close

Вход

Забыли?

вход по аккаунту

?

ICONE25-67936

код для вставкиСкачать
Proceedings of the 2017 25th International Conference on Nuclear Engineering
ICONE25
July 2-6, 2017, Shanghai, China
ICONE25-67936
MIXED OXIDE LWR ASSEMBLY DESIGN OPTIMIZATION USING DIFFERENTIAL
EVOLUTION ALGORITHMS
Alan J. Charles
University of Cambridge
Department of Engineering
Cambridge, United Kingdom
ajc289@cam.ac.uk
Geoffrey T. Parks
University of Cambridge
Department of Engineering
Cambridge, United Kingdom
gtp10@cam.ac.uk
Keywords: fuel assembly design, optimization, differential evolution, genetic algorithms.
a Genetic Algorithm (GA), a more conventional algorithm often
used as a benchmark. GAs [1] are one of the more popular
methods used in nuclear engineering design optimization. These
evolutionary algorithms mimic the processes of natural
selection and reproduction in an attempt to create successive
populations of solutions which converge on an optimal solution.
For this work, the Multi-Objective Alliance Algorithm (MOAA)
[2] was chosen as it has previously demonstrated its
effectiveness in nuclear engineering fuel assembly design
problems, producing solutions which are superior to previous
‘expert designs’ and performing better than other GAs [3], thus
making it a good algorithm against which to compare
performance.
In contrast, DE algorithms [4] are a relatively newer type of
evolutionary algorithm that work in a similar fashion to GAs but
feature key differences in the way the new population is
generated. In addition, the selection process is generally more
stringent than with GAs (where inferior solutions have a
probability of remaining in the population). This allows DE
algorithms to offer potentially faster convergence (which is
useful for computationally expensive problems, such as those
faced in nuclear engineering) at the risk of premature
convergence on a non-optimal solution [5]. DE algorithms have
previously been successfully applied to core design
optimization problems [6]. However, they do not yet appear to
have been applied to nuclear fuel assembly optimization
problems, thus making this investigation both novel and a useful
step in examining DE’s applicability to solving such problems.
For this work, new multi-objective forms of the DE algorithms
JADE [7] and μJADE [8] are developed. JADE was chosen as
it had been shown in the literature to exhibit superior
performance over classic DE algorithms through its use of a
variation on the classic DE mutation strategy, and the inclusion
ABSTRACT
Two new multi-objective differential evolution (DE)
algorithms are used to optimize heterogeneous low-enriched
uranium + mixed oxide fuel assemblies for use in a pressurized
water reactor. The objectives were to maximize plutonium
content and minimize the power peaking factor. A performance
comparison to a genetic algorithm is used to evaluate the
applicability of DE algorithms to nuclear fuel assembly design
optimization problems. Results show that DE performs highly
competitively against a more established algorithm and can
arguably better represent the trade-off between both objectives
through greater variety in the number of different pin
arrangements explored and a higher reliability in finding the
‘true’ Pareto-front.
INTRODUCTION
In mixed oxide (MOX) fuel assemblies, there is a clear
trade-off between in-core fuel performance (achieved through
variation of fuel properties in the assembly) and fabrication cost
(due to the increased complexity in the fuel). A capability to
explore this trade-off rigorously and systematically would be a
helpful aid to decision-making, as the non-linear interaction of
variables means design optimization becomes too difficult to
achieve through conventional engineering judgement alone.
Whilst there are a large variety of optimization methods
available, few performance comparisons have been made to
determine which methods are most suitable for these problems,
and a lack of suitable optimization methods has a negative
impact on the design process.
This study was conducted to determine the suitability of a
relatively new class of algorithms called Differential Evolution
(DE) in optimizing a typical nuclear fuel assembly design
problem, with performance evaluated by comparing results with
1
Copyright © 2017 ASME
Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/25/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
All three algorithms used the reactor physics code
WIMS10a [10] to solve the neutron transport equation and
obtain the PPF for each tested fuel assembly design, using an
arbitrary power level. Assemblies were modelled using a
standard CORAIL assembly layout of 264 fuel pins, which can
be divided using octant symmetry to give 39 unique fuel pin
positions. These pins are labelled as fuel types 1, 2, 3 (MOX
type 1, MOX type 2, and LEU, respectively). The quantities N1,
N2, and N3 are therefore the total number of each respective pin
type, where N1 + N2 + N3 = 39. Some pins are weighted by 0.5
due to octant symmetry. LEU enrichment is fixed at 5% U235
and reactor grade Pu composition is assumed. Two %Pu MOX
pin types are allowed (W1, W2). Constraints affecting this
problem include a requirement that the minimum number of
LEU pins must be half the total, and a maximum amount of Pu
within the MOX pins (20%). This gives a total of 39 integer
variables, 2 continuous variables, with the mentioned
constraints of N3 ≥ 16.5 (264/8) and 0 ≤ W1, W2 ≤ 20. The Pu
content of an assembly is therefore MOXT = W1 × N1 + W2 × N2.
