BWR control rod design using tabu search

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<ul><li><p>Abstract</p><p>* Corresponding author. Tel.: +52 55 53297233; fax: +52 55 53297340.</p><p>E-mail addresses: (J.A. Castillo), (J.J. Ortiz), galonso</p><p> (G. Alonso), (L.B. Morales), edmundo@nuclear.esfm.</p><p>(E. del Valle).1 Also a Ph.D. Student at Universidad Autonoma del Estado de Mexico.2 COFAA-IPN Fellow</p><p>Annals of Nuclear Energy 32 (2005) 741754</p><p></p><p>annals ofNUCLEAR ENERGY0306-4549/$ - see front matter 2005 Elsevier Ltd. All rights reserved.An optimization system to get control rod patterns (CRP) has been generated. This system</p><p>is based on the tabu search technique (TS) and the control cell core heuristic rules. The system</p><p>uses the 3-D simulator code CM-PRESTO and it has as objective function to get a specic</p><p>axial power prole while satisfying the operational and safety thermal limits. The CRP design</p><p>system is tested on a xed fuel loading pattern (LP) to yield a feasible CRP that removes the</p><p>thermal margin and satises the power constraints. Its performance in facilitating a power</p><p>operation for two dierent axial power proles is also demonstrated. Our CRP system is com-</p><p>bined with a previous LP optimization system also based on the TS to solve the combined LP-</p><p>CRP optimization problem. Eectiveness of the combined system is shown, by analyzing an</p><p>actual BWR operating cycle. The results presented clearly indicate the successful implementa-</p><p>tion of the combined LP-CRP system and it demonstrates its optimization features.</p><p> 2005 Elsevier Ltd. All rights reserved.BWR control rod design using tabu search</p><p>Jose Alejandro Castillo a,1, Juan Jose Ortiz a,Gustavo Alonso a,c,*, Luis B. Morales b, Edmundo del Valle c,2</p><p>a Instituto Nacional de Investigaciones Nucleares, Km 36.5 Carretera Mexico-Toluca,</p><p>Ocoyoacac 52045, Edo. de Mexico, Mexicob Universidad Nacional Autonoma de Mexico, Instituto de Investigaciones en Matematicas Aplicadas y en</p><p>Sistemas, Apartado Postal 70-221, Mexico, D.F. 04510, Mexicoc Instituto Politecnico Nacional, Escuela Superior de Fsica y Matematicas, Unidad Profesional Adolfo</p><p>Lopez Mateos, ESFM, Edicio 9, C.P. 07738, D.F. Mexico</p><p>Received 18 August 2004; received in revised form 7 December 2004; accepted 7 December 2004</p><p>Available online 23 February 2005doi:10.1016/j.anucene.2004.12.004</p></li><li><p>specic power prole 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). The</p><p>742 J.A. Castillo et al. / Annals of Nuclear Energy 32 (2005) 741754objective of the second stage is to obtain a control rod pattern (CRP) that provides</p><p>sucient thermal margin and a satisfactory axial power prole at any time during</p><p>the reactor cycle.</p><p>In a previous paper (Castillo et al., 2004), the rst stage was solved using the tabu</p><p>search technique (TS), the LP obtained generates more energy than those designed</p><p>by using engineer expertise. However, the optimized LP needs to be tested to know</p><p>if 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-</p><p>ists the possibility that it cannot be controlled by any CRP during the whole cycle, in</p><p>this case it will not be considered as a feasible LP.</p><p>Then, in this work, an optimization system based also on TS is introduced to</p><p>design a CRP that satises the thermal and operational constraints. This system</p><p>can be applied independently of the way that the LP was obtained or it can be</p><p>combined with our LP system (Castillo et al., 2004). Our combined system can</p><p>be 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 Turinsky</p><p>and Parks (1999). We will assess an actual BWR operating cycle to show the eec-</p><p>tiveness of the combined system. The results will be compared to those obtained</p><p>from engineer expertise.</p><p>Historically, the design of CRP is usually based on trial and error techniques. Re-</p><p>cently, this problem has been automated using IF-THEN rules (Lin and Lin, 1991),</p><p>heuristic rules and common engineering practices (Karve and Turinsky, 1999), gene-</p><p>tic algorithms (GA) (Montes et al., 2004), and fuzzy logic and heuristics (Francoiset al., 2004).</p><p>To test our CRP independent system, we search for the CRP for two dierent</p><p>problems, the rst one is an operating cycle with a specic loading pattern, for this</p><p>problem two dierent axial power proles are considered. The results are compared</p><p>against those obtained from a GA search (Montes et al., 2004) and engineering</p><p>expertise. The second problem is a loading pattern for an equilibrium cycle (Montes</p><p>et al., 2001), the cycle length obtained using the optimized CRP is compared against</p><p>the results of GA given by Montes et al. (2004) for the same problem and thoseobtained by using engineer expertise.