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Optimization of Fuel Rod Enrichment Distributionto Minimize Rod Power Peaking throughout Lifewithin BWR Fuel AssemblyYasushi HIRANO a , Kazuki HIDA a , Koichi SAKURADA a & Munenari YAMAMOTO aa Nuclear Engineering Laboratory , Toshiba Corp. , Ukishima-cho , Kawasaki-ku ,Kawasaki , 210Published online: 15 Mar 2012.

To cite this article: Yasushi HIRANO , Kazuki HIDA , Koichi SAKURADA & Munenari YAMAMOTO (1997) Optimization ofFuel Rod Enrichment Distribution to Minimize Rod Power Peaking throughout Life within BWR Fuel Assembly, Journalof Nuclear Science and Technology, 34:1, 5-12, DOI: 10.1080/18811248.1997.9732050

To link to this article: http://dx.doi.org/10.1080/18811248.1997.9732050

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Journal of NUCLEAR SCIENCE and TECHNOLOGY, Vol. 34, No. 1, p. 5-12 (January 1997)

Optimization of Fuel Rod Enrichment Distribution to Minimize Rod Power Peaking throughout Life

within BWR Fuel Assembly

Yasushi HIRANOt , Kazuki HIDA, Koichi SAKURADA and Munenari YAMAMOTO

Nuclear Engineering Laboratory, Toshiba COT.*

(Received June 19, 1996)

A practical method was developed for determining the optimum fuel enrichment distribution within a boiling water reactor fuel assembly. a combinatorial optimization problem grouping fuel rods into a given number of rod groups with the same enrichment, and a problem determining an optimal enrichment for each fuel rod under the resultant rod- grouping pattern. In solving these problems, the primary goal is to minimize a predefined objective function over a given exposure period. The objective function used here is defined by a linear combination: C1X+C2X~, where X and X G stand for a control variable to give the constraint respectively for a local power peaking factor and a gadolinium rod power, and C1 and C, are a user-definable weighting factor to accommodate the design preference.

The algorithm of solving the combinatorial optimization problem starts with finding the optimal en- richment vector without any rod-grouping, and promising candidates of rod-grouping patterns are found by exhaustive enumeration based on the resulting fuel enrichment ordering, and then the latter problem is solved by using the method of approximation programming. The practical application of the present method is shown for a contemporary 8 x 8 Pu mixed-oxide fuel assembly with 10 gadolinium-poisoned rods.

The method deals with two different optimization problems, 2.e.

KEYWORDS: algorithms, Optimization, fuel enrichment distribution, combinatorial opti- mization problem, rod grouping pattern, approximation programming, linear programming, exhaustive enumeration, mixed oxide fuels, fuel assemblies, B WR type reactors

I. INTRODUCTION A variety of constraints are imposed in the nuclear

design of boiling water reactor (BWR) fuel assembly due to the requirements on safety, operation and econ- omy. Among those demanded are reactivity for opera- tion, reactivity coefficients of void and Doppler, control rod worth, etc. These fundamental reactivity charac- teristics are mostly determinable by a couple of key pa- rameters representing geometry and material conditions, e.g. average fuel density and enrichment, hydrogen-to- heavy-metal ratio, fuel rod size and pitch, control poison materials and geometry and so on, being little affected by other details, so that they can be settled within an acceptable range in the early step of design process.

In the succeeding detailed nuclear design, therefore, resolution of remaining demands is the main task to be achieved. They include shut-down margin (SDM) and thermal margins (e.g. maximum linear heat generation rate: MLHGR) during operation. These are not deter- minable by a fuel assembly design alone; both a fuel assembly design and a core design significantly interact with each other. Thus, the detailed design of BWR fuel

* Ukishima-cho, Kawasaki-ku, Kawasaki 210. Corresponding author, Tel. $81-44-288-8132, Fax. +8 1-44-270- 1806, E-mail: hirano@rcg.nel.rdc.toshiba.co.jp

assembly needs to be made simultaneously taking into account the above interaction, being a sort of problems requiring expert knowledge and experience, having been performed by a trial-and-error approach so far. More- over, recen: design evolution also makes it more labo- rious due to higher demand for its economical perfor- mance. To these demands, on the one hand, available measures in design are fuel loading pattern in a core, gadolinium (Gd) inventory (number of rods and den- sity) and fuel enrichment distribution in a fuel assembly. Since the SDM demand is strictly fulfilled ahead of the others, it gives some limitations to the fuel loading pat- tern and Gd inventory. As to the fuel enrichment dis- tribution, number of different enrichments and available enrichment range are restricted due to the manufactur- ing requirements. Finally, the design of fuel enrichment distribution has to be performed under all such restric- tions.

