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need to improve the plant availability enhancing the performance for Nuclear Power Plants (NPPs) + a more and more ability to follow grid demands Classic control

dimensional or lumped parameter approach is sufficient Innovative control techniques Example: Optimal control Additional information regarding spatial distribution of

2 need to improve the plant availability enhancing the performance for Nuclear Power Plants (NPPs) + a more and more ability to follow grid demands Classic control techniques Example: Proportional-Integral- Derivative controller Require a simple knowledge of the system, i.e., total thermal power or average temperature Zero dimensional or lumped parameter approach is sufficient Innovative control techniques Example: Optimal control Additional information regarding spatial distribution of neutron flux and temperature may be necessary Accurate and detailed spatial models are required Developing specific simulation tools allows improving the control system design

3 Fast-running, low order, ODE (Ordinary Differential Equation) based, easy to linearize, comprehensive behaviour of the system, possibility to couple with the control system simulator How to enhance the level of detail and accuracy of the simulation tool in order to employ advanced control strategies without a strong computational burden?

4 Courtesy of T. Antoulas (Rice university), Model reduction of large-scale systems, YY Fest, Kyoto University, March 2010

. Input-output relationships have to be preserved.

Computationally efficient Compromise between accuracy

, 2003, "Creating Compact Models for Electronic

5 . Input-output relationships have to be preserved. Guarantee the stability of the ROM. Computationally efficient Compromise between accuracy and efficiency Shapiro, B., 2003, "Creating Compact Models for Electronic Systems: An Overview and Suggested Use of Existing Model Reduction and Experimental System Identification Tools", IEEE Transactions on Components, Packaging, and Manufacturing Technology - Part A, 26 (1), pp

6 . Real time context (parameter estimation, control, inverse problem). Many-query context (design optimization, multi-model simulation). Uncertainty quantification. CFD. Solid Mechanics. Fluid Mechanics. Heat Transfer. Multiphysics. Nuclear (Neutronics). Etc

7 . Modelling assumption (i.e., RANS). Projection framework (Galerkin but not only) Eigenvectors, POD, PGD, CVT, Reduced Basis, balanced truncation, Kyrlov subspaces. Courtesy of L. Daniel (MIT), Introduction to Compact Dynamical Modeling,. System identification / Data fitting Polynomial response surface, Kriging modelling

1) Collect few high-fidelity PDE solutions (snapshots) 2)

(Galerkin) projection. Never try to reduce the irreducible.

Exploit the known structure of the solutions.

Divide et impera whenever possible Taken form Rozza et al.

8 1) Collect few high-fidelity PDE solutions (snapshots) 2) Calculate the basis where project the governing equations 3) (Galerkin) projection. Never try to reduce the irreducible. If it is not in the snapshots, it is not in the ROM. Exploit the known structure of the solutions. Exploit the known physics of the problem. Divide et impera whenever possible Taken form Rozza et al., Reduced Basis Approximation and a Posteriori Error estimation for Affinely Parametrized Elliptic Coercive PDEs. Arch Comput Methods Eng (2008) 15:

9 The accuracy vs. the computational requirements can be tuned in according to the own goal (dynamics or control) Simplicity Accuracy

Simplified evaluation of the temperature feedbacks (constant coefficient) dn dt ρ β = Λ n + λ ic i + q 8

10 Neutronics usually modeled by point-wise kinetics 0-D description of the main physics of a nuclear reactor. No info on spatial distribution of the flux. Simplified evaluation of the temperature feedbacks (constant coefficient) dn dt ρ β = Λ n + λ ic i + q 8 i=1 dc i dt = β i Λ n λ ic i i = 1,, 8 ρ t = ρ K D ln T f eff t T f0 eff + α Z T f t T f0 taking into account the reactivity and flux spatial distribution

11 From a Full Order Model to a Reduced order model Transform the partial differential equations (PDEs) of the neutron diffusion into a set of ODEs involving only the time-dependent coefficient: φ 1 V t = D φ Σ a φ Σ s φ + 1 β χ p F T φ + λ j χ d C j j C j t = λ jc j + β j F T φ j = 1 8 Ansatz or God s bounty theorem* N φ r, t ψ i r n i t i=1 Spatial basis where the flux is projected * P. Ravetto. Politecnico di Torino. Time-dependent coefficients

