study of predictor-corrector methods for monte carlo burnup...

Study of Predictor-corrector methods

for Monte Carlo Burnup Codes

Supervisor

By Dan Kotlyar

Dr. Eugene Shwageraus

Serpent International Users Group Meeting

Madrid, Spain, September 19-21, 2012

Introduction – High Conversion LWRs

1

0.90

0.95

1.00

1.05

1.10

1.15

0 200 400 600 800 1000 1200

Time, days

Eig

en

va

lue

2D (no T-H)

Introduction – High Conversion LWRs

1

0.90

0.95

1.00

1.05

1.10

1.15

0 200 400 600 800 1000 1200

Time, days

Eig

en

va

lue

3D (T-H)

2D (no T-H)

Introduction – Advisor’s response

2

What ?

What?

Why ?

Programming ?

Physical (model) ?

Numerical (coupling)?

Cause

How ?

Solution !

Is this problem common for all MC based codes ? …

0.00

0.05

0.10

0.15

0.20

0.25

0 50 100 150 200 250 300 350

Height, cmP

ow

er

sh

are

in

seed

T [i]

T[i+1]

Outline

3

Simplify the model

Generic PWR assembly in 3D

No T-H feedback, but non-uniform coolant density fixed in time

Study efficient schemes for coupling MC with multi-physics

feedbacks

Identify origins of the observed problem

Focus of this work:

Examine MC-burnup coupling schemes (& t size)

Was this problem identified earlier?

Recent burnup schemes sensitivity studies

Description of the test case (3D) & Results

Conclusions & future work

Introduction

4

Examples of Integrated MC-burnup codes:

MOCUP (Moore et al., 1995)

MCODE (Xu et al., 2002)

MONTEBURNS (Trellue, 2003)

SERPENT (Leppänen, 2007)

BGCore (Fridman et al., 2008)

MCNPX (Hendricks et al., 2008)

What about Integration of MC-burnup-TH

BGCore

SERPENT

Issues related to Integrated MC codes

Accuracy

Computation requirements

Objectives

5

Investigate stability and accuracy of current depletion calculations

Monte-Carlo/Burnup coupling scheme

Depletion timestep size

Commonly used methods

Explicit Euler predictor method

Euler predictor-corrector method

Propose an extension to the modified Euler predictor-corrector

Accuracy as the predictor-corrector method

Computation requirements as the predictor method

Possible multi-physics coupling schemes

6

Thermal feedback

Depletion module

Neutronics MC

Power & BU

Distribution

Temp & density

Distribution

(r, E)

Beginning of Step analysis (predictor)

Total exe time (Depletion/TH)≈ n·Tmcnp

New concentration for next step

(x 1)

(x n)

Integration scheme 1

Inner loop: TH-neutronic

Outer loop: Depletion

Integration scheme 2

Inner loop: Depletion

Outer loop: TH-neutronuc

Important issues

Combination of #1 and #2

Calculation time

Source convergence

Distribution of errors

Variable convergence tolerances

Recent sensitivity studies

7

Yamamoto et al., 2008 – Projected predictor corrector method

Linear correlation between the number density and the microscopic RR

Tested for the Gd-bearing 2D fuel assembly

Carpenter et al., 2010 (Bettis Atomic Power Laboratory)

Modified Log Linear correlation (Yamamoto)

Tested for the Gd-bearing 2D fuel assembly

Isotalo et al., 2011 – Higher order methods

Use more (previous) BU points

Tested on PWR pin cell and seed/blanket assembly

Saadi et al., 2012 – Burnup sensitivity analysis

Tested on 1D UO2 PWR unit cell

Dufek et al., 2009 – Numerical stability of MC-burnup codes

Instability demonstrated on infinitely reflected 3D unit cell

Burnup coupling scheme

methodology

Extended Predictor-Corrector

Euler Predictor-Corrector

Explicit Euler Predictor

Formulation of the burnup problem

Burnup calculations

8

Eigen-value

transport problem

Burnup equation

Matrix Exponential solution

trNdEEtEr

dt

trdN,,,

,

0

0,,,,1

,

tErtrNF

ktrNL

n n+1 n+2 ……… EOC

nnnn ttrNrN 11 exp

Burnup calculations

8

Eigen-value equation

Burnup equation

Mat. Exp. solution

Objective: coupled space-energy-time dependent solution:

