Parametrization for optimization of reload patterns for boiling water reactors

Ann. Nucl. Eneryy, Vol. 20, No. 4, pp. 237-249, 1993 Printed in Great Britain. All rights reserved

0306-4549/93 $6.00+0.00 Copyright © 1993 Pergamon Press Ltd

P A R A M E T R I Z A T I O N F O R O P T I M I Z A T I O N OF R E L O A D PATTERNS FOR BOILING WATER REACTORS D. P. BURTE a n d S. G. VAIDYA Theoretical Physics Division, Bhabha Atomic Research Center, Trombay, Bombay, India (Received 16 September 1992)

Abstract--Parametrization of the reload patterns for BWRs is attempted with the aim of optimizing them. This is done in two stages. The first stage involves an algorithm for the construction of a reload pattern out of a given set of fuel bundles. It is designed to construct patterns which feature low leakage loading (LLL) and a chequerboard arrangement of fresh and exposed fuel bundles in the central region. The characteristics of the reload patterns can be manipulated by means of only two input parameters. The dependence of the Haling power peaking and the cycle energy of the "biparametric reload patterns (BRP)" on the two input parameters is shown through case studies. The ranges of these characteristic quantities and their mutual relation are given by case studies for both general as well as the optimum BRPs. A preliminary optimization procedure for BRPs is presented. This method is augmented by a multi-parametric algorithm to reshuffle the radial exposure distribution interactively so as to exhaust any possibility of improvement in a given reload pattern. When tested against this procedure, the optimum BRPs are seen to have only a limited scope for improvement, showing that they are very close to the optimum reload patterns. The extent of possible improvement is illustrated. The entire procedure is incorporated in a 2-D code CORECOOK. The computer time needed for optimization of a reload pattern is comparable to that required for one routine 3-D core followup calculation. This algorithm has been used for obtaining the reload patterns for all five cores in the BWRs at Tarapur (India) since 1988. i. INTRODUCTION

very clear t h a t a higher value o f Ken at a given core b u r n u p , a higher cycle energy, a higher fuel discharge b u r n u p a n d better fuel utilization are all mutually consistent. The low leakage loading (LLL) concept aimed at m i n i m i z a t i o n o f n e u t r o n leakage is consistent with the m a x i m i z a t i o n o f cycle energy ( D o w n a r a n d Sesonske, 1988). Thus, the concepts discussed above, viz. lower enrichments, higher cycle energy, higher fuel discharge b u r n u p , higher fuel utilization a n d the L L L concept are all m u t u a l l y consistent. According to D o w n a r (1986), L L L also provides accelerated fuel depreciation a n d quick returns on the fuel investment. The H a l i n g power peaking is a n o t h e r i m p o r t a n t q u a n i t y which is a function of a given reload pattern. Its value is required to be low in the interests of fuel integrity in steady-state operation. This requirement, however, is inconsistent with the m u t u a l l y consistent functions a n d their requirements considered above. Thus, we have two mutually inconsistent requirements. Therefore, one o f the two has to be optimized While the o t h e r has to be constrained. M a x i m i z a t i o n is consistent with applying a lower b o u n d a n d vice versa. Thus, one m a y maximize cycle energy or p u t a lower b o u n d o n it if one wants to c o n s t r a i n it. Similarly, one m a y either minimize Haling power peaking or p u t a n u p p e r limit on it. G i v e n a set o f values for the decision variables, a calculational model is required to evaluate the objec-

The design o f a reload p a t t e r n involves decisions regarding the assignment of fuel bundles to locations in a core. A reload p a t t e r n for a reactor core which has n fuel locations involves n such decisions. Thus, there are n decision variables for a reload pattern. The p r o b l e m o f o p t i m i z a t i o n o f a reload p a t t e r n for such a core involves m a k i n g these decisions so as to extremize a c h o s e n objective function u n d e r the given constraint(s). T h e r e can be several candidates for an objective function. It could be, for example, m i n i m i z a t i o n o f the core e n r i c h m e n t or n e u t r o n leakage. Alternatively, it could also be m a x i m i z a t i o n o f the cycle energy or the K~n at the assumed end o f cycle ( E O C ) or fuel discharge b u r n u p . One m a y note t h a t these extremizations o f the respective objective functions are consistent with each other. All o f them cater to efficient fuel utilization a n d reduction o f the n e u t r o n fluence in the reactor vessel. Thus, Suzuki a n d Kiyose (1971a) have d e t e r m i n e d that m i n i m u m core enrichm e n t c a n be o b t a i n e d for the reload core configuration by maximizing the E O C core Ken. H u a n g a n d Levine (1978) also have s h o w n t h a t " i f the mechanical design of the fuel assembly remains fixed, the m i n i m u m core e n r i c h m e n t can be o b t a i n e d for the reload core configuration by maximizing the E O C core K~fr". It is 237

