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IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 53, NO. 5, OCTOBER 2006 2957

An Evolutionary Optimization of the Refueling

Simulation for a CANDU Reactor Quang Binh Do, Hangbok Choi, and Gyu Hong Roh

Abstract—This paper presents a multi-cycle and multi-objective optimization method for the refueling simulation of a 713 MWe Canada deuterium uranium (CANDU-6) reactor based on a genetic algorithm, an elitism strategy and a heuristic rule. The proposed algorithm searches for the optimal refueling patterns for a single cycle that maximizes the average discharge burnup, minimizes the maximum channel power and minimizes the change in the zone controller unit water fills while satisfying the most important safety-related neutronic parameters of the reactor core. The heuristic rule generates an initial population of individuals very close to a feasible solution and it reduces the computing time of the optimization process. The multi-cycle optimization is carried out based on a single cycle refueling simulation. The proposed approach was verified by a refueling simulation of a natural uranium CANDU-6 reactor for an operation period of 6 months at an equilibrium state and compared with the experience-based automatic refueling simulation and the generalized perturbation theory. The comparison has shown that the simulation results are consistent from each other and the proposed approach is a reasonable optimization method of the refueling simulation that controls all the safety-related parameters of the reactor core during the simulation.

Index Terms—Fuel management, genetic algorithm, optimization, zone controller unit.

FOR a Canada deuterium uranium (CANDU) reactor, the primary objective of the fuel management is to determine the fuel loading pattern and fuel replacement strategies to operate the reactor in a safe and reliable fashion while keeping the total unit energy cost low. An important feature of the CANDU reactor is the on-power refueling operation, which is carried out to maintain the core criticality by refueling a few fuel channels everyday. At the CANDU nuclear power plant (NPP), the refueling channels are selected such that a variation of the core performance parameters from the reference core condition is minimized when the refueling operation is performed. In order to control the excess reactivity and local power perturbation caused by the refueling operation, a 713 MWe CANDU reactor (CANDU-6) has 14 zone controller units (ZCU) which are filled with light water. The ZCU has two functional requirements: a bulk control to achieve the target reactivity and a spatial control

Manuscript received March 30, 2006; revised May 25, 2006 and July 25, 2006. This work was supported by the RCA Regional Office, Korea. This work has been carried out under the Nuclear Research and Development Program of Korea Ministry of Science and Technology.

Q. Binh Do is with the Korea Atomic Energy Research Institute,

Yuseong, Daejeon 305-600, Korea and also with the Vietnam Atomic Energy Commission, Hochiminh City, Vietnam.

H. Choi and G. Hong Roh are with the Korea Atomic Energy Research Institute, Yuseong, Daejeon 305-600, Korea (e-mail: choih@kaeri.re.kr).

Digital Object Identifier 10.1109/TNS.2006.882369 to achieve the reference power distribution. Therefore, the ZCU water level is a key parameter that should be considered when selecting the refueling channels.

Recently an automatic refueling channel selection program

(AUTOREFUEL) and a deterministic program (GENOVA) were developed, which were based on reactor operation experiences and a generalized perturbation theory, respectively [1], [2]. Both programs were designed to keep the ZCU water level within a reasonable range during a continuous refueling simulation. However, a global optimization of the refueling simulation, that includes constraints on the discharge burn-up, maximum channel power (MCP), maximum bundle power (MBP), channel power peaking factor (CPPF) and the ZCU water level, was not achieved and thus remains as an ultimate goal. Therefore a stochastic method such as a genetic algorithm [3]–[6], which has been used for the light water reactor, was applied to the CANDU-6 reactor in order to consider multiple objectives of the fuel management. In this study, an evolutionary algorithm, which is indeed a hybrid method based on a genetic algorithm, an elitism strategy and the heuristic rules for a multi-cycle and multi-objective optimization of a refueling simulation has been developed for the CANDU-6 reactor.

