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

US 7,224,761 B2
Filed: 11/19/2004
Issued: 05/29/2007
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 in accordance with a set of predetermined engineering requirements, 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 which include constraints representing the neutron physics of said nuclear reactor, including nonlinear effects of shuffling said fuel assemblies and of loading said burnable absorbers into said reactor;

d. employing said system of equations as a constraint matrix for a solver and running said solver, without first requiring multiple reactor core simulations to be run to map a response domain; and

e. repeating steps b through d, updating the constraints and including objective functions which can be represented as linear constraints within said constraint matrix, in order to satisfy said specified engineering requirements and establish a global, nonlinear 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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11 Claims

1. A method of establishing a nuclear reactor core loading pattern for loading fuel assemblies and burnable absorbers into a nuclear reactor in accordance with a set of predetermined engineering requirements, 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 which include constraints representing the neutron physics of said nuclear reactor, including nonlinear effects of shuffling said fuel assemblies and of loading said burnable absorbers into said reactor;
  
  d. employing said system of equations as a constraint matrix for a solver and running said solver, without first requiring multiple reactor core simulations to be run to map a response domain; and
  
  e. repeating steps b through d, updating the constraints and including objective functions which can be represented as linear constraints within said constraint matrix, in order to satisfy said specified engineering requirements and establish a global, nonlinear 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 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 c include shuffling non-linearity;
    - and wherein said shuffling non-linearity is represented in said system of equations.
  - 5. The method of claim 4 wherein said fuel assemblies have a flux and nuclear parameters;
    - wherein said system of equations has a term comprising the nuclear parameters of said fuel assemblies times the flux of said fuel assemblies;
      
      wherein said flux of said fuel assemblies is represented as a cross term comprising the product of assigned locations of said fuel assemblies in said core;
      
      wherein step c further includes deriving said system of equations using a continuous relaxation of the term in the equations of the 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 fuel assemblies which have been shuffled have a shuffled pattern fast flux;
    - wherein said reference core depletion has a fast flux;
      
      wherein said method further comprises calculating a plurality of coefficients for said system of equations by selecting and depleting a number of said fuel assemblies which have been shuffled and a number of said burnable absorbers; and
      
      wherein said fuel assemblies and said burnable absorbers are selected and depleted using said reference core depletion fast flux as an approximation to the shuffled pattern fast flux.
  - 7. The method of claim 1 wherein step c 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 e 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;
    - andwherein step e includes the steps of;
      
      selecting at least one of said specified engineering requirements,representing at least one of said objective functions and said constraint as a linear combination of a plurality of nuclear parameters selected from the group consisting of neutron flux, power, burnup, core reactivity, core critical boron concentration, assembly location and 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, andrepeating the steps of updating the constraints and including objective functions until an optimum core loading pattern which meets all of said specified engineering requirements is achieved.
  - 10. The method of claim 1 wherein said reference core depletion has a reference fast flux;
    - and 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
      
      D^Adepends on the characteristics of the adjacent nodes,υ
      
      ⁿis the neutrons per fission in the n^thgroup,Σ
      
      _fⁿis 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,Σ
      
      _s¹²is the macroscopic down scatter cross section from the fast group to the thermal group,D¹is the diffusion coefficient for the fast group,B²is the buckling for the thermal group,ref refers to the reference calculation value,i and j are radial indices over nodes in the nuclear model,J represents the neutron current,φ
      
      represents the neutron flux, and∀
      
      _imeans for every i.When one includes the effects of shuffles, expression 1 also has the following term;
      
      $\begin{matrix} D_{ℓ}^{d} ϕ_{ℓ}^{n} \approx ϕ_{ℓ}^{n} [\sum_{a} X_{ℓ a} D_{a}^{od} + \sum_{b = 1}^{n} {\overline{P}}_{t}^{b} {step}^{b} \sum_{a} X_{ℓ a} \frac{ⅆ D^{d}}{ⅆ bu} ❘_{a}^{b}] & (4) \end{matrix}$ where;
      
      b=the burn-up step,d ∈
      
      (S,A) such that D^drepresents both D^Sand D^Aas defined in expression (3),n=the current step,bu=burnup,P¹=the power in the 1^thnode,φ
      
      represents the neutron flux,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.Expression 1 has the following term;
      
      $\begin{matrix} ϕ_{ℓ}^{n} \sum_{ℓ}^{n} = ϕ_{ℓ}^{n} [\sum_{a} X_{ℓ a} \sum_{a}^{o} + \sum_{b = 1}^{n} {\overline{P}}_{ℓ}^{b} {step}^{b} \sum_{a} X_{ℓ a} \frac{ⅆ Σ}{ⅆ bu} ❘_{a}^{b}] & (5) \end{matrix}$ where the only new notation is the following;
      
      Σ
      
      =Σ
      
      _R¹(k_∞−
      
      1)
      
      (6)Both of these (expressions 4 and
      
      5) have the following term;
      
      $\begin{matrix} \sum_{b = 1}^{n} {\overline{P}}_{ℓ}^{b} {step}^{b} \sum_{a} X_{ℓ a} \frac{ⅆ Σ}{ⅆ bu} ❘_{a}^{b} \equiv term & (7) \end{matrix}$ where;
      
      the Σ
      
      may represent the definition in expression (6), or it may represent D^d,step^bis the depletion step in mega watt days per metric ton uranium (MWD/MTU) for the core, anddΣ
      
      /dbu for the b step for the a assembly is the change in Σ
      
      during the depletion step.The average power during the b^thdepletion step is;
      
      $\begin{matrix} {\overline{P}}_{ℓ}^{b} = \frac{P^{b - 1} + P^{b}}{2} & (8) \end{matrix}$ Using the reference 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 andfor the thermal group with a superscript 2,D₂is the thermal diffusion coefficient, andB₂is the thermal buckling,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_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, andwherein 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.

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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.
Primary Examiner(s)
Palabrica; Ricardo J.

Application Number

US10/993,249
Publication Number

US 20060109944A1
Time in Patent Office

921 Days
Field of Search

376/267, 376/261, 376/260, 434/218, 703/6
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

First Claim

9 Assignments

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Abstract

52 Citations

11 Claims

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

First Claim

9 Assignments

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0 Petitions

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Accused Products

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Abstract

52 Citations

11 Claims

Specification

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Solutions

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