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D.P. BURTEand S. G. VAIDYA

tive function as well as the functions to be constrained. In the case of a reload pattern, we require knowledge of the nuclear properties of each of the n fuel bundles and a model to calculate these functions. The cycle energy and the Haling power peaking factor for the given reload pattern is evaluated by performing Haling calculation. If we assume that each of the n fuel bundles is different then we have n! possible reload patterns. If we assume that all the fresh bundles are identical then this number reduces to n!/r!, where r is the number of fresh bundles. This is still an enormously large number. It is impracticable to constitute and evaluate so many reload patterns before selecting the optimum one. The solution to this problem calls for a threepronged approach, viz. the use of standard optimization methods, the use of fast models for performing the Haling calculations and a reduction in the number of decision variables. The major portion of the work directed at solving this problem, to date, has been on the use of optimization methods. They involve heuristic methods as well as non-linear programming and linear programming methods. Non-linear programming usually involves the use of second derivatives of the system equations along with some sort of searching technique. In this approach the original system and the constraint equations are preserved, but are solved by using algorithms that are much less efficient and less well-proven than those for linear programming. Hoshino (1978) applied a non-linear programming method to the multi-cycle optimization problem with apparent success (Downar and Sesonskc, 1988). Linear programming uses first derivatives relating the decision variable to the optimization function to improve the next decision. This is obtained from the results of previous iterations. The principal problem encountered with linear programming is the inability to distinguish a local optimum from the global optimum, since the algorithm employs only the first derivatives of the objective functions. Therefore, different starting points for the algorithm will sometimes result in different optimum solutions. Suzuki and Kiyose (1971b) applied linear programming to determine the optimum refuelling schemes for LWRs described by very simple core models. The optimization methods employing simulated annealing techniques significantly reduce the probability of getting trapped in a local minimum and allow one the freedom to start from any initial configuration. Kropaczek and Turinsky (1991) combine this optimization technique with a computationally efficient core physics model based on second-order-accurate generalized perturbation theory. Perturbation theory

is also used by Mingle (1975). Dynamic programming reduces an N-dimensional problem to N 1-D problems. It uses the "principle of optimality" (Bellman, 1957) to guide the search process by noting that whatever the initial decision, the resulting decisions must constitute an optimal policy with regard to the state resulting from the initial decision. This technique was successfully used by Civita and Fornaciari (1972) for the multi-cycle optimization problem. Stout and Robinson (1973) also have used dynamic programming for this purpose. Variational methods are used to obtain the derivatives more easily. Integer programming methods are used, since the decisions involved are regarding the assignment of fuel bundles to locations and both of them can be represented by integers. Apart from how one constructs the successive reload patterns, faster models for evaluating them are desirable. This can be achieved by performing Haling calculations in 2-D instead of 3-D and also by using the symmetries of the reload patterns if there are any. The number of possible reload patterns and the complexity of the problem increases rapidly with the number of decision variables. Therefore, a reduction in this number is highly desirable. This makes application of the optimization methods to the problem practical. The reduction in the number of decision variables depends upon the scope of the fuel management problem to be solved. Many authors have chosen to study less ambitious and simpler problems, e.g. the problems in optimization of a reload pattern with only two or three regions (Downar, 1987 ; Suzuki and Kiyose, 1971a,b ; Ahn and Levine, 1985 ; Matoda, 1971 ; Sauar, 1971). This could be a study in itself to gain insight into the problem. It could also be the first stage of a multi-stage process to tackle the actual problem. The size of the problem of constructing one reload pattern can also be reduced by deciding to follow some kind of symmetry in the core. Thus, in the traditionally followed quadrant symmetric cores (QSC), the number of fuel locations reduces by a factor of 4. In the sector symmetric cores (SSC) suggested by Burte and Vaidya (1986) a "sector" contains more than one-quarter the number of bundles in the core, because here the fuel bundles appear on the axes of symmetry and they are counted in both sectors sharing a symmetry axis. It is enough to solve the problem of the reload pattern for such a symmetric part, say a "quadrant" or a "sector", of the core. This solution can easily be expanded to the solution for the full core by use of the assumed symmetry. One can refer to many other works reported in this field. Notable amongst them are Motoda et al. (1975), Wall and Fenech (1965), Melice (1969), Suh and

Optimization of reload patterns for BWRs Levine (1990), Morita et al. (1986), Kim et al. (1987), Ho and Sesonske (1982) and Galperin and Kimhy (1991). The problem is still considered to be a challenge (White and Avila, 1990). Tahara et al. (1991) have recently proposed a "computer aided system for generating fuel shuffling configurations based on knowledge engineering". According to them, "the conventional way of generating an appropriate fuel loading pattern is a method of trial and error, which is laborious and requires much computer time". Downar and Sesonske (1988) said the following in their review article with regard to recent methods to optimize the reload patterns which had been modified using LLL strategies and the use of burnable poison : "While overcoming some of the shortcomings of previous techniques, these methods are still being tested and have yet to receive widespread acceptance for actual core reload design."

2. T H E APPROACH

The emphasis here is on reducing the decision variables without compromising the complexities of the problem of optimization. We attempt to achieve this by incorporating the traditional guidelines. 2.1. Guidelines on optimization

Normally the fuel bundle locations are arranged periodically with constant pitch. This means that the fuel bundles are uniformly distributed throughout the core. In other words, the "fuel density" is uniform throughout the core. The optimization is to be achieved by the exposure distribution. The relevant guidelines are in the form of the ouWin and in-out schemes ; these refer to the direction in which the exposures increase as one goes inwards, i.e. away from the periphery, or outwards, i.e. towards the periphery, respectively. The former generally leads to power flattening at the expense of cycle energy and vice versa in the latter case. An application of a combination of these two guidelines in the central (or non-peripheral) and peripheral region, respectively, leads to what is called the low leakage loading (LLL) guideline. The power density in the peripheral region is expected to be quite low, in view of the leakage of neutrons outside the core. In the non-peripheral region, however, it tends to peak. LLL aims to flatten the power where it tends to peak by increasing the exposure towards the centre and reduce the leakage of neutrons outside and, thus, contributes to the cycle energy by increasing the exposures towards the periphery.

