Core loading pattern optimization for research reactors

Pergamon

Ann. Nucl. Energy, Vol. 24, No. 7, pp. 509-514, 1997 © 1997 Elsevier Science Ltd P I I : S0306-4549(96)00042-4 Printed in Great Britain. All rights reserved 0306-4549/97 $17.00 + 0.00

CORE LOADING PATTERN OPTIMIZATION FOR RESEARCH REACTORS Y. P. MAHLERS Institute for Nuclear Research, Prospect Nauki 47, Kiev 252022, Ukraine (Received 19 March 1996)

Abstract--The algorithm based on successive mixed-integer linear programming and backward diffusion calculation is developed to determine the best locations of fuel assemblies in the core of research reactors. To determine optimum core loading patterns, the successive mixed-integer linear programming technique is applied. The backward diffusion calculation theory based on expression of fuel property in terms of power shape is used to lower computational cost of the algorithm by decreasing computer time necessary to calculate sensitivity coefficients. The algorithm developed is applied for real 2-D problems of the fuel management optimization for the V V R - M research reactor of the Kiev Institute for Nuclear Research (Ukraine). © 1997 Elsevier Science Ltd. All rights reserved. 1. INTRODUCTION The most effective way to increase utilization of a research reactor is to optimize placement of fuel assemblies in the core thus maximizing neutron flux densities in the reactor channels used for neutron physics research, radioisotope production and neutron transmutation doping of silicon. Although heuristic approaches exist and help to generate improved core loading patterns, they cannot provide the high quality of the solutions obtained. Fortunately, successive mixed-integer linear programming is valid for core loading pattern optimization of such reactors because of small fuel depletion during the cycle and, hence, possibility to restrict alterations in core characteristics arising during fuel shuffling to apply the linearity approximation. To decrease the computer time necessary to calculate sensitivity coefficients, backward diffusion calculation can be used (Chao et al., 1986; Mahlers, 1987). 2. RESEARCH REACTOR MODEL

We use the three-group diffusion model for neutronic simulation of the thermal research reactor with water coolant and water moderator in the following matrix form (Garusov and Petrov, 1972): 509

510

Y.P. Mahlers

= !a

Ll~i

}

keff

(1) (2)

L2qb2 = ~1

~, L3~3 + A Q = qb2

(3)

where kefr is the effective multiplication factor; CI)1,2,3--(~1,2,3, _ 1
(1,

(4)

~3 = T(kerf)Q,

where T(kefr) = (L1L2L3) -1 ~eff - L1L2"4

)

and I = unity matrix.

Matrix (L1L2L3) -1 exists because equation ( L I L z L 3 ) ~ 3 = 0 has only a trivial solution. Since ~[ = Qi/ui, we can express u in terms of Q: ui = Q i / { T Q } i ,

(5)

i = 1,2 . . . . . N.

Note that a similar relation can be obtained using the corewise Green's function method (Stillman et al., 1989).

3. O P T I M I Z A T I O N T E C H N I Q U E We consider the following optimization problem: Place N1 assemblies having fuel property Yl, N2 assemblies having fuel property Y2,..., m

Nm assemblies having fuel property ym( Z

Ark = N ) at the N geometrical regions of the

k=l

reactor to minimize some objective function F[u,Q,~3] subject to the constraints: N

ui = Q i / ~

~.~" Qj

for i = 1,2 . . . . . N

j=l N

Oi ~

Omax

for i = 1,2 . . . . . N,

P = Z

ViQi ~ Pmax,

i=1

where 0 < yl < y2 < ' " < Ym, Xij ~ {T(keff)}o., m is the number of distinct fuel property values in the fuel inventory, Vi is the volume of the ith geometrical region, P is the total reactor power and ui is the fuel property at the ith geometrical region. We use one node

Core loading pattern optimization

511

per assembly and assume that it is unnecessary to restrict changing of the effective multiplication factor. The objective function F can vary from problem to problem. ui can be written as follows: m

ui ----~_,Yk~ik, where ~ik E {0, 1}; i=1 N

~_,~ik=Nk for k = 1,2 . . . . . m;

~ik= 1 for i = 1,2 . . . . . N.

i=l

i=l

Variation of u i can be written in the following forms: 8U i :

EykS~ik and 8ui = E l z o S Q j + i=l

Pit~

(6)

j=l

where 3ui ]J~i ~

1,

8ij ------ 0,

i =j 1 76j

3Qj

8qui -- Xiju 2 --

0Ui

I;i -- 3(1/ken')

'

(7)

Qi

2X--'N OX~ Q; Ui

O(1/k~rf) Qi"

Note that 8ui = 0 if 8Qi = CQi for i -- 1,2 . . . . . N, where C is a constant (Mahlers, 1992). Indeed, we can use expressions (5) and (7) and obtain the following equations for this case: N

~Ui = E

[d.ijSQi = C ( u i - u~ / u i ) = O.

j=l

Using the expression for the total reactor power, we can add the relationship s 8P -= ~ ViSQi to equations (6) and obtain the set of equations describing the unique i=l

correspondence between 8Ql, ~Q2 . . . . . ~QN, 8(1/ken`) and ~Ul, 8u2 . . . . . 3us, 8P. Then we apply the successive linear programming method. The optimization procedure has the following form: We start with some feasible initial values ~ik for i = 1,2 ..... N and k = 1,2,...,m and determine u, Q, ken` and/z. Then we solve the following mixed-integer linear programming problem: \ Find ~Qi for i = 1,2 ..... N, 8P, 8( I ) and integer values ~ik for i = 1,2 ..... N and kEei that N

minimize 8F = E

fli subject to the constraints:

i=l

--~ik<~8~ik<~ 1--~ik for i = 1 , 2 . . . . . N and k c e i ;

