Spatio-temporal correlations in fuel pin simulation : prediction of true uncertainties on local neutron flux (preliminary results)

Transcription

1 Spatio-temporal correlations in fuel pin simulation : prediction of true uncertainties on local neutron flux (preliminary results) Anthony Onillon Neutronics and Criticality Safety Assessment Department EGAMCT meeting Paris, july 07, 2015

2 Summary: 1. Introduction : - Context - Estimation of statistical uncertainties by Monte Carlo codes 2. Estimation inter-cycles correlations - The fuel pin model - methodology 3. Results for temporal autocorrelation matrix 4. Prediction of true uncertainties 5. Results for spatio-temporal correlation matrix 6. Conclusion and perspectives

3 Summary: 1. Introduction : - Context - Estimation of statistical uncertainties by Monte Carlo codes 2. Estimation inter-cycles correlations - The fuel pin model - methodology 3. Results for temporal autocorrelation matrix 4. Prediction of true uncertainties 5. Results for spatio-temporal correlation matrix 6. Conclusion and perspectives

4 1 - Introduction : Context Underestimation of statistical uncertainties on local scores by Monte Carlo codes true tally variance can be calculated by performing N repeated Monte Carlo simulations with different random number seeds and calculating the variance of this group of N tally Simulation of a quarter core in infinite medium (z = 20cm) with MORET (R1 case) n src = n AC = Neutron flux map Uncertainties estimated by MORET True uncertainties Underestimation is depending on the simulated system mean free path of neutrons, dimension, reflexion conditions, statistical parameters 1/20

5 1 - Introduction : Monte-Carlo simulation for critical systems Monte Carlo simulation for critical systems Neutron source (position/energy) unknown Probabilistic resolution of neutron transport by the power iteration method - Simulation divided i - Initial neutron source (energy/position), number of neutrons per cycle and number of cycles set by user 1 st part : inactive cycles neutron source convergence 2 nd part : active cycles tallies computation Power iteration processus in a critical system [F. Brown LA-UR ,LANL-2005]. For a simulation with n part : number of neutrons per cycle : number of active cycles Tally results : x = 1 N N x i i=1 N = n part : total number of simulated neutrons (histories) Std dev : S x 2 = S2 N with S x 2 = N i=1 (x i x) 2 N 1 Central limit theorem : If N is high enough, x normaly distributed and S x can be used to create the confidence intervals : intervals at 1σ, 2σ, 3σ intervals at 68,2%, 95,5%, 99,7% Hypothesis: No correlations of histories betwees 2/20

6 1 - Introduction : Monte-Carlo simulation for critical systems Simulation of thermal system : mean free path of neutrons low vs system dimension Spatial and temporal correlations of neutron source distributions betwees Illustration for a simulation of fuel pin in infinite homogeneous medium : «temporal» correlations Clustering effect Spatial correlations 53 Neutron source distribution in a PWR fuel pin (3.8 m infinite medium) during 1000 cycles for et neutrons per cycle Initiale source : point source in (0,0,0) Need to take into account the temporal correlations in the calculation of uncertainties 3/20

7 1 - Introduction : Monte-Carlo simulation for critical systems Without inter-cycle correlations : S x = S x N Propagation of uncertainty y = f(x 1, x 2,.., x n ) With correlations : S x = S x = S x i S x = S x i ρ xi x j S x i,j=1 n part ρ xi x j i,j=1 σ 2 y = n i,j=1 V ij = cov x i, x j y y x i x j V ij x= = ρ xi x j σ xi σ xj Dependence of uncertainties to correlations and distribution of simulated histories between and n part Illustration of the effect of correlations on a gaussian distribution µ = 10 σ = 2 ρ i,i+1 = 0 µ = 10 σ = 2 ρ i,i+1 = 0,995 Random normal sampling (no correlation) Random normal sampling with correlations between each sample 4/20