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2.2. The case o f two categories o f bundles--chequerboard arrangement

Guidelines are always to be used with discretion. The guidelines on the radial distribution of fuel bundles in the form of the in-out, out-in schemes are no exception. Loading all the fresh bundles, whether in the central or peripheral region depending upon the in-out or out-in scheme, may lead to undesirably large power peaking or large neutron leakage. As a part of such discretion it is advisable to categorize the fuel bundles according to their important nuclear properties and intersperse them with each other to soften the effects of these properties. Thus, the power densities of bundles from the category with higher reactivity will be reduced by loading fuel bundles from the category with lower reactivity near them. The power densities in the latter bundles will be increased due to that of the former bundles. This is desirable. We have seen, irrespective of the categorization of the fuel bundles, that they are distributed uniformly with constant pitch. It is desirable to apply a similar consideration to the distribution of the fuel bundles within each category also. The nuclear properties depend mainly upon the fuel exposure and their design. It is common to categorize fuel bundles by the number of cycles they have experienced. However, the group of fresh bundles stands clearly apart on the exposure histogram of the fuel bundles to be loaded in a core. Therefore, division of the fuel bundles into the categories of fresh and exposed fuel bundles is appropriate. We can recognize that the traditionally recommended chequerboard arrangement of fresh and exposed fuel bundles indeed distributes the bundles of each of the two categories as desired. The area over which the chequerboard arrangement is possible is determined by the size of the smaller group of bundles, which normally is the fresh bundles. It is possible to cover a larger area of the core by chequerboard arrangement if fewer batch refuelling is followed. The LLL philosophy requires that the fresh bundles be loaded away from the periphery of the core. This means that the chequerboard arrangement should be employed in the centralmost region of the core.

2.3. Cylindrical cores and azimuthal symmetry

Reactor cores are generally cylindrical. Though the problem of the reload pattern appears to be a 3-D one there is no freedom to shuffle the fuel axially within a fuel bundle. Therefore, the z-axis cannot be considered in this problem; only the radial and not the

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azimuthal dimension is relevant to the exposure distribution schemes. The radial distribution of fuel properties has to be equally good for each value of the azimuthal coordinate and hence has to be azimuthally symmetric. Strict azimuthal symmetry is possible only if the reactor is a perfect cylinder and can be divided into annular cylinders of fuel with uniform properties. It also requires that it should be possible to manipulate the properties of these annular cylinders as per the requirements of the optimization. This is not possible. At best one can hope to design reload patterns in such a way that the relevant quantities are independent of the azimuthal angle. Secondly, it is only an approximation that the reactors are considered to be cylindrical. Therefore, we have to consider the importance of a particular location which may depend not only upon its distance from the centre but also on its distance from the periphery. Therefore, we have to sequence the fuel locations as a function of this importance rather than as a function of radius. In fact, we suggest that this importance can be determined by calculating the power distribution in a core which uses identical fuel bundles. Although strictly we may use this importance of a location instead of its radius, for convenience of discussion we will take the liberty of using phrases such as radial distribution. 2.3.1. Quadrant symmetric core (QSC). The traditional reload patterns settle for the next best to azimuthal symmetry, i.e. quarter-core symmetry. The traditionally chosen axes of reflective symmetry run parallel to the blades of the cruciform control rods. This symmetry may be called "quadrant symmetry" and a core loaded following this symmetry is a QSC. Figure 1(a) illustrates a QSC. Many BWRs, including those at Tarapur Atomic Power Station (TAPS), have a control rod at the centre of the core. They have even numbers of rows and columns of fuel bundles. In this type of reactor the axes of symmetry, as per the QSC, pass between the rows and columns of fuel bundles. This has two effects :

(l) If the neighbouring bundles are symmetrical to each other, their properties are similar; otherwise they can be different. Thus, even for nearly the same radial distance, the properties of neighbouring fuel bundles can be similar or dissimilar. This brings the azimuthal coordinate into the picture; the possibility of a reduction in the number of decision variables on account of azimuthal symmetry is therefore ruled out.

