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Optimization of Axial Enrichment andGadolinia Distributions for BWR Fuelunder Control Rod ProgrammingKazuki HIDA a & Ritsuo YOSHIOKA ba NAIG Nuclear Research Laboratory , Nippon Atomic Industry GroupCo.Ltd. , Ukishima-cho, Kawasaki-ku, Kawasaki , 210b Isogo Engineering Center , Toshiba Corp. , Shinsugita-cho, Isogo-ku,Yokohama , 235Published online: 15 Mar 2012.

To cite this article: Kazuki HIDA & Ritsuo YOSHIOKA (1989) Optimization of Axial Enrichment andGadolinia Distributions for BWR Fuel under Control Rod Programming, Journal of Nuclear Science andTechnology, 26:5, 492-500, DOI: 10.1080/18811248.1989.9734338

To link to this article: http://dx.doi.org/10.1080/18811248.1989.9734338

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journal of NuCLEAR SCIENCE and TECHNOLOGY, 26(5), pp. 492-500 (May 1989).

Optimization of Axial Enrichment and Gadolinia Distributions for BWR Fuel under

Control Rod Programming

Kazuki HIDA,

NAIG Nuclear Research Laboratory, Nippon Atomic Industry Group Co., Ltd.*

Ritsuo YOSHIOKA

!sago Engineering Center, Toshiba Corp.**

Received November 29, 1988

The axial enrichment and gadolinia distributions of BWR (boiling water reactor) fuel are optimized under control rod programming. The objective of the problem is to minimize the average enrichment required to reach a planned EOC (end-of-cycle) with criticality condition and axial power peaking constraint.

A method of approximation programming is employed as the basis for the solution method. Resulting linear programming problem at each iteration step is solved by means of goal pro­gramming algorithm. The method is applied to the initial fuel for a typical BWR/5 represented by an axial one-dimensional core model

Two-region analysis leads to the conclusion that the core bottom should be depleted during the cycle so that the power shifts to the core top at EOC. The enrichment and gadolinia distributions are determined to maximize EOC power peaking within a limit. The optimal solution of a 24-region fuel with a power peaking limit of 1.4 saves 10.6% in uranium ore compared with a uniform fuel depleted with a Haling power shape. Half the saving comes from an optimal natural uranium blanket implementation.

KEYWORDS: optimization, BWR type reactors, axial enrichment distribution, axial gadolinia distribution, control rod programming, approximation programming, linear programming, goal programming, uranium utilization, burnup shape optimi­zation

I. INTRODUCTION

As the electricity generated by nuclear power plants has grown up, it has become important to improve the uranium utilization and thereby reduce the fuel cycle cost. Spec­tral shift0 l via coolant flow control in BWR (boiling water reactor) is a widely known oper­ating mode that takes advantage of the coolant boiling to improve the uranium utilization. In this mode, the flow rate is reduced from BOC (beginning-of-cycle) to MOC (middle-of-cycle) to facilitate 238U fast fission and plutonium breed­ing under a hard neutron spectrum due to an increased coolant void content. Preserved 235U

and bred plutonium, in turn, are effectively burned at EOC (end-of-cycle) under a soft neu­tron spectrum with an increased flow rate.

The same effects can be achieved without recourse to the coolant flow control. Appro­priate control rod programming could realize a bottom-peaked power shape from BOC to MOC to increase the coolant void content. Resulting bottom-peaked burnup shape at EOC would naturally shift the power shape to the core top and reduce the coolant void content. Jachicc2

J optimized control rod programming to maximize the cycle length, and showed * Ukishima-cho, Kawasaki-ku, Kawasaki 210. ** Shinsugita-cho, Isogo-ku, Yokohama 235.

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that an appreciable extension is attainable along this line. This depletion strategy is sometimes called BS0'1> (burnup shape optimi­zation).

Axially distributed enrichment and burn­able poison of the fuel also have a potential to improve the uranium utilization. Distributed enrichment plays a decisive role in determin­ing the importance and fissile distributions at EOC, which are most responsible for the ura­nium utilization. However, single use of dis­tributed enrichment would fail to reach an optimal EOC state, since increased enrichment results in increased burnup and hence de­creased reactivity at EOC. It is essential to control the power shape during the course of depletion in terms of poison management.

