Multi-Objective Core Optimization Framework for Advanced Reactors
Kaiyue Zeng, Jason Hou
North Carolina State University
Nicolas Stauff, T.K Kim
Argonne National Laboratory
NUC workshopInnovations in Advanced Reactor Design, Analysis, and Licensing
NC State UniversitySept 17-18, 2019
Research expertise
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Nuclear Reactor
Core & Plant Simulator
Verification & Validation
Design & Optimization
Uncertainty Analysis
Advanced Reactors
Novel Neutronics Methods
Iterative procedures
• Multi-physics modeling and simulation (M&S): neutronics, thermal-hydraulics, and fuel mechanics
• Multiple constraints must be applied to satisfy safety requirements
• Reflect a balance between economic and safety performance
Classical approaches for reactor design optimization
• Different design stages, each corresponding to one of the physics
• Assume variables among different physics to be independent
• Rely on expertise and in-depth knowledge of the problem
• Limited due to complex correlations of design parameters, strong coupling of performance outputs, and nonlinearities between input & outputs
Global optimization methods are desired
• Multiples constraints from multi-physics perspective
• Especially for advanced reactors
Nuclear reactor design
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Constraint annealing method for solution of multiconstained nuclear fuel cycle optimization
LWR loading pattern problem
Formosa (NCSU)
Breed-and-burn (B&B) sodium-cooled fast reactor capable for 3-D fuel shuffling
Automated search algorithm for shuffling scheme using Simulated Annealing
Global optimization methods have been successfully applied to loading pattern & shuffling scheme design
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Kropaczek et al., Nucl. Technol. (2019) Hou et al., Nucl. Engi. Des. (2016)
A generalized optimization framework was developed by combining
• Sensitivity analysis
• Multi-objective optimization method
• Acceleration techniques
Optimal ABTR core designs
• Balance between performance and computational cost
Conclusion and future work
Outline
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Major objectives
• Support development of prototype Advanced Burner Reactor
• Demonstrate benefit of transmutation technologies and closed fuel cycle
• Qualify TRU-containing fuel and advanced structural materials
Pool type SFR fueled with metal alloy fuel U-TRU-Zr
TRU CR = 0.65
Advanced Burner Test Reactor (ABTR)
6[1] Y. I. Chang, et, al. “Advanced Burner Test Reactor Preconceptual Design Report”, ANL, Sept. 2006.
Reference ABTR core configuration [1]
Parameters Value[1]
Reactor power (MWt) 250
Plutonium weight fraction required 20.70%
Peak assembly power (MW) 5.21
Power density (W/cm3) 258
Cycle length (month) 4
Burnup reactivity swing (pcm) 1200
External feed (kg/year) 946
Avg. fuel discharge burnup (GWd/t) 98
Peak fast flux (1015 n/cm2-s) 2.8
Peak fast fluence (1023n/cm2) 3.3
Delayed neutron fraction (pcm) 0.0033
Sodium void (100% void) ($) 1.75
Reference design
NEAMS Workbench allows different codes to interact in a unique platform
Direct physics calculations
REBUS-equilibrium cycle calculation: search for fuel enrichment to ensure criticality at EOC
Parallel DIF3D and PERSENT calculations
A simplified core thermal-hydraulic model
Need surrogate models to accelerate optimization
Simulation performed using ARC integrated NEAMS Workbench coupled with Dakota through PyARC interface
