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# Uncertainty Quantification for Lattice Physics Calculations

Practice with Prezi using real presentation material for a nuclear engineering uncertainty quantification seminar.

by

Tweet## William Wieselquist

on 27 April 2011#### Transcript of Uncertainty Quantification for Lattice Physics Calculations

Experiments Integral Evaluation Target materials

single nuclides

high energy

thermal

resonance integrals reactor physics

criticality safety

isotopics (PIE)

shielding IRPhE - reactor physics

EXFOR - 40 yrs xs & yields data

SINBAD - shielding experiments

ICSBEP - 2000+ critical configurations

Databases http://www.oecd-nea.org/science/irphe/ Plant Core Lattice Multigroup Data Evaluated

Nuclear

Data ENDF

(American) JEFF

(European) JENDL

(Japanese) CENDL

(Chinese) BROND

(Russian) 1. convert to pointwise ENDF

2. combine resonance+"smooth" parts

3. Doppler broaden (adds T dep.)

4. unres. resonance treatment (adds sigz dep.)

5. energy group approx., specify:

intragroup flux -or-

slowing down theory parameters produces groupwise library

parametrized by

- temperature (T)

- background cross section (sigz) 1. assemble macro. xs in pin cells

2. derive equivalent hom. mixture (CP)

in pin cells to interpolate multigroup

library (586g) as function of (sigz,T)

3. calculate self-shielding factors

(resonance and Dancoff)

4. assembly macro. xs

5. collapse groups (586g->95g)

6. perform neutron/gamma transport 1. construct core from

library of 2D/2g lattice "slices"

assemblies from previous cycles

2. solve coarse-mesh diffusion for flux shape and eigenvalue with

polynomial intra-node flux bases

interfacial disconinuity factors

3. incl. simple thermal-hydraulics feedback and parameter adjustment to yield k=1, e.g. control rods (B) and sol. boron (P) Simulation Code[1] Code[2] Code[3] linear map linear map nonlinear map Our concern is with the simulation of nuclear power plants. Although the UQ techniques are by no means limited to these problems, we will attempt to highlight and discuss the particular difficulties with UQ in this system. In particular, we use a chain of codes to overcome the vast dimensionality in time, space, energy, and angle at scales from the nucleus to the power plant. In[2] In[1] Out[1] Out[2] In[3] Out[3] (Our models are usually nonlinear

and never analytic.) Uncertainty Quantification (UQ) is a necessary element of a "best estimate" code system. Not only does it indicate the true precision of a calculation but can help reveal biases as well. bimodal distribution UQ Methods Statistical Sampling (SS)

Analysis of Variance (ANOVA) Simplified Equations/Models

Neural Networks Reduced Order bad: validation problems (both ways)

good: fast, maybe even analytic bad: hard to use properly, training sets must be very large

good: easy to use, resultant network is portable Global bad: no SA

good: black box method, non-linear, global UQ with ~100 code runs bad: extremely costly (evaluate n-dim integrals for n inputs),

only analyze variance

good: black box method, non-linear systems, global SA+UQ Forward and Adjoint

Perturbation Theory (FPT/APT)

Direct Perturbation (DP) Local bad: first-order/significant implimentation difficulty

good: most efficient, local sensitivity analysis (SA) bad: first-order/inefficient

good: black box method, can estimate curvature of response, local SA Conclusions UQ in STARS UQ Data nuclear models accelerator & detectors data

synthesis parametrization produces macroscopic xs (2g),

kinetics param, discontinuity factors

- average fuel temp.

- average mod. density

- average void (B) or mod. dens. (P)

- burnup T

A

P

E F

I

L

E

(MF) MAT/MT MAT/MT elastic capture (n,g) calculation sequence (NJOY) calculation sequence (CASMO-5) produces power/burnup history, pin/assembly peaking factors, assembly-average isotopics, simulates in-core detectors

used primarily in safety analysis

and core performance optimization

also needed for transient "initial conditions" calulational sequence (SIMULATE-3) Core transients Plant transients (SIMULATE-3K) (TRACE) stability analysis

reactivity initiated transients

can provide "core neutronics kernel" for system codes full plant safety analysis, e.g. LOCA, LOFA, pump trip GRS Preliminary Transient

Results from UAM-5 rod withdrawal at HZP (OECD/NEA MOX/UO2 3D core transient benchmark)

with cross section uncertainty could use NJOY ERRORR/ERRORJ on to generate variance/covariance matrices

but variance/covariance matrix (VCM) data is unavailable for many nuclides/reactions (MAT/MT)

various projects have filled in gaps (AFCI, BNL Lo-fidelity, ORNL SCALE) Black box tools make the most sense in STARS.

