Using a Combination of Weighting Factors, Genetic Algorithm

and Ant Colony Methods to Speed up the Reloading Pattern Optimization

of VVER-1000 Reactors

Yashar Rahmania , Ali Pazirandeh

b, Mohammad B. Ghofrani

c, Mostafa Sadighi

d

aDepartment of Physics, Faculty of Engineering, Sari Branch,

Islamic Azad University, Sari, Iran

bDepartment of Nuclear Engineering, Science and Research Branch,

Islamic Azad University, Tehran, Iran

cDepartment of Energy Engineering, Sharif University of Technology,

Tehran 8639-11365, Iran dOfogh Consulting Engineers, No.3, Separ Alley, Africa Blvd.,

Postal code: 1518716713, Tehran, Iran

ABSTRACT This paper discusses a new and innovative method, which significantly speeds up the reloading

pattern optimization of VVER-1000 reactors. To reduce the range of possible searchable loading

patterns and consequently speed up calculations for the reloading pattern optimization of the

BUSHEHR VVER-1000 reactor (second cycle), first, according to the values of weighting factors of

the 16 applicable fuel assemblies (in this cycle), these assemblies are categorized into 6 limited groups.

Then, the optimum arrangement was obtained in the 6-group state by using the genetic algorithm

method. Finally, the position of each subset fuel assembly in the 6 compacted groups was determined

by using the weighting factor method and ant colony algorithm. This way, the re-transformation process

of the optimum arrangement was performed from the 6-group to the real 16-group state. It should be

noted that the arrangement obtained from the re-transformation process was completely in compliance

with the optimum reloading pattern achieved from the 16-group genetic algorithm calculations, which

verifies the methods applied in this research and also the accuracy of the re-transformation calculations.

By using the method discussed in this paper, identical optimum arrangement can be achieved in nearly

half of the required runtime for conventional stochastic optimization methods. Of course, the proposed

method is not limited to use the genetic algorithm and ant colony in the optimization process; it can

accelerate the reloading pattern optimization of the reactor using other stochastic algorithms. In this

research, to increase the accuracy of the objective parameters computations in each of the arrangements,

thermo-hydraulic calculations were made and the effectiveness of temperature feedbacks in the cell and

neutronic calculations was discussed. Furthermore, all the structural constraints are observed in this

method. As no research or data have ever been reported on the reloading pattern optimization of

BUSHEHR VVER-1000 reactor in the second cycle, it was impossible to make comparisons.

Therefore, by using the coupling of WIMSD5-B, CITATION-LDI2 and WERL codes, the time-

dependent thermo-neutronic computations were made in the second cycle to ensure the safety and

desirability of the proposed pattern during the cycle.

Keywords— Reloading pattern optimization, VVER-1000 reactor, genetic algorithm, weighting

factor method, ant colony algorithm

INTRODUCTION

In the calculation of the optimum loading pattern for a nuclear reactor, several parameters, including

the multiplication factor maximization, the raising of the desired safety threshold during the cycle

length and the flattening of the radial power peaking factor should be taken into account.

Several studies have been conducted to calculate the reloading pattern of VVER-1000 reactors; among

them, one can refer to:

1. Genetic algorithms [1,2,3]

2. Particle swarm optimization [4,5]

3. Neural network & simulated annealing [6]

4. Cellular automata & simulated annealing [7]

5. Cellular automata [8]

6. Perturbation theory [9]

7. Artificial Bee Colony [10]

However, most of these studies have been conducted in the first or equilibrium cycles, which have

less fuel variety compared with the second cycle of BUSHEHR VVER-1000 reactor.

As this paper aimed to achieve a method for the reloading pattern optimization of VVER-1000

reactors in transient cycles, first, the fuel compositions were calculated for each remaining assembly

from the first cycle. After determining the desired cycle length (300 days), a quick estimation was made

for specifying the type and quantity of the loadable fresh assemblies to provide the required excess

reactivity for passing the cycle length with the reused fuel assemblies remaining from the previous

cycle.

After this stage, an approach was designed using a combination of the weighting factor method and

the genetic algorithm to quickly optimize the reloading pattern of the BUSHEHR VVER-1000 reactor

(in the second cycle).

