Method and algorithm for searching and optimizing nuclear reactor core loading patterns

US 20060109944A1
Filed: 11/19/2004
Published: 05/25/2006
Est. Priority Date: 11/19/2004
Status: Active Grant

First Claim

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1. A method of establishing a nuclear reactor core loading pattern for loading fuel assemblies and burnable absorbers into a nuclear reactor, said method comprising the steps of:

a. providing nuclear data representing fuel assemblies and burnable absorbers in a nuclear reactor core;

b. depleting said nuclear data to form a reference core depletion;

c. incorporating said nuclear data into a system of equations of a nuclear design quality neutron flux solution;

d. defining said system of equations to include constraints which accurately represent the neutron physics of said nuclear reactor, including effects of shuffling said fuel assemblies and of loading burnable absorbers into said reactor;

e. employing said system of linear equations as a constraint matrix for a solver and running said solver in order to find an optimum core pattern solution; and

f. repeating steps b through e, updating the constraints and objective functions which can be represented as linear constraints within said constraint matrix, in order to satisfy specified engineering requirements and establish an optimum core loading pattern.

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Abstract

A method is for establishing a nuclear reactor core loading pattern (LP) for fuel assemblies and burnable absorbers (BAs). The method establishes an optimum LP through the steps of: a) providing nuclear data representing fuel assemblies and BAs in a nuclear reactor core; b) depleting the nuclear data to form a reference core depletion; c) incorporating the nuclear data into a system of linear equations of a nuclear design quality flux solution method; d) defining the system of linear equations to include constraints which accurately represent the neutron physics of the reactor; employing the equations as a constraint matrix for a MIP solver to find an optimum core pattern solution; f) repeating steps b) through e) updating the constraints and objective functions to satisfy specified engineering requirements and establish an optimum core loading pattern. An algorithm for deriving the system of equations is also disclosed.

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1. A method of establishing a nuclear reactor core loading pattern for loading fuel assemblies and burnable absorbers into a nuclear reactor, said method comprising the steps of:
- a. providing nuclear data representing fuel assemblies and burnable absorbers in a nuclear reactor core;
  
  b. depleting said nuclear data to form a reference core depletion;
  
  c. incorporating said nuclear data into a system of equations of a nuclear design quality neutron flux solution;
  
  d. defining said system of equations to include constraints which accurately represent the neutron physics of said nuclear reactor, including effects of shuffling said fuel assemblies and of loading burnable absorbers into said reactor;
  
  e. employing said system of linear equations as a constraint matrix for a solver and running said solver in order to find an optimum core pattern solution; and
  
  f. repeating steps b through e, updating the constraints and objective functions which can be represented as linear constraints within said constraint matrix, in order to satisfy specified engineering requirements and establish an optimum core loading pattern.
- View Dependent Claims (2, 3, 4, 5, 6, 7, 8, 9, 10, 11)
- - 2. The method of claim 1 including loading the core of said nuclear reactor in accordance with said optimum core loading pattern.
  - 3. The method of claim 1 wherein said nuclear design quality neutron flux solution from which said system of equations is derived is the Analytic Course Mesh Finite Difference (ACMFD) method;
    - wherein said solver is a mixed integer program (MIP) solver; and
      
      wherein the system of equations derived from said ACMFD method are incorporated directly into said constraint matrix for said MIP solver.
  - 4. The method of claim 1 wherein said effects of shuffling said fuel assemblies in step d include shuffling non-linearity;
    - and wherein said system of equations represent said shuffling non-linearity directly without approximation as a set of linear equations and auxiliary variables.
  - 5. The method of claim 4 wherein step d further includes deriving said system of equations, in part, using a continuous relaxation of the term in the equations of assembly nuclear parameters times assembly flux;
    - and wherein said continuous relaxation segregates partial fluxes and partial assembly parameters by eliminating the cross term, thereby reducing the order of the equations in said system of equations.
  - 6. The method of claim 5 wherein said system of equations are derived, in part, by calculating certain coefficients;
    - and wherein said coefficients are calculated by depleting candidate shuffled fuel assemblies and candidate burnable absorbers using said reference core depletion fast flux as an approximation to the shuffled pattern fast flux.
  - 7. The method of claim 1 wherein step d includes representing the depletion of said reactor core in said system of equations.
  - 8. The method of claim 1 further including the step of generating a feed pattern to be analyzed after step b of depleting said nuclear data to form a reference core depletion.
  - 9. The method of claim 1 wherein said specified engineering requirements of step f are selected from the group consisting of fuel cycle cost, power constraints on peak power in said core, burnup constraints on peak burnup of one or more fuel assemblies, and custom user supplied engineering requirements;
    - and wherein step f includes the steps of;
      
