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North Carolina State University, Department of Nuclear Engineering. Electric Power Research Center, P. 0. Box 7909. Raleigh, North Carolina 27695-7909.



KEYWORDS: generalized perturbation theory, nodal expansion method, fuel management optimization

G. IVAN MALDONADO* and PAUL J. TURINSKY North Carolina State University, Department of Nuclear Engineering Electric Power Research Center, P. 0. Box 7909 Raleigh, North Carolina 27695-7909 Received August II, I994 Accepted for Publication December I6, I994

The determination of the family of optimum core loading patterns for pressurized water reactors (PWRs) involves the assessment of the core attributes for thousands of candidate loading patterns. For this reason, the computational capability to efficiently and accurately evaluate a reactor core's eigenvalue and power distribution versus burnup using a nodal diffusion generalized perturbation theory (GPT) model is developed. The GPT model is derived from the forward nonlinear iterative nodal expansion method (NEM) to explicitly enable the preservation of the finite difference matrix structure. This key feature considerably simplifies the mathematicaljormulation of NEM GPT and results in reduced memory storage and CPU time requirements versus the traditional response-matrix approach to

NEM. In addition, a treatment within NEM GPT can account for localized nonlinear feedbacks, such as that due to fission product buildup and thermal-hydraulic effects. When compared with a standard nonlinear iterative NEM forward flux solve with feedbacks, the NEM GPT model can execute between 8 and 12 times faster. These developments are implemented within the PWR in-core nuclear fuel management optimization code FORMOSA -P, combining the robustness of its adaptive simulated annealing stochastic optimization algorithm with an NEM GPT neutronics model that efficiently and accurately evaluates core attributes associated with objective functions and constraints of candidate loading patterns.


ery 12 to 24 months, with the management of fuel reloads spanning the plant's entire lifetime. Traditionally, the overall fuel management problem is divided into two tightly coupled subproblems whose solutions greatly impact the plant's operating expenses through nuclear fuel cycle costs:

I.A. Pressurized Water Reactor Reload Core Optimization Problem

Although significant advances have been made in recent years to address the pressurized water reactor (PWR) fuel management optimization problem, it continues to challenge today's nuclear engineers and their computational resources. A typical PWR is refueled ev*Current address: Iowa State University, Department of Mechanical Engineering, Nuclear Engineering Program, 105 Nuclear Engineering Laboratory, Ames, Iowa 500II-2241. 198

1. Out-of-core fuel management is the long-term strategy devised to purchase fissile and control material in the most economical manner. For each cycle in the planning horizon, out-of-core decisions entail selecting the number and enrichments of fresh fuel assemblies to be loaded, the types and amounts of burnable poisons, the cycle length, and the fuel assemblies to be reinserted. NUCLEAR TECHNOLOGY

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2. In-core fuel management is the selection of an optimum physical arrangement of the available partially depleted and fresh fuel assemblies (with or without burnable poisons) within the reactor core. It is usually addressed on an individual cycle basis. Both components of the overall fuel management decision (i.e., in-core and out-of-core) attempt to optimize the reactor's performance while satisfying all operational and safety constraints. In terms of the mathematical complexity and computational burden, in-core fuel management is very demanding because the validation of each loading pattern requires the assessment of a number of core attributes, which is accomplished by solving the few-group neutron diffusion equation depleted over the cycle. Thus, assessing the feasibility of candidate loading patterns constitutes an engineering and computationally intensive task. By way of example, consider the refueling of a standard four-loop PWR, during which approximately one-third of its 193 assemblies are discharged and replaced by fresh assemblies of one or more enrichments. Note that a number of these fresh assemblies may or may not be loaded with variable numbers and concentrations of burnable poisons and that all depleted assemblies entering the new cycle are each physically unique because of their irradiation history. Consequently, the number of arrangements (or loading patterns) mathematically possible under the described circumstances can exceed 10 100 , a search space too vast to be efficiently explored in an exhaustive or manual fashion. 1.8. FORMOSA·P Code

Historically, the determination of PWR loading patterns has been made by experienced engineers utilizing a well informed trial-and-error approach. Within recent years, progress has been made on automating the loading pattern determination by the employment of mathematical optimization methods. 1- 5 One such effort has resulted in the development of the FORMOSA-P computer program (Euel Optimization for Reloads: Multiple Objectives by Simulated Annealing-_EWR). This code treats the loading pattern determination problem as a combinatorial optimization with generalized nonlinear objectives and constraints, without the necessity of employing heuristic rules to reduce the extent of the search space. For example, physical restrictions, such as discrete burnable poisons under control rods, are not heuristic rules and are handled as active operational constraints. Disallowing fresh fuel neighboring pairs is an example of a heuristic rule. An in-core fuel management optimization software package can be viewed as having two principal components: the optimization algorithm and the core physics evaluator. Early evolution of FORMOSA-P successfully coupled an adaptive simulated annealing optimization algorithm to a higher order accurate generalized NUCLEAR TECHNOLOGY