(2) It causes a violation of the chequerboard arrangement of fresh and exposed fuel bundles in this type of core. We find that the symmetric locations in the region which is nearest to the centre of the core are closer to each other. Therefore, loading a fresh bundle near to the centre brings fresh bundles closer to each other in the central region. This leads to an increase in the Haling power peaking. Therefore, fresh bundles cannot be loaded in the central region. They cannot be loaded close to the periphery, as this will increase the neutron leakage and lead to a reduction in the cycle energy. Therefore, they may only be loaded in the ring-like annular region. Thus, instead of dividing the core into only peripheral and non-peripheral regions, such a core has to be divided into inner, outer and annular regions. This also increases the number of decision variables. 2.3.2. Sector symmetric core (SSC). The SSC concept, illustrated in Fig. 1(b), involves a new set of axes of reflective symmetry for loading the fuel. These axes are obtained by rotating the axes of reflective symmetry used in the QSC through 45 '~. Now the axes of reflective symmetry pass through bundles (diagonally), rather than between pairs of bundles as in quadrant symmetry. The bundles through which the axes of symmetry pass become their own symmetries. Reflective symmetry with this choice of axes may be called sector symmetry. A core following this symmetry may be called an SSC for convenience. The SSC was introduced for a different purpose (Burte and Vaidya, 1986 ; Burte, 1990), viz. for mitigation of the problem of large reactivity worths of control rods. It was also shown that the SSC is either superior or equivalent to the QSC in other respects. Its characteristics are useful for our present purpose also. The SSC is seen to accommodate the chequerboard arrangement of fresh and exposed fuel bundles rigorously. There is no dependence on the azimuthal coordinate. The number of regions can be two only : the peripheral and non-peripheral. Thus, the SSC satisfies the azimuthal symmetry as well as the uniform distribution of the pitch of the fuel bundles of each category in each of the two regions. This simplicity helps to reduce the number of decision variables for construction of the SSC and LLL reload patterns. 3. BIPARAMETRIC RELOAD PATTERNS (BRPs)

Our approach to the construction of reload patterns is based upon the traditional insistence, for good reasons, on the chequerboard arrangement of fresh and exposed fuel bundles and optimization based upon traditional guidelines, such as LLL on the radial

Optimization of reload patterns for BWRs

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Fig. l(a). Exposure (GWD/STU) map ofa QSC reload pattern (Unit-I BOC-10 of TAPS).

distribution of the fuel exposures. The SSC concept is chosen for this approach since it suits the purpose well. It turns out that this approach helps reduce the number of decision variables to a mere two. The full T A P S core contains 284 fuel locations and its sector contains 78 o f these. We will consider the problem of assigning only the 78 fuel bundles which are properly chosen out of the given 284 fuel bundles to the 78 locations as far as the construction is concerned, and assume that we have the algorithm to work out the full SSC reload pattern using the available 284 fuel bundles. The approach is not restricted to only the T A P S cores. Application o f the LLL scheme requires the delineation of the boundary between the peripheral and non-peripheral regions of the core. The extent of the chequerboard arrangement of fresh and exposed fuel bundles in the centralmost region is an important consideration regarding the boundary between the peripheral and non-peripheral regions. In fact, for a three-batch refuelling of T A P S reactors we have chosen the chequerboard region itself as the non-

peripheral region. Also note that the chequerboard arrangement decides the positions of the fresh bundles, except the trivial decision whether the fresh ones go into white or black locations. Having thus decided the boundary of the non-peripheral region and the positions of the fresh bundles, what remains to be decided is the distribution of the exposed fuel bundles. The philosophy of LLL implies only qualitatively that the exposure distribution will be a valley along a boundary and the exposures will rise as we move away from this boundary either towards the centre or towards the periphery of the core. In this section we will first describe a procedure to construct a reload pattern based only on these guidelines. Then we will turn to reducing the remaining number decision variables to only two. The next task is to use these two parameters to find an optimum BRP.

3.1. SSC mapping procedure We will first describe our procedure to map the fuel bundles on the fuel locations. F o r this purpose, the n

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5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 Fig. l(b). Exposure (GWD/STU) map of an SSC reload pattern (Unit-2 BOC-11 of TAPS).

fuel bundles are arranged in an array. On the other hand, importance numbers are assigned to the n fuel locations of the core to rank their importance. The importance numbers normally increase as we go to radially more distant fuel locations. The core is divided into two regions, the peripheral and nonperipheral. The non-peripheral region comprises a given number of the fuel locations of highest importance, while the peripheral region comprises the remaining locations of least importance. Let L represent the number of fuel locations in the non-peripheral region and n - L , that in the peripheral region. We intend to follow the chequerboard arrangement of fuel locations in the non-peripheral region. Therefore, we categorize the fuel locations in this region into black and white categories, as in a chequerboard. We assume that the first L/2 bundles belong to one category and the subsequent L/2 bundles belong to another category. The bundles of these categories are to be arranged according to the chequerboard arrangement in the non-peripheral region. For com-

pleteness of the description, let us call the category of the fuel locations in the peripheral region the grey category. Now we introduce a procedure to assign these bundles, referred to by their sequence numbers in the array, to the fuel locations, referred to by their importance numbers. We take the fuel bundles sequentially from their array and assign them to the fuel locations as follows. We assign the first L/2 bundles to the locations belonging to say, the white category, starting with the location of highest importance and proceeding to those of lower importance. We assign the next L/2 fuel bundles to the fuel locations belonging to the black category, starting from the location with the least importance and proceeding towards those of higher importance. We assign sequentially the remaining n - L fuel bundles which appear last in the array of fuel to the grey locations, i.e. the locations in the peripheral region, starting from the location of highest importance and proceeding to those of lower importance.