12 1) The Ansatz has to be substituted in the Diffusion equation 2) Multiplying by the test functions ξ i and integrating (inner product) N N 8 RV im n m = L im δl im + 1 β M im + δm im n m + λ j c ij m=1 m=1 N j=1 cij = β j X M im + δm im n m λ j c ij j = 1 8 m=1 RV im = ξ i V 1 ψ m dω L im = ξ i D + Σ a + Σ s ψ m dω δl im = ξ i δ D + Σ a + Σ s ψ m dω M im = ξ i χ p F T ψ m dω δm im = ξ i δ χ p F T ψ m dω c ij = ξ i χ d C j dω X = χ 1 d /χ1 p χ 7 d /χ7 p

13 δl im T = ξ i D T z + Σ a T z + Σ s T z ψ m dω z = z = D T z ξ i ψ m dω z + Σ a T z ξ i ψ m dω z + Σ s T z ξ i ψ m dω z + z Affine representation +γ r ξ i ψ m ds r Ω r + γ a ξ i ψ m ds a Ω a. # of spatial basis N. Affine representation. Selection of the spatial basis ψ i. Selection of the test functions ξ i They have an impact in terms of efficiency, accuracy and computational time

14 Spatial basis choice: eigenfunctions associated to the neutron diffusion equation D + Σ a + Σ s ψ i = λ i χ p F T ψ i Lψ i = λ i Mψ i POD optimal basis that minimizes N φ r, t ψ i r n i t i=1 2 dt Procedure: 1. Snapshot computation Representative of the simulated conditions 2. Spatial basis calculation Singular Value Decomposition S snapshot matrix S = UΣV U basis matrix info = Σ singular values matrix l i=1 σ i n i=1 σ i 3. Projection of spatial basis Set of ODEs (reduced order model)

15 Which test functions can be selected? [4] The same functions which constitute the spatial basis ξ i = ψ i The test functions can be different from the spatial basis modes Due to the specific meaning in the neutronics field (and for other interesting properites), a possible candidate are the adjoint functions of the spatial basis! ξ i = ψ i Method for the spatial basis Test function Function of spatial basis Adjoint functions MM Case A Case B POD Case C Case D

16 OFFLINE procedure: the computation is made only once ONLINE calculation: ODE set can be run as many times as required

Reactor) LFR embodies Gen-IV key concepts of economics,

17 . 300 MW th 125 MW e. Pool reactor 8 loops. Scaled-down demonstrator for ELFR (European Lead Fast Reactor) LFR embodies Gen-IV key concepts of economics, sustainability, safety & reliability, proliferation resistance and physical protection

19 (b) (c) Average Average Channel Average Channel Channel Fuel pin design parameter Value Average pin power, kw 13.7 Coolant inlet temperature, C 400 Average coolant outlet 480 temperature, C Average coolant velocity, m s Fuel type MOX Average enrichment as Pu/(Pu+U), wt% Cladding Ti Fill gas He Active length, mm 600 Cladding outer diameter, mm 10.5 Cladding inner diameter, mm 9.3 Fuel pellet outer diameter, mm 9 Fuel pellet inner diameter, mm 2 Pin pitch, mm 13.86

20 SERPENT model of the ALFRED reactor

21 lat The geometry consists of 6 levels: Level 0 size: max 6 zones Level 1 size: max 529 zones Level 2 size: max 3 zones Level 3 size: max 3 zones Level 4 size: max 225 zones Each FA has been modelled separately in order to calculate the group constants Level 5 size: max 36 zones V 1 D Σ a Σ s χ p F T χ d β j λ j Upper Lower Group boundary boundary MeV 2.23 MeV MeV 0.82 MeV MeV 0.30 MeV MeV kev kev kev kev 0.75 kev kev 0.00 kev

23 Reactivity (pcm) SERPENT POLIMI ERANOS ENEA CR insertion (cm) SERPENT Uncertainty (σ) ERANOS Doppler constant (pcm) Lead expansion coefficient (pcm/k) Axial fuel expansion (pcm/k) Axial cladding expansion (pcm/k) Grid expansion (pcm/k) Axial wrapper expansion (pcm/k) Calculated considering all the lead inside the inner vessel. 2 Calculated for the whole height of the fissile subassemblies.