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perturbation theory (OPT) model based on a finite difference discretization of the neutron diffusion equation. This work is attributable to Kropaczek and Turinsky 6 and was built on earlier developments by Hobson and Turinsky. 7 For all the objectives and constraints studied, it was shown that the developed adaptive simulated annealing algorithm was capable of finding multiple loading patterns substantially better than the reference (starting) design in runs requiring between 10 4 to 10 5 loading pattern evaluations. Studies of the convergence properties of implementations of the simulated annealing algorithm have shown that, although convergence to the global optimum (or extremum in constrained problems) can only be guaranteed in the somewhat impractical limit of infinite loading pattern sampling, solutions found in finite runs can be expected, with a degree of statistical certainty dependent on the loading pattern sampling size, to lie close (in terms of objective value) to the global optimum. 8 The higher order OPT model plays a key role in providing the capability to efficiently evaluate the core attributes of each sampled loading pattern at a fraction of the computational cost required to solve the forward neutron diffusion equation. Thus, the combined capabilities of simulated annealing and OPT within FORMOSA-P brought the computational requirements of the loading pattern determination problem well within the capability of modern engineering workstations. Further details about the FORMOSA-P code package fall beyond the scope of this paper; the reader is directed to the cited references and to additional publications available. 9 I.C. Research Objectives

Within the context of this section, accuracy is defined as the ability of a OPT prediction to agree with a calculation performed by the forward model from which the OPT was derived. On the other hand,jidelity is viewed as a measure of agreement between either a forward or OPT calculation and a production quality core simulator that has been extensively benchmarked against plant data. Naturally, both accuracy and fidelity are essential if FORMOSA-Pis going to be useful in determining core loading patterns. The original higher order OPT model used within FORMOSA-P has been proven to be sufficiently accurate to handle the full range of perturbations that can be encountered during a fuel shuffling analysis. In fact, the types of fuel exchanges that rule out the adequacy of standard first-order perturbation theory techniques have been used to demonstrate the validity of higher order OPT for this application. To span the search space for families of optimal loading pattern solutions, the simulated annealing optimization algorithm in FORMOSA-Pis structured to perform alternating global and local searches, which correspond to simulated annealing cooling cycles that employ larger and 199

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smaller assembly shuffling reactivity allowances, respectively. Also, the optimized loading pattern from each simulated annealing cooling cycle becomes the reference loading pattern for the following cycle and so forth. In this manner, very high GPT accuracy is achieved. The weakness of the early forward and GPT models within FORMOSA-P originated because of the employment of a coarse-mesh finite difference (CMFD) discretization of the neutron diffusion equation, which was used as an intermediate "proof of principle" model and could accommodate realistic large-scale optimizations within the practical bounds of a modern engineering workstation. The numerical grid was limited to a CMFD scheme of four radial nodes per fuel assembly in a two-dimensional geometry layout of a PWR, which produced unacceptably large numerical truncation errors leading to misdirection of the loading pattern optimization. In addition, the original model did not treat the effects of nonlinear local feedbacks (i.e., thermal hydraulic and fission products) on cross sections, which further degraded fidelity. In summary, for FORMOSA-P to become truly useful in determining core loading patterns, the fidelity of its neutronic model must be raised to a level comparable to that of licensed codes. This must be accomplished without substantially increasing the demand for computational resources, as measured in terms of memory and CPU time requirements. 1.0. Advanced Nodal Methods

Limiting the spatial truncation error to an acceptable magnitude and capturing the "pin-by-pin" heterogeneities of a PWR lattice are achievable via employment of a fine-mesh finite difference (FMFD) method. However, the computational resources required would be excessive for the current application requiring numerous loading pattern evaluations. Advanced nodal methods utilizing higher order and consistent methodologies have been developed to address the drawbacks of fine- and coarse-mesh finite difference methods. With the employment of a pin-power reconstruction method, the heterogeneous pin power distribution can be obtained from the coarse-mesh nodal result. Advanced nodal methods do demand greater computational resources than a CMFD calculation with the same mesh discretization. Nevertheless, the additional computational requirements of a nodal calculation are well within practical limitations and are thus attractive for use in the loading pattern determination application. Numerous advanced nodal methods exist for the forward solution of the neutron diffusion equation. Reviews by Lawrence 10 and Koebke and Timmons 11 summarize many alternative methods. We elected to employ the nodal expansion method (NEM). Although other methods also satisfied the basic requirements imposed, NEM was selected because of its rich history of 200

success, adaptability, and popularity and because of the availability of extensively documented research. 12- 14 The classical approach to NEM, as described in detail by Lawrence 10 and Bandini, 15 is known as the response-matrix formulation. Smith 16 subsequently developed the alternative nonlinear iterative approach to the forward nodal solutionY· 18 In this treatment, nonlinear nodal corrections are applied to the finite difference coupling coefficients in an outer-nested iterative fashion similar to that of a standard feedback update. By preserving the finite difference matrix structure and flux as the principal unknown, this approach offers significant computational advantages in solving the NEM equations. These advantages of the nonlinear iterative approach are also applicable to the solution of the nodal GPT problem within the framework of FORMOSA-P. I.E. Nodal GPT