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Pre-Processing
REBUS-equilibrium CalculationPre-generated
MCC XS
Parallel DIF3D
BOC MOC EOC
Parallel PERSENT
beffNa void worth
TH Calculation
Post-Processing
0 min – 0 sec
7 min – 37 sec
8 min – 13 sec
13 min – 26 sec
13 min – 29 sec
13 min – 30 sec
1. Define / refine optimization problem
• Determine optimization objectives and constraints
• Discretize input space
• Filter input / output variables through sensitivity analysis
2. Choose optimization method
• Gradient-based local
• Gradient-based global
• Derivative-free local
• Derivative-free global
3. Determine acceleration methods (if direct physics calculations are expensive)
• Surrogate models
• Hybrid methods
4. Select final optimal solution
• Adopt weights of each objective (based on expert judgement)
• Perform high-fidelity calculation
Calculation flow for core optimization
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Optimization problem setup
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Design constraints• Reactivity swing (< 2500 pcm)
• Peak fast flux > (2.0⨉1015 n/cm2-s)
• External Pu feed per year < (500 kg/year)
• Maximum Pu content < (30 w%)
• Peak fast fluence < (4.0⨉1023 n/cm2)
• Peak discharged burnup < (200 GWD/T)
• Pressure drop along fuel pin < (0.5 MPa)
• Peak cladding temp. < (650 ℃ with 3 s HCF)
• Peak fuel temp. < (850 ℃ without 3 s HCF)
• Sodium void worth < (2$)
Optimization objectives(-) Reactivity swing
(-) Core power
(-) Core volume
(-) External Pu feed per year
(+) Peak fast flux
Design parameters• Height of driver fuel column
• Radius of inner cladding surface
• Radius of wire-wrap structure
• No. of rings of fuel pins per FA
• No. of inner core batches
• Cycle length
Results
“Pareto front” showing
trade-offs between
responses R1 and R2
Number of designs in input space > 11 billions
Cannot exhaust all permutations
Refine optimization options by discretizing input space
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Input variables Max. Int. Min. Int. Ref. Range Size
Discrete variables
Number of inner core batch 1 1 12 3 - 12 10
Number of pin rings 1 1 9 7 - 11 5
Discretized continuous variables
Height of driver fuel column (cm) 1 1 80 50 – 100 51
Radius of inner cladding surface (cm) 0.002 0.002 0.3480 0.3306 – 0.3354 73
Radius of wire-wrap structure (cm) 0.001 0.0005 0.0515 0.05 – 0.07 24
Core total power (MW) 5 5 250 100 – 300 41
Cycle length (days) 5 1 120 50 – 300 61
Continuous variables discretized
A global sensitivity analysis is performed to generate
• Sensitivity coefficients of output vs. input
• Correlations between outputs
• Identify strongly correlated outputs
• (temporarily) remove core volume and reactivity swing from objectives
Sensitivity analysis: reduce number of objectives
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Input Parameter Objective
Driver
fuel
height
Inner
Clad
Radius
Wire
Wrap
Radius
Core
Power
Cycle
length
Inner
core
batch
Number
of fuel
pins△⍴
External
Pu Feed
Core
Power
Core
Volume
Peak
Fast Flux
Objective
△⍴ -0.28 -0.33 -0.01 0.38 0.55 0.02 -0.48 1.00
External Pu Feed 0.12 0.13 0.03 0.03 -0.58 -0.53 0.28 -0.45 1.00
Core Power -0.01 0.00 0.00 1.00 0.01 0.00 -0.01 0.38 0.03 1.00
Core Volume 0.30 0.45 0.08 0.00 -0.01 -0.01 0.82 -0.59 0.33 0.00 1.00
Peak Fast Flux -0.26 -0.30 -0.05 0.68 -0.04 -0.02 -0.58 0.70 -0.16 0.68 -0.66 1.00
Constraint
Peak Fast Fluence -0.13 -0.17 -0.03 0.39 0.55 0.51 -0.32 0.79 -0.59 0.39 -0.37 0.50
