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BWR control rod design using tabu search

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BWR control rod design using tabu search
Jose´ Alejandro Castillo
a,1
, Juan Jose´ Ortiz
a
,Gustavo Alonso
a,c,*
, Luis B. Morales
b
, Edmundo del Valle
c,2
a
Instituto Nacional de Investigaciones Nucleares, Km 36.5 Carretera Me´ xico-Toluca,Ocoyoacac 52045, Edo. de Me´ xico, Me´ xico
b
Universidad Nacional Auto´ noma de Me´ xico, Instituto de Investigaciones en Matema´ ticas Aplicadas y enSistemas, Apartado Postal 70-221, Me´ xico, D.F. 04510, Me´ xico
c
Instituto Polite´ cnico Nacional, Escuela Superior de Fı´ sica y Matema´ ticas, Unidad Profesional ‘‘AdolfoLo´ pez Mateos’’, ESFM, Ediﬁcio 9, C.P. 07738, D.F. Me´ xico
Received 18 August 2004; received in revised form 7 December 2004; accepted 7 December 2004Available online 23 February 2005
Abstract
An optimization system to get control rod patterns (CRP) has been generated. This systemis based on the tabu search technique (TS) and the control cell core heuristic rules. The systemuses the 3-D simulator code CM-PRESTO and it has as objective function to get a speciﬁcaxial power proﬁle while satisfying the operational and safety thermal limits. The CRP designsystem is tested on a ﬁxed fuel loading pattern (LP) to yield a feasible CRP that removes thethermal margin and satisﬁes the power constraints. Its performance in facilitating a poweroperation for two diﬀerent axial power proﬁles is also demonstrated. Our CRP system is com-bined with a previous LP optimization system also based on the TS to solve the combined LP-CRP optimization problem. Eﬀectiveness of the combined system is shown, by analyzing anactual BWR operating cycle. The results presented clearly indicate the successful implementa-tion of the combined LP-CRP system and it demonstrates its optimization features.
2005 Elsevier Ltd. All rights reserved.
0306-4549/$ - see front matter
2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.anucene.2004.12.004
*
Corresponding author. Tel.: +52 55 53297233; fax: +52 55 53297340.
E-mail addresses:
jacm@nuclear.inin.mx (J.A. Castillo), jjortiz@nuclear.inin.mx (J.J. Ortiz), galonso
@nuclear.inin.mx (G. Alonso), lbm@servidor.unam.mx (L.B. Morales), edmundo@nuclear.esfm. ipn.mx
(E. del Valle).
1
Also a Ph.D. Student at Universidad Auto´noma del Estado de Me´xico.
2
COFAA-IPN FellowAnnals of Nuclear Energy 32 (2005) 741–754www.elsevier.com/locate/anucene
annals of
NUCLEARENERGY
1. Introduction
Optimization of BWR fuel reloads is a two stages task: The ﬁrst one comprisesthe allocation of fuel assemblies in the core to get maximum cycle length under aspeciﬁc power proﬁle and it is a two edges calculation, at the beginning and at theend of the cycle. This array of assemblies is called the loading pattern (LP). Theobjective of the second stage is to obtain a control rod pattern (CRP) that providessuﬃcient thermal margin and a satisfactory axial power proﬁle at any time duringthe reactor cycle.In a previous paper (Castillo et al., 2004), the ﬁrst stage was solved using the tabusearch technique (TS), the LP obtained generates more energy than those designedby using engineer expertise. However, the optimized LP needs to be tested to knowif it is feasible to operate without violating the thermal and safety operational limitsat any time during the whole operating cycle. For each LP obtained by TS there ex-ists the possibility that it cannot be controlled by any CRP during the whole cycle, inthis case it will not be considered as a feasible LP.Then, in this work, an optimization system based also on TS is introduced todesign a CRP that satisﬁes the thermal and operational constraints. This systemcan be applied independently of the way that the LP was obtained or it can becombined with our LP system (Castillo et al., 2004). Our combined system canbe considered as a tool to tackle the combined LP-CRP optimization problem,which, for a BWR, is a tightly coupled problem as it was noted by Turinskyand Parks (1999). We will assess an actual BWR operating cycle to show the eﬀec-tiveness of the combined system. The results will be compared to those obtainedfrom engineer expertise.Historically, the design of CRP is usually based on trial and error techniques. Re-cently, this problem has been automated using IF-THEN rules (Lin and Lin, 1991),heuristic rules and common engineering practices (Karve and Turinsky, 1999), gene-tic algorithms (GA) (Montes et al., 2004), and fuzzy logic and heuristics (Franc¸ois
et al., 2004).To test our CRP independent system, we search for the CRP for two diﬀerentproblems, the ﬁrst one is an operating cycle with a speciﬁc loading pattern, for thisproblem two diﬀerent axial power proﬁles are considered. The results are comparedagainst those obtained from a GA search (Montes et al., 2004) and engineeringexpertise. The second problem is a loading pattern for an equilibrium cycle (Monteset al., 2001), the cycle length obtained using the optimized CRP is compared againstthe results of GA given by Montes et al. (2004) for the same problem and thoseobtained by using engineer expertise.