From this, both objectives to be minimized are therefore PPF
and –MOXT. Algorithms were run on the ‘Ray’ computer cluster
used by the University of Cambridge’s Department of
Engineering, with specifications shown in Table 3.
of an archive and self-adapting control parameters. Similarly,
μJADE was chosen as it was shown to be effective when
working on multimodal problems, but uses a significantly
smaller population size. These features should help the
algorithm perform well in a multi-objective environment, where
diversity and convergence rates are important.
The problem investigated concerns optimization of a
‘CORAIL’ assembly [7] containing both low-enriched uranium
(LEU) and plutonium MOX pins, with the objectives of
minimizing the power peaking factor (PPF) and maximizing
plutonium content. Plutonium and LEU fuel have different
reaction probabilities and produce a wide range of neutron
energies, potentially resulting in an uneven neutron flux and
subsequent fuel temperature problems. Optimization of
plutonium distribution by varying the position of MOX pins and
the amount of plutonium contained within them can result in
improved thermal margins, whilst increasing the overall
plutonium content above that of the ‘CORAIL’ design.
METHODOLOGY
Multi-Objective JADE (MOJADE) and μJADE
(MOμJADE) were created using C++ and are based on the
JADE and μJADE algorithms as described by the originators in
[8,9], with the following modifications implemented to allow
them to operate in a multi-objective environment. First,
selection and ranking are no longer done based on one
objective, and the ‘best’ solutions are now a list of nondominated solutions which represents the current Pareto-front.
These are determined from the current population. Secondly,
the archive was changed to accept solutions from the population
that have been dominated by new solutions, and an additional
archive was added to accept new solutions that are Paretoequivalent to the existing population. The pseudocode for
MOJADE and MOμJADE can be seen in Annex A. Control
parameters used for MOAA and MOJADE / MOμJADE are
given below in Table 1 and Table 2, respectively.
TABLE 3. RAY COMPUTER CLUSTER SPECIFICATIONS
Processor
RAM
Video Card
Hard Drive
RESULTS
Each algorithm was run 30 times, with a limit of 1600
solution evaluations per run. Analysis of the results involved
comparing Pareto-fronts found by the three algorithms using a
number of indicators and statistical tests to determine the
performance of each algorithm. The indicators used were the
epsilon and hypervolume indicators. The epsilon indicator [11]
represents the minimum distance required to translate all the
points of the found Pareto-front from a given algorithm to
weakly dominate the reference set (which is the overall Paretofront constructed from all solutions found by all algorithms).
The hypervolume indicator [12] is the difference between the
hypervolume of the objective space dominated by the Paretofront found by a particular algorithm and the hypervolume of
the objective space dominated by the reference set, using the
least-optimal solution found from all three algorithms as a
reference point. In both cases smaller values indicate better
performance. The Kruskal-Wallis test [13] was then used to
determine the statistical significance of these values. For this
work, the Kruskal-Wallis test results represent the probability
that the given indicator values are not a true representation of
the algorithm’s relative performance against another, and are
instead the result of random chance.
TABLE 1. MOAA CONTROL PARAMETERS
Number of tribes
Probability 1 for the creation of tribes
Probability 2 for the creation of tribes
Initial standard deviation
Final standard deviation
Probability 3 for the creation of alliances
Alliance standard deviation
Total number of Pareto-optimal solutions
Factor for evaluation neighbourhood
6
0.5
0.2
0.3
0.01
2 / variables
0.1
100
10
TABLE 2. MOJADE AND MOμJADE CONTROL
PARAMETERS
Parameter
Rate of parameter adaptation
Greediness of mutation strategy
Population
Generations
MOJADE
0.1
0.05
32
50
Dual Intel Xeon Processor E5-2650 (8-core
hyper threading, 2.6 GHz Turbo with 20 MB
cache
64 GB (8×8 GB) 1866 MHz DDR3
512 MB AMD FirePro 2270
1TB 7,200 rpm
MOμJADE
0.05
3 / population
8
200
2
Copyright © 2017 ASME
Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/25/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
FIGURE 1. RESULTS OF MOAA, MOJADE AND MOµJADE OPTIMIZATION OF MOX FUEL ASSEMBLIES
FIGURE 2. COMPARISON OF NON-DOMINATED SOLUTIONS FOUND USING MOAA, MOJADE AND MOµJADE ALGORITHMS
TO OPTIMIZE MOX FUEL ASSEMBLIES
3
Copyright © 2017 ASME
Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/25/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
A graph showing the results from all 30 runs for each of the
three algorithms can be seen in Figure 1. Filtering these to show
just the Pareto-Optimal (PO) solutions for each algorithm
produces Figure 2.