8 1 - Introduction : Monte-Carlo simulation for critical systems Without inter-cycle correlations : S x = S x N Propagation of uncertainty y = f(x 1, x 2,.., x n ) With correlations : S x = S x = S x i S x = S x i ρ xi x j S x i,j=1 n part ρ xi x j i,j=1 σ 2 y = n i,j=1 V ij = cov x i, x j y y x i x j V ij x= = ρ xi x j σ xi σ xj Dependence of uncertainties to correlations and distribution of simulated histories between and n part Illustration of the effect of correlations on a gaussian distribution µ = 10 σ = 2 ρ i,i+1 = 0 µ = 10 σ = 2 ρ i,i+1 = 0,995 Random normal sampling (no correlation) n = 5000 Random normal sampling with correlations between each sample n = /20

9 1 - Introduction : Monte-Carlo simulation for critical systems Without inter-cycle correlations : S x = S x N Propagation of uncertainty y = f(x 1, x 2,.., x n ) With correlations : S x = S x = S x i S x = S x i ρ xi x j S x i,j=1 n part ρ xi x j i,j=1 σ 2 y = n i,j=1 V ij = cov x i, x j y y x i x j V ij x= = ρ xi x j σ xi σ xj Dependence of uncertainties to correlations and distribution of simulated histories between and n part Illustration of the effect of correlations on a gaussian distribution µ = 10 σ = 2 ρ i,i+1 = 0 µ = 10 σ = 2 ρ i,i+1 = 0,995 Random normal sampling (no correlation) n = Random normal sampling with correlations between each sample n = /20

10 Summary: 1. Introduction : - Context - Estimation of statistical uncertainties by Monte Carlo codes 2. Estimation inter-cycles correlations - The fuel pin model - methodology 3. Results for temporal autocorrelation matrix 4. Prediction of true uncertainties 5. Results for spatio-temporal correlation matrix 6. Conclusion and perspectives

11 1 - Intro : Context - 2 Fuel pin model Estimation of true uncertainties on local neutron flux in a multi-cells geometry through the computation of temporal inter-cycles correlation/covariance matrix on local neutron flux Generic calculation : Influence of histories distribution between the number of cycles and the number of neutrons per cycle Model : 1,325 cm UO 2 fuel rod : 3,8 m Infinite medium (reflexion on all surfaces : x-/x+/y-/y+/z-/z+) homogeneous flux along z 380 fuel volumes (L cell = 1 cm) Reflective surfaces Cell 1 Reduction to 20 cells for the analysis of temporal correlations (L cell = 19 cm) H 2 O (without bore) UO 2 ( 235 U : 4.3%) Cell 10 Cell 2 Cell 11 MORET 5.B.2 Simulation 4500 independant simulations with different random seed Uniforme initial neutron source distribution spatialy converged source neutrons/cycle, 2000 cycles limitation of clustering effect Pour all cells c, cycle i, run k : Extraction of neutron flux c i,k Cell 20 5/20

12 1 - Intro : Context - 2 Computation of autocorrelation coefficients : Bravais-Pearson estimator Temporal autocorrelation : Same variable (flux), same cell i, different temporal lags k ρ m i, m+k = cov( i m, m+k i ) = i 2 σ i N k ( i m i )( i m+k i ) m=1 N k m=1 ( m i i ) 2 N k m=1( m+k i i ) 2 ρ i m, i m+k = Autocorrelation in the same cell i for a temporal lag of k. i : mean value of in a cell i N : number of cycles i : cell number m : iterator over the cycles k : temporal lag Notation : ρ i m, i m+k = R i [k] Notation for a simulation with N cycles i 1 i 2 i 3 i i i i i i i i i i i 14 i N Cycle k = 1 k = 1 k = 1 k = 1 k = 1 k = 1 k = 1 k = 1 k = 1 k = 1 k = 1 k = 1 k = 1 k = 2 k = 2 k = 2 k = 2 k = 2 k = 2 k = 2 k = 2 k = 2 k = 2 k = 2 k = 2 Temporal lag k k = 3 k = 3 k = 3 k = 3 k = 3 k = 3 k = 3 k = 3 k = 3 k = 3 k = 3 Simulation with N cycles : (N-k) iterations available for a temporal lag of k 6/20