Peak Burnup -0.23 -0.25 0.01 0.33 0.46 0.41 -0.35 0.84 -0.50 0.33 -0.44 0.54
Peak Pu Content -0.40 -0.48 0.04 0.19 0.25 0.20 -0.55 0.86 -0.41 0.19 -0.74 0.70
Peak Pressure Drop 0.09 -0.28 -0.21 0.52 0.00 0.00 -0.61 0.62 -0.16 0.52 -0.58 0.84
Peak Clad Temp. -0.49 -0.05 0.03 0.59 -0.01 -0.02 -0.59 0.69 -0.18 0.59 -0.62 0.92
Peak Fuel Temp. BOC -0.37 -0.01 -0.01 0.61 -0.02 -0.03 -0.66 0.66 -0.18 0.61 -0.64 0.93
Peak Fuel Temp. EOC -0.37 0.00 -0.01 0.61 -0.04 -0.03 -0.66 0.65 -0.17 0.61 -0.63 0.93
Sodium Void Worth 0.69 0.47 -0.11 0.04 0.08 0.07 0.51 -0.51 0.22 0.04 0.82 -0.58
Optimization method: Genetic Algorithm
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Optimization using Dakota capabilities
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Dakota
Sampling
Py
AR
C.d
riv
er
Py
AR
C.d
riv
er
Dakota
Post-processing
Optimization
Algorithm
PyARC
Pre-processing
ARC code Execution
Post-processing
PyARC
Pre-processing
ARC code Execution
Post-processing
PyARC
Pre-processing
ARC code Execution
Post-processing
PyARC
Pre-processing
ARC code Execution
Post-processing
PyARC
Pre-processing
ARC code Execution /
Surrogate Model
Post-processing
Parameter
FileOutput File
Parameter
File
ARC-based optimization using Multi-Objective Genetic Algorithm (MOGA)
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3 optimization objectives for improved convergence
• Core power
• External Pu feed
• Peak fast flux
• Reactivity swing (constraint 2500 pcm)
• Core volume
Optimization process
• Number cases > 25,000
• Generations = 73
• Calc. time > 5,400 hours
Optimization: pareto front formation
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Determine perfect core performance available in final generation
Calculate weighted distance of a core design to the perfect point
Weights determined based on expert opinion
Core design with shortest distance to perfect design is selected as optimal case
Transport calculation will be performed on the optimal case designs
Select final optimal design from a set of near optimal solutions
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Criterion External Pu feed Core power Peak fast flux
Opt. 1 1 0 0Opt. 2 0 1 0
Opt. 3 0 0 1Opt. 4 1/2 1/4 1/4
Opt. 5 1/4 1/2 1/4Opt. 6 1/4 1/4 1/2Opt. 7 1/3 1/3 1/3
Perfect core design = {min ExtPuFeed ,min Power ,max(PFF)}
7 candidate designs selected based on various weights selection
Transport calculation carried out for optimal designs
A set of near-optimal designs reached
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Output parameters Ref. * Opt. 1 Opt. 2 Opt. 3 Opt. 4 Opt. 5 Opt. 6 Opt. 7
Reactivity swing (pcm) 1415 2200 1430 2326 2318 1430 2326 1542
Core power (MW) 250 165 150 220 180 150 220 160
Core volume (m3) 9.7 5.4 5.0 5.9 5.4 5.0 5.9 4.8
Pu external feed (kg/year) 178 126 160 177 132 160 177 175
Peak fast flux (1015 n/cm2-s) 2.8 2.9 2.9 3.6 3.1 2.9 3.6 3.1
Pu weight fraction required (wt%) 0.21 0.30 0.29 0.30 0.30 0.29 0.30 0.30
Peak fast fluence (1023 n/cm2) 3.6 3.7 2.5 3.6 4.0 2.5 3.6 2.5
Peak discharge burnup (GWd/t) 138 182 129 168 191 129 168 130
Pressure drop along fuel pin (MPa) 0.24 0.36 0.28 0.47 0.43 0.28 0.47 0.42
Peak cladding temp. w/ 3 s HCF (K) 641 645 650 649 649 650 649 646
BOC Peak fuel temp. w/o 3 s HCF (K) 730 746 753 750 762 753 750 746
EOC Peak fuel temp. w/o 3 s HCF (K) 728 742 750 745 757 750 745 743
7 candidate designs selected based on various weights selection
Transport calculation carried out for optimal designs
A set of near-optimal designs reached (cont.)
