NUCLEAR SCIENCE AND ENGINEERING: 121, 312-325 (1995). Employing Nodal Generalized Perturbation Theory for the Minimization of Feed Enrichment ...
NUCLEAR SCIENCE A N D ENGINEERING: 121, 312-325 (1995)
Employing Nodal Generalized Perturbation Theory for the Minimization of Feed Enrichment During Pressurized Water Reactor In-Core Nuclear Fuel Management Optimization G. Ivan Maldonado,* Paul J. Turinsky, and David J. Kropaczek North Carolina State University, Department of Nuclear Engineering Electric Power Research Center, P.O. Box 7909, Raleigh, North Carolina, 27695-7909 and Geoffrey T. Parks Cambridge University Engineering Department, Trumpington St. Cambridge CB21PZ, United Kingdom Received October 17, 1994 Accepted March 21, 1995
Abstract - The computer code FORMOSA-P (Fuel Optimization for Reloads Multiple Objectives by Simulated Annealing—PWR) has been developed to address pressurized water reactor (PWR) in-core nuclear fuel management optimization. Until recently, the optimization objectives available to the user included minimization of relative power peaking throughout the cycle, maximization of the end-ofcycle reactivity, and maximization of region-average discharge burnup. In addition, during an optimization, various core attributes (including the preceding objectives) can be optionally activated as constraints via penalty functions or to directly reject sampled loading patterns that violate established design limits. The underlying theoretical framework that enables the accurate and efficient calculation of objective and constraint values within the FORMOSA-P code is its higher order, nodal generalized perturbation theory (GPT) neutronics model. The utility of the FORMOSA-P code has been extended to include a traditionally out-of-core decision variable, namely, the fresh (i.e., feed) reload fuel enrichment. This is accomplished by formulating the feed enrichment as a GPT variable that can be adjusted concurrently with changes in the core loading pattern to enforce a target cycle length. This provides a reload designer with the capability to minimize feed enrichment during an in-core optimization while enforcing all other constraints (e.g., power peaking limit, cycle energy requirement, degree of eighth-core power tilt, discharge burnup limit, and moderator temperature coefficient limit).
I. INTRODUCTION I.A.
Background
As evident from a number of recent publications on the subject, mathematical optimization methods to *Current address: Department of Mechanical Engineering, Nuclear Engineering Program, Iowa State University, 105 Nuclear Engineering Laboratory, Ames, Iowa 50011-2241.
automate the determination of pressurized water reactor (PWR) loading patterns continues to be an active area of research. 1 - 6 The computer code F O R M O S A - P (Fuel Optimization for Reloads Multiple Objectives by Simulated Annealing — PWR) has been developed to determine the family of near-optimum loading patterns for PWRs. Historically, the earliest version of FORMOSA-P is attributable to Kropaczek and Turinsky, 7 who built upon earlier developments by Hobson and Turinsky. 8
Most recently, the work by Maldonado et al. 9 and Maldonado and Turinsky 1 0 led to the development of an efficient and accurate nonlinear nodal generalized perturbation theory (GPT) neutronics model. This model was built directly from the original FORMOSA-P finite difference G P T model by employing Smith's nonlinear iterative approach 1 1 applied to the nodal expansion method 12 " 14 (NEM). In addition, the nodal G P T technique was further refined to include important nonlinear effects such as local thermal-hydraulic and fission product feedbacks. 15 In summary, the combination of efficiency and fidelity within a higher order advanced nodal G P T neutronics model coupled to a robust and adaptive simulated annealing optimization algorithm has produced, via utilization in the reload design process, a useful in-core nuclear fuel management optimization tool for P W R loading pattern design. 16 Further details about the FORMOSA-P code fall beyond the scope of this study; thus, the reader is referred to the references provided in this section. I.B. Motivation
for
Research