The incentive and objective of the present study is to develop an optimization method that can be used in the final design stage of fuel enrichment distribution. The method focuses in a problem determining an opti- mal fuel enrichment distribution under given constraints; problems seeking an optimal design for fuel loading pat- tern, core-wide zoning and/or Gd poisoning are out of the scope of this study.

This paper describes an optimization method appli-

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cable to the aforementioned nuclear design of BWR fuel assembly and its application results. A similar study has been performed by Lim and Leonard"), but their study was limited to an optimization at the beginning-of-life (BOL) only, i. e., time-dependent depletion effects were not considered, and in addition, a commonly used Gd rod was left out of the sight. Thus, the optimal solu- tions by their method are often insufficient to the real design. For example, since the optimal enrichment of Gd rod is the maximum value by minimization of local power peaking factor (LPF) at the BOL only, the Gd rod is a power peaking rod after the Gd burns away. This is undesirable because of the lower thermal conductivity of Gd rod. Moreover, when the optimal PuOz enrichment distribution for Pu mixed-oxide (MOX) fuel assembly is made in the same way, the LPF sometimes tends to in- crease at the latter exposure period.

In contrast to their method, our present study con- siders all such effects together with realistic constraints. To this end, the present method is constructed with two separate steps. The first step is to group fuel rods into a given number of rod groups with the same enrichment, and then to search the best rod-grouping pattern that yields the minimum power peaking factor over a given exposure period. The second step is to determine an optimal enrichment value for each fuel rod group under the resultant rod-grouping pattern. The problem for the latter step is a t first discussed in Chap.11, since the method elaborated for the former problem employs the same technique as parts of the solution algorithm. The former problem is described next in Chap.111. In Chap.IV, the results of application are given.

n . OPTIMIZATION OF FUEL ENRICHMENT VALUES UNDER GIVEN ROD-GROUPING PATTERN

As mentioned previously, an ordinary BWR fuel assembly consists of several different "fuel types" that are distinguished by initial fuel compositions (fuel en- richment value and Gd-poisoning). In point of optimiza- tion, the present subjects include two different problems, i.e. seeking for a distributing pattern of fuel type (rod- grouping pattern) and determining a value of enrichment for each fuel type. We consider here the latter problem, namely a method for determining an optimal enrich- ment value for each fuel type under given rod-grouping pattern. The optimum condition we are seeking is to minimize the MLHGR value during a given exposure period under several practical constraints. Considera- tion of burnup effects makes this problem be of the min- max type. We divide a given exposure period into L burnup steps, so burnup dependent quantities are ap- proximately treated by those at such discretized points.

Consider a BWR fuel assembly which consists of NT fuel rods including NG Gd-poison rods; the remain-

ders, N(= NT-NG) are a normal fuel rod with no Gd- poisoning. Let Ns be the number of fuel types, the enrichment vector to be determined is represented by e=(el, e2,. . . , ei, . . . , eNs)t . In addition, we introduce two more variables; X and X G , as defined later on. Thus, the control variables are given by a vector u:

(1) 21 = (e , x, XG)t .

Our optimization problem is formally to find a vector u that minimizes the objective function defined by

A ( X , X G ) = cix + C ~ X G , (2) where C1 and C2 is a user-definable weighing factor for X and X G , respectively, to accommodate the design pref- erence. The constraints imposed are

WlPjl(e) 5 X , for normal rods,

Pzd(e ) 5 XGLPFl(e) , for Gd rods,

j = 1 , - - . , N , 1 = l , . . . , L , (3)

l c = l , . . . , N ~ , 1 = 1 , . . . , L , (4)

a/100 5 XG, (5) subject to the conditions;

, NS, (7 ) emin 5 ei 5 emax, i = 1, . . . where Wl: Burnup step 1 dependent weighting factor

Pjl(e): Relative power of normal rod j at burnup

X : Control variable of the LPF constraints for

simulating the nodal power in a core

step 1

normal rods Pzd(e) : Local power of Gd rod Ic at burnup step 1

X G : Control variable of the local power con- straints for Gd rods

LPFl(e): Local peaking factor at burnup step 1 a: Power suppression factor for Gd rods.