The coupling of WIMSD5-[11] and CITATION-LDI2[12] codes with the thermo-hydraulic

calculations were used to calculate the effective multiplication factor and the radial power peaking

factor distribution of the reactor core. As there were no similar studies or reported data for reloading

pattern optimization of VVER-1000 reactor in the second cycle, it was not possible to compare the

results; however, in this paper, in order to ensure the desirability of the proposed arrangement, the

thermo-neutronic behavior of the reactor core (with this arrangement) was simulated during the cycle.

The evaluation of the results indicated that the proposed arrangement enjoys a desirable safety margin.

GENERAL INTRODUCTION TO THE MAIN CHARACTERISTICS OF THE BUSHEHR VVER-1000

REACTOR

The BUSHEHR VVER-1000 reactor falls into the category of pressurized light water reactors. The

core of this reactor is composed of 163 hexahedral fuel assemblies [13]. Ten groups of control rod

assemblies are used to control and maintain the reactor’s safety. In the second operational cycle, 16

types of fuel assemblies are used including 12 types that remain from the first cycle (109 fuel

assemblies). Chromium diboride has been used as a burnable absorber in two of the 16 types of

assemblies. Table.1 describes the main characteristics and working conditions of the BUSHEHR

VVER-1000 reactor [13].

Fig.1 also outlines the general layout of the core of BUSHEHR VVER-1000 reactor and the

arrangement of the fuel assemblies in the first operational cycle [13].

TABLE 1. THE MAIN TECHNICAL CHARACTERISTICS OF VVER-1000 (V-446) REACTOR

3000 Reactor nominal thermal power, MW

30 Coolant heating in the reactor, Co

4 Number of loops, pieces.

4.14 Flow area of the core,m2

1.44 Maximum allowable radial power

peaking factor

Hexahedral

prism

Fuel assembly geometry

3.53 Fuel height in the cold state, m

pellets

UO2

Fuel

)( 232

4

TiOODy

CB + Absorbing material

0.236 Pitch between fuel assemblies, m

31075.12 −× Pitch between the fuel rods, m

Fig.1 The arrangement of the VVER-1000 reactor core in the first cycle

BURNUP CALCULATION PROCESS

Because the purpose of this study is to describe the process of the optimum reloading pattern

proposition in the second cycle of BUSHEHR VVER-1000 reactor, burnup calculations should be

conducted to determine the fuel compositions of the reused fuel assemblies.

Therefore, a coupling of neutronic and thermo-hydraulic calculations was used. To do this, the

physical group constants of the fuel assemblies and reflectors were calculated using the WIMSD5-B

code.

Furthermore, to obtain the time-dependent changes in the fuel composition and calculate the rate of

burnup in each fuel assembly, the computational capabilities of the WIMSD5-B code were utilized. By

inserting the physical group constants obtained from the WIMSD5-B code into the input file of the

CITATION-LDI2 code and also defining the geometry of the reactor core, the effective multiplication

factor and the three-dimensional distribution of the reactor’s thermal power were calculated.

In this study, a thermo-hydraulic software was designed using the Enveloped Pin method [14],[15].

Furthermore, the Dittus-Boelter [15], Ross-Stoute [15],[16] and Lee-Kesler [17] models were used in

the calculations of the heat transfer coefficient of coolant, gap conductance coefficient and gap

pressure, respectively. In addition, to estimate the concentration of the released gaseous fission products

into the gap space of fuel, the Weisman [18] model was used.

By using the results of the thermo-hydraulic calculations, the temperature and density of the fuel, clad

and coolant elements (in each fuel assembly) were applied to the neutronic calculations and thus, a

continuous sequence of neutronic and thermo-hydraulic calculations was created. Fig.2 provides a

schematic description of the applied computational flowchart.

Fig.2 Schematic description of the applied computational flowchart.

THE CALCULATION OF THE TYPE AND QUANTITY OF THE LOADABLE FRESH FUEL ASSEMBLIES IN

THE SECOND CYCLE

The estimation of the rate of fuel depletion in the 163 assemblies of the reactor core (in the first

cycle), specified that 109 fuel assemblies of the first cycle can be reused in the second cycle and 54

fresh fuel assemblies together with those 109 assemblies should be used to complete reloading of the

reactor core in the second cycle. The calculation in this section aims to determine the type and quantity

of these fresh fuel assemblies, which is be used in the reloading of the second cycle. Generally, the

designer proposes 11 types of fuel assemblies in VVER-1000 reactors, which can be used in all of the

operating cycles to complete the reloading of reactor core. In Table.2 the characteristics of these fuel

assemblies has been discussed.