A key aspect of the mathematical optimization strategy within FORMOSA-P is the capability to select from a flexible set of objective and constraint formulations, thereby providing the user with control over the loading pattern search without the need to impose heuristic rules. In the past, the FORMOSA-P optimization strategy included three objective functions: minimization of relative power peaking (i.e., assembly quadrant FAh), maximization of end-of-cycle (EOC) reactivity (i.e., ktff C ), and maximization of region-average discharge burnup. When not performing a relative power peaking minimization, the FAh limit becomes an active constraint. Similarly, various constraints can be handled directly through rejection of a sampled loading pattern [e.g., a violation of the moderator temperature coefficient (MTC) limit] or in the form of penalty functions (e.g., degree of eighth-core power tilt and discharge burnup). At the completion of a maximization of k ^ c , the nuclear reload engineer may conclude that the optimal loading pattern arrangement implies a longer cycle length. In practice, however, the cycle length is usually fixed a priori via out-of-core fuel management decisions. Thus, if the engineer adjusts the feed enrichment to match the target cycle length, then the power distribution is perturbed, which, in turn, impacts . The implication is that a loading pattern optimization is required following each feed enrichment adjustment, and this process must be iteratively continued until the cycle energy production and all other constraints are simultaneously satisfied. Of secondary concern during an optimization is the fact that FORMOSA-P utilizes input values of soluble boron versus cycle burnup, with the E O C value being the target value. Consequently, the critical boron concentration (PPMcrit) throughout
the cycle is not known as a function of sampled loading pattern, which introduces some errors in evaluating power distributions and MTCs during the optimization. To overcome these two shortcomings, one must accomplish two tasks simultaneously for each sampled loading pattern: 1. adjust the feed enrichment such that the cycle length is constrained to a known target value 2. ensure that each sampled loading pattern is evaluated and depleted at the critical boron concentration throughout the cycle. This study presents the development and results of a technique (employing nodal GPT) by which the two aforementioned adjustments (criticality and feed enrichment) are implemented during the course of a FORMOSA-P reload optimization and in concurrency with changes in the core loading pattern, thus satisfying the imposed cycle energy constraint for each loading pattern being sampled. The calculation of feed enrichment in this manner has the advantage of allowing an accurate determination of the core power distribution during the optimization process, thus avoiding the impractical iterative process noted earlier. From an optimization viewpoint, if the feed enrichment is known for each sampled loading pattern, then it can be utilized as either an objective function to be minimized or as a constraint function that can be imposed. Needless to say, the ability to minimize a given cycle's feed enrichment can directly translate into significant savings in fuel cycle costs. 17
II. METHODOLOGY To establish the notation to be employed in describing this study, we define the standard matrix form of the few-group neutron diffusion equation as follows: = X 0> ,B 0i ,* 0> , ,
(1)
where the subscripts in Eq. (1) denote an unperturbed (reference) state o at depletion step t. The tilde notation simply highlights the presence of nonlinear NEM coupling corrections to the bands of the standard finite difference matrix f o r m . 3 II.A. Linear Superposition
Framework
As described in detail within previous publications, 9 ' 10 - 16 the nodal G P T strategy utilized by FORMOSA-P to evaluate global and local core attributes during an optimization relies on a two-step approach. a A major highlight of the nonlinear iterative approach to NEM is that it allows full preservation of the finite difference system's coefficient matrix structure."
Initially, first-order-accurate estimates of the flux, eigenvalue, and matrix operators are calculated by linear superposition of single-assembly perturbation calculations. Subsequently, standard variational formulations of the Rayleigh quotient 18 and of a G P T power response functional 19 utilize the first-order estimates to produce higher-order-accurate estimates of the perturbed eigenvalue and power distribution, respectively. For example, the expression employed to boot up the accuracy of the first-order perturbed eigenvalue estimate is x ^ d > = KT
+
(2) Similarly, the higher-order-accurate perturbed power response for location k is Arp(2nd) _ /tp - ? n , , [ A < : r > - x ]*£?>> , o> where the subscript p denotes a perturbed response, is the homogeneous adjoint flux, and kT*o t is the generalized adjoint function for the power response at location k.b Traditionally, the single-assembly perturbations correspond to discrete changes (shuffle, reorientation, or burnable poison loading) in the loading pattern. However, in the case of feed enrichment, its value is not discrete and thus can fall anywhere within a realistically bounded but continuous range. Therefore, feed enrichment single-assembly perturbations are performed to determine derivative information instead of actual changes in the response sought. For example, consider the following expressions for the first-order-accurate eigenvalue and flux at a depletion step t within the cycle: Apjf = X 0j , + 2 (^)'
,v
1.45
-
•. i
1.4
''.