24 Diffusion model of the ALFRED reactor Finite Element φ V 1 t = D φ Σ a φ Σ s φ + 1 β χ p F T φ + λ j χ d C j j C j t = λ jc j + β j F T φ j = 1 8. Radial and axial albedo boundary conditions n D g φ g = γ a φ g n D g φ g = γ r φ g

and slice Σ T = Σ 0 + α log T T 0 5 for the

Production XS (Group 7) (1/m) Removal XS

25 . Temperature dependence of the cross sections (from SERPENT) for each coarse zone and slice Σ T = Σ 0 + α log T T 0 5 for the inner fuel, 5 for the outer fuel, 4 for the SRs, 5 for the CRs and 3 for the dummy elements. Production XS (Group 7) (1/m) Removal XS (Group 1) (1/m) Production XS (Group 4) (1/m) Diffusion coeff. (Group 4) (1/m) Removal XS (Group 5) (1/m)

27 Reactivity (pcm) SERPENT Transport COMSOL Diffusion CR insertion (cm) CR insertion (pcm) SERPENT COMSOL Error k Δρ (pcm) k Δρ (pcm) k Δρ(pcm) % % -5.00% % 3.50% % 6.37% % 7.13% % 7.72%

position and characteristic temperature).

28 The snapshots have been collected from the eigenfunctions related to different configurations of the ALFRED core (CR position and characteristic temperature). # ψ i for Case A,B ξ i for Case A ξ i for Case B ψ i for Case C,D ξ i for Case C ξ i for Case D 1 2 3

29 n = irv A 1,np + A 1,p n + A 2 c c = A 3,np + A 3,p n A 4 c Rho_in Reactivity Modal Neutron_flux Neutronics Th_Power T_Lead_in Pow er_connector Temperature_connectors W_Lead_in FuelPin_Lead (1) w 0 T p0 P T Flow _rate_source FuelPin_Lead (2) Sink FuelPin_Lead (3)

30 Reactivity comparison Neutronics modeling approach Reactivity inserted (pcm) Case 1 Case 2 Case 3 Multi-group diffusion PDEs (reference) Spatial Neutronics ROM (SN-ROM) Point Kinetics (PK) SN-ROM/PK error 0.51 % / 3.3 % 0.39 % / 25 % 0.10 % / 7.6 % From stationary case: 1. Uniform temperature decrease, T f1 = T f2 = T f3 = T l =-50 K; 2. Temperature enhancement in a single pin and in the 5 th axial slice, i.e., T f1 =+400 K, T f2 =+300 K, T f3 =+200 K, T l =+100 K; 3. Shutdown scenario: all the temperature are set equal to the inlet lead temperature, i.e., T= K.

31 Pin power (kw) 40.5 Transient comparison Increase of the inlet lead temperature (20 C) Neutronics modeling approach Multi-group diffusion PDEs (reference) Spatial Neutronics ROM Computational time 40 h N=1 N=3 N=5 N=10 10 s 23 s 45 s 143s The SN-ROM run with a laptop (2.20 GHz, 8 GB RAM). The neutron diffusion PDE solved with a workstation (8 x 2.8 GHz, 64 GB RAM). Pin power (kw) Object-oriented Multi-group diffusion model PDEs (present model work) [22] Multi-group Object-oriented diffusion model PDEs model [22] Time (s) Time [s] Time (s) Temperature Temperature variation variation ( C) ( C) Temperature variation ( C) Time (s) Inner fuel Central fuel Outer fuel Lead Inner fuel Central fuel Outer fuel Lead -5 x Object-oriented model (present work) Multi-group diffusion PDEs model [22] Time (s)

32 SERPENT (transport) COMSOL (diffusion) Error (%) Inner zone 128.1± Outer zone 206.7± Case A (N=5) Case B (N=5) Case C (N=5) Case C (N=7) Case D (N=5) COMSOL (reference) Inner zone Outer zone Reactivity (pcm) Case B Case C Case D COMSOL (reference) # of functions Reactivity (pcm) Case B Case C 160 Case D COMSOL (reference) # of functions

33 Doppler effect. Spatial reactivity (pcm) variation in each assembly in the active zone and along the axial direction.

34 SERPENT COMSOL Error (transport) (diffusion) (%) Lead density -262± Case A Case B Case C Case C Case D COMSOL (N=5) (N=5) (N=5) (N=7) (N=5) (reference) Reactivity (pcm) Case B Case C Case D COMSOL (reference) # of functions

35 Lead density effect. Spatial reactivity (pcm) variation in each assembly in the active zone along the axial direction.