</p><p>2. BWR control rod pattern</p><p>The reactor control system must be capable of compensating for all reactivity</p><p>changes that take place throughout a reactor operating cycle, and to do this at1. Introduction</p><p>Optimization of BWR fuel reloads is a two stages task: The rst one comprises</p><p>the allocation of fuel assemblies in the core to get maximum cycle length under aa rate that roughly matches that of the reactivity changes. The movable control</p></li><li><p>J.A. Castillo et al. / Annals of Nuclear Energy 32 (2005) 741754 743rods have a great eect on power distribution, and the interaction between control</p><p>rod arrangement and power distribution must be considered throughout the reac-</p><p>tor cycle.</p><p>Design of a CRP involves the control rod allocation in the core. An optimal CRP</p><p>will compensate for excess reactivity during the entire cycle, respecting the thermaland operational constraints with a minimal reduction in cycle length.</p><p>In this study, a BWR core with 444 fuel assemblies is analyzed to get a CRP that</p><p>satises operational constraints. The core has 109 control rods, and each one of these</p><p>can be placed in 25 dierent axial positions. A typical analysis of an operating cycle</p><p>is divided into 10 burnup steps (Total_Burnup_Steps), yielding ((25)109)10 possible</p><p>control rod patterns. If one assumes eighth core symmetry, this number is reduced</p><p>to ((25)19)10. Moreover, exploiting the control cell core (CCC) technique commonly</p><p>Fig. 1. BWR control rod distribution and its 1/8 classication.used 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 (see</p><p>Fig. 1).</p><p>The control rod axial positions are labeled as [00,02,04,06, . . ., 44,46,48]. Posi-tions 0018 are considered deep positions, positions 2030 are considered inter-</p><p>mediate positions, and positions 3248 are considered shallow positions. The</p><p>intermediate positions are forbidden during normal operation because if they are</p><p>used the axial power distribution shape is deformed (Almenas and Lee, 1992). There-</p><p>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 1064.</p><p>The minimal critical power ratio (MCPR) and linear heat generation rate</p><p>(LHGR) must be satised. Furthermore, the reactor core must be critical, in this</p><p>case, the eective multiplication factor must be adjusted to an eective multiplication</p><p>factor target for each burnup step through the whole cycle, and the axial power dis-</p><p>tribution must be adjusted to a target axial power distribution. Thus, the objective</p><p>function that we propose is a function of the eective multiplication factor (ke),</p><p>axial power prole (P), linear heat generation rate (LHGR), and minimal critical</p></li><li><p>MCPRt Minimal critical power ratio 1.45</p><p>744 J.A. Castillo et al. / Annals of Nuclear Energy 32 (2005) 741754power ratio (MCPR). Therefore, the CRP design problem can be formulated as the</p><p>following optimization problem:</p><p>minimizeF x w1 keff ;x keff ;tj j w2X25</p><p>i1Px;i P t;ij j w3 LHGRx LHGRtj j</p><p> w4 MCPRx MCPRtj j;where ke,x is the eective multiplication factor of the control rod pattern x; ke,t, tar-</p><p>get eective multiplication factor; Px,i, axial power distribution for node i of the con-</p><p>trol rod pattern x; Pt,i, target axial power distribution for node i; LHGRx, linear heat</p><p>generation rate of the control rod pattern x; LHGRt, maximum linear heat genera-</p><p>tion rate value permitted; MCPRx, minimal critical power ratio of the control rod</p><p>pattern x; MCPRt, minimal critical power ratio value permitted; and w1, . . .,w4 arecalled weighting factors and wiP 0, for i = 1, . . ., 4.</p><p>If the thermal limits LHGR and MCPR are satised, the corresponding weighting</p><p>factors will be zero, otherwise they will be the ones given by the user penalizing the</p><p>objective function. Thus, the objective function will have only the contribution due</p><p>to the eective multiplication factor and the axial power prole, when the thermal</p><p>limits are satised. Table 1 shows the MCPRt, LHGRt and kt values. Two dierent</p><p>target axial power distribution will be assessed, one obtained from the Haling calcu-</p><p>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 following</p><p>constraints:</p><p>keff ;x keff;tj j 6 d; d &gt; 0;</p><p>Px;i P t;ij j 6 eP t;i; e &gt; 0; for i 1; . . . ; 25;where d and e are the convergence criteria for the multiplication factor and the powerTable 1</p><p>Target and limits parameters</p><p>Symbol Meaning Limit value</p><p>kt Eective multiplication factor 1.0 (target)</p><p>LHGRt Linear heat generation rate 439 w/cmprole, respectively.</p><p>3. Methodology</p><p>TS is an iterative heuristic procedure for optimization. It has been designed to</p><p>overcome local optimality. It is distinguished from other methods because it incor-</p><p>porates a tabu list of length t of moves that forbids the reinstatement of certain attri-</p><p>butes of previously visited solutions, this tabu list is called short term memory,</p></li><li><p>J.A. Castillo et al. / Annals of Nuclear Energy 32 (2005) 741754 745because it stores information on the t most recent moves. These forbidden moves are</p><p>called tabu. For a more detailed presentation of TS, see Glover (1989).