The literature available on GPT applications to the finite difference form of the neutron diffusion equation is quite extensive and essentially complete. 19 By contrast, the extension of perturbation-based techniques into advanced nodal methodologies has only produced a handful of publications during the past decade. Lawrence 20 proposed a similarity transformation matrix approach to efficiently extract the mathematical adjoint from the physical adjoint solution to the response matrix NEM equations used within the DIF3D code. Subsequent research by Yang, 21 Yang et ai., 22 Taiwo et al., 23 and others has built upon this work. Concurrently, Taiwo and Henry 24 and Taiwo 25 derived perturbation theory expressions for reactivity within the framework of the analytical nodal method. Recently, Downar 26 •27 developed the depletion perturbation theory for the DIF3D code. Although successful within their specific applications, none of the aforementioned contributions were deemed applicable for the envisioned nodal GPT model required by the FORMOSA-P code for the following reasons: 1. All the contributions utilized the response matrix versus the nonlinear iterative formulation of NEM. 2. None of the existing contributions considered higher than first-order nodal GPT accuracy in their perturbation methodologies. Unfortunately, it has been confirmed by computational experiments that first-order accuracy is inadequate to handle the range of perturbations in a loading pattern determination application. 3. The type of responses studied involved global/ integral quantities (e.g., keff) and not localized distributions of core attributes (e.g., flux and power with feedbacks), which must be evaluated in our application. Hence, it was necessary to develop an NEM GPT methodology to address these three points. NUCLEAR TECHNOLOGY

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The underlying neutronics formulation utilized corresponds to a standard Cartesian geometry implementation of NEM to the two-group, steady-state neutron diffusion equation. 10 •15 The principal characteristics adopted for this polynomial nodal method are quartic expansions of the one-dimensional transverse-integrated flux and quadratic fits to the transverse leakages. The numerical solution scheme is the nonlinear iterative technique 16 applied to NEM, which is the principal basis leading to the NEM GPT model. For completeness, a detailed description of the nonlinear iterative NEM is presented in Appendix A. II.A. Matrix Notation Adopted

A convenient matrix notation is herein adopted to acknowledge the fact that the nonlinear iterative NEM strategy preserves the banded finite difference matrix structure. Consider the energy and space discretized steady-state neutron diffusion eigenvalue problem (at a fixed state during the depletion cycle) expressed in the following matrix form: 1 At>= k 84> .


The tilde notation in Eq. (1) is employed to denote a modified finite difference (FD) operator due to the nonlinear NEM treatment. In contrast to a finite difference approach, the operator on the left is defined as follows: (2)

where the matrix DNEM isolates the nonlinear NEM coupling corrections defined by Eq. (A.6). To illustrate specific details of the modified matrix structure, consider a two-dimensional five-point stencil of nodes with the convention of axes and labels shown in Fig. 1. Accordingly, label the finite difference coupling coefficient amtthe nonlinear NEM coupling correction coefficients in Eq. (A.5) for the interface between the center node C and its north neighbor N as dC,FD and JC.NEM respectively. g,N g,N ' Similarly, after applying Eq. (A.5) to the remaining interfaces, substitution into the nodal balance Eq. (A. 1) [via Eq. (A.2)] reveals the details of one individual row from the modified system described by Eq. (1) corresponding to node C and group g, as follows: _ (dC,FD _ g,N


g, N


_ (dC,FD g, S

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It is clear that the modified system retains the familiar three-, five-, or seven-banded form for one-, two-, or three-dimensional geometry, respectively. However, unlike the finite difference matrix, the nonlinear NEM corrections cause the upper and lower bands to slightly differ, implying an unsymmetric matrix. In theory, many of the convergence proofs for iterative techniques rely on symmetric positive definite structures. In practice, the loss of symmetry due to the nonlinear NEM corrections has proven to be sufficiently mild to not be of numerical concern in light water reactor (LWR) applications.


The NEM GPT model is derived directly from the nonlinear NEM system of equations denoted by Eq. (1). Indicating the unperturbed state as the reference condition o while suppressing the burnup dependency for clarity, the associated matrix system is

(4) where

1 Ao = ko Similarly, the perturbed condition p counterpart of Eq. (4) is

+ JC,NEM)cpS g,S g

- (di·~P- Ji·/:EM)c/J:'- (di'[D + difEM)cp: + ((dftD + di~EM) + (df{D- Ji-fEM) + d-c,NEM) + (dC,FD _ a.cE,NEM) + (dg,C,FD W g, W g,E g, + (A_f~xC~yC)]cp.f = Q[~xC~yC . (3) NUCLEAR TECHNOLOGY

Fig. 1. Five-point two-dimensional Cartesian stencil of nodes.

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A.Pt>P = 'ApBp+p ,


+ ~A.p , Bp = 8 0 + ~BP, f>p = f>o + ~f>p '



Ap = Ao


(8) 201

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For the in-core fuel management application, the perturbed matrix operators !l.AP and !l.Bp correspond to perturbations in the cross-sectional data arising from fuel shuffling. The key attributes being sought are 'Ap and ~P' which must be accurately and efficiently solved for under any combination of multiple perturbations from the reference condition. Since during the course of an optimization, thousands of loading patterns are examined, solving the perturbed forward system [Eq. (5)] for each loading pattern would lead to long computer execution times. The strategy employed within the FORMOSA-P code to achieve both accuracy and efficiency relies on a two-step approach. Initially, first-order accurate results are generated by applying the principle of linear superposition, and, subsequently, higher order accurate results are obtained by employing GPT in a manner that takes advantage of the first-order estimates to eliminate the dominant second-order errors (see Appendix B).