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Optimal-to-reference ratio
Constraint (2500 pcm)
Direct physics calculation
Reactor core diffusion calculation using ARC code suite
• REBUS equilibrium (~7 mins) per case
• PERSENT (~4 mins) per case
25,000 function evaluations
5,400 hours (cannot parallelize calculations among generations)
Non-physics-based surrogate models to improve effectiveness of optimizers
Interpolation or regression of data generated from original model
• Gaussian process
• Neural network
• Polynomial response surfaces
• Splines
• …
Acceleration methods
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Hybrid method
• Surrogate + direct ARC calculation
• Global optimization search + local optimization search
• …
3 optimization objectives (same as before)
Optimization process
• Surrogate model buildup = 513 hours
• Optimization time = 1,152 hours
• Samples = 27,578
Direct physics calculation performed for chosen designs
Surrogate-based core optimization
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• Gaussian process regression• 2000 data points (from sensitivity analysis)
used to build surrogate• Rest data points used for error estimation
Performance parameters
Optimal-to-reference ratio
Can we improve further the effectiveness?
1. ARC-based MOGA
2. Surrogate-based MOGA
3. Surrogate-based global optimization (SBGO)
• ARC calculations performed during optimization
• Results returned to train surrogate model
• Reduce design space as quickly as possible
• Fine tune final solutions
Optimization options: improved efficiency and performance
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Comparison of optimization options: summary
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Option ARC-MOGA Surro-MOGA SBGO
Optimization method
MOGA MOGA MOGA
Objective representation
Multiple objectives Multiple objectives Multiple objective
Core simulation tool
ARC Surrogate ARC + Surrogate
Computational time* (hour)
> 5400 513 + 1152 481†
Convergence sample size
> 25000 27578 1943 + 23057
Advantage Accuracy Efficiency Efficiency
Disadvantage CostHigh requirement on surrogate
modelRelatively low accuracy
* Estimated if job is executed serially† Does not require surrogate model reloading
Solution selection
• 3 objectives: equal weights
• 5 objectives: equal weights
All results based on transport solution
Comparison of optimization options: performance
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Output Quantities of Interests Ref. ARC-Moga Surro-Moga SBGOARC-Moga
(5Vars)
Reactivity swing (pcm) 1415 1542 1635 2044 1581
Core power (MW) 250 160 170 230 160
Core volume (m3) 9.7 4.8 4.7 5.1 4.9
Pu external mass feed (kg/year) 178 175 173 176 157
Peak fast flux (1015 n/cm2-s) 2.8 3.1 3.3 3.3 3.0
Pu weight fraction required (wt%) 0.21 0.30 0.30 0.28 0.30
Peak fast fluence (1023n/cm2) 3.6 2.5 2.7 1.0 2.8
Peak discharge burnup (GWd/t) 138 130 145 199 143
Pressure drop along fuel pin (MPa) 0.24 0.42 0.46 0.15 0.39
Peak cladding temp. w/ 3 s HCF (K) 641 646 649 642 646
BOC Peak fuel temp. w/o 3 s HCF (K) 730 746 759 728 746
EOC Peak fuel temp. w/o 3 s HCF (K) 728 743 774 719 743
Comparison of optimization options: performance (cont.)
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An optimization framework was developed by combining
• Sensitivity analysis
• Multi-objective optimization method
• Acceleration techniques
Optimal ABTR core designs obtained by trying various options
• Better suited computational resources are required to performed ARC-simulation based optimization
• Surrogate models are desired with limited computational resources
• Both optimization and acceleration methods require efforts and expertise for correct problem set up
This framework can be applied to other types of LMRs (e.g. LFRs)
Future work
• More efficient global optimization algorithms
• More accurate surrogate models
• Machine learning approaches
Conclusion and future work
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Thank you.Questions?
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