Inequality (3) gives the constraint on the LPF for normal rods. The burnup step dependent factor Wl is introduced to simulate a LHGR by the product of Wl and LPF.

As to Gd rods, Inequalities (4) and ( 5 ) are imposed such that the maximum value of Gd rod power, P E d ( e ) , does not exceed the product of X G and LPF at any bur- nup point. These constraints are to suppress the tem- perature of Gd rod below that of the peak power rod at all time; to compensate the defect due to the less thermal conductivity of Gd rod. The variable X G is in- troduced to avoid the numerical infeasibility that would be often encountered when X G were replaced by a/100 in Inequality ( 4 ) . Otherwise, the solution would be often infeasible when the a-value is too small or a guessed en- richment is undesirable at MAP stage as described later on. In case of need, this method permits a Gd rod power to be greater than a% of LPF.

If we can minimize the objective function (2), X and

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Optimization of Fuel Rod Enrichment Distribution within BWR Fuel Assembly 7

X G can be suppressed simultaneously, then the products of Wl and Pjl(e) for all normal rods are suppressed less than X , and the ratios of the relative power for Gd rods to the LPF are suppressed less than XG, which in turn is minimized down to a%. As a result, both MLHGR and LHGR of Gd rods are simultaneously suppressed within the prescribed limits. The priority between two con- straints is controlled by the choice of weight C1 and C2;

a larger weight denotes the constraint higher priority. The condition of Eq.(6) is to keep the average enrich-

ment e unchanged during the solution, and Eq.(7) gives the lower and upper bounds of the allowable enrichment value.

The problem [(l) through (7)] is a sort of non- linear programming problems, being solved by using the method of approximation programming (MAP")). In the MAP algorithm, the nonlinear functions are locally linearized, and the resulting linear programming (LP) problem is solved with a standard LP algorithm. So, the nonlinear state variables Pjl ( e ) and PZd ( e ) are assumed to be locally linear around a guess enrichment vector eo. The constraint (3) and (4) are then converted into a lin- ear form:

Wl(Pjl(e0) + ( e - e ~ ) ~ V P j l ( e o ) ) 5 X ,

Pgd(eo) + ( e - eo)tVPZd(eo) 5 X G L P ~ ( ~ O ) , j = 1 , . . . , N , Z = l , . . . , L , (8)

k = l , " ' , N G l z=1,." ,L . (9)

Here, LPFl in the right-hand side of the constraint (9) is assumed to be independent on e to make it be a LP prob- lem. While this assumption is crude at the early MAP stages, it would be acceptable as the solution converges. In addition, to ensure the validity of thus linearity as- sumption, the following constraint is further imposed:

lei -eiol 5 6, i = I , . . . , NS, (10) where 6 gives the maximum allowable change in enrich- ment per one time.

The gradients of the state variables, VPjl(eo), can be approximately estimated by using the sensitivity cc- efficients based on the first order approximation (see APPENDIX). For example, the ( i , j ) element of the gradi- ent matrix is represented by a summation of the sensi- tivity coefficients over all rods belonging to group i:

-- dPj1 apj1 - c -> 8% m E g ( i ) a&,

2 = l , . . . , N S , j = 1 , . . . , N , 1 = 1,. . . , L , (11)

where g ( i ) is number of rods in group i. The gradients VPzd(eo) are also represented in the same form.

This LP problem is repeatedly solved until the solu- tion converges. If the solution converges, the constraint (4) is satisfied in spite of the assumption made on LPFL in Eq.(9). Note that, in this method, introducing the variable X G is a key factor of generating a feasible solu-

tion at each MAP stage.

JJI. OPTIMIZATION OF FUEL ENRICHMENT DISTRIBUTION INCLUDING DETERMINATION OF ROD-GROUPING PATTERN

In the previous section, we described the method for determining an optimal fuel enrichment value for each fuel rod under given rod-grouping pattern. It is natu- ral, in the strict sense, that an optimal solution is de- pendent on rod-grouping pattern adopted. Here, we describe a method for optimizing fuel enrichment dis- tribution including the determination of rod-grouping pattern. Choice of rod-grouping pattern is a combinato- rial problem, so this problem is a sort of nonlinear mixed integer programming problems.

The algorithm elaborated is constructed with three successive steps, i. e. choice of promising rod-grouping patterns, solving the LP problem for each resulting rod-grouping pattern and finding the best rod-grouping pattern among them, finally making an optimization calculation for the best rod-grouping pattern chosen. Namely, the algorithm is as follows.