Therefore, to complete the arrangement of the reactor core in the second cycle, some fresh fuel

assemblies should be selected among these 11 types of assemblies.

As the positions of the fuel assemblies in the reactor core are divided into six and twelfth-fold

symmetric cells (Fig.5), these 54 fresh fuel assemblies should be made using 6 and 12 fuel packages.

Studying the types of these 109 reused assemblies shows that 6 sets of the whole 9 symmetric six-

fold cells in the reactor core must be reserved for these types of assemblies. Therefore, to achieve the

whole of the possible states of loading 54 fresh fuel assemblies, we have to consider the states, in which

not more than three sets of six-fold assemblies are used.

It should be noted that, 54 required fresh fuel assemblies could be made in four different states.

Table.3 describes these 4 different states.

TABLE .2THE SPECIFICATIONS FOR DIFFERENT TYPES OF

APPLICABLE FUEL ASSEMBLIES IN VVER-1000 REACTORS

Quantity of fuel rods

(enrichment U-235,%

weight)

Burnable absorber rod

(BAR) characteristics

FA type U-235,

% weight

Fuel rod

Type 1

Fuel rod

Type 2

existence Number content

natural,

g/cm3 1.6% 1.6 311

(1.6)

-

2.4% 2.4 311

(1.6)

-

2.4%

B0.02

2.4 311

(1.6)

+ 18 0.02

2.4%

B0.036

2.4 311

(1.6)

+ 18 0.036

3.62% 3.62 245

(3.7)

66

(3.3)

-

3.62%

B0.02

3.62 245

(3.7)

66

(3.3)

+ 18 0.02

3.62%

B0.036

3.62 245

(3.7)

66

(3.3)

+ 18 0.036

4.02% 4.02 245

(4.1)

66

(3.7)

-

4.02%

B0.02

4.02 245

(4.1)

66

(3.7)

+ 18 0.02

4.02%

B0.036

4.02 245

(4.1)

66

(3.7)

+ 18 0.036

4.02%

B0.050

4.02 245

(4.1)

66

(3.7)

+ 18 0.050

TABLE .3 THE ARRANGEMENT OF 54 FRESH FUEL ASSEMBLIES IN THE REACTOR CORE

According to the reasons mentioned before, states III and IV never occur. Therefore, among the

different loading states of 54 fresh fuel assemblies, only states I and II should be considered, which is

illustrated in Fig.3

Fig.3 Allowed placements for fresh fuel assemblies

After creating all the possible states for loading of fresh fuel assemblies, it was realized that only

5005 states could occur. To avoid spending too long time for testing the desirability of each possible

state, ranking is performed for each of these possible states using the weighting factor method. Based

on that, a loading state would be searched which has a desirable effective multiplication factor at the

end of a 300-day cycle.

APPLYING WEIGHTING FACTORS FOR EACH POSSIBLE STATE OF FRESH ASSEMBLIES LOADING

To search for the appropriate loading state of the fresh fuel assemblies, the time-dependent

calculations should be made for estimating the fuel composition and subsequently the core’s excess

reactivity at each 5005 possible loading states (at the end of the cycle).

To do so, WIMSD5-B code was used for the time-dependant calculation of fuel consumption in each

assembly .Therefore, at each of the possible state, first, the fuel composition of the fresh assemblies (at

that state) and the reused 109 assemblies were averaged; then, designing a virtual fuel assembly was

begun based on the averaged fuel composition.

To perform the calculations of fuel consumption in the virtual fuel assembly, the thermal power was

considered about 18.4049 megawatt (163

3000MW ) and the cycle length was considered to be 300 days.

After calculating the fuel composition of each state and obtaining the physical group coefficient of the

virtual assembly at the end of the cycle (using WIMSD5-B code), the data were linked to the

CITATION-LDI2 code through which the multiplication factor of the reactor core was calculated for

this possible state.