2. BWR control rod pattern
The reactor control system must be capable of compensating for all reactivitychanges that take place throughout a reactor operating cycle, and to do this ata rate that roughly matches that of the reactivity changes. The movable control
742
J.A. Castillo et al. / Annals of Nuclear Energy 32 (2005) 741–754
rods have a great eﬀect on power distribution, and the interaction between controlrod arrangement and power distribution must be considered throughout the reac-tor cycle.Design of a CRP involves the control rod allocation in the core. An optimal CRPwill compensate for excess reactivity during the entire cycle, respecting the thermaland operational constraints with a minimal reduction in cycle length.In this study, a BWR core with 444 fuel assemblies is analyzed to get a CRP thatsatisﬁes operational constraints. The core has 109 control rods, and each one of thesecan be placed in 25 diﬀerent axial positions. A typical analysis of an operating cycleis divided into 10 burnup steps (Total_Burnup_Steps), yielding ((25)
109
)
10
possiblecontrol rod patterns. If one assumes eighth core symmetry, this number is reducedto ((25)
19
)
10
. Moreover, exploiting the control cell core (CCC) technique commonlyused in BWR plants, only control rods that have no fresh fuel are used for reactorcontrol. This reduces the number of possible control rod patterns to ((25)
5
)
10
(seeFig. 1).The control rod axial positions are labeled as [00,02,04,06,
. . .
,44,46,48]. Posi-tions 00–18 are considered ‘‘deep positions’’, positions 20–30 are considered ‘‘inter-mediate positions’’, and positions 32–48 are considered ‘‘shallow positions’’. Theintermediate positions are forbidden during normal operation because if they areused the axial power distribution shape is deformed (Almenas and Lee, 1992). There-fore, only 19 of the possible 25 positions are allowed. Thus, the total number of pos-sibilities to generate all control rod patterns is ((19)
5
)
10
10
64
.The minimal critical power ratio (MCPR) and linear heat generation rate(LHGR) must be satisﬁed. Furthermore, the reactor core must be critical, in thiscase, the eﬀective multiplication factor must be adjusted to an eﬀective multiplicationfactor target for each burnup step through the whole cycle, and the axial power dis-tribution must be adjusted to a target axial power distribution. Thus, the objectivefunction that we propose is a function of the eﬀective multiplication factor (
k
eﬀ
),axial power proﬁle (
P
), linear heat generation rate (LHGR), and minimal critical
Fig. 1. BWR control rod distribution and its 1/8 classiﬁcation.
J.A. Castillo et al. / Annals of Nuclear Energy 32 (2005) 741–754
743
power ratio (MCPR). Therefore, the CRP design problem can be formulated as thefollowing optimization problem:minimize
F
ð
x
Þ¼
w
1
k
eff
;
x
k
eff
;
t
j jþ
w
2
X
25
i
¼
1
P
x
;
i
P
t
;
i
j jþ
w
3
LHGR
x
LHGR
t
j jþ
w
4
MCPR
x
MCPR
t
j j
;
where
k
eﬀ,
x
is the eﬀective multiplication factor of the control rod pattern
x
;
k
eﬀ,
t
, tar-get eﬀective multiplication factor;
P
x
,
i
, axial power distribution for node
i
of the con-trol rod pattern
x
;
P
t,
i
, target axial power distribution for node
i
; LHGR
x
, linear heatgeneration rate of the control rod pattern
x
; LHGR
t
, maximum linear heat genera-tion rate value permitted; MCPR
x
, minimal critical power ratio of the control rodpattern
x
; MCPR
t
, minimal critical power ratio value permitted; and
w
1
,
. . .
,
w
4
arecalled weighting factors and
w
i
P
0, for
i
= 1,
. . .