Coupling scheme: Independent neutronics and depletion solvers

trNdEEtEr

dt

trdN,,,

,

0

0,,,,1

,

tErtrNF

ktrNL

n n+1 n+2 ……… EOC

nnnn ttrNrN 11 exp

Euler predictor-corrector methods

9

1. Explicit Euler Predictor method

Reaction rate (RR) calculation (neutronics - ) at BOT

Depletion with BOT RR’s Nn+1

2. Modified Euler Predictor-Corrector method

Reaction rate calculation ( ) with predicted Nn+1 values

Depletion with EOT (n+1) reaction rates to obtain corrected Nn+1 values

Finally, the predicted and corrected ND are averaged

Used as initial values for the following step

n n+1 Time/BU

Nn Nn+1

const RRn

const RRn+1

Nn Nn Nn Nn+1 Nn

Why is the P-C method is not enough ? t !

10

Quote: “Predictor-corrector methods are numerically explicit”

The stable size may not be much larger than of that of the predictor

Numerical methods for engineers and scientists, Joe D. Hoffman

0.00

0.01

0.02

0.03

0.04

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0.06

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0.08

0.09

0 50 100 150 200 250 300 350 400

Height, cm

No

rmalized

co

mm

ula

tive e

nerg

y (

50 d

ays)

5 d

10 d

25 d

50 d

0.000

0.020

0.040

0.060

0.080

0.100

0.120

0.140

0 50 100 150 200 250 300 350 400

Height, cm

Po

we

r S

ha

re

Predictor

Corrector

50,0t

i ttPEt

50 d 0

50 days

Suggested E-P-C method

11

3. Extended Predictor-Corrector method (E-P-C)

Assume (power, RR, σ, N) are known for the previous [tn-1,tn] interval

Deplete with previous average RR Nn+0.5 ( =Nnexp[-RRn-1·t0.5] )

Update transport solution (RR calculation ) at MOT (tn+0.5)

Deplete with MOT RR’s Nn+1

Reaction rate calculation ( ) with predicted Nn+1 values

Depletion with EOT (n+1) reaction rates to obtain corrected Nn+1 values

Finally, the predicted and corrected ND are averaged

n 100d

n+0.5 125d

Time/BU

Nn Nn+0.5

const RRn+0.5

const RRn+1 n+1 150d

Nn+1 Nn-1

n-1 75d

Different coupling

scheme results

3D assembly test case

12

+20,000 Radial layout

Cylindrical pins

With guide tubes

Axial layout / Coolant density profile

21.5 cm

0.6

9

0.7

0

0.7

1

0.7

2

0.7

3

0.7

4

0.7

5

0.7

6

Coolant Density, gr/cm3

BGCore vs. SERPENT comparison

13

1.10

1.15

1.20

1.25

1.30

1.35

1.40

0 50 100 150 200 250 300 350 400

Time, days

Eig

en

valu

e

BGCore

SERPENT

0.0E+00

1.0E-05

2.0E-05

3.0E-05

4.0E-05

5.0E-05

6.0E-05

7.0E-05

8.0E-05

9.0E-05

1.0E-04

0 50 100 150 200 250 300 350 400

Time, days

Co

ncen

trati

on

of

Pu

239, #/b

·cm

BGCore

SERPENT

Maximum difference,

SERPENT vs. BGCore Parameter

~70 pcm k-eff

~0.5 % Xe135

~0.4 % U235

~0.4 % Pu239

Method : predictor-corrector

Timestep : 5 days

Burnup coupling scheme method & T

14

Description Case

no. Designated T,

days Method

P-C-0 (25d) 25 Predictor 1

P-C-1 ( 5d) 5 Predictor-Corrector a 3

P-C-1 (25d) 25 Predictor-Corrector 3


E-P-C (50d) 50 Extended-Predictor-Corrector 5

a. Chosen to be the reference case


14

-200

0

200

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600

800

1000

1200

0 50 100 150 200 250 300 350 400

Time, days

Dif

fere

nce in

eig

en

valu

e, p

cm

P-C-0 (25d)