Optimization of reload patterns for BWRs By the chequerboard arrangement we normally mean a chequerboard arrangement of fresh and exposed bundles. In order that all the fresh bundles, say r in number, are covered by this chequerboard arrangement we must have L = 2r or L > 2r. We take 2r as the default value of L. We assume for our discussion that this default value is always used. The appearance of fresh bundles at the beginning of the array is convenient and suits our purpose and procedure. The division of the core into the peripheral and non-peripheral regions thus turns out to be based upon the number of fresh fuel bundles; normally L < n. It is because of this that we can divide the core into peripheral and non-peripheral regions. 3.2. The parameterless or basic L L L S S C reload pattern Given an array of fuel bundles, the above procedure yields a unique SSC reload pattern in which the fresh and exposed fuel bundles are arranged in a chequerboard pattern in the non-peripheral region and only the exposed fuel bundles are loaded in the peripheral region. There are no decision variables involved. The radial exposure distribution of the exposed bundles depends entirely upon how they are ordered in an array which is to be mapped on the fuel locations according to the mapping procedure described above. Let us define "basic array" as the ordered array of the n fuel bundles whcih starts with all the (r number of) fresh fuel bundles followed by the exposed fuel bundles arranged according to their increasing exposures. The mapping procedure operating upon the basic array of a given set of fuel bundles results in a unique reload pattern, whose exposure distribution follows the LLL guidelines as described below. Therefore, this reload pattern will be called the basic SSC LLL or simply the "basic" reload pattern. The fresh bundles in the basic SSC LLL pattern are loaded only in the central or non-peripheral region and they are arranged according to the chequerboard arrangement with the exposed fuel bundles. We note the peculiarities of this pattern with reference to their expected effects on the Haling power peaking factor and cycle energy : (a) The non-peripheral region contains the least exposed of the fuel bundles. This is expected to enhance the power peaking in the central region. (b) Fresh bundles are supposed to be loaded in the white locations in the non-peripheral region. This is expected to increase the Haling power peaking in the central region. However, following the out-in scheme for assigning the

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exposed bundles to the black locations is expected to reduce the Haling power peaking in the central region. (c) The leakage of neutrons in the peripheral region already reduces the power factors there. Therefore, we need not apply the out-in scheme in this region. On the other hand, we stand to gain some cycle energy by applying the in-out scheme in the peripheral region. Therefore, the exposed bundles are arranged with their exposures increasing outwards. The exposures of the consecutive fuel bundles in the basic array vary by small amounts. This leads to rather a graded radial variation except at the boundary between the peripheral and non-peripheral region. Observations (a) and (c) show the in-out character. On the other hand, observation (b) points to the o u t in character of the basic pattern. However, if all the fresh bundles are assumed to be identical, their distribution is neutral regarding the in-out or out-in character. With these balancing effects, on the whole, the power peaking for the basic pattern is rather high and occurs in the centre and the power factor slowly reduces outwards. 3.3. Biparametric modification The basic SSC LLL pattern is unique with its own Haling power peaking. The probability of it being the reload pattern which fulfils the particular requirements of the problem of optimization is insignificant. This obviously calls for the introduction of some flexibility into the algorithm. In order to clarify our ideas about the nature of the required flexibility we note that, as mentioned above and also as per our experience, the basic SSC LLL pattern is biased towards a high value of the Haling power peaking. Therefore, the flexibility in the algorithm should make it possible to modify the basic pattern, so as to reduce the Haling power peaking to the desired extent while obtaining the optimum cycle energy corresponding to such power peaking. We attempt to achieve such flexibility by introducing some parameters which modify the basic array via a suitable procedure. These parameters, in fact, are the decision variables and their number should be kept as low as possible. If the Haling power peaking factor exceeds the allowed value, the bundle where the power peaks may not be the only one where the limit is exceeded. There may be more bundles which cross it and a parameter has to be provided in the algorithm to specify the number of such bundles. We may assume that it is

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the centralmost region where the powers are to be reduced. Therefore, the number of such bundles is related to the extent of the central region where it is desired to reduce the power factors. We also need to specify the extent to which it is desirable to reduce the power factors. This specification also requires a minimum of one parameter. In order to reflect these two requirements, we introduce only the minimum required number of decision variables. We have only two input parameters, which may be referred to as M and N ; hence the name biparametric. The effect is achieved only by modifying the basic array as per the input parameters M and N as follows : slice off a segment, say S1, of the N bundles starting from the serial number L - M + 1 ; push upwards the segment, say $2, which comprises the subsequent M bundles so that they are placed immediately after the position L - M in the array; plug in the segment SI subsequent to segment $2 which has already been pushed up as described above. This modification affects the order in the array of only the bundles whose sequence numbers are from L - - M + 1 to L + N inclusive. Figure 2 illustrates the modification. Here the bundle with L as its serial number is identified as " h " . The illustrative values of M and N are 8 and 5, respectively. The bundle whose serial number is L - - M + 1 is " a " . We see that the 5 bundles a, b, c, d and e form the segment S1 which is sliced off. The next 8 bundles f, g, h, i, j, k, 1 and m form segment $2. In the unmodified array Sl is followed by $2. In the modified array, $2 is followed by SI without disturbing the internal order of these segments. Mapping this modified array on the core using the

same mapping procedure as described above yields a modified reload pattern. Its exposure distribution is found to be modified according to the desired bias; the exposures in the central region are increased. The size of this region is governed by M and the size of the increase in the exposures is governed by the parameter N. We now consider the number of possible permutations of these two parameters. The two decision variables M and N in the biparametric procedure take non-negative integer values. We note here that if any one of the two parameters is zero, the biparametric modification leads back to the basic pattern irrespective of a non-zero value of the other parameter. Therefore, the two decision variables M and N take only positive integer values for modifying the basic pattern according to the biparametric procedure. Assuming the default value 2r for L, the parameters M and N can take only r and n - 2r non-zero values, respectively. Thus, the total number of possible BRPs become only r(n--2r) apart from the basic pattern. Thus, the biparametric algorithm reduces the number of possible reload patterns from n!/r! to only r(n-2r) apart from the basic pattern.

