Optimization Algorithm (QDPSO) (J.Sun et
al, 2004). Besides we use the results found by others
quantum inspired algorithms such as Quantum
Evolutionary Algorithm (QEA) (Nicolau et al, 2012)
and Quantum PBIL (QPBIL)(Da Silva et al, 2011) to
show the performance of this kind of technique in to
solve hard optimization problems as NRROP.
QDPSO uses the philosophy of “collective learning”
of Particle Swarm optimization (PSO) (Kennedy and
Eberhart, 1995) and are inspired on different theory
of quantum mechanics. Uses quantum theory of
mechanics to govern the movement of swarm
302
Nicolau A. and Schirru R..
Study of Nuclear Reactor Reload Using Different Approaches of Quantum Inspired Algorithms.
DOI: 10.5220/0005141703020307
In Proceedings of the International Conference on Evolutionary Computation Theory and Applications (ECTA-2014), pages 302-307
ISBN: 978-989-758-052-9
Copyright
c
2014 SCITEPRESS (Science and Technology Publications, Lda.)
particles, thus the quantum state of a particle is
depicted by wave function instead of the velocity
and position functions which are in PSO. Inspired by
analysis of convergence of the traditional PSO,
assume that an individual particle moves in a Delta
potential well in search space, of which the center is
point p.
QEA is based on the most important concepts of
Quantum computation: Q_bits and interference of
quantum states. Different from QDPSO it uses the
philosophy of Evolutionary Computation, more
specifically on Genetic Algorithm (GA). QEA uses a
population characterized by a chromosome formed
by Q_bits, instead of a conventional binary
representation as GA. Unlike GA which uses, for
instance, the operator mutation and crossover, the
population in QEA evolves based upon a variation
operator known as Q-gate.
QPBIL is a new version of PBIL (Machado MD,
2005) that uses some basic concepts of quantum
computing: Q_bit and the linear superposition of
quantum states as QEA. In QPBIL as well as PBIL a
whole population is created every generation from
the vector probability distribution P. However, the
vector P of QPBIL consists of Q_bits.
This article is structured as follows: in the next
section, we will present a brief summary of QDPSO,
QEA and QPBIL algorithms compared with his
classical counterpart PSO, GA and PBIL
respectively. Section 3 describes the Nuclear
Reactor Reload Optimization Problem. Section 4
presented the computational results and in Section 5
is presented the conclusion of this study.
2 METHODOLOGY
2.1 QDPSO Algorithm
QDPSO belongs to the class of Quantum-inspired
algorithms that uses the philosophy of “collective
learning” of Particle Swarm optimization (PSO) was
proposed by Sun Jun, et al., 2004 and is based on the
quantum theory of mechanics to govern the
movement of swarm particles. Thus, the quantum
state of a particle is depicted by wave function
instead of the velocity and position which are in
PSO. According to the statistical significance of the
wave function, the probability of a particle’s
appearing in a certain position can be obtained from
the probability density function. And then the
probability distribution function of the particle’s
position can be calculated through the probability
density function. Inspired by analysis of
convergence of the traditional PSO, assume that an
individual particle moves in a Delta potential well in
search space, of which the center is point p.
In Quantum Mechanics, the state of a particle
with momentum and energy can be depicted by its
wave function
),( tx
. For this, in QDPSO each
particle is in a quantum state and is formulated by its
wave function
),( tx
instead of the position and
velocity which are in PSO. Thus, the probability of a
particle's appearing in a certain position can be
obtained from the probability density function
2
),( tx
. And then the probability distribution
function of the particle's position can be calculated
through the probability density function.
According to Sun Jun, et al., 2004, the wave
function of the particle is defined as:
L
|xp|
e
L
1
)x(
(1)
and the probability density function is defined by:
L
xp
e
L
xxQ
||
2
1
|)(|)(
(2)
where,
L
is the most important variable, which
determines search scope of each particle.
Due of quantum nature of these equations the
measurements using classical computers should
utilized the Monte Carlo method. The position of the
particle can be defined by:
)1,0(),/1ln(
2
randuu
L
Px
(3)
where,
u
is a randon number uniformly distributed
in (0,1). L is defined as L=(1/g)|x
k
-p| and g is a
parameter that is constrained by g=ln((2)
1/2
).
QDPSO procedure is described in Sun Jun, et al.,
2004.