      selecting at least one of said specified engineering requirements, representing said objective function or constraint as a linear combination of neutron flux, power, burnup, core reactivity, core critical boron concentration, assembly location or BA assignment, supplying said linear representation to the constraint matrix of said solver, running said solver to find an optimum satisfying said specified engineering requirements, and repeating the steps of adding additional constraints or objectives until an optimum core loading pattern which meets all of the desired requirements is achieved.
  - 10. The method of claim 1 wherein said equations incorporated into said constraint matrix are determined by the approach embodied in the following expressions (1) through (18):
    - $\begin{matrix} \sum_{j \in adjacent} J_{ij}^{'} + ϕ_{i}^{1} \sum_{R}^{1} (h_{\infty i} - 1) = 0 \forall_{i} & (1) \end{matrix}$ where k_∞ for the i^thnode is;
      
      $\begin{matrix} k_{\propto i} = {[\frac{υ^{1} \sum_{f}^{1}}{\sum_{R}^{1}} + \frac{υ^{2} \sum_{f}^{2}}{\sum_{R}^{1}} \frac{\sum_{s}^{12}}{\sum_{a}^{2} + D^{1} B^{2^{2}} ❘ ref} - 1]}_{i} & (2) \end{matrix}$ and the equation for the currents between the i^thnode and the j neighbors of i are;
      
      $\begin{matrix} J_{ij} = - \sum_{j \in adjacent} (D_{i}^{s} {\overline{ϕ}}_{i} - D_{j}^{A} {\overline{ϕ}}_{j}) & (3) \end{matrix}$ where;
      
      S stands for “
      
      self” and
      
      D^Sdepends on the characteristics of the i^thnode, A stands for “
      
      adjacent” and
      
      DA depends on the characteristics of the adjacent nodes, ν
      
      ^mis the neutrons per fission in the n^thgroup, Σ
      
      _f^mis the macroscopic fission cross section for the n^thgroup, Σ
      
      _R¹is the macroscopic removal cross section for the fast group, Σ
      
      _a²is the macroscopic absorption cross section for the thermal group, D²is the diffusion coefficient for the thermal group, B²is the buckling for the thermal group, i and j are radial indices over nodes in the nuclear model, J represents the neutron current, and φ
      
      represents the neutron flux. When one includes the effects of shuffles, expression 1 also has terms like the following;
      
      $\begin{matrix} D_{ℓ}^{d} ϕ_{ℓ}^{n} \approx ϕ_{ℓ}^{n} [\sum_{a} X_{ℓ a} D_{a}^{od} + \sum_{b = 1}^{n} {\overline{P}}_{t}^{b} {step}^{b} \sum_{a} X_{ℓ a} \frac{ⅆ D^{d}}{ⅆ bu} ❘_{a}^{b}] & (4) \end{matrix}$ where;
      
      b=the burn-up step, n=the current step, P₁=the power in the 1^thnode, the subscript “
      
      1”
      
      is the index over location, the subscript “
      
      a”
      
      is the index over assembly, X_lais the [0,1] variable that is 1 only if assembly a is assigned to location 1, The “
      
      0”
      
      superscript refers to quantities that are from the reference distribution, and “
      
      step”
      
      refers to the length of the burnup step in the depletion. When one includes the effects of shuffles, expression 1 also has a term like the following;
      