In order to calculate the effective multiplication factor using the CITATION-LDI2 code, the fuel

composition of the virtual assembly was assumed for all the 163 fuel assemblies within the reactor core.

In this research to overcome the phenomena such as samarium locking, it was supposed that the

effective multiplication factor of the reactor should be considered a little greater than the critical value

(approximately 1.02) at the end of the 300-day cycle.

Making time-dependant calculations for fuel depletion in each 5005 possible state would take a lot of

time. Therefore, to avoid spending a long time on calculations, an innovative approach based upon the

weighting factor method was used.

In this method, first, the fuel composition of each 11 types of the fresh fuel assemblies was calculated

at the end of the 300-day cycle. Next, the fuel-composition-dependent weighting factors were

calculated for each of these assemblies (at the end of the cycle).

With respect to the type and quantity of fresh fuel assemblies at each of these 5005 possible states,

the equivalent weighting factor was calculated.

Strictly speaking, the equivalent weighting coefficient (for each of the 5005 possible states) was

calculated through averaging the weighing factors of the fuel assemblies in that state.

This way, instead of making time-consuming calculations for estimating the fuel depletion in each of

the possible states, the weighting factors of each state were estimated through making calculations for

only 11 types of the fresh fuel assemblies. Later, the ascending grading of these 5005 states was

performed through the weighting factors of each possible loading state, as the state with minimum

weighting factor is placed in the first row and the state with the maximum weighting factor is placed in

the last row of the 5005 states.

It was evident that the possible states with high weighting factors, due to their higher fuel enrichment,

enjoyed higher effective multiplication factor than the states with lower weighting factor. Therefore, the

Bisection computational method was used in this section.

In this regard, first, fuel depletion calculations were made for each of the upper and lower boundaries of

the possible states. Next, calculating the average weighting factors in both domains, the state with the

lowest difference between its weighting factor and the averaged weighting factor was searched. Then

the effective multiplication factor was calculated in that middle state. If the effective multiplication

factor was lower than the desired quantity, calculations of the interval division should be continued at

the lower half of the states set which has higher weighting factors as compared with the earlier state;

otherwise, the upper half would be selected. According to the Bisection method, the span of possible

states would be halved in the following steps so that the searching process would reach the desired

result.

The important point which was considered as the second condition of selection in these calculations

was the choice of the possible state with the lower effective multiplication factor at the beginning of the

cycle within the sets of states with an equal multiplication factor at the end of the cycle. This is because

lower effective multiplication factor at the start of a cycle, equivalently, leads to more stability and

safety in the reactor’s core against positive reactivity feedbacks induced by boric acid dilution.

Furthermore, it would enjoy a higher safety margin as far as the rest of the thermo-neutronic parameters

are concerned.

Fig.4 schematically describes the computational flowchart applied in this section.

Meanwhile, Table.4 describes the results of the calculations and the offer made for the type and

quantity of the fresh fuel assemblies applicable in completing the BUSHEHR VVER-1000 reactor

reloading pattern in the second cycle.

Fig.4 A schematic description of the applied computational flowchart to

determine the type and quantity of the fresh fuel assemblies in the second cycle

TABLE .4 TYPES AND QUANTITIES OF THE FRESH FUEL ASSEMBLIES APPLICABLE IN THE SECOND CYCLE

Fuel assembly

type

Quantity

1.6% 0

2.4% 0

2.4%B0.02 0

2.4%B0.036 0

3.62% 12

3.62%B0.02 6

3.62%B0.036 12

4.02% 24

4.02%B0.02 0

4.02%B0.036 0

4.02%B0.050 0

USING THE COMBINATION OF WEIGHTING FACTOR METHOD AND GENETIC ALGORITHM FOR THE

RELOADING PATTERN OPTIMIZATION OF THE VVER-1000 REACTOR

After estimating the types and quantities of the loadable fuel assemblies in the second cycle, the fuel-

composition-dependent weighting factors of each assembly were calculated.

Then, according to the weighting factor of each type of assemblies, the assemblies with close

weighting factors were categorized in the compacted groups, so that the number of varieties of the fuel

assemblies was reduced to six.