13 1 - Intro : Context - 2 Computation of autocorrelation coefficients : Bravais-Pearson estimator This study : 4500 simulations with 2000 cycles R i [k] = N k 4500 p= N k ( i m,p i )( i m+k,p i ) m=1 N k 4500 p=0 ) 2 ( m,p m=1 p=0 i i ) 2 ( m+k,p m=1 i i m : iterator over cycles p : iterator over simulations inconveniant : fast increase of σ R i [k] with the lag k by comparison with a study with only 1 simulation et cycles (same history number : ) p (N-k) iterations for a lag of k Advantage : allow the construction of cycle to cycle correlation / covariance matrix between the simulations Illustrations of i m, i m+k distributions for different values of ρ i m, i m+k ρ m i, m+k = 0 ρ m i, m+k i ρ m i, m+k = 0,25 ρ m = 0,75 ρ m i i i, m+k i, m+k = 0,5 = 1 i i 7/20

14 Estimated uncertainties (1σ) [%] True uncertainties (1σ) [%] 1 - Intro : Context - 2 Comparison of true and estimated uncertainties on local neutron flux 1 4 set of statistical parameters : identical n part and different 3000 independant simulations with different randoms seeds Computation of uncertainties on local neutron flux : standard deviation of the group True uncertainties along z 1/ decrease not respected temporal correlations Axial uncertainties not constant spatial correlations Uncertainties estimated by MORET quasiconstant along z et 1/ 1 Uncertainties estimated by MORET along z 8/20

15 True_uncertainties /MORET_uncertainties True uncertainties (1σ) [%] 1 - Intro : Context - 2 Comparison of true and estimated uncertainties on local neutron flux 1 4 set of statistical parameters : identical n part and different 3000 independant simulations with different randoms seeds Computation of uncertainties on local neutron flux : standard deviation of the group True uncertainties along z 1/ decrease not respected temporal correlations Axial uncertainties not constant spatial correlations Factor of underestimation [21-73] [16-38] Uncertainties estimated by MORET quasiconstant along z et 1/ Strong underestimation of uncertainties Dependance of the underestimation to the number of used cycles [6-13] [2-5] Ratio of true uncertainties to uncertainties estimated by MORET along z 9/20

16 Summary: 1. Introduction : - Context - Estimation of statistical uncertainties by Monte Carlo codes 2. Estimation inter-cycles correlations - The fuel pin model - methodology 3. Results for temporal autocorrelation matrix 4. Prediction of true uncertainties 5. Results for spatio-temporal correlation matrix 6. Conclusion and perspectives

17 1 - Intro : Context - 2 Fuel pin model / Pearson coefficient 3 - Résultats pour 3 Temporal les matrices correlation de corrélation matrix results temporelle : Cell : n 1 Cellule n 1 (1) Diagonal : ρ i, i = 1 ρ = 0.25 ρ = 0.5 ρ = 0.9 (2) off-diagonal : ρ i, i 1 Cellule 1 (extremity of the fuel pin) (1) Coefficients along the principal diagonal : Correlation of one cycle with itself ρ i, i = 1 (2) Off-diagonal coefficients : Correlation of one cycle with one another ρ i, j 1 10/20

18 1 - Intro : Context - 2 Fuel pin model / Pearson coefficient 3 - Résultats pour 3 Temporal les matrices correlation de corrélation matrix results temporelle : Cell : n 1 Cellule n 1 (1) Diagonal : ρ i, i = 1 (4) Decrease of correlation with the increase of temporal lag ρ = 0.25 ρ = 0.5 ρ = 0.9 ρ 1200, 1800 ρ 1200, 1500 ρ 1200, 1200 ρ 1900, 1500 ρ 1600, 1200 ρ 1300, 900 (3) Identical correlations (same temporal lag : 400) (2) off-diagonal : ρ i, i 1 Cellule 1 (extremity of the fuel pin) (1) Coefficients along the principal diagonal : Correlation of one cycle with itself ρ i, i = 1 (2) Off-diagonal coefficients : Correlation of one cycle with one another ρ i, j 1 (3) Coefficients along a same descending diagonal : Identical correlations same temporal lag (4) For a give, decrease of correlation when the temporal lag increase 10/20