Need lots of glue. fast text processing

good for scripting

functional/modular

object-oriented

60% of the time, it works every time

better than GIBIANE PERL automated, robust sensitivity coefficient calculation

sensco

STARS::Sensco CASMO-5M with cross section perturbation

cas5mx

STARS::Cas5mx CASMO-5M sensitivity coefficients for xs driver

cas5mx_sc

STARS::Sensco

STARS::PertXS

STARS::Cas5mx generation of xs perturbation files

pertxs

STARS::PertXS variance/covariance data lookup/manipulation

cvx

STARS::Coverx cross section sampling

sharkx error propagation (sandwich rule)

uqx

STARS::Coverx

STARS::MiM automated NJOY GROUPR calculations and "scattering fraction" extraction

njoyhelpr TOOLS First-order Local UQ

in a Nutshell Write a Taylor series.

Apply definition of mean/expectation.

Apply definition of variance.

Extend to multiple inputs.

Extend to multiple outputs. SS Efficiency

independent of number of inputs/outputs (order statistics)

dependent on correlation of output PDFs

moments of 1-output pdfs (mean, variance) ~100 runs for 95/95

moments of 2-output pdfs (covariance, correlation) ~1000 runs PT Efficiency

outputs > inputs = forward approach

inputs < outputs = adjoint approach

DP Efficiency

roughly 1/2 as efficient as forward Is transport linear? Yes No & The operators (streaming, absorption, scattering) in the transport equation are linear with respect to the (angular) flux. Leads to

existence and uniqueness

(if operators are not singular)

boundnedness with respect to perturbations The output (flux) is NOT a linear map of the input (cross sections). Note: self-shielding and depletion also makes the system nonlinear non-linearity is well-known to be fairly small

experience has shown first-order UQ methods are acceptable SCALE 6.0 44-group VCM LIBRARY

~400 nuclides

MTs 1,2,4,16-17,18,102-109,452,1018 U 238 3-group cross sections (barns) and relative std (%) total size

281 600 x 281 600

but very sparse correlation matrix interpolation (44g -> 3g)

does not preserve anti-correlation Statistical Sampling UQ

in a Nutshell theoretical foundation based on order statistics and provides bounds on output PDF, independent of distribution Wilk's formula yields N number of runs/samples for specified a/b 1-sided (e.g. max or min) or 2-side tolerance limits (e.g. variance). Only valid with random sampling.

In words:

"Given N runs, there is a b% chance that the N+1 run would produce a result outside the a% quantile." Wilk's Formula one-sided two-sided normal distribution sampling (MATLAB mvnrnd) Simple Random

Latin Hypercube (MATLAB lhsnorm) UQ Results Future Work UQ in STARS developed toolset to perform UQ with respect to nuclear data (xs)

active participation in UAM benchmark

observations so far perturb CASMO-5M proprietary 586-group ENDF-B/VII library

perturbations can be provided in any group structure

supported microscopic xs perturbations inelastic/elastic scattering

capture

neutrons per fission (prompt, delayed, and average)

fission spectrum (prompt, delayed, and average)

(n,2n) reaction k unc. ~0.5% UO2/~0.6% MOX

dominated by U238 cap. (n,g)

1-group xs uncertainty ~1% to ~3%

dominated by U238 inel. (n,n')

macroscopic xs uncertainty ~1%

dominated by U238 inel. (n,n') UQ elsewhere macroscopic xs 1% for fast group and 0.5% for thermal