Afterwards, the fuel composition of the subset assemblies was averaged in each of the compressed

groups. Then, as a sample of that group, a virtual assembly with the averaged fuel composition was

created.

Besides, the optimum arrangement at the six-group state was obtained using the genetic algorithm.

Then, the re-transformation of the optimal arrangement from 6-group state into the real 16-group was

performed using the radial distribution of its maximum power peaking factor and the estimated

weighting factor. Fig.8 schematically describes the computational flowchart applied in this section.

Calculating the Weighting Factor in Each Type of Fuel Assemblies

To calculate the weighting factor of each of the 16 types of fuel assemblies based on their fuel

compositions, the WIMSD5-B and CITATION-LDI2 codes will be coupled with thermo-hydraulic

computations to calculate the produced thermal power of each fuel assembly per equal neuron flux.

This weighting factor (factor C) is obtained by dividing the produced thermal power of each fuel

assembly by the thermal power of a typical assembly with a fuel composition averaged from the entire

fuel assemblies (Formula 1).

ave

i

iP

PC = (1)

Where Ci is the weighting factor value, i is the type of fuel assemblies, Pi is the produced thermal

power in each type of fuel assembly and Pave is the produced power of the fuel assembly with the

averaged enrichment.

In order to calculate this factor, we assumed an equal neutron flux in numerator and denominator of

this formula; as a result, neutron flux parameter was removed from both the numerator and

denominator of this correlation.

Table.5 shows the calculated weighting factor values that are based on the fuel compositions of each

the 16 type of fuel assemblies.

TABLE .5 THE WEIGHTING FACTOR BASED ON THE

FUEL COMPOSITION OF EACH ASSEMBLY (“C” FACTOR)

“C” factor Fuel

assembly

type

number

0.8267205 24-2 1

0.8266882 24-2 2

0.8250063 24-2 3

0.8250063 24-2 4

0.8268361 24-2 5

0.8475394 24B20-2 6

0.8551277 24B36-2 7

0.8531355 24B36-2 8

1.185350 36-2 9

1.196011 36-2 10

1.183665 36-2 11

1.205054 36B36-2 12

1.231149 36-1 13

1.128669 36B20-1

14

1.074982 36B36-1

15

1.359755 40-1 16

1 FA

(average enrichment)

The calculated values of weighting factors in the 16 types of fuel assemblies show that this

parameter’s values for some assemblies are similar (Table.5). Thus, these assemblies are classified into

fewer groups based on their weighting factor values. For this purpose, fuel assemblies of 1 to 8 are

assigned to group one; 9 to 12 to group two; and 13, 14, 15, and 16 to groups three, four, five, and six,

respectively .Table.6 indicates the C factor values, and the quantity of assemblies in each of the limited

six groups.

TABLE .6 THE NUMBER AND THE VALUES OF C FACTOR

FOR ASSEMBLIES IN EACH OF THE SIX LIMITED GROUPS

The types of fuel

in sextet groups

Burnup C factor Quantity

1 (group 2.4%)

reused fuel

12.7-14.56 0.8403266 67

2 (group 3.6%)

reused fuel

9.6-12.17 1.201491 42

3( group 3.6%)

fresh fuel

0 1.231149 12

4 ( group 3.6%B0.02)

fresh fuel

0 1.128669 6

5( group 3.6%B0.036)

fresh fuel

0 1.074982 12

6( group 4.02%)

fresh fuel

0 1.359755 24

Description of the reloading pattern optimization of BUSHEHR VVER-1000 reactor using the

genetic algorithm

Having done the coding process and defined the arrangement of the reactor core as a chromosome

with 19 genomes, the random generation of the initial population of the chromosomes was performed in

the following step. It should be noted that in the present research, the initial population consists of 20

chromosomes. After evaluating the initial population and selecting 10 chromosomes as the initial

parents, producing the next generation was carried out using the mutation and crossover operators. In

this research, the number of offsprings for each generation was considered to be 7 times bigger than the

number of the parents’ population (10).

In the present paper, the optimization calculations were made in two states using two different fitness

functions. In the first state, the optimality condition was considered as per the access to an arrangement

with flattened power distribution. In the second state, the definition of the fitness function was offered

in order to achieve an arrangement with the minimum “maximum power peaking factor” and maximum

“effective multiplication factor”.