Both figures plot solutions in PPF against (–MOXT) space,
so that both objectives are to be minimized, and thus the
bottom-left corner represents an ideal solution. It can be seen
from these two graphs that MOJADE and MOμJADE are
clearly performing comparably to MOAA, and both
significantly contribute to the highlighted overall Pareto-front
shown in Figure 2. MOAA appears to dominate the Pareto-front
at both the extremes of plutonium content, with MOJADE and
MOμJADE being most effective around the middle of the
Pareto-front. A small amount of clustering is also present in the
PO solutions of MOAA, with clear gaps in the Pareto-front that
are populated by MOJADE and MOμJADE solutions.
The clustering of MOAA results can be explained by
looking at the patterns themselves. Each MOAA run tends to
converge on a single pin pattern with one or two changes in the
pins, and the non-dominated solutions therefore show the effect
of increasing or decreasing the values of W1 and/or W2 within
the same pin pattern. This results in a number of solutions that
have very similar values for MOXT and PPF. In contrast, both
MOJADE and MOμJADE produce many different patterns
within each run and thus arguably better explore the search
space of different pin arrangements.
Results from each of the 30 runs for each algorithm were
used to compute the mean and standard deviation of the
hypervolume and epsilon indicators along with their
corresponding p-values from the Kruskal-Wallis test. These can
be seen in Tables 4 – 6 respectively.
Table 4 indicates that MOJADE is most consistent at
producing results which dominate the entirety of the known
search space. MOμJADE performs slightly worse and also has a
large standard deviation. This is likely due to the small
population of MOμJADE, making its performance highly
influenced by the number of times the algorithm is run (more so
than the other algorithms), in order to provide more data. Table
5, however, suggests that whilst MOμJADE results do not give
as much data as MOJADE as to the size of the search space,
they are more likely to be closer to the ‘true’ Pareto-front.
Table 6 gives the p-value results of the Kruskal-Wallis test
for the hypervolume and epsilon indicators for both DE
algorithms versus the GA, as well as against each other. Against
MOAA, hypervolume test results show that MOJADE performs
better due to methodology and not by chance. However, for the
epsilon indicator, the result shows a statistical likelihood that
differences in epsilon performance between MOAA and
MOJADE are by chance. For MOμJADE, the results suggest
that the DE algorithm outperforms the GA on both indicators
due to the methodology. Finally, when comparing MOμJADE
against MOJADE, the Kruskal-Wallis test results indicate that
the lower mean hypervolume indicator for MOJADE and the
lower mean epsilon indicator for MOμJADE are due to
differences in their methodology and not by chance.
TABLE 4. HYPERVOLUME INDICATOR VALUES
Algorithm
MOAA
MOJADE
MOμJADE
Mean
1.6664
0.7672
1.1267
Standard Deviation
0.5169
0.1047
0.7723
TABLE 5. EPSILON INDICATOR VALUES
Algorithm
MOAA
MOJADE
MOμJADE
Mean
0.3897
0.3941
0.3320
Standard Deviation
0.1478
0.1204
0.1081
TABLE 6. KRUSKAL-WALLIS TEST RESULTS
Algorithm comparison
MOJADE vs MOAA
MOμJADE vs MOAA
MOμJADE vs MOJADE
Hypervolume
3.879E-11
8.513E-07
9.497E-05
Epsilon
9.528E-01
7.363E-02
5.650E-02
CONCLUSIONS
These results suggest that DE is able to find solutions
comparable in quality to those found by MOAA and arguably
better examines the search space of pin patterns. Both DE
algorithms show good performance in this exploratory
optimization problem, despite the algorithms originally being
designed for single-objective optimization. Although further
optimization of algorithm control parameters, as well as
determining the robustness and scalability of this method, are
still required, this study demonstrates that DE has been shown
to be an effective method alongside a proven GA when looking
at the optimization of nuclear fuel assembly designs.