Distributed burnable poison more efficiently controls the power shape than control rod programming, since the control rods are in­serted into the core only from the bottom upward. In addition, optimal control rod programming is generally complicated and hence unfavorable for practical application. It is therefore desirable to minimize control rod pattern change during the cycle, and leave the power shape control to distributed burn­able poison loading.

In this paper, the axial enrichment and burnable poison distributions of the BWR fuel are optimized to maximize the uranium utili­zation within the framework of the axial one-dimensional core model. Control rod pro­gramming is taken into account by assuming a minimum pattern change during the cycle. This is an extension of the previous work' 3>, in which the axial enrichment distribution is optimized under the Haling depletion strate­gyc4>. While the assumption of this strategy enables us to determine the enrichment distri­bution without addressing the poison manage­ment, a constant Haling power shape fails to make use of the coolant boiling as discussed above.

ll. FORMULATION

1. Problem Definition Various objective functions may be defined

as a performance index to the uranium utili-

493

zation. While the EOC kett and the operating cycle length are often used to be maximized, the average enrichment is adopted here to be minimized with the condition that a planned EOC be reached. The uranium saving is calculated directly from it.

When designing a BWR fuel in the axial direction, one has to determine the number of regions into which the fuel is divided, the boundary locations, and the enrichment and burnable poison content of each region. In BWR fuels, gadolinia is used as a lumped burnable poison. Let the fuel be divided into I regions and the region i consist of li nodes, then the decision variables are the enrichment ei and the gadolinia content gi of each region i (i=1, ···,I). Here the axial one-dimensional diffusion calculation is carried out by dividing the core into K nodes of equal thickness.

The operating cycle is divided into ] burn­up steps, where j=l denotes BOC and j=] EOC. In order to minimize control rod pat­tern change, the number of inserted rods is held fixed throughout the cycle, and the depths ri of inserted rods at the burnup steps j U=1, · ·· , ]) are selected as the decision vari­ables. We shall not assume that all the control rods are entirely withdrawn from the core at EOC, though it is customarily assumed.

Constraints are imposed on criticality and axial power peaking, which must be satisfied at each burn up step. While the MCPR c5> (minimum critical power ratio) constraint was imposed in the previous work' 3 >, it is not constrained but the calculated values are only given in this paper. It will be shown that the MCPR and power peaking constraints are redundant, since the MCPR depends on the axial power shape.

2. Gadolinia Poisoning Model It is essential to parametrize gadolinia

poisoning effects by means of least variables in order for our optimization problem to be tractable. The gadolinia poisoning effects are completely determined by the number of poisoned rods per assembly, the gadolinia concentration and the allocation of poisoned rods within the assembly. Although the effect of the allocation cannot be neglected at all,

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it is smaller than those of the remaining two parameters. We therefore selected the num­ber n of poisoned rods and the gadolinia concentration c as the decision variables. Here the variable c should be understood to repre­sent the average gadolinia concentration of n poisoned rods. Consequently, the variable g; introduced in the preceding section to denote the gadolinia content of region i results in a two-component vector g;= (n;, c;)T.

In order to predict the burnup dependence of the gadolinia worth from these two vari­ables, it is convenient to represent the gado­linia worth by an analytic function of burnup. Let it be represented by a quadratic function which smoothly connects with an exponential function at some joint burnup. We determined the coefficient values for a variety of fuels designed so far, and found that they are systematically expressible by these two vari­ables. Physical considerations complete the systematics ; for instance, the gadolinia worth integrated over its life depends solely on the total number nc of gadolinium isotopes initially present. The coolant void dependence of the coefficients is also taken into account.

Various combinations of the number of poisoned rods and the gadolinia concentration will appear in the course of optimization process. Their gadolinia worth is predicted by using the systematics of the coefficients derived above. We have confirmed that the predicted gadolinia worth is well within 1% Llk deviation from the detailed lattice physics calculation. While an optimal solution may result in a fractional number of poisoned rods, it can be realized by suitably allocating the poisoned rods of a next integer number within the assembly.