The chromosomes of each generation were evaluated using the determined fitness function.

Meanwhile, nine superior chromosomes and one of the worst chromosomes were selected.

It should be noted that the reason for choosing a bad chromosome as one of the 10 parents selected in

each generation was to prevent the trapping of the search process in a series of local optimum results.

In addition, in order to prevent the loss of the desired chromosomes of the earlier generation and the

probable reduction of optimality during the process of searching, the selecting of superior chromosomes

in each generation is performed among the offsprings of that generation and the selected chromosomes

of the previous generation.

Therefore, this process is performed successively in each of the generations to achieve convergence.

Finally, to determine the appropriate time to finish the calculations, the fitness function values of the

selected parents of the last three generations are compared. In case of coincidence or negligible

differences among them, the calculations will be ended.

Coding of the VVER-1000 Arrangements

The hexagonal structure of the core of VVER-1000 reactors has a one-twelfth symmetry. In addition,

as the fuel assemblies of this reactor are hexagonal, the 163 positions of the reactor core are divided

into 9 twelve-fold sets, 9 six-fold sets, and one central cell.

Hence, the core of the VVER-1000 reactor is divided into 19 positions. By knowing the type of the

fuel assemblies in these 19 positions, it is possible to generate the arrangement of the reactor core. Fig.5

describes the classifying of the reactor core positions into six and twelve-fold cells.

Fig.5 The classifying of the reactor core positions

into 6- and 12-piece cells

Therefore, the geometrical structure of the reactor core is coded as a 19-genome chromosome. Fig.6

describes this chromosome and quantity of the relevant cells.

Fig.6 the coding of a reloading pattern as a 19-genome chromosome

The Evaluation and Selection of Chromosomes

In this research, the calculation of reloading pattern optimization was made in two states.

To estimate the quantity and type of the fresh fuel assemblies, the calculation was made in a way to

keep the criticality at an optimal level during the cycle. Therefore, in the first state, the optimization

was conducted so as to achieve an arrangement by flattened thermal power distribution.

When the criticality of the reactor is assured during the cycle (at a desirable level), the reactor core

with a flattened power distribution, results in more reduced fluctuations caused by the boron dilution

phenomenon and temperature feedbacks.

Therefore, to evaluate the generated chromosomes in each generation, chromosomes with a flatter

power distribution (minimum SPPE factor) were selected.

∑=

−=163

1

)1(m

mk PPFSPPE

(2)

where PPFm is the power peaking factor in each of the assemblies and SPPEk indicates the flattening

level of the radial power peaking factor distribution of each chromosome.

In the second state, the evaluation of chromosomes was carried out as per the access to a reloading

pattern with the maximum effective multiplication factor and the safe maximum power peaking factor.

To do so, the optimization condition was considered in maximizing the value of the fitness function

offered in the following formula.

)()1(),( maxmax PPFPPFYKXPPFKF uleffeff −+−= (3)

Where effk is the multiplication factor of the reactor and

maxPPF and ulPPF are the maximum power

peaking factor of the reactor and the greatest possible value of this factor (5), respectively. X and Y are

the weighting coefficients, which have been considered as 268 and 27, respectively.

It is noteworthy that the second state calculations were only performed to ensure the desired and rapid

performance of the research’s proposed method at multi-objective optimization states.

3. Using Crossover and Mutation Operators to Generate Offsprings of Each Generation

After the processes of selecting the parent chromosomes of the next generation, offspring

chromosomes are generated using operators such as crossover and mutation.

In this research, the population of the offspring chromosomes was considered to be 7 times bigger

than the parent chromosomes and the probabilities of the mutation (Pm)and crossover (Pc) operators

were considered as 0.7 and 0.3, respectively. Therefore, in a 70-piece population of the offspring

chromosomes, 49 chromosomes are generated through the mutation operator and the 21 remaining

chromosomes are generated through the crossover operator.

In this research, a one-point mutation operator and a four-point crossover operator were used. Fig.7

schematically describes the procedure of these operators. In the process of generating offspring

chromosomes, it is necessary to observe the structural constraints including matching quantity of each

type of fuel assembly applied in each arrangement with its permissible quantity and prohibition of

placing the fuel assemblies with burnable poison at the specified positions for entering control rods.