The objectives of the optimization are a maximization of the average discharge burnup traced back to previous refueling operations, a minimization of the MCP and a minimization of the maximum change in the ZCU water levels. Because a genetic algorithm uses the information of a fitness function when selecting the fittest individuals of the current generation for the next generation and the fitness function is required to be nonnegative, it is necessary to map the objective function to a fitness function form [7]. In our problem, the fitness function is written as a linear weighted sum of the objective functions according to the weighting method of the multi-objective optimization [6], [8], [9]:

where x denotes the refueling pattern, and are weighting factors, is an input coefficient that is chosen such that the MCP is always lower than is an input coefficient such that the maximum change in the ZCU level is always lower than is the discharge burnup of fuel bundle j in fuel channel i, is the refueling scheme (the number of refueling bundles per channel) for channel i, is

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2958 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 53, NO. 5, OCTOBER 2006 the refueling rate for cycle k, L is the number of the current cycle and is the number of the previous cycles. The number of cycles is incorporated in the fitness function because one objective of the optimization is the maximization of the average discharge burnup traced back to previous refueling operations.

The weighting factor ( and ) is considered as a measure of the significance of each objective function in the multi-objective optimization process by following the weighting method. For the multi-objective model, all the weighting factors are positive. By writing the fitness function as (1), a minimization problem is transformed to a maximization problem.

The constraints include the discharge burnup, MCP, MBP,

CPPF and the ZCU water level, which are very important from the view points of a reactor operation, safety and economy.

where and are the numberoffuel channelsandZCU in the reactor core, respectively. The constraints define a feasible region and any point in that region is a feasible solution, which represents a feasible refueling pattern that satisfies all the constraints and can be chosen for a practical refueling operation. The constraints used for the refueling simulation of the CANDU-6 reactor are summarized in Table I.

Decision variable vector is a refueling pattern, which composes of a refueling rate and refueling channels. The refueling rate is the number of refueling channels per day, which is fixed for a single cycle. In a genetic algorithm, the decision variable vector is coded into a binary string of numbers 0 and 1 and vice versa. For the application of a genetic algorithm to a fuel loading pattern optimization, a decision variable can be coded by arrays of decimal numbers or a string of bits [10], [1]. In this study a one-dimensional binary chromosome is relevant to the refueling strategy of the CANDU-6 reactor.

A. Selection Strategy for the Optimal Solution

The overall procedure of the optimal refueling channel selection by a genetic algorithm can be classified into four stages: the preparation of an initial population, a selection of the best solution, a crossover and the mutation. At the beginning of the search process, a pool of candidate refueling channels is created for a single cycle under the condition that the reactor channel and bundle powers are kept below specified limiting values. When creating a candidate pool, the algorithm utilizes the heuristic method of the AUTOREFUEL program that considers the channel power distribution and the zonal reactivity requirement. Then an initial population is produced by randomly selecting refueling channels from the candidate pool. This process is still probabilistic though it is based on a set of deterministically selected channels. However a procedure that creates the initial population in this way significantly reduces the computing time owing to the heuristic feature of the candidate channels.

Selection is a process that chooses the fittest individuals in the current generation to create a breeding pool for the next generation. The elitism strategy is incorporated into the genetic search by creating an archive that stores the best non-dominated solutions found during the search process [8]–[10]. The fittest solutions in terms of the fitness in the archive are directly transferred to the next generation. For a three-objective optimization problem, for example, a solution x with the average discharge burnupF11,theMCPF12andthemaximumchangeintheZCU level F13 is dominated by a solution y with the average discharge burnup F21, the MCP F22 and the maximum change in the ZCU level F23 if and and there exists at least or . Any solution that is not dominated by others is regarded as a non-dominated solution. All the non-dominated solutions satisfying the archival conditions, given in Table I, are considered to be stored in the archive.