3. Mathematical Statement The decision variables are summarized in

a vector as

u=(e, ··· e1n, ··· n1c, ··· c1r, ··· rJ)T

=(u1 ••• u,I+J)r. ( 1 )

Using this definition, our optimization problem can be formulated in a mathematical form : Find a vector u that minimizes the objective

]. Nucl. Sci. Techno/.,

function

( 2)

subject to the constraints

..:1./u)=At, i=l, ... 'J ( 3)

Pk/u)-;;;.Pmax, k=l,···, K; i=l, ... 'J ( 4)

rj-;;;.K, i=l, ... 'J ( 5)

ef-;;;.e;-;;;.er, i=l, ···,I. ( 6)

Equality ( 3 ) is the criticality condition, which states that the keff, ..:1./u), should be equal to a target value At. The axial power peak­ing constraint is given by a collection of inequalities ( 4 ), each representing that the power Pk/u) of the node k (k=l, ···, K) should not exceed a limit Pmax· This representation is needed because the power peaking does not necessarily stay at the same node when a decision variable changes its value. Constraint ( 5 ) represents a limitation on the depth of the control rods, which is given in nodes. These constraints must be satisfied at all burnup steps. In constraint ( 6 ), lower and upper bounds are imposed on the enrichment.

4. Solution Method We shall solve the nonlinear programming

problem (1) through (6) by the method devel­oped in the previous work<'l. Following MAP<•l (method of approximation programming)

prescription, nonlinear state variables ).j(u)

and Pk/u) are iteratively linearized, and the resulting LP (linear programming) problem is solved with a goal programming algorithm <7l. The procedure is repeated until the solution converges.

The goal programming algorithm is intro­duced to solve infeasible LP problems in early MAP stages, since it is hard to find a feasible solution in a few shots which satisfies con­straints ( 3) and ( 4) simultaneously. Other­wise, a feasible solution must be separately searched for prior to the optimization calcu­lation. This infeasibility is more serious in the present problem than in the previous one, because ] burnup steps must be considered whereas the assumption of the Haling strategy reduces the number of burnup steps to unity,

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i.e. EOC only. By defining the following linear functions

f/u)=A./u")+(u-u")TV' A./u")-A.t,

}=1, ... 'J ( 7)

g ki(u)= Pkiu")+(u-u")Tfi' Pkiu")- Pmax,

k=1,···,K;j=l,···,], (8)

constraints ( 3 ) and ( 4 ) are expressed as

f/u)=O, i=l, ... 'J ( 9)

k=l, ···, K; j=l, ···, ], (10)

under additional constraints which ensure the validity of these linearity assumptions :

lum-u~l;;;;om, m=l,···,3l+], (11)

where u" stands for a reference state and o is a small variation.

To formulate the linearized problem by means of linear goal programming, we define the deviational variablescn by

A.j=[lf/u)l ±f/u)]/2,

}=l, ... 'J (12)

P:i= [I g kj(u) I± g kj(u)]/2,

k=l, ···, K; j=l, ···, ]. (13)

Then the LP problem in an MAP step is reduced to find the decision variables ( 1 ) as well as the deviational variables (12) and (13) that minimize the augmented objective func­tion

subject to the constraints

A.j-lj=f/u),

i=l, ... 'J (15)

Pti-P"ki=gkj(u),

k=1, ···, K; }=l, ···,] (16)

where

m=l, ··· 3!+], (17)

u~==e~, u~==e~, u~=O, u:;.=+oo,

u~=O, u:;.=K,

m=l, ···,I

m=I+l, ···, 3! m=3I+l, ···, 3!+].

495

Here constraints ( 5 ), ( 6) and (11) are put together into constraint (17). Equations (15) and (16) are derived from respective defini­tions (12) and (13) of the deviational variables.

The weights w A and wp in the augmented objective function (14) must be determined so that the original constraints ( 3 ) and ( 4) be satisfied in a converged solution even if they are not at early MAP stages. We use the inverse sensitivity of the functions fj(u) and g kj(u) with respect to the enrichment vari­ation to equalize the importance of the constraint satisfaction with that of the en­richment reduction. Otherwise, the average enrichment would be unlimitedly reduced at the sacrifice of the constraints. Then they are multiplied by hundreds or thousands to give the weights w A and Wp in order to reach first a feasible solution of the original LP problem.

M. APPLICATION

1. Calculational Assumptions The optimization method is applied to the

initial fuel of 8x8 lattice for a typical BWR/5 described in Table 1. The core is represented in an axial one-dimensional model. A modified one-group diffusion equation coupled with a thermal-hydraulic feedback model is solved using the computer code DIFUSECBl. While the method is applicable to the reload fuel as well, the evaluation of the reload core perform­ance requires a time consuming three-dimen­sional calculation. A rapid calculational model would be necessary to carry out the optimi­zation of the reload fuel.