• :'•;!*
'm*
v*
1.35 1.3 0
1000
i
i
i
2000
3000
4000
I
5000
6000
7000
All Accepted Histories (total=2107) Fig. 6. Quadrant FAh for simulated accepted feed enrichment histories.
We now examine the P T and G P T results at BOC and EOC on a per-assembly basis. The magnitude of the spatial material perturbations are illustrated by the change in each assembly's k ^ relative to the reference
TABLE II RMS and Maximum Absolute GPT Errors in Assembly Power During Feed Enrichment Minimization Assembly Average Relative Power Errors RMS Error Core Burnup (MWd/tonne U) 0 500 1500 2 500 3 500 4 500 5 500 6 500 7 500 8 500 9 500 10500
loading pattern (i.e., optimum minus reference). These assembly A k x perturbations at BOC and E O C are given in Tables III and IV, respectively. The perturbations shown contain the effects of material property changes due to the reduced enrichment of the feed assemblies as well as the superimposed effects of assembly shufflings, burnable poison changes, nonlinear local feedbacks, and soluble boron (only at BOC). Tables V and VI display the corresponding performance of P T and G P T in predicting assembly average power responses. The results once again confirm both the adequacy of using the higher order accurate nodal GPT for this application and the lack thereof displayed by its first-order-accurate P T counterpart.
Maximum Error
PT Error
GPT Error
PT Error
GPT Error
0.008 0.008 0.009 0.010 0.011 0.012 0.013 0.014 0.015 0.016 0.017 0.018
0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002
0.018 0.015 0.017 0.021 0.026 0.029 0.032 0.035 0.038 0.040 0.042 0.044
0.005 0.004 0.004 0.005 0.005 0.005 0.006 0.006 0.006 0.006 0.007 0.007
I II.C. Cost-of-Margin
Trends
The example shown in the previous section provides a satisfactory verification of the optimization algorithm employed and of the fidelity performance of the nodal G P T method developed. In an effort to gain further insight from a nuclear fuel management perspective, additional tests were performed to study the cost-ofmargin trade-off relationship between the quadrant power peaking (FAh) limit and a minimized feed enrichment. To accomplish this, three additional optimization cases were added to supplement the results generated for the case already discussed. We denote the example already discussed as case 1 and label the additional cases as cases 2, 3, and 4. Table VII identifies
TABLE III semb
BOC Aic™ >y (Optimized - Reference) During Feed Enrichment Minimization H
G
F
E
D
C
B
A
8 9 10 11
0.0007 -0.0893 0.0017 -0.0601
-0.0886 -0.0010 0.0204 0.1060
0.0017 0.0204 0.0842 -0.0430
-0.0601 0.1061 -0.0430 0.0030
-0.0013 -0.0116 -0.1055 -0.0107
0.0350 0.0295 -0.0120 0.0877
0.0516 0.0537 0.0440 -0.0424
-0.0417 -0.1013 -0.0418 -0.1086
12 13 14 15
-0.0014 0.0350 0.0516 -0.0417
-0.0116 0.0295 0.0537 -0.1013
-0.1055 -0.0120 0.0440 -0.0418
-0.0107 0.0877 -0.0424 -0.1086
-0.0010 0.0535 -0.0184
0.0535 -0.0025 -0.0036
-0.0184 -0.0036
the constraints imposed upon each individual feed enrichment minimization case, namely, target cycle length and FAh limit. Note that all other characteristics of the optimization were left unchanged (i.e., 12 depletion steps, enrichment minimization). Table VII also tabulates the minimized feed enrichment determined for each case with all constraints imposed satisfied. These results can be collectively interpreted as direct quantifications of the cost of thermal operating margin (F A h ) in terms of fuel enrichment cost as a function of cycle energy requirement. For cases 1 and 2, associated with a 10 500 M W d / t o n n e U cycle energy constraint, feed enrichment was found to increase strongly with decreasing power peaking limit constraint. For cases 3 and 4, associated with a 12000 MWd/tonne U cycle energy constraint, a much weaker correlation between feed enrichment and power peaking constraint was found. To better understand the cost-of-margin behavior, consider the global behavior displayed by the thousands of loading patterns sampled and accepted along the pathway to the optima tabulated in Table VII. Figure 7 plots the feed enrichment versus accepted histories for cases with the same cycle energy requirement (10 500 M W d / t o n n e U) but of different FAh lim-