NOMENCLATURE
DE
Differential Evolution
GA
Genetic Algorithm
LEU
Low-Enriched Uranium
MOAA
Multi-Objective Alliance Algorithm
MOJADE Multi-Objective JADE
MOμJADE Multi-Objective μJADE
MOX
Mixed Oxide
PO
Pareto-Optimal
PPF
Power Peaking Factor
REFERENCES
[1] Holland, J.H., 1992, Adaptation in Natural and
Artificial Systems, MIT Press, Cambridge, Massachusetts.
[2] Lattarulo, V., and Parks, G.T., 2012, “A Preliminary
Study of a new Multi-objective Optimization Algorithm”, Proc.
IEEE Congress on Evolutionary Computation, IEEE, Brisbane,
pp. 1–8.
[3] Lattarulo, V., Lindley, B.A., and Parks, G.T., 2014,
“Application of the MOAA for the Optimization of CORAIL
Assemblies for Nuclear Reactors”, Proc. IEEE Congress on
Evolutionary Computation, IEEE, Beijing, pp. 1413–1420.
4
Copyright © 2017 ASME
Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/25/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
[4] Storn, R., and Price, K., 1997, “Differential Evolution –
A Simple and Efficient Heuristic for Global Optimization over
Continuous Spaces”, J. Global Optim., 11, (4), pp. 341–359.
[5] Zio, E., and Viadana, G., 2011, “Optimization of the
Inspection Intervals of a Safety System in a Nuclear Power
Plant by Multi-Objective Differential Evolution (MODE)”,
Reliab. Eng. Syst. Safe., 96, (11), pp. 1552–1563.
[6] Sacco, W.F., Henderson, N., Rios-Coelho, A.C., Ali,
M.M., and Pereira, C.M.N.A., 2009, “Differential Evolution
Algorithms Applied to Nuclear Reactor Core Design”, Ann.
Nucl. Energy, 36, (8), 1093–1099.
[7] Youinou, G., Zaetta, A., Vasile, A., Delpech, M., Rohart,
M., and Guillet, J.L., 2001, “Heterogeneous assembly for
plutonium multi recycling in PWRs: The CORAIL concept”,
Proc. International Conference on Back-End of the Fuel Cycle:
From Research to Solutions (GLOBAL 2001), Paris, pp. 1–5.
[8] Zhang, J., and Sanderson, A.C., 2009, “JADE: Adaptive
Differential Evolution with Optional External Archive”, IEEE
Trans. Evol. Comput., 13, (5), pp. 945–958.
[9] Brown, C., Jin, Y., Leach, M., Hodgson, M., 2013,
“μJADE: Adaptive Differential Evolution with a Small
Population”. Soft Comput., 1, pp. 1–10.
[10] Lindley, B.A., Newton, T.D., Hosking, J.G., Smith,
P.N., Powney, D.J., Tollit, B., and Smith, P.J., 2015, “Release of
WIMS10: A Versatile Reactor Physics Code for Thermal and
Fast Systems”, Proc. International Congress on Advances in
Nuclear Power Plants (ICAPP 2015), ANS, Nice, pp. 1793–
1801.
[11] Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C.M.,
and Grunert Da Fonseca, V., 2003, “Performance Assessment of
Multiobjective Optimizers: An Analysis and Review”, IEEE
Trans. Evol. Comput., 7, (2), pp. 1–22.
[12] Knowles, J., Thiele, L., Zitzler, E., 2006, “A Tutorial
on the Performance Assessment of Stochastic Multiobjective
Optimizers”, TIK Report 214, Swiss Federal Institute of
Technology (ETHZ), Zurich.
[13] Kruskal, W.H., Wallis, W.H., 1952, “Use of Ranks in
One-criterion Variance Analysis”, J. Am. Stat. Assoc., 47, (260),
583–621.