      $\begin{matrix} ϕ_{ℓ}^{n} \sum_{ℓ}^{n} = ϕ_{ℓ}^{n} [\sum_{a} X_{ℓ a} \sum_{a}^{o} + \sum_{b = 1}^{n} {\overline{P}}_{ℓ}^{b} {step}^{b} \sum_{a} X_{ℓ a} \frac{ⅆ Σ}{ⅆ bu} ❘_{a}^{b}] & (5) \end{matrix}$ where the only new notation is the following;
      
      Σ
      
      =Σ
      
      _R¹(K_∞−
      
      1)
      
      (6) Both of these (expressions 4 and
      
      5) have a term like the following;
      
      $\begin{matrix} \sum_{b = 1}^{n} {\overline{P}}_{ℓ}^{b} {step}^{b} \sum_{a} X_{ℓ a} \frac{ⅆ Σ}{ⅆ bu} ❘_{a}^{b} \equiv term & (7) \end{matrix}$ where;
      
      the Σ
      
      may represent the definition in expression, or it may represent D^d. The average power during the b^thdepletion step is;
      
      $\begin{matrix} {\overline{P}}_{ℓ}^{b} = \frac{P^{b - 1} + P^{b}}{2} & (8) \end{matrix}$ where;
      
      step b is the depletion step in mega watt days per metric ton uranium (MWD/MTU) for the core, and dΣ
      
      /db for the b step for the a assembly is the change in Σ
      
      during the depletion step. Using the reference pattern fast flux for depletion gives;
      
      $\begin{matrix} {\overline{P}}_{l}^{b} = {\overline{ϕ}}_{l}^{b, ref} \sum_{a} X_{l a} (K^{1} \sum_{f}^{1} + \frac{\sum_{s}^{12} K^{2} \sum_{f}^{2}}{{\sum_{R}^{1} + D_{2} B_{2}^{2} \rangle}_{ref}}) ❘_{a}^{b} & (9) \end{matrix}$ where;
      
      K represents the energy per fission for the fast group with a superscript 1 and for the thermal group with a superscript 2, which makes;
      
      $\begin{matrix} {{term = \sum_{b = 1}^{n} {\overline{ϕ}}_{l}^{b, ref} \sum_{a} X_{la} K \sum_{f} \rangle}_{a}^{b} \sum_{a^{'}} X_{{ta}^{'}} \frac{ⅆ \sum}{ⅆ bu} \rangle}_{a^{'}}^{b} & (10) \end{matrix}$ The equation becomes;
      
      $\begin{matrix} term \to \sum_{b = 1}^{n} {\overline{ϕ}}_{l}^{b, ref} \sum_{a} {X_{la} (K \sum_{f} \rangle}_{a}^{b}) {(\frac{ⅆ \sum}{ⅆ bu} \rangle}_{a}^{b}) & (11) \end{matrix}$ For X_la=1, the equations are identical. This makes the neutron current terms;
      
      $\begin{matrix} D_{l}^{d} ϕ_{l}^{n} \approx ϕ_{l}^{n} [\sum_{a} X_{la} D_{a}^{od} + \sum_{b = 1}^{n} {\overline{ϕ}}_{l}^{b, ref} {step}^{b} \sum_{a} {X_{la} (K \sum_{f} \rangle}_{a}^{b}) {(\frac{ⅆ \sum}{ⅆ bu} \rangle}_{a}^{b})] & (12) \end{matrix}$ This makes the multiplication and absorption term;
      
      $\begin{matrix} \sum ϕ_{l}^{n} = ϕ_{l}^{n} [\sum_{a} X_{la} \sum_{a}^{o} + \sum_{b = 1}^{n} {\overline{ϕ}}_{l}^{b} {step}^{b} \sum_{a} {X_{la} (K \sum_{f} \rangle}_{a}^{b}) {(\frac{ⅆ \sum}{ⅆ bu} \rangle}_{a}^{b})] & (13) \end{matrix}$ The non-linear terms in the above equation are;
      
      φ
      
      _lⁿX_la
      
      (14) The linearization of this product is;
      