Therefore, each offspring chromosome caused by these operators was examined. If the constraints are

not matched with the generated chromosomes, they are eliminated and the operators are used

continuously to achieve the population, which is matched with the constraints.

As a result, using these operators in each generation, the offspring chromosomes are generated and

this cyclic process is carried out repeatedly until optimum result is achieved.

Fig.7 The performance of operators

a) one-point mutation; b) four-point crossover

To explain more clearly, Fig.8 schematically describes the computational flowchart applied in this

section.

Fig.8 A schematic description of the computational flowchart

applied in the optimization section using the genetic algorithm

RE-TRANSFORMATION PROCESS

Re-transformation of the optimum 6-group Arrangement to the real 16-group pattern by using

weighting factor concept

The process of re-transformation is based on calculating the radial power-peaking-factor distribution of

the proposed six-group arrangement. Then, based on the power peaking factor for the fuel assemblies in

groups one and two, the fuel assemblies will be positioned based on their C weighting factors while

observing the structural constraints.

It is performed in such a way that the assemblies with the highest C factor are located in places with the

lowest power peaking factors. This process will continue in this way until the last fuel assembly with

the lowest C factor is located in the place with the highest power peaking factor.

Thus, the units in groups one and two are divided into their subset fuel assemblies, and the optimum 6-

group arrangement is converted to 16 groups.

Re-transformation of the optimum few group patterns to the real mode by using ant colony

algorithm

As a second solution, the re-transforming process has been done by using the ant colony algorithm.

In this method, by using the η parameter (in the algorithm), the structural constraints can be easily

considered during the loading pattern optimization.

In order to escape from the local optimum solutions, the random values (between 0-1) were generated

for each position of the condensed groups.

In continuation, if the value of the random number in each position was lesser than 0.2, then the

uniform pheromone was assigned for the possible states of that position and subsequently the random

type of fuel assembly was selected for it.

To prevent from retreating of the optimization process, the value of the objective function of the

mentioned ant was compared with its previous value and in order of superiority of the previous state, it

is remained.

In this section, the initial population consists of 10 ants which have been generated randomly from the

6-group optimum pattern.

Fig.9 schematically describes the computational flowchart applied in this section.

Fig.9 A schematic description of the computational Flowchart applied

in the optimization section using the ant colony algorithm (in re-transforming process)

RESULTS

Figs.10 to 12 compares the rate of changes of the fitness function for the first state optimization

model (minimizing SPPE factor) in the 6 and 16-group calculations.

051015202530354045505560657075808590

0 1 2 3 4 5 6 7 8 9 10

Generation

Fitness function (SPPE)

Fig.10 Changes in the rate of first fitness function (SPPE)

during the generation 1 to10 in the 16-group state

11.9511.975

1212.02512.0512.07512.1

12.12512.1512.17512.2

12.22512.2512.27512.3

12.32512.3512.37512.4

11 12 13 14 15 16 17 18 19 20 21 22

Generation

Fitness function (SPPE)

Fig.11 Changes in the rate of first fitness function (SPPE) during

the generation 11 to 21 in the 16-group state

0

10

20

30

40

50

60

70

80

90

100

0 1 2 3 4 5 6 7 8 9 10 11 12 13

Generation

Fitness Function (SPPE)

Fig.12 Changes in the rate of first fitness function (SPPE)

in the 6-group state

By making the optimization calculations at the 16-group state, the optimum reloading pattern is

obtained after 21 generations. In spite of the convergence of the 6-group genetic algorithm calculations

in the 12th generation, this method reached the optimum arrangement equivalent to the 16-group state

from the 8th generation on.

Figures 13 and 14 describe the changes of the fitness function in the second re-transformation model(

by using ant colony algorithm).

In the second mode of optimization (using formula no.3),by making the optimization calculations at

the 16-group state, the optimization process is converged after 26 generations and in the 6-group

genetic algorithm calculations this method reached the optimum arrangement equivalent to the 16-

group state in the 11th generation .