2.2 QEA Algorithm
In QEA, similarly to Genetic Algorithm – GA,
where a genetic individual of the population is
represented by a string of bit, a quantum individual
(
i
q
) is represented by a string of Q_bit. We can say
that a Q_bit is a quantum representation of classical
bit, where a generic Q-bit
, might be represented
not by an exact representation, but by a linear
combination of the vectors
0
and
1
that assumes
the values 0 and 1 simultaneously.
In such way that,
10
, where, α and
StudyofNuclearReactorReloadUsingDifferentApproachesofQuantumInspiredAlgorithms
303
β are complex numbers that satisfy
1
22
.
The information stored in
is a combination of
all the possible states of
0
and
1
. And a set of N
Q-bits may be put in a superposition of 2
N
. But,
when
is measured, it is possible to find a
unique state, on the other words, it is possible to find
the state
0
with a probability
2
or the state
1
with a probability
2
.
The individual of the population is represented in
two distinct phases. In the first phase, it is fully
quantum, represented by a individual
)t(q
i
where
his chromosome consisting of Q_bits, and assumes a
superposition of states
0
and
1
. After
observation of quantum individual, creates a
classical individual
)(tX
i
represented by a classic
chromosome, which will be evaluated.
The population of solutions is represented by
)}t(q),...,t(q),t(q{)t(Q
n21
, where n is the
size of the population, m is the number of Q_bits,
and
)t(q
i
is the quantum chromosome defined by:
)(
)(
...
...
)(
)(
)(
)(
)(
2
2
1
1
t
t
t
t
t
t
tq
m
i
m
i
i
i
i
i
i
(4)
where,
1)t()t(
2
ij
2
ij
(5)
In this way, any
0
i
q can represent the linear
superposition of all the possible states with the same
probability. In addition, the linear superposition of
the Q_bits provides good diversity in the evolution
process.
Classic individual
)(tX
n
is derived from the
observation of quantum individual. This is a
characteristic of the evolutionary algorithms of
quantum computing adopting the theories of
quantum mechanics. The classical population
represented by:
)}t(X),...,t(X),t(X{)t(P
n21
(6)
and the candidate solutions
)t(X
i
with m bits,
which will be evaluated by the fitness function, are
represented by:
)]t(x),...,t(x),t(x[)t(X
im2i1ii
(7)
where )(tx
ij
is the observed bit.
The best candidate solution of
)t(P at each
iteration
t
is stored in )t(B , that is,
)]t(b)...t(b)t(b[)t(B
m21
, where
)t(b
j
represents the bits of the best solution.
Every bit of the binary string is obtained
observing the step for construction of the population
)t(P . When all the states of )t(Q are observed, the
value
0|)(|
tx
ij
or
1|)(| tx
ij
, from )t(P is
determined by the probability
²|)(| t
ij
. The
pseudo-code for production of
)t(P is according to
Nicolau et al, 2012.
Unlike GA, which uses for instance the operators
mutation and crossover, the population evolves
based upon a variation operator known as Q-gate.the
quantum gate operator, defined by the rotation
matrix
)(
ij
U
, which is applied to each one of the
columns of each individual’s Q_bit. In practice, each
pair of values
ij
and
ij
is treated as a bi-
dimensional vector and rotated using
)(
ij
U
in
such a way that
)(
)(
))(cos(
))((
))((
))(cos(
)1(
)1(
t
tsen
sent
t
ij
ij
ij
ij
ij
ij
ij
ij
(8)
with,
ijijijij
S
),()(
(9)
Where the sign function
),(
ijij
S
represents the
direction of rotation and the pass
ij
represents
the magnitude of the angle of rotation. Both
ij
and
),(
ijij
S
are obtained in accordance with
Nicolau et al, 2012.
2.3 QPBIL Algorithm
The QPBIL is a new version of the original PBIL
that uses some basic concepts of quantum computing
as QEA, like Q_bit and the linear superposition of
quantum states.
According to Da Silva et al, 2011, as well as
PBIL a whole population is created every generation
from the vector probability distribution P. However,
the vector P of QPBIL consists of Q_bits. Such
vector corresponds to the Q_bit individual described
by [2], and for this reason, it shares all the quantum
characteristics related to that one.
m
m
P
...
...
2
2
1
1
(10)
QPBIL acts in a binary space which turns it
ECTA2014-InternationalConferenceonEvolutionaryComputationTheoryandApplications
304
capable of solving optimization problems in
continuous search space. For this to be possible,
however, it is necessary to covert the notation of
Q_bits to conventional binary form, i.e, a string of
zeros and ones. This is done from the observation of
the probability distribution vector, P that generates
valid solutions shown in the form of binary strings.