its: 1.35 and 1.375, respectively. This figure clearly displays the cost of margin as captured by the higher feed enrichment required at a lower power peaking limit. However, as the cycle energy requirement is increased, the display of the cost of margin (Fig. 8) indicates that it is not discernible. From this study, we conclude that employment of engineering intuition rules (heuristics), such as "higher power peaking limits always lead to lower feed enrichment requirements," should be exercised with great caution. IV. CONCLUSIONS The results shown illustrate an accurate and successful implementation of a technique based on nodal G P T that provides the reload designer with the unique capability to simultaneously adjust feed enrichment (to match a target cycle energy requirement) and to maintain criticality (via soluble boron adjustment) in concurrency with the sampling of each loading pattern during an optimization. By this approach, every loading pattern examined (and archived) during a FORMOSA-P optimization is guaranteed to meet the target cycle length — clearly a difficult requirement to satisfy without employing a G P T technique.
TABLE IV sembly
EOC Ak™
(Optimized — Reference) During Feed Enrichment Minimization
H
G
F
E
D
C
B
A
8 9 10 11
0.0091 -0.0684 0.0029 -0.1024
-0.0681 0.0030 -0.0206 0.0833
0.0029 -0.0205 0.0646 -0.0327
-0.1024 0.0833 -0.0327 0.0034
-0.0034 -0.0280 -0.0344 -0.0265
0.0323 0.0197 -0.0286 0.0657
0.0310 -0.0081 0.0312 -0.0291
-0.0279 -0.0855 -0.0218 -0.0973
12 13 14 15
-0.0034 0.0323 0.0310 -0.0279
-0.0280 0.0197 -0.0081 -0.0856
-0.0344 -0.0286 0.0313 -0.0218
-0.0265 0.0658 -0.0291 -0.0972
-0.0048 -0.0053 -0.0184
-0.0053 -0.0067 -0.0049
-0.0184 -0.0049
TABLE V GPT Performance in Predicting BOC Assembly Average Relative Power During Feed Enrichment Minimization H
G
F
E
D
C
B
A
8
0.613 -0.003 0.014
0.746 -0.005 0.013
1.084 -0.001 0.014
1.153 -0.003 0.014
1.202 0.000 0.003
1.360 -0.002 -0.018
1.150 -0.002 -0.011
0.712 -0.001 -0.003
9
0.746 -0.005 0.013
0.988 -0.002 0.013
1.314 0.000 0.009
1.219 0.003 0.011
1.334 0.001 0.003
1.086 -0.001 -0.006
1.214 -0.002 -0.007
0.508 -0.001 0.000
10
1.084 -0.001 0.014
1.315 0.000 0.009
1.198 0.002 0.008
1.236 0.001 0.003
1.366 0.002 -0.003
1.283 0.000 -0.003
1.077 0.000 -0.003
0.627 0.000 -0.001
11
1.153 -0.003 0.014
1.219 0.003 0.011
1.235 0.001 0.002
1.210 0.001 -0.003
1.319 0.001 0.003
1.144 0.000 -0.001
0.956 0.001 -0.005
0.302 0.001 -0.002
12
1.202 0.000 0.003
1.334 0.001 0.003
1.366 0.002 -0.003
1.318 0.001 0.003
1.113 0.000 -0.005
1.082 0.000 -0.007
0.473 0.000 -0.001
13
1.360 -0.002 -0.018
1.086 -0.001 -0.006
1.283 0.000 -0.003
1.144 0.000 -0.001
1.082 0.000 -0.007
0.627 0.000 -0.003
0.239 0.000 -0.001
14
1.150 -0.002 -0.011
1.214 -0.002 -0.007
1.077 0.000 -0.003
0.956 0.001 -0.005
0.473 0.000 -0.001
0.239 0.000 -0.001
15
0.712 -0.001 -0.003
0.508 -0.001 0.000
0.627 0.000 -0.001
0.302 0.001 -0.002
- Forward (FWD) -» (GPT) - (FWD) -> (PT) - (FWD)
Maximum Power Error
RMS Power Error
-0.005 -0.018
0.002 0.008
F-delta-h Limit: 1.375 F-delta-h Limit: 1.350 + + + + +
f j j i f
V
3.1 •< 0
1000
2000
#
'"-Vr 3000
4000
5000
6000
7000
All Accepted Histories Fig. 7. Feed enrichment minimization accepted histories for 10 500 MWd/tonne U cycle length (cases 1 and 2).