Table 1 Description of reference BWR/5 core

Electrical output Thermal output Active fuel length Power density

1, 100MWe 3,293 MWth

3. 71 m 50.04 kW /I

Nuclear parameters appearing in the diffu­sion equation are prepared with the lattice physics computer code TGBLA csl for a variety of fuels of different enrichments containing no gadolinia burnable poison. The parameters for arbitrary enrichment are obtained by inter-

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polation. Then the gadolinia poisoning effects are superposed on the resulting kint'S using the method described in Sec. IT -2.

The assumptions made for the optimization calculations are listed in Table 2. The oper­ating cycle length is 10 GWd/t, which is divided into six burnup steps. The core is depleted from the burn up step j to j + 1 with a constant power shape at the step j. The density of inserted control rods is fixed at 10%, which corresponds to an excess reac­tivity of 2% L1k. Remind that the control rods are not necessarily withdrawn from the core at EOC. The lower bound of the en­richment is set to be 0. 711 '% (natural uranium),

but no upper bound is imposed.

Table 2 Calculational assumptions

Number of axial nodes K=24 Operating cycle length 10. 0 GW d/t Number of burnup steps ]= 6

[Cycle burnup=O.O. 0.2, 2.5,5.0, 7.5, 10.0 GW d/t] Inserted control rod density 10. 0% Lower bound of enrichment Channel power peaking R-factor Tails assay

e1= 0. 7ll:Yb 1.4 1. 08 0. 2:Yb

Instead of iterative MCPR calculation, the critical to equilibrium steam quality ratioc•J

]. Nucl. Sci. Techno!,

is calculated for each node and its minimum is used in substitution for MCPR. Assump­tions on the channel power peaking and the R-factorc•J are given in Table 2.

The uranium ore requirement is calculated by assuming a tails assay of 0.2'% enrichment, from which the uranium saving can be obtain­ed by comparing with a reference fuel to be defined. Since, however, the initial fuels are discharged from the core after first through third cycles, a rigorous uranium utilization should be determined by the average discharge burnup. The uranium utilization calculated from the enrichment requirement as defined above only gives a performance index to the optimization problem.

2. 2-region Fuel As the simplest design, a two-region fuel

divided by the mid plane is optimized by varying the value of Pmax· This division is chosen not because it is optimal but because it simplifies the analysis of the solutions. Results are summarized in Table 3, in which the average and difference are given for the enrichment, the number of poisoned rods and the gadolinia concentration. Here the differ­ence is defined as the value in the top half less that in the bottom half. Also given in the table is the minimum MCPR during the cycle.

Table 3 Results of two-region analysis

Maximum Enrichment Number of Gadolinia

power poisoned rods concentration Uranium

peaking (%) (%) saving

Pmax AvY Dif_t2 Av. Dif. Av. Dif. (%)

1. 2214 2.046 3.12 3.74

1.2 2.046 +0.154 3.12 -0.11 3.74 -0.38 0.0

1.4 2.013 +0.140 3.05 +0.23 3.32 -0.75 1.8

1.7 1. 969 +0.127 2.94 +0.80 2.95 -1.02 4.2

2.0 1. 932 +0.107 2.89 +1.24 2.42 -1.21 6.2

+oot5 1. 823 +0. 226 3.27 +3.88 0.95 -0.88 12.1

tl Average of top and bottom halves. t 2 Difference between top and bottom halves. t 3 Minimum MCPR during the cycle, which appears at EOC for all the optimal solutions. t 4 Uniform fuel depleted with Haling power shape. t 5 Maximum axial power peaking of 2.8 appears at EOC.

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Minimum critical power ratiot•

1. 30

1.29

1. 27

1. 21

1.16

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As a reference, the enrichment of a uni­form fuel is determined by depleting with a Haling power shape. The gadolinia content is assumed to be the same as the average content of the optimal solution with Pmax=L2. The uranium saving, which is calculated by taking the enrichment of this uniform fuel as a base, increases linearly with increasing

Pmax• The burnup process is simply depicted in

Fig. 1, which shows the relationship of the average burnup between the top and bottom halves. Each closed circle represents the burnup shape. The slope of an arrow which connects two circles shows the power shape at the starting point of the arrow. Figure 1 indicates that the power shape changes from bottom to top peaking as burnup proceeds, and that this tendency is more remarkable for larger Pmax· Detailed EOC burnup and power distributions are shown in Fig. 2.