EOL isotopics increasing with burnup 0.1% U8 and 1% U5 decreasing with burnup 1.5% P9 and >2% other MA

assembly pin powers 0.6%

core radial power dist. avg. 3% range 0.5-5%

Beta-effective 3-5% Note: TMI-1 PWR CORE

continue submitting UAM results

address delayed neutron yield uncertainty and Beta-effective

use SS to generate SIMULATE-3 libraries (~100 libraries)

implement/research/develop uncertainty decomposition techniques for SS

Full transcriptsingle nuclides

high energy

thermal

resonance integrals reactor physics

criticality safety

isotopics (PIE)

shielding IRPhE - reactor physics

EXFOR - 40 yrs xs & yields data

SINBAD - shielding experiments

ICSBEP - 2000+ critical configurations

Databases http://www.oecd-nea.org/science/irphe/ Plant Core Lattice Multigroup Data Evaluated

Nuclear

Data ENDF

(American) JEFF

(European) JENDL

(Japanese) CENDL

(Chinese) BROND

(Russian) 1. convert to pointwise ENDF

2. combine resonance+"smooth" parts

3. Doppler broaden (adds T dep.)

4. unres. resonance treatment (adds sigz dep.)

5. energy group approx., specify:

intragroup flux -or-

slowing down theory parameters produces groupwise library

parametrized by

- temperature (T)

- background cross section (sigz) 1. assemble macro. xs in pin cells

2. derive equivalent hom. mixture (CP)

in pin cells to interpolate multigroup

library (586g) as function of (sigz,T)

3. calculate self-shielding factors

(resonance and Dancoff)

4. assembly macro. xs

5. collapse groups (586g->95g)

6. perform neutron/gamma transport 1. construct core from

library of 2D/2g lattice "slices"

assemblies from previous cycles

2. solve coarse-mesh diffusion for flux shape and eigenvalue with

polynomial intra-node flux bases

interfacial disconinuity factors

3. incl. simple thermal-hydraulics feedback and parameter adjustment to yield k=1, e.g. control rods (B) and sol. boron (P) Simulation Code[1] Code[2] Code[3] linear map linear map nonlinear map Our concern is with the simulation of nuclear power plants. Although the UQ techniques are by no means limited to these problems, we will attempt to highlight and discuss the particular difficulties with UQ in this system. In particular, we use a chain of codes to overcome the vast dimensionality in time, space, energy, and angle at scales from the nucleus to the power plant. In[2] In[1] Out[1] Out[2] In[3] Out[3] (Our models are usually nonlinear

and never analytic.) Uncertainty Quantification (UQ) is a necessary element of a "best estimate" code system. Not only does it indicate the true precision of a calculation but can help reveal biases as well. bimodal distribution UQ Methods Statistical Sampling (SS)

Analysis of Variance (ANOVA) Simplified Equations/Models

Neural Networks Reduced Order bad: validation problems (both ways)

good: fast, maybe even analytic bad: hard to use properly, training sets must be very large

good: easy to use, resultant network is portable Global bad: no SA

good: black box method, non-linear, global UQ with ~100 code runs bad: extremely costly (evaluate n-dim integrals for n inputs),

only analyze variance

good: black box method, non-linear systems, global SA+UQ Forward and Adjoint

Perturbation Theory (FPT/APT)

Direct Perturbation (DP) Local bad: first-order/significant implimentation difficulty

good: most efficient, local sensitivity analysis (SA) bad: first-order/inefficient

good: black box method, can estimate curvature of response, local SA Conclusions UQ in STARS UQ Data nuclear models accelerator & detectors data

synthesis parametrization produces macroscopic xs (2g),

kinetics param, discontinuity factors

- average fuel temp.

- average mod. density

- average void (B) or mod. dens. (P)

- burnup T

A

P

E F

I

L

E

(MF) MAT/MT MAT/MT elastic capture (n,g) calculation sequence (NJOY) calculation sequence (CASMO-5) produces power/burnup history, pin/assembly peaking factors, assembly-average isotopics, simulates in-core detectors

used primarily in safety analysis

and core performance optimization

also needed for transient "initial conditions" calulational sequence (SIMULATE-3) Core transients Plant transients (SIMULATE-3K) (TRACE) stability analysis

reactivity initiated transients

can provide "core neutronics kernel" for system codes full plant safety analysis, e.g. LOCA, LOFA, pump trip GRS Preliminary Transient

Results from UAM-5 rod withdrawal at HZP (OECD/NEA MOX/UO2 3D core transient benchmark)

with cross section uncertainty could use NJOY ERRORR/ERRORJ on to generate variance/covariance matrices

but variance/covariance matrix (VCM) data is unavailable for many nuclides/reactions (MAT/MT)

various projects have filled in gaps (AFCI, BNL Lo-fidelity, ORNL SCALE) Black box tools make the most sense in STARS.