[Step- 11 In the first step of choosing promising rod-grouping

patterns, the optimization problem of obtaining the op- timal rod-by-rod fuel enrichment value without any rod- grouping is firstly solved by the method described in the previous section. The resulting fuel enrichment values are arranged in a monotonically decreasing order. This ordering provides us with the basis for grouping fuel rods into a given number of rod groups. In other words, each rod group is consisting of one or more rods with consec- utive orded in this ordering. This is an approximation, since the interaction effect due to the enrichment change in neighboring rods might disturb the ordering. The choice of candidate rod-grouping patterns is made based on the evaluated value of objective function A(X , X,) in Eq.(2) for all possible patterns ( N ~ - ~ C N ~ - ~ patterns), where the enrichment of each rod group is taken to be the average value of all fuel rods in the group. Thus, the best fifty candidate rod-grouping patterns are chosen for the next step.

[Step-2] In the second step, the LP problem is solved for

each of fifty candidate rod-grouping patterns to opti- mize the enrichment value of each group, and the best rod-grouping pattern that has the lowest value of the ob- jective function is selected. When solving the LP prob- lems, the value of LPFl(e) estimated at [Step-1] is used in Inequality (9), and the constraint (10) is removed to find a feasible solution in larger search space to save the burnup calculation for each of fifty rod-grouping pat- terns. In addition, the sensitivity coefficients obtained at [Step-1] are used to save the calculation time.

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Rod-by-Rod Enrichment Distribution

Y. HIRANO et al.

[ S t ep-31 Finally, for the best rod-grouping pattern determined

at [Step2], the optimal enrichment distribution is ob- tained by using the method described in the previous sec- tion. The sensitivity coefficients are calculated at each MAP stage to obtain rigorous optimal solution.

Now, we summarize the algorithm of optimizing the enrichment distribution including determination of the rod-grouping pattern in Fig. 1.

IV. APPLICATION 1. Calculational Conditions The present method was applied to the MOX fuel

assembly shown in Fig. 2, which is a contemporary

Initial guess : Uniform Enrichment Distribution I

[Step21 : Determination of Rod-grouping Pattern

2-1 : Rough Evaluation of Objective Function All Possible Patterns- Fifty Candidate Patterns

Fifty Patterns - Best Pattern 2-2 : LP Problem

I

[Step31 : Final Optimization I (Rod-grouping Pattern : Best Pattern obtained at [Step21 )

I Enrichment Value of Each Group

I Optimal Enrichment Distribution

Fig. 1 Flow chart of overall algorithm

1-16: Normal MOX rods Gl-G3: Gd-poisoned rods

W: Water rods

Fig. 2 MOX fuel assembly

8x8 BWR lattice with two water rods in center. The design parameters preestablished by the preceding requirements are given in Table 1. The average PuOz enrichment is 4.25 wt%, corresponding to a target dis- charge exposure of 30 GWd/t. The specifications on Gd rods are fixed as shown in the table due to the reactivity requirements, being kept unchanged in our optimization problem.

Various conditions for the optimization calculations are listed in Table 2. The lattice physics computer code TGBLA(4) is employed as a module evaluating rod- by-rod power distribution and doing a burnup calcula- tion. The LP problem is solved with the revised simplex method. In the case with no rod-grouping, number of variables is reduced to twenty-one by considering assem- bly symmetry as shown in Fig. 2. Although the burnup step dependent weight function Wl is originally to be a maximum power of the nodes containing the fuel as- sembly in question, the infinite multiplication factor is used here as a substitute for it. If the three-dimensional core calculation for a reference core was already per- formed, the result should be used for the weight function. The Gd rods’ power suppression factor cr is taken to be go%, by which the maximum power of Gd rods is limited

Table 1 Preestablished design parameters

Average PuO, enrichment 4.25 wt% Number of PuO, enrichments 4 Plutonium isotopic composition^(^)

239Pu 57.80wt%, 240Pu 24.77wt%, 241Pu 12.70wt%, 242Pu 4.73wt%

Total number 10 Concentration of Gd,03 2 wt% Positions See Fig. 2.