Top

~ ~ 10

~ a. :I c .. :I lXI

a. 5 ~

Fig. 1

'""' ""'· -~ycle Burn~. "'-.(GWd/t) ~max

"' 10.0 . 1.2 -~ 1.4 1.7

+CII . 2.0

497

10 15 Bottom Burnup (GWd/t)

Average burn ups of top and bottom halves for two-region fuels (solid arrows) and for uniform fuel depleted with Haling strategy (broken arrows)

Bottom~--~~--~~~~ ~~~~~--~--~~--~----------~ 0 2 3

Burnup (GWd/t) Relative Power

Fig. 2 Axial burnup and power distributions at EOC for two-region fuels (solid lines) and uniform fuel depleted with Haling strategy (broken lines)

When Pmax=L2, the differences of the enrichment and the number of poisoned rods are positive and negative, respectively. They flatten the power shape throughout the cyclec10' rather than reduce the enrichment require­ment. The positive enrichment difference shifts the EOC power shape to the core top more than the Haling one in spite of flatter burnup shape. As Pmax increases, the differ-

ence of the number of poisoned rods turns from negative to positive so that the bottom of the fuel be sufficiently burned with decreasing but even positive enrichment dif­ferencec11). When the power peaking is uncon­strained, a great uranium saving of 12.1% is achieved with an extreme top peaking of 2.8. This power peaking depends more largely on the enrichment difference than on the burnup

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shape. The gadolinia concentration difference is negative in all the solutions, and decreases with decreasing burnup difference.

Figure 3 shows how the kin£ is controlled by the poison management during the cycle. In the figure, open circles represent the un­controlled average kint's of the top and bot­tom halves, which are controlled by gadolinia and control rods as is shown by the dashed lines and then result in the controlled kint'S shown by closed circles. When Pmax=l.4, the kin£ is uniformly controlled by both gado­linia and control rods. The amount of the kin£ controlled by the control rods is almost constant during the cycle, which indicates a constant excess reactivity. The controlled kin£ does· not move appreciably in the figure, resulting in a small change of the power shape during the cycle. When the power peaking is unconstrained, the gadolinia mainly controls the top kin£, while the control rods uniformly control the kin£· The controlled kin£ indicates that the power shape changes markedly as burnup proceeds.

!' 'E

~ .:ic Q.

1.2,....--~--.---~- --,--~----,r---,

2.5

1.1

0.2 Cycle Burnup IGWd/t}

7.5

0.2

' i I I ' f Gadolinia

'f I ,/' i ·'? 1.0 I

/'' Control Rods

Pmax=1.4 Pmax=+=

0.9!-=---~--:-L;--~- -~-~---:-'':---~ 1.0 1.1 1.0 1.1

Bottom k-infinity

Fig. 3 Average controlled (closed circles) and un­controlled (open circles) kint'S of top and bottom halves for typical two-region fuels

The minimum MCPR during the cycle, which appears at EOC for all the solutions, shows strong negative correlation with Pmax, since the top-peaked power shape reduces the MCPR. The MCPR constraint can therefore be replaced by the power peaking constraint,

]. Nucl. Sci. Techno!.,

and would bring us no other information. Criticality and power peaking constraints

are not imposed at BOC in these optimization calculations. At BOC of all the optimal solutions, the core is subcritical and the power shape is bottom peaked beyond Pmax by in­serting the control rods as deeply as the EOC power peaking allows. At EOC, the control rods are entirely withdrawn from the core as expected. These results at BOC and EOC prove that our optimization method works successfully.

3. 24-region Fuel Optimization of a 24-region fuel (/=24), in

which the enrichment and gadolinia content of each node can differ from one another, is tried in this section. While such a fuel is hardly manufactured, it tells us an upper bound of the uranium saving attainable by means of enrichment and gadolinia distribu­tions.

The optimal solution with Pmax=L4 is shown in Fig. 4. The average enrichment is 1.851%, which amounts to 10.6% uranium saving. The minimum MCPR of 1.24 appears at EOC. It is characteristic that natural uranium is implemented at the top two nodes and the bottom one node, and that no gado­linia is constained at the top four nodes and the bottom two nodes.