In order to create an archive, the archival conditions are initially set up a bit tighter than the license limits for selecting the solutions to be stored in the archive.All the non-dominated solutions that satisfy the archival conditions are stored in the archive when the archive is not full. In case the archive is full, each individual in the current population is compared with the archive. First of all, if it is dominated by any member in the archive, it is not stored. If it is not dominated by any member of the archive but dominates a member of the archive, it replaces that member. If it is not dominated by the archive and does not dominate any

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DO et al.: AN EVOLUTIONARY OPTIMIZATIONOF THE REFUELING SIMULATION FOR A CANDU REACTOR 2959 member of the archive but its fitness value is higher than the fitness value of some members of the archive, it replaces the worst member with the lowest fitness in the archive.

In addition to the solutions selected from the archive, the new generation consists of the solutions of the current population by the roulette wheel spin method [7]. The number of archive solutions that enters the new generation is typically chosen as a quadrant of the population size [3].

Crossover is the most important operator in a genetic algorithm. With the crossover operation, a genetic algorithm can obtain more information and therefore the search space is extended. In this study, the crossoveris performed by the one-point method which mixes parts of the two parent solutions in the breeding pool to create two off-springs [7]. Then a mutation is performed to maintain the diversity of the population by randomly altering the value of a string position with a small probability. The mutation helps to avoid the loss of information at a particular position. In this study, the mutation probability is set relatively high as 0.01.

A flexible terminating condition is used in this study to reduce the computing time for the refueling simulation. The search process for a single cycle terminates whenever a target solution, which satisfies all the target conditions, is found. The target conditions are composed of constraints that the solution of the problem is to attain. They are set up based on the operational experience of the CANDU-6 reactor as given in Table I. In case when a target solution is not found at the final generation, the non-dominated solution with the highest fitness value of the archive is chosen as the final solution of the problem.

B. Multi-Cycle Algorithm

The multi-cycle optimization is carried out through a single cycle optimization performed for one full power day (FPD). During the continuous refueling simulation, the average ZCU water level is used as an indicator of the excess reactivity that the reactor core needs to maintain its criticality. If the average ZCU water level drops below the typical operating value of 50%, the excess reactivity should be provided by the fresh fuels. In this study, the average ZCU water level is maintained simply by adjusting the minimum or maximum limit of the refueling rate for the next cycle when the average ZCU water level in the current cycle is lower or higher than the specific value. For the CANDU-6 reactor, the reactivity loss per one FPD is mk and the reactivity increment by one channel refueling (8-bundle shift scheme) is 0.2–0.3 mk on average. In this study, the refueling rate is set as 1 or 2 channels/day if the average ZCU level is above 50%, while it is adjusted to be 2 or 3 when the average ZCU level is below 50%. Though the reactivity increment by the refueling does not exactly balance with the reactivity loss due to a fuel irradiation for a single cycle, the deficit of the reactivity can be accommodated for by the subsequent refueling operations.

C. Computer Program

A computer program based on the proposed method has been developed as schematically shown in Fig. 1. A CANDU core physics design and analysis code RFSP [12] is used to evaluate the core performance parameters of the refueling patterns for a

Fig. 1. Schematic flowchart of the optimal simulation.

single cycle simulation. The AUTOREFUEL program is used to create a set of refueling channel candidates at the beginning of the search process. At first, the genetic program sets up the constraints for the whole simulation period. Then the genetic program randomly selects channels from the pool of candidates to form the initial population, evaluates individuals and updates the archive with better non-dominated solutions of the current population.

If a target solution is found, the genetic program takes it as the final solution for the current cycle and continues the simulation for the next cycle. If a target solution is not found, the search process performs the genetic operation until the time limit is reached. In this case, the non-dominated solution of the archive with the highest fitness value is chosen as the optimal solution for the current cycle. Once the refueling simulation for the current cycle is completed, the limits of the refueling rate are adjusted in order to provide a reactivity sufficient for the next cycle and to maintain the average ZCU water level within the typical range.