Il.B.l. First-Order Linear Superposition Framework Consider the setS, representing a group of perturbations that distinguish a perturbed loading pattern from the reference loading pattern. This set could include any number of fuel assembly exchanges, rotations, changes in discrete or integral burnable poison loadings, feed enrichment, etc., relative to the reference loading pattern. The changes in eigenvalue and flux can be estimated to first-order accuracy via linear superposition of single-assembly perturbation responses i, which have been evaluated and stored a priori. These first-order estimates are given by 'A11st)

= Aa + L




and (11)

II.B.2. Nonlinear NEM Effects The effects of NEM are only embedded within the left~side operator of Eq. (1). Recall Eq. (2) and explicitly denote the nonlinear dependencies involved: A(4»,A) =Am+ DNEM(~,'A) .


For any given perturbed conditionp at a depletion step t (with this dependence suppressed for notational clarity), the exact expression for the perturbed operator appearing in Eq. (6) is 202

= A(~p.'Ap)- A(~ 0 ,'Aa)

= [AFD,p +


- [AFD,o + DNEM(~o,Ao)]

= !l.AFo,p +

!l.DNEM(~p,Ap) .


This change accounts for cross-sectional changes originating directly from fuel shuffling, which in turn perturbs power and thus changes burnup. Note, in FORMOSA-P, depletion effects within the forward or GPT models are handled explicitly by solving the ordinary differential equation associated with burnup, a forward difference approximation that is computationally inexpensive. Within FORMOSA-P, the higher order perturbation theory and GPT formulations are based on a firstorder estimate of the change in the operator described by Eq. (13). From Eq. (A.5) and the original NEM linear matrix system expressed in terms of node-average flux and transverse-integrated flux expansion coefficients, one can show to first-order accuracy that the nonlinear NEM coupling corrections A DNEM ( ~P, 'Ap) can be expressed as a linear function of the loading pattern perturbations (see Appendix C and Ref. 28), the same dependence as that exhibited by the finite difference matrix operator !l.AFo,p. This linear functional dependence in turn implies that the principle of linear superposition applies, thus suggesting the following reconstruction strategy to approximate Eq. (13) and to generate a first-order accurate estimate for Eq. (6):

For Eq. (7), which does not involve NEM, the equivalent statement is given by s~st)

= Ba + L !l.Bi



II.B.J. Higher Order Boot Up Equations (10), (11), (14), and (15) provide the framework needed to more accurately evaluate the perturbed eigenvalue by utilizing a standard perturbation theory form of the Rayleigh quotient to boot up the first-order result: )\(2nd)_



A(1st)+ P



[A(lst) _ p








~here all quantities have been defined with the exceptiOn of the homogeneous adjoint flux vector~~. It is calculated from the mathematical adjoint counterpart to Eq. (4) at the reference (e.g., unperturbed) conditions, namely,


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Similarly, one can employ the same strategy to boot up localized responses (i.e., flux or power) to higher order accuracy by employing GPT functional expressions. 19 Specifically, the higher order accuracy power response for assembly k is given by kp(2nd) _ P


kp(lst) _

(kl'* [A(lst) _ f._(2nd)B(Ist)]+(lst))








(18) where the kr~ is the assembly power response generalized adjoint function for response location k. It is also evaluated at the reference condition by solving the following expression: ([AoJT- A 0 [B 0 ]T)kl'~



= ks; ,


s; is the generalized adjoint source defined by

kS;= kH0



(kH~+o>] [Kl;f]o



The response vector kHo is defined to yield the reference average power for assembly k when integrated with the flux vector over the volume of the core V, denoted kpo = (kHo,+o> .


Also, the following condition of biorthogonality must be satisfied to guarantee the uniqueness of the solution to the singular problem posed by Eq. (19): (ki'~,Bo+o> = 0 .


Equation (22) is utilized along with the homogeneous adjoint flux to remove the homogeneous contamination that inevitably accumulates when solving Eq. (19). Thus, to ensure a unique particular solution to that system, the following operation must be performed to remove any component of the homogeneous contamination appearing within the generalized adjoint function: noncontaminated

homogeneous contamination


feedback corrections, the reference eigenvalue (nonlinearly obtained via the power method), and any depletion adjustments to cross sections for cases not at the beginning of cycle (BOC). Thus, unlike the response matrix formulation to NEM GPT, the similarity matrix transformation approach20-23 to obtain the mathematical adjoint from its physical adjoint counterpart is not required or employed in this work. This follows since only flux related adjoint quantities appear due to the nonlinear iterative approach (with linear superposition to obtain aDNEM), which eliminates the need to introduce either NEM expansion coefficients or flux moments related adjoint quantities into the GPT formulation. Hence, without any loss of rigor, the levels of complexity and of computational resources required are considerably reduced.