19 1 - Intro : Context - 2 Fuel pin model / Pearson coefficient 3 - Résultats pour 3 Temporal les matrices correlation de corrélation matrix results temporelle : Cell : n 1 Cellule n 1 Autocorrelation coefficients converged after i 350 Stabilisation of correlations Cellule 1 (extremity of the fuel pin) (1) Coefficients along the principal diagonal : Correlation of one cycle with itself ρ i, i = 1 (2) Off-diagonal coefficients : Correlation of one cycle with one another ρ i, j 1 (3) Coefficients along a same descending diagonal : Identical correlations same temporal lag (4) For a give, decrease of correlation when the temporal lag increase 10/20

20 1 - Intro : Context - 2 Fuel pin model / Pearson coefficient 3 - Résultats pour 3 Temporal les matrices correlation de corrélation matrix results temporelle : Cell : n 1 Cellule n Cellule 1 (extremity of the fuel pin) zoom Convergence of autocorrelation coefficients R 1 [k] for different temporal lags k in the cell 1. Cell 1 Cell 10 ~ 350 cycles required to have fully converged autocorrelation coefficients and flux per cycle distribution 400 first cycles excluded of our analysis Convergence of σ i (std dev of flux distributions per cycle) for the first 10 fuel volumes of the rod 11/31

21 1 - Intro : Context - 2 Fuel pin model / Pearson coefficient 3 - Résultats 3 pour Temporal les matrices correlation de corrélation matrix results temporelle : Cell n 1-10 : Cellule n 1 Cell 1 Cell 2 Cell 3 Autocorrelation coefficients for a temporal lag of 1 : Cell Cell Cell Cell Cell 4 Cell 5 Cell 6 Cell Cell Cell Cell Cell Cell Cell 7 Cell 8 Cell 9 Cell 10 12/31

22 1 - Intro : Context - 2 Fuel pin model / Pearson coefficient 3 - Résultats 3 pour Temporal les matrices correlation de corrélation matrix results temporelle : Cell n 1-10 : Cellule n 1 Cell 1 Cell 2 Cell 3 Autocorrelation coefficients for a temporal lag of 1 : Cell Cell Cell Cell Cell 4 Cell 5 Cell 6 Cell Cell Cell Cell Cell Cell Cell 7 Cell 8 Cell 9 Cell 10 Identical "structures" : correlated results as coming from the same data set Reduction of intercycles correlations when we get closer the center of the rod Effect of the mirror reflexion at the top & bottom surfaces (spatial effect) : Cycle i : cell 1 : neutrons from C1 i-1 + C2 i-1 cell 2: neutrons from C1 i-1 + C2 i-1 + C3 i-1 cell 3: neutrons from C2 i-1 + C3 i-1 + C4 i-1 13/31

23 1 - Intro : Context - 2 Fuel pin model / Pearson coefficient 3 - Résultats pour les matrices de corrélation temporelle : Cellule n Autocorrelation & autocovariance fonction results Autocorrelation & autocovariance in fonction of the temporal lag k cov i, j = ρ i j σ i σ j R i [k] = cov( i m, m+k i ) 2 σ i Temporal lag k Temporal lag k Autocorrelation expanded over ~ 700 cycles for the most correlated fuel cells different "order" of curves because σ k i different : σ 1 i, σ 2 i,, σ 10 i = 4.28 %, 3.95 %, 3.65 %, 3.36 %, 3.08 %, 2.84 %, 2.62 %, 2.45 %, 2.32 %, 2.26 % Cell 1 Cell 2 Cell 3 Cell 4 Cell 5 Cell 6 Cell 7 Cell 8 Cell 9 Cell 10 R x < 0, R x < 0, Correlation cut-off (arbitrary limits) 14/20

