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

2

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

3

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

4

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

5

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

7

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

8

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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