P-C-1 (25d)

P-C-1 (50d)

Description Case

no. Designated T,

days Method








14

-200

0

200

400

600

800

1000

1200

0 50 100 150 200 250 300 350 400

Time, days

Dif

fere

nce in

eig

en

valu

e, p

cm

P-C-0 (25d)

P-C-1 (25d)

P-C-1 (50d)

0.0E+00

2.0E-09

4.0E-09

6.0E-09

8.0E-09

1.0E-08

1.2E-08

0 50 100 150 200 250 300 350 400

Time, days

Co

ncen

trati

on

of

Xe-1

35

P-C-0 (25d)

P-C-1 (25d)

P-C-1 (50d)

P-C-1 (5d)

Description Case

no. Designated T,

days Method








14

-200

0

200

400

600

800

1000

1200

0 50 100 150 200 250 300 350 400

Time, days

Dif

fere

nce in

eig

en

valu

e, p

cm

P-C-0 (25d)

P-C-1 (25d)

P-C-1 (50d)

0.0E+00

2.0E-09

4.0E-09

6.0E-09

8.0E-09

1.0E-08

1.2E-08

0 50 100 150 200 250 300 350 400

Time, days

Co

nc

en

trati

on

of

Xe-1

35

P-C-0 (25d)

P-C-1 (25d)

P-C-1 (50d)

P-C-1 (5d)

0.0E+00

5.0E+13

1.0E+14

1.5E+14

2.0E+14

2.5E+14

3.0E+14

3.5E+14

4.0E+14

0 50 100 150 200 250 300 350 400

Height, cm

Flu

x d

istr

ibu

tio

n, n

/s·c

m2

T(i)

T(i+1)

Description Case

no. Designated T,

days Method







The effect of non-uniform coolant density

15

0.80

0.90

1.00

1.10

1.20

1.30

1.40

0 200 400 600 800 1000 1200

Time, days

Eig

en

valu

e

P-C-0

P-C-1

Averaged coolant density along the z-axis

1.10

1.15

1.20

1.25

1.30

1.35

1.40

0 50 100 150 200 250 300 350 400

Time, days

Eig

en

valu

e

P-C-0 (25d)

P-C-1 (25d)

P-C-1 (50d)

Realistic coolant density along the z-axis

Comparison between P-C and E-P-C

16

1.10

1.15

1.20

1.25

1.30

1.35

1.40

0 50 100 150 200 250 300 350 400

Time, days

k-e

ff

-50

50

150

250

350

450

550

Dif

fere

nc

e [

pc

m]

to P

-C-1

-25

d

P-C-1 (25d)

P-C-1 (50d)

E-P-C (50d)

DIFF (P-C-1)

DIFF (E-P-C)

4.5E-04

5.0E-04

5.5E-04

6.0E-04

6.5E-04

7.0E-04

7.5E-04

8.0E-04

8.5E-04

0 50 100 150 200 250 300 350 400

Time, days

Co

ncen

trati

on

of

U-2

35

-3.0%

-2.5%

-2.0%

-1.5%

-1.0%

-0.5%

0.0%

0.5%

1.0%

Rela

tive d

iffe

ren

ce t

o P

-C-1

-25d

P-C-1 (25d)

P-C-1 (50d)

E-P-C (50d)

DIFF (P-C-1)

DIFF (E-P-C)

Maximum difference (compared to P-C-1) in different core parameters

Maximum difference,

E-P-C (50d) vs. P-C-1 (25 d) Maximum difference,

P-C-1 (50d) vs. P-C-1 (25 d) Parameter

~20 pcm ~500 pcm k-eff

~0.5 % ~2.0 % Xe135

~0.1 % ~1.9 % U235

~0.2 % ~2.5 % Pu239

Conclusions

&

future work

Summary

17

Sensitivity studies of different coupling schemes were performed

Explicit Euler predictor method may be unstable for some 3D problems

Predictor-corrector method with Δt ~50d does not resolve the problem

EPC method allows using larger timesteps

Maintains sufficient accuracy

Future work:

How/When is the Thermal-hydraulic feedback should be applied?

Thank You for your attention!

study of predictor-corrector methods for monte carlo burnup...

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