5
Copyright © 2017 ASME
Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/25/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
ANNEX A
PSEUDOCODE OF MOJADE AND MOµJADE
MOJADE
NOMENCLATURE
µCR is the adaptive crossover probability
µF is the adaptive mutation probability
A1 is the archive used for dominated solutions
A2 is the archive used for Pareto-equivalent solutions
c is the rate of parameter adaptation
D is the number of dimensions (variables)
G is the number of generations
meanA is the arithmetic mean
meanL is the Lehmer mean
NP is the last member of the population
p is the greediness of the mutation strategy
P is the population
randn is a normal distribution
randc is a Cauchy distribution
SCR is the set of successful crossover factors
SF is the set of successful mutation factors
Begin
Set µCR = 0.5; µF = 0.5; A1, A2 = 0
Create random initial population {xi, 0|i = 1, 2, …, NP}
Evaluate and rank population, determine 100p% best vectors
For g = 1 to G
SF = 0, SCR = 0
For i = 1 to NP
CRi = randni (µCR, 0.1), Fi = randci (µF, 0.1)
Randomly choose xp_best from 100p%
Randomly choose xr1 =/= xi from P
Randomly choose xr2 =/= xr1 =/= xi from P ∪ A1 + A2
vi = xi + Fi · (xp_best - xi) + Fi · (xr1 - xr2)
Generate jrand = randint(1, D)
For j = 1 to D
If j = jrand or rand(0, 1) < CRi
ui,j = vi,j
Else
ui,j = xi,j
End If
End For
If f(ui) dominates f(xi)
xi → A1 (replaces random member of A1 if A1 is full)
xi = ui
CRi → SCR, Fi → SF
Else
If f(ui) is Pareto-equivalent to f(xi)
&& f(ui) is NOT dominated by f(A2)
Remove members of A2 that are dominated by ui
ui → A2
End If
End If
Rerank 100p% best vectors
End For
µCR = (1 – c) · µCR + c · meanA(SCR)
µF = (1 – c) · µF + c · meanL(SF)
End For
End
vi is the ith test vector following mutation
ui is the ith test vector following crossover
xi is the ith member of the population
6
Copyright © 2017 ASME
Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/25/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
MOµJADE
NOMENCLATURE
Begin
Set µCR = 0.5; µF = 0.5; A1, A2 = 0
Create random initial population {xi, 0|i = 1, 2, …, NP}
Evaluate and rank population, determine 100p% best vectors
For g = 1 to G
SF = 0, SCR = 0
For i = 1 to NP
CRi = randni (µCR, 0.1), Fi = randci (µF, 0.1)
Randomly choose xa =/= xi from P
Randomly choose xb =/= xa =/= xi from P
Randomly choose xp_best =/= xa from 100p%
Randomly choose xc from P ∪ A1 + A2
vi = xi + Fi · (xp_best - xa) + Fi · (xb - xc)
Generate jrand = randint(1, D)
For j = 1 to D
If j = jrand or rand(0, 1) < CRi
ui,j = vi,j , bi,j = 1
Else
ui,j = xi,j , bi,j = 0
End If
End For
For j = 1 to D
If rand(0, 1) ≤ 0.005
ui,j = low_lim + rand(0, 1) · (up_lim – low_lim)
bi,j = 0
Else
ui,j = ui,j, bi,j = bi,j
End If
End For
CRi = ∑ b / D
If f(ui) dominates f(xi)
xi → A1 (replaces random member of A1 if A1 is full)
xi = ui
CRi → SCR, Fi → SF
Else
If f(ui) is Pareto-equivalent to f(xi)
&& f(ui) is NOT dominated by f(A2)
Remove members of A2 that are dominated by ui
ui → A2
End If
End If
Rerank 100p% best vectors
If ui ∪ 100p% best vectors
BIR = BIR + 1
End If
End For
If mod(g, max(100, 10D) = 0
µCR = (1 – c) · µCR + c · meanA(SCR)
µF = (1 – c) · µF + c · meanL(SF)
End If
If mod(g, max(1000, 100D) = 0
If BIR== 0
Reinitialize pop, include random member of 100p%
BIR = 0
End If
End If
End For
End
µCR is the adaptive crossover probability
µF is the adaptive mutation probability
A1 is the archive used for dominated solutions
A2 is the archive used for Pareto-equivalent solutions
BIR is a restart variable used if no improvement has been made
c is the rate of parameter adaptation
D is the number of dimensions (variables)
G is the number of generations
meanA is the arithmetic mean
meanL is the Lehmer mean
NP is the last member of the population
p is the greediness of the mutation strategy
P is the population
randn is a normal distribution
randc is a Cauchy distribution
SCR is the set of successful crossover factors
SF is the set of successful mutation factors
up_lim / low_lim are the limits set by the variable constraints
bi is used to repair the crossover rate after crossover and perturbation
vi is the ith test vector following mutation
ui is the ith test vector following crossover and perturbation
xi is the ith member of the population
7
Copyright © 2017 ASME
Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/25/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Документ
Категория
Без категории
Просмотров
1
Размер файла
400 Кб
Теги
icone25, 67936
1/--страниц
Пожаловаться на содержимое документа