      Ψ
      
      _la=X_la
      
      (15)
      Ψ
      
      _la−
      
      MX_α≦
      
      0
      
      (16)
      −
      
      φ
      
      _l+Ψ
      
      _la≦
      
      0
      φ
      
      _l−
      
      Ψ
      
      _ka+MX_la≦
      
      M where;
      
      M is the upper bound for X and thus, also for Ψ
      
      _la, and wherein expressions 15-18 represent new constraint equations to be added to said constraint matrix.
  - 11. The method of claim 10 including the steps of:
    - programming expressions 15-18 directly into the constraint matrix of said MIP solver, thereby minimizing the number of design code flux solutions required; and
      
      running said MIP solver to find an optimal LP.

12. An algorithm for optimizing nuclear reactor core loading patterns for loading fuel assemblies and burnable absorbers into a nuclear reactor wherein said loading pattern is established in accordance with a method comprising the steps of:
- a. providing nuclear data representing fuel assemblies and burnable absorbers in a nuclear reactor core;
  
  b. depleting said nuclear data to form a reference core depletion;
  
  c. incorporating said nuclear data into a system of equations of a nuclear design quality neutron flux solution;
  
  d. defining said system of equations to include constraints which accurately represent the neutron physics of said nuclear reactor, including effects of shuffling said fuel assemblies and of loading burnable absorbers into said reactor;
  
  e. employing said system of equations as a constraint matrix for a solver and running said solver in order to find an optimum core pattern solution;
  
  f. repeating steps b through e, updating the constraints and objective functions which can be represented as linear constraints within said constraint matrix, in order to satisfy specified engineering requirements and establishing an optimum core loading pattern. wherein said algorithm defines said system of equations incorporated into said constraint matrix and is embodied in the approach of the following expressions (1) through (18);
  
  $\begin{matrix} \sum_{j \in adjacent} J_{ij}^{1} + ϕ_{i}^{1} \sum_{R}^{1} (h_{\infty i} - 1) = 0 \forall_{i} & (1) \end{matrix}$ where k_∞ for the i^thnode is;
  
  $\begin{matrix} k_{\infty i} = {[\frac{v^{1} \sum_{f}^{1}}{\sum_{R}^{1}} + \frac{v^{2} \sum_{f}^{2}}{\sum_{R}^{1}} \frac{\sum_{s}^{12}}{\sum_{a}^{2} + D^{1} B^{2^{2}} \langle ref} - 1]}_{i} & (2) \end{matrix}$ and the equation for the currents between the i^thnode and the j neighbors of i are;
  
  $\begin{matrix} J_{ij} = - \sum_{j \in adjacent} (D_{i}^{s} {\overline{ϕ}}_{i} - D_{j}^{A} {\overline{ϕ}}_{j}) & (3) \end{matrix}$ where;
  
  S stands for “
  
  self” and
  
  D^Sdepends on the characteristics of the i^thnode, A stands for “
  
  adjacent” and
  
  D^Adepends on the characteristics of the adjacent nodes, ν
  
  ^mis the neutrons per fission in the n^thgroup, Σ
  
  _f^mis the macroscopic fission cross section for the n^thgroup, Σ
  
  _R¹is the macroscopic removal cross section for the fast group, Σ
  
  _a²is the macroscopic absorption cross section for the thermal group, D²is the diffusion coefficient for the thermal group, B²is the buckling for the thermal group, i and j are radial indices over nodes in the nuclear model, J represents the neutron current, and φ
  
  represents the neutron flux. When one includes the effects of shuffles, expression 1 also has terms like the following;
  
  $\begin{matrix} D_{l}^{d} ϕ_{l}^{n} \approx {ϕ_{l}^{n} [\sum_{a} X_{la} D_{a}^{od} + \sum_{b = 1}^{n} {\overline{P}}_{t}^{b} {step}^{b} \sum_{a} X_{la} \frac{ⅆ D^{d}}{ⅆ bu} \rangle}_{a}^{b}] & (4) \end{matrix}$ where;
  
  b=the burn-up step, n=the current step, P₁=the power in the Ith node, the subscript “
  