To determine in which quantum state the Q_bit will
collapse, or what state it will be observed (0 or 1), a
random number is generated according to the
equation:
2
2
0
1
randif
randif
I
ij
(11)
where
ij
I is the jth bit of the ith individual.
The goal is to make the new generated
individuals to be increasingly look like the best
individual and decreasingly as the worst one. The
updating process is done by means of the quantum
rotation gate
).(
j
R
)cos()sin()sin()cos()(
jjjjj
R
(12)
where
j
represents how the Q_bits will approach
the best individual. This gate works as follows: first,
for each Q_bit j, is given a rotation that brings it to
the best individual is given. The process is showed
with more detail in Da Silva et al, 2011.
Then, a new quantum gate is used for the
application of the angle
j
in order to remove the
next gate generation groups of the worst individual
in accordance with Da Silva et al, 2011. Pseudocode
of QPBIL is described in Da Silva et al, 2011.
3 NRROP
Started after the operation of the plant, the
concentration of fissile material (U
235
) fuel elements
begins to decrease. After a time period, called
operation cycle, it is not possible to maintain the
NPP operating at the nominal power. The Fuel
Assemblies (FA’s) with low concentrations of U
235
are replaced by new fuel elements and along with
other FA’s of the previous cycle compose the core of
the subsequent cycle (Nicolau et al, 2012).
NRROP consists in searching for the best
loading pattern of FA’s in the core, aiming to
determine the permutation of FA’s that optimizes the
uranium utilization, with objective function
evaluated according to specific criteria and methods
of nuclear reactor physics. Thus, NRRP can be seen
as a combinatorial problem: a number n of FA’s are
permuted in n positions of the core.
Although presenting a simple formulation the
NRROP is a NP-Complete problem, whose difficulty
grows exponentially with the number of FA’s in the
reactor core. The Nuclear Power Plant of Angra 1,
for instance, contains 121 FA’s and gives rise to
approximately 8.09 x 10
200
(121!) loading patterns.
However, due to 1/4 and 1/8 core symmetries and
also to rules of placement of the FA’s in the nucleus,
this number falls to approximately 10
25
loading
patterns. This number is extremely high to solve this
problem by enumeration. It would take
approximately 5.8 x 10
19
years to test all these
combinations with the Reactor Physics codes and
today's computers, making it infeasible to check all
these combinations to find the best. Besides these
difficulties, this problem has nonlinear
characteristics with discontinuities and multiple
optima in the solutions search space.
For safe operation of a nuclear plant is necessary
a loading pattern thoroughly being examined. For
such, reactor physics codes are used, with
implementations of the numerical resolution of
Neutrons Transport or Diffusion models (Chapot et
al, 2000). The direct use of these codes in an
optimization process of reloading makes the process
very slow. In this paper was used the Reactor
Physics code RECNOD (Chapot et al, 2000).
Combination of these attributes: high-level
combinations, nonlinear objectives and constraints,
multimodality and high computational cost describe
NRRP, which is challenging the traditional
optimization methods and encouraging researchers
to develop and implement more "intelligent"
methods optimization in order to solve this problem.
4 COMPUTATIONAL RESULTS
For benchmarking QDPSO, the 7th reload cycle for
Angra 1 NPP, PWR, designed by Westinghouse and
operated by Eletronuclear, located at the Southeast
of Brazil, has been selected. Angra 1 core gives 121
FA’s and two main axis of symmetry dividing the
core into four regions that, are called 1/4 (one-
fourth) symmetry axis. These axis and two
secondary diagonal axes divide the core into eight
regions. Figure 1 shows Angra1 core (view from
top) and the representation of 1/8 core symmetry
(view from top).
In fact with 1/8 symmetry we reduce the
complexity of the problem and works with 21 FA’s:
1 at the center of the core, 10 over the lines of
StudyofNuclearReactorReloadUsingDifferentApproachesofQuantumInspiredAlgorithms
305
Figure 1: Representation of Angra 1 core and 1/8 core
symmetry.
symmetry and 10 between the symmetry lines. In 1/8
core symmetry the quartets can only occupy the
positions 1-10 and the octets must occupy only the
positions 11-20. The central element is considered
fixed and not part of the optimization process, as
well as others approach in literature.