A refueling simulation of a CANDU-6 reactor loaded with natural uranium fuel was performed for 6 months at an equi-

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2960 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 53, NO. 5, OCTOBER 2006

Fig. 2. Comparison of the initial population randomly generated, the initial population generated from the AUTOREFUEL program and the archive after a 40-generation search starting from the random initial population for the MCP.

Fig. 3. Comparison of the initial population randomly generated, the initial population generated from the AUTOREFUEL program and the archive after a 40-generation search starting from the random initial population for the ZCU level change.

librium state, in which the global power distribution and the average fuel burnup reached steady-states. The constraints and other parameters used for the optimal refueling simulation were summarized in Table I. These parameters were determined based on a suggestion that a good performance is achievable for the genetic algorithm with a high crossover probability, a low mutation probability and a moderate population size [7]. The weighting factors ( and ) and the input coefficients ( and ) were determined based on the preliminary simulation results of a single cycle optimization. Figs. 2 and 3 show the initial population randomly generated, the initial population generated based on the refueling channel candidates obtained by the AUTOREFUEL program and the archive after a 40-generation search starting from the random initial population for the MCP and maximum ZCU level change, respectively. These distributions show how significantly the AUTOREFUEL approach saves on the computing time.

Fig. 4. Variation of the average discharge burnup versus full power day.

The search process usually requires at least several generations to find the target solution. The maximum number of generations in this calculation is set as 15 to limit the computing time for each single cycle. The computing time is min on an HP 9000 workstation for a 15-generation optimization search. So it is possible that the final solutions in several cases exceed the attainable ranges. When the search process cannot find the target solution, the best non-dominated solution of the archive is chosen as the optimal solution. With the archival conditions set up in Table I, there will always be feasible solutions stored in the archive after a few generations of the search. This means that the search algorithm will find the optimal solution such that the reactor operates in a nominal condition. During the 6-month refueling simulation, the final solutions satisfy the target conditions of the MCP, MBP and CPPF. However, the target conditions of the discharge burnup and ZCU level are not satisfied for a very few cases, which can be attributed to the convergence to a local optimum when starting from an initial population of poor candidates as well as the number of generations not sufficient for the genetic search to escape from the local optimum. An extended study is recommended to clarify this hypothesis in the future.

Table I summarizes the core performance parameters of the simulation and compares them with those obtained by other methods. In general, the performance parameters of the four simulations are very close to each other. For the Wolsong NPP 3, the discharge burnup was estimated by the simulation, while others were from the operation history in the year 2000. Note that the simulation by the generalized perturbation theory (GENOVA) was specifically designed to satisfy the ZCU level of 0.2–0.8 while others such as the MCP and MBP were free to change. Therefore the MCP and MBP of the GENOVA were higher than those of other simulations. The simulation results of the genetic program are very close to those of the AUTOREFUEL simulation except that the average ZCU level was tuned at 47%. As was discussed earlier, the genetic program has multiple objectives including the discharge burnup, which is shown in Fig. 4. The average discharge burnup obtained by the genetic program is the same as that by the other methods. This means that it is not possible to drastically improve a single performance parameter such as the discharge burnup under the

DO et al.: AN EVOLUTIONARY OPTIMIZATIONOF THE REFUELING SIMULATION FOR A CANDU REACTOR 2961

current fuel management strategy. In other words, the current fuel management adopts an 8-bundle shift refueling scheme with two different refueling regions (inner and outer core). Under this fixed fuel management strategy, it is not easy to change either the axial or radial power distribution of the core by selecting a different combination of the refueling channels. Conversely speaking, the genetic algorithm performs the refueling simulation consistent with the other methods including the operation data. More specifically, the deterministic approach is superior to the genetic algorithm from the viewpoint of the computational efficiency. However, the genetic algorithm is capable of simultaneously controlling all the safety-related neutronic parameters of the reactor core during the simulation.

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