146

146.2146.4

146.6146.8147

147.2

147.4

147.6147.8148

148.2

148.4148.6

148.8

0 1 2 3 4 5 6 7 8 9 10 11

Iteration

Averaged fitness function

Fig.13 the changes of the averaged fitness function of ants in each iteration

148.35

148.4

148.45

148.5

148.55

148.6

148.65

148.7

148.75

148.8

148.85

148.9

148.95

0 1 2 3 4 5 6 7 8 9 10 11

Iteration

The fitness value's of best Ant

Fig.14 Changes of the fitness function of the best ant in each iteration

It is observed that the number of evaluations in whole of 10 iterations of ant colony algorithm is much

lesser than the required evaluations for 2 generations of the genetic algorithm method (140 genomes).

Figs.15-16 shows the optimum reloading patterns in 6 and 16-group states of the second optimization

mode. It should be noted that, after re-transformation computations in the 6-group optimum

arrangement (by using the ant colony algorithm), exactly the same 16-group optimum arrangement was

obtained.

Fig.15 The proposed optimum reloading pattern of

the BUSHEHR VVER-1000 reactor (6-groups) in second mode of optimization

Fig. 16. The optimum reloading pattern of the VVER-1000 reactor in 16-group in the second mode of

optimization (obtained from the re-transformation process by ant colony algorithm and also the 16-

group genetic algorithm calculations)

As in the earlier state, it is observed that for an equal value of the fitness function, the 6-group

calculations enjoy a higher convergence velocity as compared with the 16-group model.

Table.7 describes the value of objective parameters in 16 and 6-group optimum reloading patterns (in

first mode of optimization).

TABLE .7 A DESCRIPTION OF THE BASIC PARAMETERS OF

THE PROPOSED OPTIMUM ARRANGEMENTS IN 6 AND 16-GROUP COMPUTATIONS

Optimization type SPPE Keff PPFmax

16 group 11.897610 1.1547 1.1539

6 group 12.502290 1.1552 1.1815

Figs.17-20 shows the optimum reloading patterns and distribution of the radial power peaking factor

of the optimum arrangements in 6 and 16-group computations(in first mode of optimization). It should

be noted that, after re-transformation computations in the 6-group optimum arrangement (by using

weighting factor method), exactly the same 16-group optimum arrangement was obtained.

Fig.17 The proposed optimum reloading pattern of

the BUSHEHR VVER-1000 (V-446) reactor (6-groups)

Fig.18 The radial power-peaking-factor distribution

of the proposed optimum reloading pattern (6-groups)

Fig. 19. The optimum reloading pattern of the VVER-1000 reactor in 16-group (obtained from the

retransformation process and also the 16-group genetic algorithm calculations)

Fig.20 The radial power-peaking-factor distribution of the proposed optimum reloading pattern (16-

groups)

Fig.21 describes the rates of change in the maximum power peaking factor of the proposed pattern

during the second cycle.

Fig. 21 The time dependent changes of maximum power peaking factor

in the proposed reloading pattern (second cycle)

DISCUSSION AND CONCLUSION

As Figs 10-14 show, using the proposed method in this research, identical optimum arrangements

can be achieved through the usual optimization method by using the genetic algorithm but with fewer

generations and, consequently, within a shorter run time.

Of course, the proposed method is not limited to use the genetic algorithm in the optimization

process; it can accelerate the reloading pattern optimization of the reactor using other stochastic

algorithms.

In addition, by comparing the re-transformed 16-group arrangement from the 6-group state and the

optimum pattern obtained from the 16-group state, it is observed that there is a complete conformity

between these two arrangements. This shows that the proposed method and also the applied

computational process in the re-transformation section are correct.

In this research, to increase the accuracy of the objective parameters computations in each of the

arrangements, thermo-hydraulic calculations were made and the effectiveness of temperature feedbacks

in the cell and neutronic calculations was discussed. Furthermore, all the structural constraints are

observed in this method, while in the other studies [3to10], the constraints were not considered.

As no research or data have ever been reported on the reloading pattern optimization of VVER-1000

reactors in the second cycle, it was impossible to make comparisons. Therefore, the time-dependent

thermo-neutronic computations in the second cycle were made to ensure the safety and desirability of

the proposed pattern during the cycle. As shown in Fig.21, the arrangement enjoys a desired safety

margin during the second cycle.

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