TABLE VI GPT Performance in Predicting EOC Assembly Average Relative Power During Feed Enrichment Minimization D
1.048 0.000 0.023
1.087 -0.004 0.019
1.136 -0.001 0.019
1.262 0.001 0.013
1.100 0.000 0.022
0.781 -0.005 0.038
0.995 0.000 0.020
1.221 0.000 0.014
1.138 0.002 0.017
1.284 -0.001 0.008
1.062 0.000 0.007
1.161 -0.001 -0.001
0.584 -0.002 0.002
1.048 0.000 0.023
1.221 0.000 0.015
1.111 0.002 0.015
1.154 0.000 0.011
1.328 -0.001 0.003
1.277 0.000 -0.012
1.086 0.000 -0.022
0.729 0.003 -0.026
11
1.087 -0.004 0.019
1.138 0.002 0.017
1.154 0.000 0.012
1.153 0.000 0.007
1.299 -0.001 -0.007
1.142 -0.002 -0.017
1.010 0.004 -0.044
0.387 0.007 -0.020
12
1.136 -0.001 0.019
1.284 -0.001 0.010
1.328 -0.001 0.006
1.298 -0.001 -0.005
1.114 -0.001 -0.006
1.108 0.000 -0.020
0.571 0.002 -0.017
13
1.262 0.001 0.013
1.063 0.000 0.011
1.277 0.000 -0.006
1.142 -0.001 -0.013
1.108 0.000 -0.018
0.736 0.000 -0.011
0.329 0.000 -0.007
14
1.100 0.000 0.022
1.161 -0.002 0.008
1.086 -0.001 -0.012
1.010 0.003 -0.037
0.571 0.001 -0.015
0.329 0.000 -0.007
Maximum Power Error
RMS Power Error
15
0.781 -0.005 0.038
0.584 -0.005 0.013
0.729 0.001 -0.015
0.387 0.005 -0.015
-> Forward (FWD) - > - (GPT) - (FWD) — > (PT) - (FWD)
+0.007 -0.044
0.002 0.018
G
8
0.713 -0.001 0.016
0.808 -0.002 0.017
9
0.808 -0.002 0.017
10
B
A
E
F
H
C
4.1 F-delta-h Limit: 1.375
4in m di
3.9
KU • •£•
*
F-delta-h Limit: 1.350 + + + + +
*
o au E J= u W
•a
[8
i
3.6 -
I 3.5 0
1000
2000
3000
4000
5000
6000
7000
All Accepted Histories Fig. 8. Feed enrichment minimization accepted histories for 12000 MWd/tonneU cycle length (cases 3 and 4). NUCLEAR SCIENCE AND ENGINEERING
VOL. 121
OCT. 1995
TABLE VII Feed Enrichment Minimization Cases for Cost-of-Margin Study Cycle Energy Constraints Case Number
FAh Limit
Cycle Burnup (MWd/tonne U)
PPM at EOC
Optimum Feed Enrichment
1 2 3 4
1.375 1.350 1.375 1.350
10500 10500 12000 12000
50 50 50 50
3.0214 3.1147 3.5552 3.5646
Allowing feed enrichment to become one of the decision variables available during an optimization provides a new link between in-core and out-of-core fuel management decisions, which brings the designers one step closer to determining truly optimum fuel inventories.
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