Gd rod specifications

Table 2 Calculational conditions

Lower and upper bounds of PuO, enrichment

Historical void fraction in burnup calculation Burnup steps (GWd/t)

Number of burnup points to impose constraints

Number of independent variables (including X , X G )

(constraint (7)) 0.0-100.0 wt% 0.40

0.0, 0.22, 2.2, 4.4, 6.6, 8.8, 11.0, 13.2, 15.4, 18.7, 22.0

L=10 (excluding 0.22 GWd/t)

Case with no rod-grouping

Case with given rod-grouping pattern 21 (N=16, N G d , ie., NT=19)

6 (Ns=4) Weight function Wl at burnup point 1

Infinite multiplication factor Gd rod power suppression factor

a=90% Weighting factor in objective function, Eq.(2)

Convergence criteria C1=1.00, Cz=5.00

Case with no rod-grouping Case with given rod-grouping pattern

6ei=0.03 wt% 6ei=0.01 wt%

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Optimization of Fuel Rod Enrichment Distribution within BWR Fuel Assembly 9

below a% of the LPF at all time. We give higher pri- ority to the constraint (4) than to the constraint (3) by taking Cz to be larger than C1. The convergence for the MAP iteration is judged by the maximum difference in fuel enrichment between two successive iterations.

2. Results of Step-1 Optimization of rod-by-rod enrichment with no rod-

grouping is performed by the method described in Chap.111. This results give a base enrichment distribu- tion which is used later on in generating a rod-grouping pattern. Starting with a uniform Pu02 enrichment dis- tribution, a pair of calculations, i. e., solving LP problem and performing burnup calculation, are repeated until the solution converges.

The converged solution is shown in Fig. 3. As natu- rally expected, enrichments are distributed inversely pro- portional to thermal flux distribution; lower enrichments are in the assembly periphery and vice versa The bur- nup step dependent products of the weight function and relative power, WlPjl(e), for each normal rod are given in Fig. 4, showing that all the peak values agreed with each other and well converged to X-value of 1.16. This uniformity of the peak values verifies the optimality in the enrichment distribution. The ratios of the Gd rods power to the LPF are shown in Fig. 5. The peak val- ues are suppressed below X G , but show a visible margin to the limit-value of 0.90. This is due to the loose conver-

Fig. 3 Optimal PuO, enrichment distribution

Product WL . PJL

1 . 3 0 r 1 , i i l i i I I

9 A

- i ~ , I

4 8 12 16 20 0 . 8 0 O l ' I ! I I 1

Assembly Average Exposure [ GWd/t ]

Fig. 4 Burnup step dependent product of weight func- tion and relative power for each normal rod (Refer to Fig. 2 on rod numbers and positions.)

Gd Ratio P,, / LPF

, I ,

1 I

: - , d

- - i ~ : I I l l l l I I 1

4 8 12 16 20 0.45;

Assembly Average Exposure [ GWd/t ]

Fig. 5 Ratio of Gd rod power to LPF (Refer to Fig. 2 on rod numbers and positions.)

gence of the enrichment for Gd rods at MAP iteration.

3. Determination of Rod-grouping Pattern The resultant enrichments are arranged in the de-

creasing order of the enrichment-value as shown in Fig. 6. According to the algorithm described in Chap.111 [Step-11, we select fifty rod-grouping pat- terns out of NT--1C~s-~(=816) patterns, and solve the LP problem for each of them by using the algorithm described in Chap.111 [Step-21. Three rod-grouping pat- terns showing the smallest values of the objective func- tion are selected as the final candidates, which are shown in Figs. 7(a), (b) and (c). The values of variables X , X G , and objective function are summarized in Table 3. The results show that the values of X G converge to the limit-value of 0.90 and all the cases satisfy the constraint for Gd rods, and that the values of X, ie., maximum val- ues of WlP'l(e), also converge close to each other. The results imply that the adjustment of enrichment was suc-

PuOn Enrichment [wP/.] 10.00 ,1 , / I

1-16 : Regular Rods i 8.00 L 6 GI -G3 : Gd Rods 2

6 . 0 0 t I

' I

0'000 1 0 20 30 40 50 60' !

Number of fuel rods

Fig. 6 Arrangement of the optimal enrichments (Refer to Fig. 2 on rod numbers and positions.)