24 Summary: 1. Introduction : - Context - Estimation of statistical uncertainties by Monte Carlo codes 2. Estimation inter-cycles correlations - The fuel pin model - methodology 3. Results for temporal autocorrelation matrix 4. Prediction of true uncertainties 5. Results for spatio-temporal correlation matrix 6. Conclusion and perspectives

25 Number of cycles Number of cycles 1 - Intro : Context - 2 Fuel pin model / Pearson coefficient 3 - Results : - Résultats pour les 4 matrices True statistical de corrélation uncertainties temporelle : Cellule n 1 ρ ij & cov i, j Map of uncertainties : uncertainties dependency to n part & Uncertainties taking into account temporal correlations : S x = ρ n ij part i,j=1 S x S 1 S = n + 2 ( i) R part i i=1 Number of neutrons per cycle Cellule 1 (extremity) Number of neutrons per cycle Cellule 10 (center) 15/20

26 Number of cycles Number of cycles 1 - Intro : Context - 2 Fuel pin model / Pearson coefficient 3 - Results : - Résultats pour les 4 matrices True statistical de corrélation uncertainties temporelle : Cellule n 1 ρ ij & cov i, j Map of uncertainties : uncertainties dependency to n part & Uncertainties taking into account temporal correlations : S x = n part i,j=1 ρ ij S 1 S = n + part i=1 2 ( i) R i σ < 10% σ < 5% σ < 1 % σ < 10% σ < 5% σ < 1 % S x Number of neutrons per cycle Cellule 1 (extremity) Number of neutrons per cycle Cellule 10 (center) Undersampling bias assumed to be zero for low statistics 15/20

27 Number of cycles Number of cycles 1 - Intro : Context - 2 Fuel pin model / Pearson coefficient 3 - Results : - Résultats pour les 4 matrices True statistical de corrélation uncertainties temporelle : Cellule n 1 ρ ij & cov i, j Map of uncertainties : uncertainties dependency to n part & Uncertainties taking into account temporal correlations : S x = n part i,j=1 ρ ij S 1 S = n + part i=1 2 ( i) R i σ < 10% σ < 5% σ < 1 % σ < 10% σ < 5% σ < 1 % S x 35% 10% 31% 9% 16% 7% Cpu-time x100 Cpu-time x100 Cpu-time x100 6% 2% Cpu-time x100 Cpu-time x100 Cpu-time x100 3% 2% Number of neutrons per cycle Cellule 1 (extremity) Number of neutrons per cycle Cellule 10 (center) "iso cpu-time" lines 15/20

28 1 - Intro : Context - 2 Fuel pin model / Pearson coefficient 3 - Results : - Résultats pour les 4 matrices True statistical de corrélation uncertainties temporelle : Cellule n 1 ρ ij & cov i, j Influence of the number of active cycles on the true statistical uncertainties σ(w/o cor. ) σ(w/ cor. ) = ρ ij i,j=1 Number of active cycles Ratio between uncertainties computed without and with temporal correlations Increase from a factor 9 (cell 10) to 20 (cell 3) of the uncertainties for an imporante number of cycles Equivalent to a reduction of the number of neutrons per cycle respectively of 80 and 400! Exemple : 2 simulations expected to have identical flux uncertainties in the cell 3 n part = = 1 N = n part = = N = /20

29 Standard deviation Standard deviation 1 - Intro : Context - 2 Fuel pin model / Pearson coefficient 3 - Results : - Résultats pour les 4 matrices True statistical de corrélation uncertainties temporelle : Cellule n 1 ρ ij & cov i, j Validation of predicted uncertainties (through correlation matrix) with the one obtained by performing N repeated Monte Carlo simulations with different random number seeds S = S i ρ n i j cycle i,j=1 MORET True uncertainties (gaussian fit of flux distribution of 3000 independant simulations) True uncertainties computed with the correlation matrix Very good agreement between the two methods! 17/20