Figure 5 shows the distributions for the power, the coolant void fraction and the control rod density at BOC (0.2 GWd;t), MOC and EOC. The distributions are also given at EOC for the burnup, the historical void fraction and the historical control rod density, where "historical" means "burnup-integrated". The regions adjacent to natural uranium blankets are most depleted at BOC, and the core bottom is depleted at BOC through MOC. As a consequence, the EOC burnup shape is bottom peaked and the power shape is top peaked within the power peaking limit. It can be viewed as another proof of the success of the present optimization method that the EOC powers reach the limit Pmax over the one-third of the core. A bimodal power shape of the bottom at BOC results from many poisoned rods at the nodes where the power

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Top

... ..c: Cl

·a; :I:

Cl>

0 u

Bottom L---'---:'---:--~ 0 1 3 0 2 4 6

Number of Poisoned Rods

8 0 2 4 6 Enrichment

(wt%)

Gadolinia Concentration (wt%)

Fig. 4 Optimal enrichment and gadolinia distributions for 24-region fuel

BOC (0.2GWd/t) MOC (5.0GWd/t) EOC (10.0GWd/t)

Top

,., I I I' I '1 I I I I I I ... I I

I ..c: I ' ,, I Cl I I

·a; 1 I I I I I

:I: I I I I I I I I I

Cl> I I I

0 I I I

I 1 / u 1 I I -, 1 ' Ll 1 I

1 ,1 /

1,' /I \

I \ r

I I _____________ ... ,'

Bottom 1

' 0.0 0.5 1.0 1.4

Relative Power /Void Fraction

Fig. 5 Axial distributions for power, coolant void fraction and control rod density of 24-region fuel at BOC, MOC and EOC (Axial distributions for burnup, historical void fraction and historical control rod density are given by broken lines at EOC)

499

is depressed. If the number of poisoned rods at these nodes is reduced, the MOC power peaking would exceed the limit.

The simplest design derivable from Fig. 4 may be the blanketed fuel which implements natural uranium at the top two nodes and the bottom one node. The enrichment of central region is determined by depleting with a Haling power shape. Here no gadolinia is contained in the blanket regions, but the total gadolinia content is the same as that of the

reference uniform fuel. The average enrich­ment results in 1.947~, which saves 5.4% in uranium ore. Note that this saving amounts to a half of that obtained by the optimal solution of the 24-region fuel.

IV. CONCLUSION

The axial enrichment and gadolinia distri­butions of the BWR fuel are optimized under control rod programming to improve the ura­nium utilization. The optimization method is

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based on MAP, and the goal programming algorithm is introduced to solve infeasible LP problems in early MAP stages. The method is successfully applied to the initial fuel divided into 2- and 24-regions to minimize the enrichment requirement subject to the power peaking constraint.

Through the two-region analysis, it is concluded that the core bottom should be depleted during the cycle as much as the power peaking constraint allows so that the EOC power shape shifts to the core top. This indicates that what is called BSO is the optimal depletion strategy. A positive en­richment difference coupled with a positive difference of the number of poisoned rods is effective to maximize EOC top peaking within a limit. This is entirely different from the solution under the Haling strategy, which showed that the negative enrichment differ­ence is optimal<•l.

Optimization of the 24-region fuel with Pmax=L4 shows that the uranium saving attainable by means of axial enrichment and gadolinia distributions amounts to 10.6%. It is shown that half the saving comes from natural uranium blanket implementation. To put this blanketed fuel to practical use, the central enriched region should be divided into at least two regions to satisfy the power peaking constraint. Further improvement is achievable by reducing the gadolinia content at the regions adjacent to the blankets.

The purpose of this paper is to study the optimal axial enrichment and gadolinia distri­butions that maximize the uranium utilization. Much more constraints must be considered to design a fuel for an operating nuclear power plant. The cold shutdown margin, which is inexpressible by the axial one-dimensional core model, is one of the most important criteria and should be noted since the EOC top peaking of the optimal solutions reduces the margin. Note that the cold shutdown

]. Nucl. Sci. Techno!.,

margin depends more on the fuel loading pattern than on the axial power shape, and its evaluation requires a three-dimensional calculation.

ACKNOWLEDGMENT

The authors would like to thank Dr. S. Kusuno of Nippon Atomic Industry Group Co., Ltd. (NAIG) for his encouragement and review of the manuscript, Mr. M. Yamamoto of NAIG for his helpful discussion, and Mr. A. Tanabe of Toshiba Corp. for his interest and support throughout this work.

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