3.4. Biparametric optimization BRPs have the following features : (i) They are azimuthally symmetric as far as possible. (ii) Fresh bundles are loaded in the centralmost region of the core along with the non-fresh bundles according to the chequerboard arrangement.

Optimization of reload patterns for BWRs (iii) The non-fresh bundles in this central chequerboard region are loaded according to the outin scheme. In the remaining peripheral region non-fresh bundles are loaded according to the in out scheme. These indeed are the guidelines traditionally prescribed for optimization of reload patterns. One would expect to find the optimum reload pattern among the patterns following these guidelines. Since the BRPs follow these guidelines systematically, their study is expected to throw light on the reload patterns following the above guidelines. It has been indicated in the above that the total number of BRPs is not enormously large. Constructing and analysing them even exhaustively appears practicable. The computer code CORECOOK was written specifically for working out reload patterns using the biparametric algorithm. Given the input values for the parameters M and N, the code constructs the corresponding reload pattern using a given set of fuel bundles. The code can compile this set using the fuel bundles from an earlier cycle and a variable number of identical fresh bundles to be given as input. The code also evaluates the pattern by performing the Haling calculations in 2-D using the TACHY formalism (Jain and Jagannathan, 1982). Normally cycle energy is given as an input and the Haling calculations yield the Kea at the EOC. CORECOOK accepts the critical value of Kc~as input. Based on this it calculates the cycle energy for the given reload pattern. In a study which is effectively exhaustive, alternate values of each of the two parameters M and N over their respective ranges were used, the range of M being from 1 to r and that of N being from 1 to n - 2 r (r and

245

n being, respectively, the number of fresh bundles and the total number of bundles in a sector as mentioned earlier). The fuel bundles of Unit-I at EOC-12 were used for this study. Figure 3 represents the relation between the RPF (Haling radial power peaking factor) and the cycle energy of the BRPs as obtained in this study. One hundred fresh fuel bundles for the full core of the new reload pattern were used in this study (corresponding to r = 28 for a sector). The same set of fuel bundles is used for the parametric study, the results of which are presented in Figs 5(a)-5(d) (see below). Each point in Fig. 3 represents a BRP. That they form a region implies several feasible values for the cycle energy for a given value of the RPF and vice versa. The only points of interest on this graph are those with the largest cycle energy for their RPF. They lie on a segment of the envelope of this region. This segment of interest extends from the point of minimum RPF to the point of maximum cycle energy. We may call this segment the optimum biparametric segment. It is worthwhile to compare this segment in Fig. 3 with the figure presented previously by Huang and Levine (1978), Fig. 4 herein, which shows the power peaking vs the Ken for the various shufftings of the EOC fuel. Each point in Fig. 4 represents the fuel arrangement whose serial number it bears. The envelope in Fig. 3 continues to the left side of the point with the minimum RPF. The points on this segment correspond to BRPs with the minimum cycle energy for a given value of RPF and are, probably, only of academic interest. For a given number of fresh bundles, three reload patterns are noteworthy: one

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246

D.P. BURTEand S. G. VAIDYA Table I. BRPs: characteristic reload patterns for different numbers (r) of fresh bundles in a core sector consisting of 78 fresh bundles Reload patterns with maximum DELE (M = N = 0)

Reload patterns with minimum DELE (M = r, N = 78--2r)

Reload patterns with minimum RPF

r

DELE

RPF

DELE

RPF

M

N

DELE

RPF

23 24 25 26 27 28 29 30 31

4.9992 5.0938 5.1767 5.3327 5.3883 5.4549 5.5625 5.5961 5.6560

1.7244 1.6925 1.6486 1.5945 1.5769 1.5450 1.5091 1.4870 1.4620

4.4463 4.5678 4.7575 5.0732 5.1414 5.2404 5.3691 5.4358 5.5079

1.5556 1.5369 1.5183 1.5117 1.5007 1.4799 1.4619 1.4498 1.4324

10 10 9 7 7 7 7 7 7

32 30 28 26 23 22 20 17 16

3.6703 3.7711 3.8833 4.0091 4.0601 4.1780 4.3200 4.4486 4.5811

1.6454 1.6307 1.6113 1.5951 1.5865 1.5705 1.5476 1.5267 1.5072

r = Number of fresh bundles in a sector consisting of 78 fuel bundle locations. D E L E = Cycle energy. R P F = Haling radial power peaking factor. M, N = The two decision variables of the BRPs.