,4.If the thermal limits LHGR and MCPR are satisﬁed, the corresponding weightingfactors will be zero, otherwise they will be the ones given by the user penalizing theobjective function. Thus, the objective function will have only the contribution dueto the eﬀective multiplication factor and the axial power proﬁle, when the thermallimits are satisﬁed. Table 1 shows the MCPR
t
, LHGR
t
and
k
t
values. Two diﬀerenttarget axial power distribution will be assessed, one obtained from the Haling calcu-lation and a second one using spectral shift (Montes et al., 2004). These axial powerdistributions depend on the fuel reload studied. Moreover, we impose the followingconstraints:
k
eff
;
x
k
eff
;
t
j j
6
d
;
d
>
0
;
P
x
;
i
P
t
;
i
j j
6
e
P
t
;
i
;
e
>
0
;
for
i
¼
1
;
. . .
;
25
;
where
d
and
e
are the convergence criteria for the multiplication factor and the powerproﬁle, respectively.
3. Methodology
TS is an iterative heuristic procedure for optimization. It has been designed toovercome local optimality. It is distinguished from other methods because it incor-porates a tabu list of length
t
of moves that forbids the reinstatement of certain attri-butes of previously visited solutions, this tabu list is called short term memory,
Table 1Target and limits parametersSymbol Meaning Limit value
k
t
Eﬀective multiplication factor 1.0 (target)LHGR
t
Linear heat generation rate 439 w/cmMCPR
t
Minimal critical power ratio 1.45744
J.A. Castillo et al. / Annals of Nuclear Energy 32 (2005) 741–754
because it stores information on the
t
most recent moves. These forbidden moves arecalled tabu. For a more detailed presentation of TS, see Glover (1989).Let us now describe how we use TS to get CRPs. In our approach, a feasible solu-tion will be an octant of a reactor core and it is represented by a vector(
c
1
,
c
2
,
c
3
,
c
4
,
c
5
), where
c
i
is the axial position of the
i
th control rod, for
i
= 1,
. . .
,5(see Fig. 1). The range of each
c
i
, avoiding intermediate positions, is2,4,
. . .
,18,32,34,
. . .
,48.In our problem, a move is a transition from one CRP to another that is deter-mined by the change of only one axial position. Whenever the
i
th rod is movedto a diﬀerent axial position, the tabu list forbids any movement of this rod to anaxial position considered in
t
preceding iterations. Formally, the tabu list consistsof vectors (
i
,
c
i
), where the
i
th rod could not be allocated in that axial position
c
i
.During the process the tabu list is updated circularly. For this study, the lengthof the tabu list was randomly selected in the range (6
6
t
6
16). The value of amove is the diﬀerence between the objective function (deﬁned in Section 2) valuebefore and after the move. At each iteration the best move is choose, even if itdoes not improve the objective function. The number of possible moves in eachiteration is 5
·
18 = 90. Since it is too expensive from a computational point of view to evaluate all the moves, only a percentage of such moves will be consid-ered in this study. The set
M
of these moves will be randomly generated, and theﬁrst move that improves the objective function is done. However, if there is nomove that improves the objective function, then one must examine the whole sub-set
M
.In this study, the set
M
corresponds to 40% of the whole moves. This value waschosen from previous experimental analysis where it was observed that for highervalues of moves there was no apparent improvement in the objective function value.The range of moves analyzed was from 10% to 100%.The long-term memory is a function that records moves taken in the past in orderto penalize those that are non-improving. The goal is to diversify the search by com-pelling regions to be visited that possibly were not explored before (Glover, 1989). Inour particular TS implementation, the long-term memory is an array, which will bedenoted
F
. The array has zeroes at the beginning of the procedure. When a controlrod is settled in an axial position at a given iteration, the array
F
changes as follows:
F
i
;
c
i
¼
F
i
;
c
i
þ
2. The entry
F
i
;
c
i
is the frequency at which the axial position
c
i
of the
i
thcontrol rod has been settled. The values of non-improving moves that switch the ax-ial position
c
i
of the
i
th control rod are then increased by
F
i
;
c
i
.The two last concepts to explain are the aspiration and the stopping criteria used.The aspiration criteria cancels the status tabu of a move when it ﬁnds a feasible solu-tion with a better function value than the best solution in the past. Our TS will bestopped if the number of iterations used without improving the best solution is greaterthan 40. This number was chosen from a statistical analysis as it was done with thepercentage of moves.As it is set in Section 2, the objective function includes the thermal safety lim-its, the axial power distribution and the eﬀective multiplication factor. TS wasimplemented along with the 3-D reactor core simulator CM-PRESTO (Scand-
J.A. Castillo et al. / Annals of Nuclear Energy 32 (2005) 741–754
745

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