II.B.5. Nonlinear Local Feedback Effects The effects of thermal-hydraulic and fission product feedbacks affect all cross sections and thus both sides of Eq. (1). Within the forward solution, the effect of feedbacks is calculated by means of an outermost iteration level, where the most recent iterative values of the flux and power distribution are utilized to calculate all appropriate cross-section corrections. The procedure is repeated until convergence is achieved. Recall the two-step approach in FORMOSA-P that provides first-order accurate flux and power estimates utilizing linear superposition. If feedback iterations are performed during the single-assembly perturbation forward calculations, then these first-order accurate results also account for feedback, which suggests a logical approach to obtaining the local feedback corrections for a perturbed condition p within the GPT formulation. Consider the no feedback (NF) representations of the first-order accurate operators given by Eqs. (14) and (15), and add to them local feedback (LF) corrections that can be obtained from the appropriate feedback expressions by employing the first-order accurate values of the flux (or power): - p(1st)) LF_ [AOst)) + ilA LF (+(1st)) [A (24) p NF p and

l/.B.4. Physical Versus Mathematical Adjoint

A key attribute of the previously described approach to NEM GPT is the straightforward manner in which the mathematical adjoint functions +~ and kr~ are defined. In essence, these vectors are the solutions to Eqs. (17) and (19), respectively, where the left-hand operator is merely the transposed version of the converged forward nodal diffusion operator at the reference/ unperturbed condition. In other words, the left-hand operator of Eqs. (17) and (19) includes all the converged values of any iteratively updated nonlinear adjustments. These include the nonlinear NEM coupling corrections to the bands of matrix A0 , the nonlinear local NUCLEAR TECHNOLOGY

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_ [B(Ist)] [ B(Ist)] p LF p NF

+ ilBLF (+(1st)) p


These first-order operators are substituted into Eqs. (16) and (18) to obtain the eigenvalue and power responses. Computational experiments have revealed that, unlike the first-order corrections to the coupling coefficients, correcting for local feedbacks utilizing first-order feedback operators degrades the GPT accuracy. Consequently, an additional refinement step to address feedbacks was deemed necessary to maintain a level of GPT accuracy comparable to that of a feedback-free secondorder accurate methodology. Consider the second term on the right side of Eq. (18), which defines the higher order correction to 203

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the power response. To generate refined estimates of the changes in the operators due to local feedbacks, and thus to evaluate a modified change in the assemblyaverage power response, the following expression is employed: k .!lft(lst)--+(ind) _ k ,1p(lst)--+(2nd) p



_ (kf* [.!lA(Ist) o•


inates due to the moderator temperature feedback effect on the removal cross section. Therefore, the effort expended in improving the feedback correction can be reduced significantly by only accounting for the effect due to removal [during the modification step given by Eq. (26)] with a negligible loss of accuracy.

II.C. FORMOSA-P Optimization Algorithm


d~~nd) ~B11st)]LF~11st)) . (26)

The caret notation denotes the modified result and modified operators based on the latest OPT power response predictions. Equation (26) defines an iterative process that could continue, but computational results29 (reported in Sec. III) have shown adequate convergence without further iterations. In practice, it has been determined that the largest component of the rightmost inner product term is, by far, that which orig-

The entire in-core fuel management optimization strategy employed by FORMOSA-P can be summarized into three flowcharts, presented as Figs. 2, 3, and 4. Figure 2 describes the main driver of the code. Within it, one should note the preliminary calculations performed at the beginning of each simulated annealing iteration (denoted as a cooling cycle). These are the forward and homogeneous adjoint reference calculations, the single-assembly perturbation calculations,




Fig. 2. The FORMOSA program flowchart. 204


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Fig. 3. Simulated annealing algorithm flowchart.

and the generation of the fuel assembly relative power GPT adjoint functions. The preceding calculations are required prior to entering each cooling cycle of the simulated annealing optimization algorithm, whose principal stages are outlined in Fig. 3. The simulated annealing algorithm relies on an efficient and accurate methodology to calculate the attributes of sampled loading patterns. These are provided by the NEM OPT algorithm presented within Fig. 4.

and without local feedback effects and only at BOC. Further results on benchmarking the forward model, including depletion effects, are reported elsewhere by Kropaczek et al. 30 Subsequently, the results from a FORMOSA-P large-scale power peaking minimization are presented to contrast the accuracy and performance of the NEM OPT model against forward NEM calculations. This test is used to also confirm that the robustness of the original optimization model remained intact. III.A. Performance of the Neutronics Model

Ill. RESULTS AND DISCUSSION Results are presented in this section. Initially, the nonlinear NEM forward model is contrasted against independently benchmarked FMFD calculations from two different sources . These tests are performed with NUCLEAR TECHNOLOGY

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Il/.A.l. Verification of NEM Model Implementation The predictions of the implemented nonlinear NEM forward model were compared against FMFD calculations obtained from the independently developed 205