36 CR position COMSOL Case C (N=7) Case D (N=7) (cm) (reference) k Δρ (pcm) k Δρ (pcm) Multiplication factor (-) COMSOL reference Case C Case D # of functions Multiplication factor (-) COMSOL reference Case C Case D # of functions

37 The best results are obtained if adjoint is used for the test functions. Thermal feedbacks δl im d, T = ψ i δl d, T ψ m dω = ψ i D d gz, T gz + Σ a d gz, T gz + Σ s d gz, T gz ψ m dω gz =. Reactivity: formulation alternative to Inverse Method gz = D d gz, T gz ψ i ψ m dω gz + Σ a d gz, T gz ψ i ψ m dω gz + Σ s d gz, T gz ψ i ψ m dω gz + gz + γ r ψ i ψ m ds r + γ a ψ i ψ m ds a Ω r Ω a ρ t = dωφ Lφ + 1 β χ p + βχ d Fφ dωφ 1 β χ p + βχ d Fφ n T A 1,np + A 1,p + βa md n ρ t = n T 1 β A mp + βa md n

38 . Perturbation theory Hφ = L λm φ = 0 Hφ = 0 H φ = 0 f Hg = H f g Adjoint definition H = H + dl φ N i=1 ψ a t Ansatz φ = φ + dε dε is the error introduced by the truncation φ H φ = φ H + dl φ + dε = = φ Hφ + φ Hdε + φ dlφ + φ dldε = = 0 + H φ dε + φ dlφ + φ dldε = = φ dlφ + φ dldε = First order perturbative term Second error term

Coolant pool usually represented by 0D/1D models The

39 Coolant pool usually represented by 0D/1D models The local heterogeneity cannot be represented by a classic Thermal-Hydraulic (T-H) model. No info on spatial distribution of the flow, turbulence mixing and 3D effects. No possibility of exploiting all the capacities of advanced control schemes (i.e., the oxygen control is crucial for LFR) From a Computational Fluid Dynamics (CFD) model of the pool to a reduced model able to catch the main T-H dynamics

40 Collaboration with Gianluigi Rozza (SISSA, Italy)

41 . Innovative techniques to improve NPP availability and to follow grid demands. Development of specific simulation tools allows improving control system design Need to develop models that are both accurate and fast-running Good agreement w/ reference model for reactivity and transient behavior w/o an excessive computational cost Preliminary results on the entire core models (Serpent, Comsol) indicate satisfactory agreement w/ literature data Best results when considering the adjoint flux as test function Model under development ALFRED (LFR concept) as reference reactor for the present general purpose approach

42 1. G. Rozza, D.B.P. Huynh, A.T. Patera. Reduced Basis Approximation and a Posteriori Error Estimation for Affinely Parametrized Elliptic Coercive Partial Differential Equations. Arch Comput. Methods Eng., 15, (2008). 2. A. Manzoni, A. Quarteroni, G. Rozza. Computational Reduction for Parametrized PDEs: Strategies and Applications. Milan Journal of Mathematics, 80, (2012). 3. Toni Lassila, Andrea Manzoni, Alfio Quarteroni and Gianluigi Rozza. Model Order Reduction in Fluid Dynamics: Challenges and Perspectives. In Reduced Order Methods for Modeling and Computational Reduction. MS&A - Modeling, Simulation and Applications, 9, (2014). 4. S. Lorenzi, A. Cammi, L. Luzzi. Spatial neutronics modelling to evaluate the temperature reactivity feedbacks in a Lead-cooled Fast Reactor. Proceedings of International Congress on Advances in Nuclear Power Plants " (ICAPP 2015). May 03-06, Nice (France), Paper S. Lorenzi, A. Cammi, L. Luzzi, R. Ponciroli. A control-oriented modeling approach to spatial neutronics simulation of a Lead-cooled Fast Reactor. Accepted for publication in ASME J. of Nuclear Rad. Sci., Paper No: NERS (2015)

45 φ φ 1 st group 4 th group 6 th group 7 th group

ANALYSIS OF THE COOLANT DENSITY REACTIVITY COEFFICIENT IN LFRs AND SFRs VIA MONTE CARLO PERTURBATION/SENSITIVITY

ANALYSIS OF THE COOLANT DENSITY REACTIVITY COEFFICIENT IN LFRs AND SFRs VIA MONTE CARLO PERTURBATION/SENSITIVITY Manuele Aufiero, Michael Martin and Massimiliano Fratoni University of California, Berkeley,