Need lots of glue. fast text processing

good for scripting

functional/modular

object-oriented

60% of the time, it works every time

better than GIBIANE PERL automated, robust sensitivity coefficient calculation

sensco

STARS::Sensco CASMO-5M with cross section perturbation

cas5mx

STARS::Cas5mx CASMO-5M sensitivity coefficients for xs driver

cas5mx_sc

STARS::Sensco

STARS::PertXS

STARS::Cas5mx generation of xs perturbation files

pertxs

STARS::PertXS variance/covariance data lookup/manipulation

cvx

STARS::Coverx cross section sampling

sharkx error propagation (sandwich rule)

uqx

STARS::Coverx

STARS::MiM automated NJOY GROUPR calculations and "scattering fraction" extraction

njoyhelpr TOOLS First-order Local UQ

in a Nutshell Write a Taylor series.

Apply definition of mean/expectation.

Apply definition of variance.

Extend to multiple inputs.

Extend to multiple outputs. SS Efficiency

independent of number of inputs/outputs (order statistics)

dependent on correlation of output PDFs

moments of 1-output pdfs (mean, variance) ~100 runs for 95/95

moments of 2-output pdfs (covariance, correlation) ~1000 runs PT Efficiency

outputs > inputs = forward approach

inputs < outputs = adjoint approach

DP Efficiency

roughly 1/2 as efficient as forward Is transport linear? Yes No & The operators (streaming, absorption, scattering) in the transport equation are linear with respect to the (angular) flux. Leads to

existence and uniqueness

(if operators are not singular)

boundnedness with respect to perturbations The output (flux) is NOT a linear map of the input (cross sections). Note: self-shielding and depletion also makes the system nonlinear non-linearity is well-known to be fairly small

experience has shown first-order UQ methods are acceptable SCALE 6.0 44-group VCM LIBRARY

~400 nuclides

MTs 1,2,4,16-17,18,102-109,452,1018 U 238 3-group cross sections (barns) and relative std (%) total size

281 600 x 281 600

but very sparse correlation matrix interpolation (44g -> 3g)

does not preserve anti-correlation Statistical Sampling UQ

in a Nutshell theoretical foundation based on order statistics and provides bounds on output PDF, independent of distribution Wilk's formula yields N number of runs/samples for specified a/b 1-sided (e.g. max or min) or 2-side tolerance limits (e.g. variance). Only valid with random sampling.

In words:

"Given N runs, there is a b% chance that the N+1 run would produce a result outside the a% quantile." Wilk's Formula one-sided two-sided normal distribution sampling (MATLAB mvnrnd) Simple Random

Latin Hypercube (MATLAB lhsnorm) UQ Results Future Work UQ in STARS developed toolset to perform UQ with respect to nuclear data (xs)

active participation in UAM benchmark

observations so far perturb CASMO-5M proprietary 586-group ENDF-B/VII library

perturbations can be provided in any group structure

supported microscopic xs perturbations inelastic/elastic scattering

capture

neutrons per fission (prompt, delayed, and average)

fission spectrum (prompt, delayed, and average)

(n,2n) reaction k unc. ~0.5% UO2/~0.6% MOX

dominated by U238 cap. (n,g)

1-group xs uncertainty ~1% to ~3%

dominated by U238 inel. (n,n')

macroscopic xs uncertainty ~1%

dominated by U238 inel. (n,n') UQ elsewhere macroscopic xs 1% for fast group and 0.5% for thermal

EOL isotopics increasing with burnup 0.1% U8 and 1% U5 decreasing with burnup 1.5% P9 and >2% other MA

assembly pin powers 0.6%

core radial power dist. avg. 3% range 0.5-5%

Beta-effective 3-5% Note: TMI-1 PWR CORE

continue submitting UAM results

address delayed neutron yield uncertainty and Beta-effective

use SS to generate SIMULATE-3 libraries (~100 libraries)

implement/research/develop uncertainty decomposition techniques for SS