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8 9 10 11 12 13 14 16

.8865 1.0699 1.0336 .8696 1.0306 1.0289 1.0714 1.0032


1.0699 .8886 1.0228 1.0714 1.0043 1.0626 1.1671 .8574

F 1.0511 1.0227 1.0410 .8891 1.0052 1.0714 .8982




.8597 1.0714 . 8873 1. 0425 1.0699 1.1904 .8610

1.0321 1.0019 1.0050 1.0699 .8989 .8627



1.0338 1.0714 1.0017 1.0518 1.1671 .8574 1.0714 . 8982 1.1904 . 8610 .8628

Fig. 5. Fuel assembly koo map used in verification of NEM model.

against the DIFPAR3D FMFD 20 x 20 mesh per fuel assembly discretization of the quarter core. Excellent agreement between NEM and FMFD is noted, as indicated by assembly average power differences below a relative error of 1OJo and eigenvalue agreement within at least five significant figures. These results are representative of a variety of problems tested, involving different types of fuel and/or plants, confirming the correct implementation of the forward NEM solution.

II/.A.2. Test with Local Feedbacks Testing of FORMOSA-P at Electricite de France (EdF) enabled BOC comparisons against EdF's COCCINELLE licensing code, version 2.2. Figure 7 shows, EVALUATE OBJECTIVE FUNCTION AND CONSTRAINTS

t ROUTINE I IRETURN TO OPTIMIZATION Fig. 4. The GPT algorithm flowchart.

DIFPAR3D code,3 1 which has been successfully benchmarked in the past,32 utilizing the International Atomic Energy Agency three-dimensional PWR benchmark problem. The only purpose of these calculations was to verify the correct implementation of the forward NEM static neutronics model. Therefore, these calculations exclusively isolated the neutronics model from any local feedback and/or depletion treatment. The calculations were performed at BOC, hot-full-power, allrods-out, and with core-average equilibrium xenon and samarium conditions. The cross-sectional data utilized correspond to that of a modern three-loop Westinghouse core and used assembly homogenized cross sections for both the NEM and FMFD calculations. The particular loading pattern arrangement utilized was arbitrarily selected and unoptimized to purposely exhibit considerable BOC reactivity gradients. Figure 5 displays the assembly-averaged koo map of the test case. a Figure 6 contrasts the FORMOSA nonlinear iterative NEM model with a 2 x 2 mesh per fuel assembly cross-sectional data are courtesy of the Carolina Power and Light Company.

a The




F G c H D E A B .786 1.224 1.194 .869 1.362 1.508 1.424 .538 .786 1.226 1.194 .857 1.348 1.500 1.413 .534 . 000 +.002 .000 - .002 -.004 -.008 - .011 -.004

.848 1.130 1.343 1.252 1.465 1.362 1.195 .323 .849 1.130 1.342 1.247 1.467 1.352 1.198 .321 +.003 +.001 .000 -.001 -.005 -.008 - .010 - .002

.560 10 1.119 1.105 1.200 .862 1.160 1.346 1.122 1. 108 1.202 .861 1.157 1.340 . 557 +.003 +.003 +.002 -.001 -.003 -.006 -.003 11

.864 1.106 1.204 1.123 .836 1.323 . 290 .866 1.107 1.202 1.118 .839 1.328 . 289 +.003 +.005 +.002 +.001 -.002 - .005 -.001

12 1.326 1.242 1.158 1.207 1.333 1.247 1.162 1.209 +.007 +. 005 +.004 +.002

.580 .580 .000

1.472 1.448 1.338 1. 121 1.479 1.455 1.343 1.123 +.007 +. 007 +.005 +.002

.297 . 297 .000


.296 .296 .000

14 1.402 1.344 .555 .288 .557 1.409 1.350 .290 +.007 +.006 +.002 +.002 15


.533 .319 ---> (A) FKFD (20x20 per FA) --- > .535 .321 ---> (B) IRK (2x2 per FA) ---> +.002 +.002 ---> Difference (8)-(A) --->

----------1.0048085 1.0047708 -.0000377

Fig. 6. Nonlinear NEM (2 x 2 grid per fuel assembly) versus FMFD (20 x 20 grid per fuel assembly). NUCLEAR TECHNOLOGY

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Maldonado and Turinsky 0.708 0.924 0.696 0.911 0.012 0.013

0.706 0.693 0.013

NUCLEAR FUEL RELOAD OPTIMIZATION i+!- cf>i) + JN~-EM] ( cJ>i+! + cJ>i) . (A.6)

For each nodal interface, the NEM current can be efficiently calculated by analytically solving small matrix systems denoted as the NEM two-node problems. This produces a spatially decoupled (i.e., without corewide coupling) NEM calculation applicable to an individual nodal interface. Numerically, the NEM currents and coupling corrections are nonlinearly updated in an outer-nested iterative fashion, which, for LWR applications, has proven to be a convergent· technique that forces the NUCLEAR TECHNOLOGY

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finite difference equations to exactly yield the higher order NEM node-average flux distribution and fundamental mode eigenvalue. Our estimates of the additional calculational overhead associated with the nonlinear updates range between 15 and 3007o relative to a reference CMFD calculation. In other words, a very small additional computational effort is required in exchange for higher order nodal accuracy. NEM TWO-NODE PROBLEM

The main goal of the so-called two-node problem is to calculate the NEM currents and coupling correction for a given interface. Again, consider the boundary between two adjoining nodes aligned in the x direction: Node I

x I --->.