For this study QDPSO was developed in MatLac
6.5 with communication interface with RECNOD
code. The simulations with the reactor physics code
RECNOD (Chapot, et. al, 2000) used a low-leakage
strategy with burnable poison. The QDPSO was
used as a tool to determine the optimal boron
concentration in 30 experiments with different
random seeds. The population was 100. The
objective function used to evaluate each individual
of QDPSO is show bellow, as is the same function
used in others woks in literature (Chapot, et. al,
1999, Machado MD, 2005, Meneses et. al, 2010, Da
Silva et al, 2011, Nicolau et al, 2012).
otherwiseP
PifC
Fitness
rm
rmB
,
395.1/1
(13)
where, C
B
is the boron concentration and P
rm
is the
Maximum Normalized Assembly Power.
Thus, the optimization of this problem is closely
related to the power plant cycle length as it
maximizes the boron concentration yielded by the
reactor physics code. At Angra 1 NPP,
approximately 4ppm of soluble boron is consumed
per Effective Full Power Day (EFPD). This relation
indicates that increasing the boron concentration by
an optimized core configuration will increase the
NPP core operational days.
To do so, a candidate solution of the QDPSO is a
vector that indicates a possible fuel rod
configuration. In this vector, it’s possible to appear
repetition of values. However, the repetition of FA’s
does not stand as a valid configuration, because the
same FA cannot be in more than one position in the
core. In this way, Random Keys (Bean, 1994) model
was used as well as in others woks in literature
(Chapot, et. al, 1999, Machado MD, 2005, Meneses
et. al, 2010, Da Silva et al, 2011, Nicolau et al,
2012). Thus, a candidate solution of the QDPSO
converted by Random Keys method is a string with
20 elements that corresponds to the positions of the
FA (where the quartets can only occupy the positions
1-10 and the octets must occupy the position 11-12).
Table 1 shows the best results of QEA
implemented by Nicolau et al, 2012, GA
implemented by Chapot, et al, 1999, PSORK
implemented by Meneses et al, 2010, PBIL
implemented by Machado MD, 2005 and QPBIL
implemented by Da Silva et al, 2011 every all
applied to the NRROP of Angra 1 in the same
conditions. The acquired results are analysed and
compared with QDPSO developed in this study
according to the value of C
B
and number of
evaluations.
Table 1: Comparison Results.
Technique C
B
Average
(C
B
)
Std*
(C
B
)
Evaluations
GA 1197 703 381,95 4000
FPBIL 1428 1353 65 430.364
PSORK 1394 1168 95 4000
QEA 1431 1385 35 70400
QPBIL 1413 1383 45 49680
QDPSO 1441 1393 26 6500
*Std = Stand deviation
According to Table 1, the best result for C
B
using
QDPSO is higher than others results of the literature
with less computational effort. The difference
between the best value of C
B
found by QDPSO
(1424) and PSORK (1394) is equal to 30 ppm of
boron. It’s corresponds to about 7 EFPD more. Also
we can observe that QDPSO found the best result
with 6.500 evaluations. Furthermore, we can observe
that all others quantum inspired algorithms reported
shows best results of the average boron
concentration and lower standard deviation that his
classical counterpart.
Figure 2, shows the evolution curve of QDPSO
implemented. We can observe that at the beginning
of the experiment QDPSO converges rapidly to
values near 9x10
-4
, at this moment the search of
QDPSO consists in find fitness values that
guarantees the restriction
395.1
rm
P
, and then goes
on to maximize the C
B
value and consequently
minimizes 1/C
B
(equation 13). In this case the best
value for the fitness occurs near generation 140.
ECTA2014-InternationalConferenceonEvolutionaryComputationTheoryandApplications
306
Figure 2: Evolution curve of QDPSO.
5 CONCLUSIONS
In this study we discuss about the efficiency of
quantum-inspired algorithm in to solve the NRROP.
In this case we implemented QDPSO algorithm in
the same conditions of other optimization techniques
presented in the literature. The results found by the
QDPSO was best than its classical counterpart and
compared with others quantum inspired algorithms
is the best in relation to the number of evaluations.
Furthermore, the results shown the superiority of
quantum inspired algorithm compared with his
classical counterpart presented in the literature for
the NRROP. According to this study we can say that
quantum inspired algorithms are well situated
among the best alternatives for dealing with hard
optimization problems that number of evaluations is
crucial due to the high computational cost of the
evaluations, such as NRROP.
ACKNOWLEDGEMENTS
The authors acknowledge FAPERJ (Fundação de
Amparo à Pesquisa do Estado do Rio de Janeiro).
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