with the maximum cycle energy ( M = N = 0); another with the minimum cycle energy (M = r, N = n - 2r) ; and a third with the minimum RPF. (The values of the parameters M and N corresponding to the third pattern are identified using the procedure described later in this section.) All the points representing these in Fig. 3 lie on the envelope. From Fig. 3 we can conclude that there are optimum and non-optimum BRPs. Secondly, the shape of the optimum segment indicates that a higher cycle energy can be expected for a higher RPF. This is also evident in Fig. 4. The envelope in Fig. 3 is characteristic of a given set of bundles. We change this set by varying the number of fresh bundles. The three characteristic reload patterns mentioned above are worked out for different numbers of fresh bundles. The values of their parameters M and N, their cycle energies and the RPF are given in Table 1. Figure 3 can serve as a reference to check the efficacy of any optimization method or reload pattern. By optimization of BRPs we mean here that the cycle energy is to be maximized while the RPF is to be constrained to be less than a given input value. A simple procedure to identify the two parameters M and N for the optimum BRP has been incorporated as an option in CORECOOK. The procedure accepts as input the RPF constraint or the maximum allowed value of the RPF. It starts with the minimum value of M and the maximum value of N. It aims first to reduce the RPF accepting the consequent reduction in the cycle energy, by increasing M. When the RPF is reduced to an acceptable value, M is frozen. Subsequently, it aims to gain as much cycle energy as

possible without violating the RPF constraint. It achieves this by reducing N to a suitable extent. We found this procedure to be quite useful. Figures 5(a)-(d) show the behaviour of the RPF and cycle energy as functions of M and N, as obtained in the study. Apart from being interesting in their own right, these results and the conclusions from them may be found useful in optimization procedures. The following qualitative conclusions were drawn from these figures : (1) As M increases, the RPF passes through its minimum value. As N increases, it may pass through the minimum of RPF for larger values of M. (2) The cycle energy reduces monotonically with M as well as N. We have used the former conclusion to improve the optimization procedure of CORECOOK. The old procedure leads the search for the proper value of M astray if the minimum for the RPF is crossed while M is increased in steps. Secondly, if the RPF constraint is less than this minimum, the procedure does not realize that the task is impossible. The improved optimization procedure takes care of these two difficulties. Firstly, it monitors whether the minimum of the RPF is crossed. If it is, it first finds the value of M where the minimum occurs and analyses the corresponding pattern. Then, if the value of the RPF constraint is higher than the minimum, it proceeds to find the minimum value of M which just satisfies the RPF constraint. Otherwise, it stops the search at the minimum. In either case, it subsequently proceeds to gain as much cycle energy as possible without violating the

Optimization of reload patterns for BWRs (a)

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R P F c o n s t r a i n t by reducing N, as is d o n e in the old procedure. W e note t h a t the trick using C O R E C O O K to find the B R P with the m i n i m u m possible R P F is therefore to ask for a reload p a t t e r n with a n u n a t tainably low R P F . This o p t i m i z a t i o n p r o c e d u r e is r a t h e r crude in t h a t it decides M a n d N separately r a t h e r t h a n in combination. Therefore, the resulting b i p a r a m e t r i c pattern c a n n o t be said to be the o p t i m u m one. These reload p a t t e r n s are represented by the points m a r k e d

by arrows in Fig. 3. T h e i r closeness to the o p t i m u m segment is indicative o f the efficacy of the present b i p a r a m e t r i c o p t i m i z a t i o n procedure o f C O R E COOK. G i v e n a n achievable R P F constraint, the a l g o r i t h m constructs a b o u t 8-10 trial reload p a t t e r n s taking a total o f a b o u t 3-4 m i n o f c o m p u t e r time o n a n N D 500 before o b t a i n i n g the " o p t i m u m p a t t e r n as per the a b o v e procedure"• U s i n g the same set o f fuel bundles (as used in the exercise for Fig. 3) reload p a t t e r n s

248

D . P . BURTE and S. G. VAIDYA

Table 2. Biparametrically optimized reload patterns : RPF and cycle energy for a given upper bound on the RPF with a total number of fresh bundles of 100 (or 28 in a sector consisting of 78 fuel locations)

BRPs is a d e q u a t e for the practical requirements o f the current cycles of T A P S reactors.

Upper Optimum biparametric pattern Improved pattern bound .... on the RPF RPF DELE M N RPF DELE 1.54 1.53 1.52 1.51 1.50 1.49 1.48

1.5400 1.5300 1.5172 1.5097 1.4969 1.4894 1.4798

5.4452 5.4140 5.3748 5.3667 5.3052 5.2949 5.2417

2 2 4 5 6 7 7

3 22 15 12 17 13 22

1.5373 1.5273 1.5146 1.5085 1.4845 1.4885 1.4801

5.4866 5.4560 5.4172 5.4086 5.3022 5.3369 5.2846

Abbreviations as in Table 1.

are o b t a i n e d by applying this procedure for different values of the R P F constraint. T a b l e 2 gives the R P F vs the cycle energy for these patterns. 4. MULTIPARAMETRIC MODIFICATION The B R P s follow the guidelines for optimization. W i t h this in mind, it is expected that the o p t i m u m one will be close to the overall o p t i m u m pattern. A multip a r a m e t r i c a l g o r i t h m involving an arbitrary n u m b e r of radial zones is presented. It allows shuffling of the fuel bundles from one zone to another. In o u r case we have chosen 7 such zones. One can use this a l g o r i t h m interactively to i m p r o v e the optimization until satisfied. This a l g o r i t h m serves two purposes. Firstly, it can evaluate the efficacy o f a given BRP. The scope for i m p r o v e m e n t by radial shuffling will depend upon how far the given p a t t e r n is from the o p t i m u m pattern. The o p t i m u m p a t t e r n will provide n o scope for i m p r o v e m e n t . A n y radial shuffling would result in a worse reload pattern. By this m e t h o d it is f o u n d that the o p t i m u m o f the B R P s is very close to the overall o p t i m u m pattern. Secondly, it enables one to improve the B R P to the greatest extent possible. T h e input c o r r e s p o n d i n g to the fruitful shuffling to i m p r o v e the o p t i m u m B R P for the T A P S reactors is now standardized. T a b l e 2 includes the data (the R P F a n d cycle energy) on the i m p r o v e m e n t over the biparametrically optimized patterns, as described above.