To determine the NEM net current, the onedimensional transverse-integrated form of the diffusion equation is solved. For example, the one-dimensional x-dependent transverse-integrated equation, derived similarly to the nodal balance Eq. (A.1) with the exception that the dependence in x is not removed by integration, is given by the following expression: ;

1 Jix(X )

+ Ai .

H = ((E_R


By cross multiplying the terms within Eq. (B.2) and collecting terms of third or higher order, one obtains H

= + - ¢;;)



(C.lO) Now Ll.c;,p and L1dp depend directly on cross sections to first-order accuracy in a linear fashion. Likewise, Ll.a;,p, .:1¢;, and .:1¢;; are obtained from the solution of a linear system of equations, which, to firstorder accuracy, is linearly dependent on cross sections. Given these points and the fact that cross sections can be expressed as linear functions of the loading pattern, Eq. (C.10) implies that Ll.dp obeys the principle of linear superposition. Consequently, for a setS representing a group of multiple perturbations that distinguish a perturbed loading pattern from the reference loading pattern, the change in the nonlinear NEM coupling corrections can be estimated to first-order accuracy via linear superposition of single-assembly perturbation responses i evaluated and stored a priori. These first-order estimates are given by a~lst)

= do +

~ L1d;



REFERENCES I. B. J. JOHANSEN, "ALPS: An Advanced Interactive Fuel Management Package," Proc. Top/. Mtg. Advances in Reactor Physics, Knoxville, Tennessee, April 11-15, 1994, Vol. 3, p. 324, American Nuclear Society (1994).

2. J. G. STEVENS, K. S. SMITH, K. R. REMPE, and T. J. DOWNAR, "Optimization of PWR Shuffling by SimNUCLEAR TECHNOLOGY

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7. G. H. HOBSON and P. J. TURINSKY, "Automatic Determination of Pressurized Water Reactor Core Loading Patterns That Maximize Beginning-of-Cycle Reactivity Within Power-Peaking and Burnup Constraints," Nucl. Techno/., 74, 5 (1986). 8. P. J. M. VAN LAARHOVEN and E. H. L. ARTS, Simulated Annealing: Theory and Applications, D. Reidel Publishing Company, Dordrecht, Holland (1987). 9. D. J. KROPACZEK, P. J. TURINSKY, G. I. MALDONADO, and G. T. PARKS, "The Efficiency and Fidelity of the In-Core Fuel Management Code FORMOSA," Proc. Int. Conf. Reactor Physics and Reactor Computations, Tel Aviv, Israel, January 23-26, 1994, p. 572, Ben Gurion University of the Negev Press (1994). 10. R. D. LAWRENCE, "Progress in Nodal Methods for the Solution of the Neutron Diffusion and Transport Equations," Prog. Nucl. En., 17, 3, 271 (1986). 11. K. KOEBKE and D. H. TIMMONS, "Overview of LWR Analysis Methods," Proc. Int. Conf. Reactor Physics, Jackson Hole, Wyoming, September 18-22, 1988, Vol. 3, p. I, American Nuclear Society (1988). 12. F. BENNEWITZ, H. FINNEMANN, and H. MOLDASCHL, "Solution of the Multidimensional Neutron Diffusion Equation by Nodal Expansion," Proc. Conf. Computational Methods in Nuclear Engineering, Charleston, South Carolina, April 15-17, 1975, CONF-750413, p. 1, American Nuclear Society (1975). 13. H. FINNEMANN, F. BENNEWITZ, and M. R. WAGNER, "Interface Current Techniques for Multidimensional Reactor Calculations," Atomkernenergie, 30, 123 (1977). 217

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14. M. R. WAGNER, K. KOEBKE, and H. J. WINTER, "A Non-Linear Extension of the Nodal Expansion Method," Proc. Conf Advances in Mathematical Methods for the Solution of Nuclear Engineering Problems, Munich, Germany, April27-29, 1981, Vol. 2, p. 43, American Nuclear Society (1981). 15. B. R. BANDINI, "A Three-Dimensional Transient Neutronics Routine for the TRAC-PF1 Reactor Thermal Hydraulic Computer Code," PhD Dissertation, Pennsylvania State University (1990). 16. K. S. SMITH, "Nodal Method Storage Reduction by Nonlinear Iteration," Trans. Am. Nucl. Soc., 44,265 (1983). 17. K. S. SMITH, D. M. VER PLANCK, and M. EDENIUS, "QPANDA: An Advanced Nodal Method for LWR Analyses," Trans. Am. Nucl. Soc., 50, 532 (1985). 18. K. S. SMITH and K. R. REMPE, "Testing and Applications of the QPANDA Nodal Model," Proc. Int. Top/. Mtg. Advances in Reactor Physics, Mathematics and Computation, Paris, France, April27-30, 1987, Vol. 2, p. 861, Organization for Economic Cooperation and Development (1987). 19. M. L. WILLIAMS, "Perturbation Theory for Nuclear Reactor Analysis," CRC Handbook of Nuclear Reactor Calculations, Vol. 3, p. 63, Y. RONEN, Ed., CRC Press, Boca Raton, Florida (1986). 20. R. D. LAWRENCE, "Perturbation Theory Within the Framework of a Higher Order Nodal Method," Trans. Am. Nucl. Soc., 46, 402 (1984). 21. W. S. YANG, "Similarity Transformation Procedure for Nodal Adjoint Calculations," Trans. Am. Nucl. Soc., 66, 270 (1992). 22. W. S. YANG, T. A. TAIWO, and H. KHALIL, "Solution of the Mathematical Adjoint Equations for an Interface Current Nodal Formulation," Nucl. Sci. Eng., 116, 42 (1994). 23. T. A. TAIWO, W. S. YANG, and H. S. KHALIL, "Direct Solution of the Mathematical Adjoint Equations for an Interface Current Nodal Formulation," Proc. Joint Int. Conf Mathematical Methods and Supercomputing in Nuclear Applications, Karlsruhe, Germany, April19-23, 1993, p. 1-507, Kernsforschungszentrum Karlsruhe GmbH, Karlsruhe (1993). 24. T. A. TAIWO and A. F. HENRY, "Perturbation Theory Based on a Nodal Model," Nucl. Sci. Eng., 92, 34 (1986).