</p><p>Let us now describe how we use TS to get CRPs. In our approach, a feasible solu-</p><p>tion will be an octant of a reactor core and it is represented by a vector</p><p>(c1,c2,c3,c4,c5), where ci is the axial position of the ith control rod, for i = 1, . . ., 5(see Fig. 1). The range of each ci, avoiding intermediate positions, is</p><p>2,4, . . ., 18,32,34, . . ., 48.In our problem, a move is a transition from one CRP to another that is deter-</p><p>mined by the change of only one axial position. Whenever the ith rod is moved</p><p>to a dierent axial position, the tabu list forbids any movement of this rod to an</p><p>axial position considered in t preceding iterations. Formally, the tabu list consists</p><p>of vectors (i,ci), where the ith rod could not be allocated in that axial position ci.</p><p>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 dierence between the objective function (dened in Section 2) value</p><p>before and after the move. At each iteration the best move is choose, even if it</p><p>does not improve the objective function. The number of possible moves in each</p><p>iteration is 5 18 = 90. Since it is too expensive from a computational point ofview to evaluate all the moves, only a percentage of such moves will be consid-</p><p>ered in this study. The set M of these moves will be randomly generated, and the</p><p>rst 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-</p><p>set M.</p><p>In this study, the set M corresponds to 40% of the whole moves. This value was</p><p>chosen from previous experimental analysis where it was observed that for higher</p><p>values of moves there was no apparent improvement in the objective function value.</p><p>The range of moves analyzed was from 10% to 100%.</p><p>The long-term memory is a function that records moves taken in the past in order</p><p>to 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). In</p><p>our particular TS implementation, the long-term memory is an array, which will be</p><p>denoted F. The array has zeroes at the beginning of the procedure. When a control</p><p>rod is settled in an axial position at a given iteration, the array F changes as follows:</p><p>F i;ci F i;ci 2. The entry F i;ci is the frequency at which the axial position ci of the ithcontrol rod has been settled. The values of non-improving moves that switch the ax-</p><p>ial position ci of the ith control rod are then increased by F i;ci .The two last concepts to explain are the aspiration and the stopping criteria used.</p><p>The aspiration criteria cancels the status tabu of a move when it nds a feasible solu-</p><p>tion with a better function value than the best solution in the past. Our TS will be</p><p>stopped if the number of iterations used without improving the best solution is greater</p><p>than 40. This number was chosen from a statistical analysis as it was done with the</p><p>percentage of moves.</p><p>As it is set in Section 2, the objective function includes the thermal safety lim-</p><p>its, the axial power distribution and the eective multiplication factor. TS wasimplemented along with the 3-D reactor core simulator CM-PRESTO (Scand-</p></li><li><p>END START</p><p>YES</p><p>SAVE BURNUP NO IS IT THE LAST STEP VALUES BURNUP STEP?</p><p>IS IT THEBURNUP STEP 1?</p><p> NO YES</p><p> YES NO END N-TH ITERATION CALCULATIONS</p><p>OR ITERATIONS? HALING</p><p>CALCULATION ASPIRATION FUNCTION EQUAL TO THE BEST</p><p> NEIGHBOR NO</p><p> CONTROL ROD PATTERN YES ASPIRATIONRANDOMLY GENERATED, FUNCTION &gt; THE BEST</p><p> ASPIRATION FUNCTION = M NEIGHBOR?</p><p> NEW CONTROL ROD AXIAL POSITION UPDATE TABU TIME ARRAY,</p><p> FREQUENCIES AND THE BESTRETURN THE CONTROL ROD PATTERN</p><p>CONTROL ROD TO ORIGINAL AXIAL YES</p><p> POSITION HEURISTIC </p><p>RULE VIOLATED OR YES NO END40 % OF THE NEIGHBORHOOD NEIGHBORHOOD</p><p> ANALYZED? SEARCH? OBJECTIVE FUNCTION</p><p> EQUAL TO THE BEST NO NEIGHBOR</p><p>NO</p><p> CM-PRESTO OBJECTIVE CALCULATIONS YES FUNCTION PREVIOUS &gt; </p><p> THE BEST NEIGHBORHOOD?</p><p> OBJECTIVE FUNCTION</p><p> CALCULATIONS RETURN THE UPDATE THE BEST CONTROL ROD NEIGHBORHOOD, TO ORIGINAL AXIALTHE BEST SAFETY POSITION</p><p> LIMITS AND THE BESTAXIAL POWER</p><p> OBJECTIVE DISTRIBUTION YES FUNCTION NO</p><p>&lt; IS IT ASPIRATION A TABU</p><p> FUNCTION? NO MOVEMENT?</p><p> YES</p><p>YES OBJECTIVE FUNCTION</p><p> EQUAL TO OBJECTIVE IS IT FUNCTION PLUS AXIAL THE BESTCONTROL ROD POSITION NEIGBORHOOD?</p><p> FREQUENCY NO</p><p>Fig. 2. Flowchart of the CRP search.</p><p>746 J.A. Castillo et al. / Annals of Nuclear Energy 32 (2005) 741754</p></li><li><p>does succeed for all the burnup steps then the process is stopped, obtaining as a re-</p><p>sult a feasible LP-CRP. A pseudo-code for our iterative LP-CRP combined system</p><p>J.A. Castillo et al. / Annals of Nuclear Energy 32 (2005) 741754 747...</p></li></ul>