  1”
  
  is the index over location, the subscript “
  
  a”
  
  is the index over assembly, X_lais the [0,1] variable that is 1 only if assembly a is assigned to location 1, The “
  
  0”
  
  superscript refers to quantities that are from the reference distribution, and “
  
  step”
  
  refers to the length of the burnup step in the depletion. When one includes the effects of shuffles, expression 1 also has a term like the following;
  
  $\begin{matrix} ϕ_{l}^{n} \sum_{l}^{n} = {ϕ_{l}^{n} [\sum_{a} X_{la} \sum_{a}^{o} + \sum_{b = 1}^{n} {\overline{P}}_{l}^{b} {step}^{b} \sum_{a} X_{la} \frac{ⅆ \sum}{ⅆ bu} \rangle}_{a}^{b}] & (5) \end{matrix}$ where the only new notation ins the following;
  
  Σ
  
  =Σ
  
  _R¹(K_∞−
  
  1)
  
  (6) Both of these (expressions 4 and
  
  5) have a term like the following;
  
  $\begin{matrix} {\sum_{b = 1}^{n} {\overline{P}}_{l}^{b} {step}^{b} \sum_{a} X_{la} \frac{ⅆ \sum}{ⅆ bu} \rangle}_{a}^{b} \equiv term & (7) \end{matrix}$ where;
  
  the Σ
  
  may represent the definition in expression (6), or it may represent D^d. The average power during the b^thdepletion step is;
  
  $\begin{matrix} {\overline{P}}_{l}^{b} = \frac{P^{b - 1} + P^{b}}{2} & (8) \end{matrix}$ where;
  
  step b is the depletion step in mega watt days per metric ton uranium (MWD/MTU) for the core, and dΣ
  
  /db for the b step for the a assembly is the change in Σ
  
  during the depletion step. Using the reference pattern fast flux for depletion gives;
  
  $\begin{matrix} {{\overline{P}}_{t}^{b} = {\overline{ϕ}}_{l}^{b, ref} \sum_{a} X_{la} (K^{1} \sum_{f}^{1} + \frac{\sum_{s}^{12} K^{2} \sum_{f}^{2}}{\sum_{R}^{1} + D_{2} B_{2}^{2} \langle_{ref}}) \rangle}_{a}^{b} & (9) \end{matrix}$ where;
  
  K represents the energy per fission for the fast group with a superscript 1 and for the thermal group with a superscript 2, which makes;
  
  $\begin{matrix} {{term = \sum_{b = 1}^{n} {\overline{ϕ}}_{l}^{b, ref} \sum_{a} X_{la} K \sum_{f} \rangle}_{a}^{b} \sum_{a^{'}} X_{{ta}^{'}} \frac{ⅆ \sum}{ⅆ bu} \rangle}_{a^{'}}^{b} & (10) \end{matrix}$ The equation becomes;
  
  $\begin{matrix} term \to \sum_{b = 1}^{n} {\overline{ϕ}}_{l}^{b, ref} \sum_{a} {X_{la} (K \sum_{f} \rangle}_{a}^{b}) {(\frac{ⅆ \sum}{ⅆ bu} \rangle}_{a}^{b}) & (11) \end{matrix}$ For X_la=1, the equations are identical. This makes the neutron current terms;
  
  $\begin{matrix} D_{l}^{d} ϕ_{l}^{n} \approx ϕ_{l}^{n} [\sum_{a} X_{la} D_{a}^{od} + \sum_{b = 1}^{n} {\overline{ϕ}}_{l}^{b, ref} {step}^{b} \sum_{a} {X_{la} (K \sum_{f} \rangle}_{a}^{b}) {(\frac{ⅆ \sum}{ⅆ bu} \rangle}_{a}^{b})] & (12) \end{matrix}$ This makes the multiplication and absorption term;
  