Table 3 Comparison of the values of X , X G and objective functions for the three final candidates

Rod-grouping X X G Objective function Fig. 7(a) 1.360 0.900 5.860 Fig. 7(b) 1.365 0.900 5.865 Fig. 7(c) 1.369 0.900 5.869

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[:,I I I I I I - ---- :Enrichments (Nt=19) - : Enrichments (Ns= 4)

- '-7-1- - - -7 -

'- 7

Puo2 Enrichment [wt%]

---- :Enrichments (Nt=19) - : Enrichments (Ns= 4)

6.00

r,,,,,,ll O-O0O 10 20 30 40 50 60

Number of fuel rods Fig. 7(a) First candidate for optimu

Fig. 7(b) Second candidate for optimum rod-grouping pattern

PuOz Enrichment ( 6 1

---- : Enrichments (Nt=19) - : Enrichments (Ns- 4)

-.. I

2.001 *-

16 26 36 & 6;' ' Number of fuel rods

Fig. 7(c) Third candidate for optimum rod-grouping pattern

cessfully performed for the individual rod-grouping pat- tern. At the moment, the best choice of rod-grouping pattern will be that of Fig. 7(a). However, if we con- sider dubious effects from the approximate treatments in the present processes, these three patterns cannot be discriminated from each other. It is more reasonable, therefore, to make a final optimization for all three pat- terns. and then to choose the best one out of them.

4. Optimal Enrichment Distribution For each of the rod-grouping patterns in Figs. 7,

a pair of calculations, i.e. solving LP problem and performing burnup calculation, are repeated, until the solution converges. The final optimal solutions are sum- marized in Table 4.

In Fig. 8 are shown the burnup variations of

Table 4 Comparison of the final enrichment value ei of each group i for the three final candidates

Rod-grouping el (wt%) e2 (wt%) e3 (wt%) e4 (wt%)

Fig. 7(a) 8.98 3.52 2.26 1.45 Fig. 7(b) 9.06 3.14 2.27 1.48 Fig. 7 ( c ) 9.14 3.51 2.39 1.38

WlPjl(e) factor for each normal rod. The thick lines show the factor for the hottest rod in each rod group, and the peak values converge to the X-value of 1.27. Again, this uniformity of the peak values assures the optimality of the current solution. The ratios of Gd rods power to the LPF are shown in Fig. 9. The peak values are fairly less than X~=0 .90 , indicating that Gd rods have never been a limiting rod in the present rod-grouping, i. e. Gd rods belong to the PuOz enrichment type 2 and 3.

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Optimization of Fuel Rod Enrichment Distribution within BWR Fuel Assembly 11

Product WL. PJL

Assembly Average Exposure [ GWdh ] 1: 8.98wt%, 2: 3.52wt%, 3: 2.26wt%, 4: 1.45wt%

Fig. 8 Burnup step dependent product of weight func- tion and relative power for each normal rod (Rod-grouping pattern: Fig. 7(a))

Similar results are seen in the cases of Figs. 7(b) and (c), while the results are not shown here. The final opti- mal solutions are shown in Table 4. The infinite multipli- cation factors at 22 GWd/t agree less than O.l%dk. The values of variables X , X G , and objective functions are summarized in Table 5. Although the solutions given in Table 4 are slightly different from those shown in Figs. 7(a), (b) and (c), the objective function corresponding to Fig. 7(a) is still the minimum among the three; the case is probably the optimal solution.

I I I I I ~ I '

0 4 a 12 16 20

i 1

Assembly Average Exposure [ GWdA

2: 3.25wt%, 3: 2.26wt%

Fig. 9 Ratio of Gd rod power to LPF (Rod-grouping pattern: Fig. 7(a))

Table 5 Comparison of the final values of X , X G and objective functions for the three final candidates

Rod-grouping X X G Objective function Fig. 7(a) 1.271 0.900 5.771

Fig. 7(c) 1.284 0.900 5.784 Fig. 7(b) 1.273 0.900 5.773

V. CONCLUSION A practical method was developed for determining

the optimum fuel enrichment distribution within a BWR fuel assembly under practical design constraints. The method deals with two different optimization problems. First, the rod-grouping pattern is determined by exhaus- tive enumeration and linear programming. Second, the enrichment values given rod-grouping pattern are opti- mized by method of approximation programming.

The method was applied for an 8x8 MOX fuel as- sembly containing 10 Gd-poisoned rods. The results were quite satisfactory, showing that the present method would be able to produce a couple of feasible candidate designs comparable to that elaborated by an expert en- gineer. This method would be equally applicable to an ordinary UO2 fuel assembly containing several Gd rods.