6. CONCLUSIONS (1) A z i m u t h a l symmetry, L L L a n d a c h e q u e r b o a r d a r r a n g e m e n t o f fresh a n d exposed fuel bundles are built into the m e t h o d presented here for constructing reload p a t t e r n s for the B W R s at TAPS. A n e n o r m o u s l y large n u m b e r of undesirable reload patterns, therefore, are removed f r o m consideration. (2) The balance o f o p t i m i z a t i o n can be shifted towards lower Haling power peaking or higher cycle energy with the help o f two i n p u t parameters. (3) The n u m b e r o f these B R P s is small e n o u g h to p e r m i t exhaustive study. The set of points representing the B R P s comprehensively, o n a plot of the radial (Haling) power peaking factor ( R P F ) vs the cycle energy, are seen to be enveloped into a region with a crescent-like shape. (4) The segment o f the envelope consisting o f the points with m a x i m u m cycle energy for a given R P F shows t h a t higher cycle energy can be o b t a i n e d for higher Haling power peaking. Secondly, we see t h a t the range o f the R P F covered by this segment is a d e q u a t e for the practical requirements o f the B W R s at TAPS. (5) The o p t i m u m BRPs are f o u n d to be very close to o p t i m u m reload patterns. However, the m e t h o d has been a u g m e n t e d with a multi-parametric a l g o r i t h m which can be used to exploit the slightest scope for further i m p r o v e m e n t . (6) The task of generating o p t i m u m reload p a t t e r n s u n d e r the relevant c o n s t r a i n t s is computerized. A code C O R E C O O K has been written primarily for this purpose. The code has been successfully used to generate the 5 most recent reload p a t t e r n s loaded into T A P S reactors a n d it is n o w in regular use.

REFERENCES

5. APPLICATION T h e b i p a r a m e t r i c a l g o r i t h m with the search procedure as well as the a l g o r i t h m for i m p r o v e m e n t of the radial exposure distribution are i n c o r p o r a t e d in the c o m p u t e r code C O R E C O O K a n d form its m a i n feature. T h e code has been successfully used to generate 5 o u t o f 6 SSC reload p a t t e r n s loaded into T A P S cores, to date, a n d is n o w in regular use. This also shows t h a t the range of the R P F s of the o p t i m u m

Ahn D. H. and Levine S. H. (1985) Nucl. Technol. 71,535. Bellman R. (1957) Dynamic Programming. Princeton Univ. Press, Princeton, NJ. Burte D. P. (1990) Ann. Nucl. Energy 17, 239. Burte D. P. and Vaidya S. G. (1986) Ann. Nucl. Energy 13, 317. Civita P. and Fornaciari P. (1972) Nucl. Technol. 14, 116. Downar T. (19863 Ann. Nucl. Energy 13, 545. Downar T. (1987) Ann. Nucl. Eneryy 14, 135. Downar T. J. and Sesonske A. (1988) Adv. Nucl. Sci. TechnoL 20, 71.

Optimization of reload patterns for BWRs Galperin A. and Kimhy Y. (1991) Nucl. Sci. Engn9 109, 103. Ho A. and Sesonske A. (1982) Nucl. Technol. 58, 422. Hoshino T. (1978) Nucl. Technol. 39, 46. Huang H. Y. and Levine S. H. (1978) Trans. Am. Nucl. Soc. 30, 339. Jain R. P. and Jagannathan V. (1982) Report BARC/I-718. Kim Y. et al. (1987) Nucl. Sci. Engn9 96, 85. Kropaczek D. J. and Turinsky P. J. (1991) NucL Technol. 95, 9. Matoda H: (1971) Nucl. Sci. Engng 46, 88. Melice M. (1969) Nucl. Sci. Engng 37, 451. Mingle J. (1975) Nucl. Technol. 27, 248.

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Morita T. et al. (1986) Trans. Am. Nucl. Soc. 52, 41. Motoda H. et al. (1975) Nucl. Technol. 25, 477. Sauar T. (1971) Nucl. Sci. Engng 46, 274. Stout S. and Robinson A. (1973) Nucl. Technol. 20, 86. Suh J. and Levine S. (1990) Nucl. Sci. Engn9 105, 371. Suzuki A. and Kiyose R. (1971a) Nucl. Sci. Engn9 44, 121. Suzuki A. and Kiyose R. (1971b) Nucl. Sci. Engn9 46, 112. Tahara Y. et al. (1991) J. Nucl. Sci. Technol. 28, 399. Wall I. and Fenech H. (1965) Nucl. Sci. Engn9 22, 285. White J. R. and Avila K. M. (1990) In Proc. P H Y S O R : Physics of Reactors, Marseille, France, Vol. 2, p. XIV-42.