512

Y.P. Mahlers

8Qi <~ Omax - Qi for i = 1,2 . . . . . N;

8P ~< Pmax - P;

ZN tzijSQ] --k vi8 ( k e ~ ) = Zm ykS~ik j=l

(8)

k=l

for i = 1,2 ..... N; N

Z

(9)

Vi,~Qi = SP;

i=l

~

3~k = 0 for i

1, 2

N;

kE~i

~.~ ~ik

= 0 for k = 1, 2 . . . . . m,

iEOk

where

OF

Uv~(

OF ~ . OF ) ;

k Eei if max{ 1,ni-M} ~
4. R E S U L T S The algorithm was applied for real problems of the fuel management optimization for the VVR-M research reactor of the Kiev Institute for Nuclear Research (Ukraine). This reactor has many experimental channels used for various purposes and contains 192 fuel assemblies located in the core asymmetrically.

Core loading pattern optimization

513

The three-group 2-D diffusion model with 1141 nodes of hexagonal spatial mesh for each group was used for neutronic simulation of the reactor. The backward diffusion calculation was examined by comparing with the forward diffusion calculation. Neutron flux densities and power densities calculated for several 2-D core configurations using the backward diffusion calculation were consistent with the results obtained by the forward diffusion calculation. Accuracy of the linearity approximation for various alterations of fuel property (/~u) was determined also. The results are shown in Table 1. For ~u ~< 0.2(ym - Yl), errors in power density alterations calculated using the linearity approximation did not exceed 3%. The optimization algorithm was applied to increase thermal neutron flux densities in the following large diameter channels which can be used for neutron transmutation doping of silicon: - - 51/58 located in the core (problem 1), - - 80/32 located in the core (problem 2), - - 29/60 located in the reflector (problem 3), all of these channels simultaneously (problem 4). -

-

The current core loading pattern of the V V R - M reactor of the Institute for Nuclear Research was chosen as the initial core loading pattern for the first iteration of the optimization procedure. The results are summarized in Table 2. To study the effect of running the algorithm from different starting points, the optimization procedure was performed for nine more variants with different initial core loading patterns chosen stochastically. For problem 4, the following set of the optimum values of the objective function was obtained: (5.372; 5.368; 5.356; 5.379; 5.370; 5.349; 5.371; 5.369; 5.363). The maximum difference in these values is about 0.6%. As shown in Table 2, it is possible to attain considerable increase in a local thermal neutron flux density, namely 39% for the channel 51/58, 43% for the channel 80/32 and 48% for the channel 29/60 if the other channels are not taken into account. If the problem is to increase the thermal neutron flux densities in all of these channels simultaneously, the optimization procedure is not so effective and only a 17% improvement in the average thermal neutron flux density can be gained. Calculations were performed on a personal computer with microprocessor Intel8 0 4 8 6 D X 2 Computational expenditures were about 10 rain per iteration. Table 1. Errors due to the linearity approximation

3ul(Ym - Yl)

0 0.1 0.2 0.3 0.4 0,5 0,6

Exact value of 3Q

0 -0.0985 -0.1834 -0.2754 -0.3756 -0.4851 -0.6048

3Q calculated using the linearity approximation

Exact value of 3kerr(10-5)

0 -0.0885 -0.1770 -0.2655 -0.3540 -0.4425 -0.5310

0 24 -45 68 -94 121 -151

~keff(10-5) calculated using the linearity approximation 0 -25 -49 74 -98 -123 -148

514

Y. P. Mahlers Table 2. Solution of core loading pattern optimization problems Value of the objective function

Number of iterations 1 2 3 4 5 6 7 8 9 10

Problem 1

Problem 2

5.196 6.154 6.928 7.212 7.212

2.917 3.487 3.803 3.961 4.026 4.078 4.122 4.158 4.169 4.169

Problem 3 4.936 6.260 6.948 7.078 7.223 7.303 7.303

Problem 4 4.570 4.797 5.020 5.135 5.196 5.245 5,298 5.326 5.352 5.352

5. CONCLUSIONS

The algorithm based on successive mixed-integer linear programming and backward diffusion calculation presents a powerful tool for determination of the best locations of fuel assemblies in the core of research reactors. The algorithm is valid for solution of real 2-D fuel management optimization problems on personal computers.

Acknowledgements--The work was partially supported by a contract with the International Atomic Energy Agency.

REFERENCES

Chao, Y. A., Hu, C. W. and Suo, C. A. (1986) Nucl. Sci. Engng 93, 78. Garusov, E. A. and Petrov, Y. V. (1972) Atom. Energ. 32, 225 (in Russian). Khromov, V. V., Kashutin, A. A. and Glebov, V. B. (1974) Atom. Energ. 36, 385 (in Russian). Mahlers, Y. P. (1987) Metody i Algoritmy v Issled. Fiziki Yadern. Reakt., p.19. Energoatomizdat, Moscow (in Russian). Mahlers, Y. P. (1991) Ann. Nucl. Energy 18, 661. Mahlers, Y. P. (1992) Atorm. Energ. 72, 624 (in Russian). Shvedov, M. O. and Kashutin, A. A. (1986) Fizika i Metody Rastcheta Yadern. Reakt., p. 19. Energoatomizdat, Moscow (in Russian). Stillman, J. A., Chao, Y. A. and Downar, T. J. (1989) Nucl. Sci. Engng 103, 321.