30 Summary: 1. Introduction : - Context - Estimation of statistical uncertainties by Monte Carlo codes 2. Estimation inter-cycles correlations - The fuel pin model - methodology 3. Results for temporal autocorrelation matrix 4. Prediction of true uncertainties 5. Results for spatio-temporal correlation matrix 6. Conclusion and perspectives

31 1 - Intro : Context - 2 Fuel pin model / Pearson coefficient 3 - Results : 4 statistical - Résultats pour les matrices 5 de - Spatio-temporal corrélation temporelle correlations : Cellule n 1 ρ ij & cov i, j uncertainties Temporal autocorrelation : Same variable (flux), same cells i, different temporal lags k ρ i m, i m+k = cov( i m, i m+k ) σ 2 = ρ i m, i m+k = R i [k] N k ( i m i )( i m+k i ) m=1 N k m=1 ( m i i ) 2 N k m=1( m+k i i ) 2 Reminder i : mean value of in a cell i N : number of cycles i : cell number m : iterator over the cycles k : temporal lag Notation for a simulation with N cycles Spatio-temporal correlations : Same variable (flux), different cells, different temporal lags k C k i,j C k i,j = cov( i m, m+k j ) = σ i σ j N k ( i m i )( j m+k j ) m=1 N k m=1 ( m i i ) 2 N k m=1( m+k j j ) 2 = Correlation between two spatial bins i and j for a temporal lag of k i : mean value of in a cell i N : number of cycles i : cell number m : iterator over the cycles k : temporal lag Notation for a simulation with N cycles 18/20

32 1 - Intro : Context - 2 Fuel pin model / Pearson coefficient 3 - Results : 4 statistical - Résultats pour les matrices 5 de - Spatio-temporal corrélation temporelle correlations : Cellule n 1 ρ ij & cov i, j uncertainties Spatio-temporal correlations for different temporal lag k : 380 cells discretisation is used Spatio-temporal correlations extended over ~350 cycles same than the convergence time fo temporal correlations Correlation of closed cells and anti-correlation of apart cells Small spatial correlations in the center and high on the edge shape in "U" of σ i 19/20

33 Summary: 1. Introduction : - Context - Estimation of statistical uncertainties by Monte Carlo codes 2. Estimation inter-cycles correlations - The fuel pin model - methodology 3. Results for temporal autocorrelation matrix 4. Prediction of true uncertainties 5. Results for spatio-temporal correlation matrix 6. Conclusion and perspectives

34 1 - Intro : Context - 2 Fuel pin model / Pearson coefficient 3 - Results : 4 statistical 5 - spatio-temporal - Résultats pour les matrices de corrélation 6 Conclusion temporelle & Perspectives : Cellule n 1 ρ ij & cov i, j uncertainties correlations Conclusions Computation of temporal & spatial correlation matrix for a fuel pin simulation with MORET «Time» to converged the autocorrelation matrix is higher than the one needed for the convergence of the neutron source : ~ 400 cycles Autocorrelations extended over ~ 700 cycles (cells close to the edges) Possible to predict true uncertainties through a computation of autocorrelation matrix important influence of the distribution of histories between n part & : maximum variation of local uncertainties by a factor 20! validation via a comparison with the «standard method» usually used to compute true uncertainties nevertheless. : Huge cpu time needed to have a precise prediction Perspectives Study still in progress Estimation of uncertainties associated to R i k (also dependent of statistical parameters) domain of validity of the predicted uncertainties : Contribution of σ R k to σ S? predictive ability of S computation through a calcul of correlations during the simulation? Comparison of MORET results with the theoretical 1D correlation fonction computed by A. Zoia and E. Dumonteil Application of this study for the R2 case of the benchmark (in progress) Thanks for your attention! 20/20