Our discussions in this paper have concentrated so far on an optimization of enrichment distribution within a single section of BWR assembly. In the practical cases, however, some of the recent BWR assembles employ an axially-zoned design. Such designs require a simulta- neous optimization of multiple sections('). Thus, the present method should be extended in future so as to deal with such a problem.

The other optimization method, such as simulated annealing and genetic algorithm may be effective to this combinatorial optimization problem. However, since the direct burnup calculations for all search steps are not practical for much calculational time, the linear approx- imation is useful to the estimation of the LPF value.

( 5 )

-REFERENCES-

Lim, E.Y., Leonard, A.: Optimal pin enrichment dis- tributions in boiling water reactor fuel bundles, Nucl. Sci. Eng., 64, 694-708 (1977). Griffith, R.E., Stewart, R.A.: A nonlinear program- ming technique for the optimization of continuous pro- cessing systems, Manage. Sci., 7 , 379 (1961). Naito, Y., Kurosawa, M., Kaneko, T.: Data book of the isotopic composition of spent fuel in light water re- actors, JAERZ-M 94-034, 157 (1994). Yamamoto, M., Mizuta, H., Makino, K., Chiang, R.T.: Development and validation of TGBLA BWR lat- tice physics methods, Proc. Topical Meeting Reactor Physics and Shielding, Chicago, Illinois, Sep. 17-19, 1984, Vol.1, p.364 (1984). Hida, K.: Burnup shape optimization for BWR cores by enrichment and gadolinia zoning, Proc. 2994 Top- ical Meeting Advances in Reactor Physics, Knoxuille, V01.111, 233 (1994).

[APPENDIX]

Approximate Calculation of Sensitivity Coefficient Matrix Elements of Rod Power to Initial Fuel Enrichment An LP problem is solved at each iteration in the MAP

algorithm, requiring that each state variable ( P : rel- ative rod power at each burnup step) must be locally

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linearized on the control variables (E: rod-by-rod initial fuel enrichment). Namely, the elements of sensitivity co- efficient matrix, 2.e. the partial derivative of P on E , need to be evaluated in some way. Physically, this is to obtain a change in rod power at a certain burnup point when some change occurred in initial fuel enrichment. This is a type of problems covered by a generalized per- turbation theory, and may be evaluated accurately by such a method. To our end, however, more approximate evaluation is sufficient, since it will be used only in the intermediate process and its accuracy will give little dif- ference to the final results. Thus, we do it, more simply and approximately, as follows.

Let PI(.) be a vector representing the rod-by-rod power at a burnup step 1, and e and E be a vector of fuel enrichment at a burnup step 1 and BOL, respec- tively. To our purpose, we need to evaluate dPI/dE for each burnup point 1, i.e.

We assume here that the partial derivative, ae/dE, be a unit diagonal matrix. This is a crude approximation, but we use it for simplicity. This substantially simplifies the problem, allowing us to evaluate dPl/dE without considering burnup effect. In addition, a rod power is approximated to be proportional to fission reaction rate. Now, we can express a rod power of a certain rod i, dropping a burnup subscript, by

where c: Energy released per a fission ( X ) i : Volume and energy integral of a quantity X

for rod i C f : Macroscopic fission cross section N : Fissile atomic number density of: Microscopic fission cross section.

A relative rod power (Pi) is then given by

(A31 Pi Pa =

j : all rods

where M is a total number of fuel rods in a fuel assembly. Consider that a small change occurred only in rod

m power, Sp,, which was resulted from a corresponding change in the fuel enrichment (fissile number density), SE,(SN,). Supposing that fluxes and microscopic cross sections stay unchanged, we can write the perturbed rod power ( P A ) as

P: = Pm + b m , (A41 = c ( C f 4 ) m + c(SNof4)m. (A4’)

One can readily obtain the resulting perturbed relative rod power for rod m as follows:

j : all rods

Similarly, the perturbed relative power for rod i(# m) can be written as

(A87 1 =pi-.

1 + P m Subtracting Eq.(A3’) from Eq.(A5’) and Eq.(A8’) yields the change in relative rod power for rod i ,

ti - Pm 1 + P m

6Pi =Pi- for i = m,

= -pi--- Pm for i # m. 1 + P m

Then, the sensitivity coefficient, dPi/dEm, can be ap- proximately evaluated through dividing Eq.(9) by SE,,

(A?O)

In the actual numerical calculations, we evaluate the sen- sitivity coefficient by taking SE, (m=1,2, . . . , N T ) to be 0.01 (1%).

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