25. T. A. TAIWO, "Mathematical Adjoint Solution to the Nodal Code QUANDRY," Trans. Am. Nucl. Soc., 55, 580 (1987). 26. T. J. DOWNAR, "Depletion Perturbation Theory Within the Framework of an Advanced Nodal Model," Trans. Am. Nucl. Soc., 66, 272 (1992). 27. T. J. DOWNAR, "Depletion Perturbation Theory Within the Framework of an Advanced Hexagonal Nodal Model," Nucl. Sci. Eng., 115, 334 (1993). 28. G. I. MALDONADO, "Non-Linear Nodal Generalized Perturbation Theory Within the Framework of PWR InCore Nuclear Fuel Management Optimization," PhD Dissertation, North Carolina State University (1993). 29. G. I. MALDONADO, P. J. TURINSKY, and D. J. KROPACZEK, "On the Treatment of Nonlinear Local Feedbacks Within Advanced Nodal Generalized Perturbation Theory," Trans. Am. Nucl. Soc., 68, 218 (1993). 30. D. J. KROPACZEK, J. D. McELROY, P. J. TURINSKY, and S. B. THOMAS, "Integration of the FORMOSA-P Fuel Management Code into Design Licensing Code Systems," to be published in Proc. Int. Conf Fuel Management and Handling, Edinburgh, United Kingdom, March 20-22, 1995, British Nuclear Energy Society. 31. S. K. ZEE, "Numeric Algorithms for Parallel Processors Computer Architectures with Applications to the Few Group Neutron Diffusion Equations," PhD Dissertation, North Carolina State University (1987). 32. P. J. TURINSKY, Z. SHAYER, H. N. SARSOUR, and P.M. AL-CHALABI, "Benchmarking of Codes on Computers with Parallel Hardware," Proc. Advances in Nuclear Engineering Computation and Radiation Shielding, Santa Fe, New Mexico, April9-13, 1989, Vol. 1, p. 40:1, American Nuclear Society (1989). 33. G. I. MALDONADO, P. J. TURINSKY, D. J. KROPACZEK, and G. T. PARKS, "Constrained Minimization of Fresh Reload Fuel Enrichment Within the FORMOSA Code's Nodal GPT and Optimization Framework," Proc. Top/. Mtg. Advances in Reactor Physics, Knoxville, Tennessee, April1l-15, 1994, Vol. 3, p. 314, American Nuclear Society (1994). 34. P. R. ENGRAND, G. I. MALDONADO, R. ALCHALABI, and P. J. TURINSKY, "Nonlinear Iterative Strategy for NEM: Refinement and Extension," Trans. Am. Nucl. Soc., 65, 221 (1992).

G. Ivan Maldonado [BS, engineering physics, University of Toledo, 1985; MNE, nuclear engineering, North Carolina State University (NCSU), 1988; PhD, nuclear engineering, NCSU, 1993] is an assistant professor of nuclear and mechanical engineering at Iowa State University (ISU). Prior to attending NCSU, he was employed at the Knolls Atomic Power Laboratory, and prior 218


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to joining the faculty at ISU, he was a postdoctoral visitor at the Center of Studies and Research of Electricite de France (Clamart). His research interests are in the area of computational reactor physics focusing on light water reactor core reload and fuel assembly design. Paul J. Turinsky (BS, chemical engineering, University of Rhode Island, 1966; MSE, 1967, and PhD, 1969, nuclear engineering, University of Michigan; MBA, business administration, University of Pittsburgh, 1979) is a professor of nuclear engineering, director of the nuclear program of the Electric Power Research Center, and faculty coordinator of the interdisciplinary graduate program in computational engineering and sciences at NCSU. Prior to joining NCSU, he was on the faculty of Rensselaer Polytechnic Institute and with the Water Reactor Divisions of Westinghouse Electric Corporation. His research interest is computational reactor physics, with emphasis on nuclear fuel management optimization, nodal kinetics, and parallel solution algorithms for the neutron diffusion and integral transport equations.


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