  $\begin{matrix} \sum ϕ_{l}^{n} = ϕ_{l}^{n} [\sum_{a} X_{la} \sum_{a}^{o} + \sum_{b = 1}^{n} {\overline{ϕ}}_{t}^{b} {step}^{b} \sum_{a} {X_{la} (K \sum_{f} \rangle}_{a}^{b}) {(\frac{ⅆ \sum}{ⅆ bu} \rangle}_{a}^{b})] & (13) \end{matrix}$ The non-linear terms in the above equation are;
  
  φ
  
  _lⁿX_la
  
  (14) The linearization of this product is;
  
  Ψ
  
  _la=X_la
  
  (15)
  Ψ
  
  _la−
  
  MX_a≦
  
  0
  
  (16)
  −
  
  φ
  
  _l+Ψ
  
  _la≦
  
  0
  
  (17)
  φ
  
  _l−
  
  Ψ
  
  _la+MX_la≦
  
  M
  
  (18) where;
  
  M is the upper bound for X and thus, also for Ψ
  
  _la, and wherein expressions 15-18 represent constraint equations to be added to said constraint matrix.
- View Dependent Claims (13, 14, 15, 16, 17, 18, 19, 20)
- - 13. The algorithm of claim 12 wherein said method includes loading the core of said nuclear reactor in accordance with said optimum loading pattern.
  - 14. The algorithm of claim 12 wherein said nuclear design quality neutron flux solution from which said system of equations is derived is the Analytic Course Mesh Finite Difference (ACMFD) method;
    - wherein said solver is a mixed integer program (MIP) solver; and
      
      wherein the system of equations derived from said ACMFD method are incorporated directly into said constraint matrix for said MIP solver.
  - 15. The algorithm of claim 12 wherein said effects of shuffling said fuel assemblies in step d include shuffling non-linearity;
    - and wherein said system of equations represent said shuffling non-linearity directly without approximation as a set of linear equations and auxiliary variables.
  - 16. The algorithm of claim 15 wherein step d further includes deriving said system of equations, in part, using a continuous relaxation of the term in the equations of assembly nuclear parameters times assembly flux;
    - and wherein said continuous relaxation segregates partial fluxes and partial assembly parameters by eliminating the cross term, thereby reducing the order of the equations in said system of equations.
  - 17. The algorithm of claim 16 wherein said system of equations are derived, in part, by calculating certain coefficients;
    - and wherein said coefficients are calculated by depleting candidate shuffled fuel assemblies and candidate burnable absorbers using said reference core depletion fast flux as an approximation to the shuffled pattern fast flux.
  - 18. The algorithm of claim 12 wherein step d includes representing the depletion of said reactor core in said system of equations.
  - 19. The algorithm of claim 12 further including the step of generating a feed pattern to be analyzed after step b of depleting said nuclear data to form a reference core depletion.
  - 20. The algorithm of claim 12 wherein said specified engineering requirements of step f are selected from the group consisting of fuel cycle cost, power constraints on peak power in said core, burnup constraints on peak burnup of one or more fuel assemblies, and custom user supplied engineering requirements;
    - and wherein step f includes the steps of;
      
      selecting at least one of said specified engineering requirements, representing said objective function or constraint as a linear combination of neutron flux, power, burnup, core reactivity, core critical boron concentration, assembly location or BA assignment, supplying said linear representation to the constraint matrix of said solver, running said solver to find an optimum satisfying said specified engineering requirements, and repeating the steps of adding additional constraints or objectives until an optimum core loading pattern which meets all of the desired requirements is achieved.

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Current Assignee
Westinghouse Electric Company LLC (Brookfield Asset Management Incorporated)
Original Assignee
Westinghouse Electric Company LLC (Brookfield Asset Management Incorporated)
Inventors
Popa, Frank D.

Granted Patent

US 7,224,761 B2
Time in Patent Office

Days
Field of Search
US Class Current

376/267
CPC Class Codes

G21C 19/205 Interchanging of fuel eleme...

Y02E 30/30 Nuclear fission reactors

Method and algorithm for searching and optimizing nuclear reactor core loading patterns

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Method and algorithm for searching and optimizing nuclear reactor core loading patterns

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