- Email: [email protected]
LONSA as a tool for loading pattern optimization for VVER-1000 using synergy of a neural network and simulated annealing
Annals of Nuclear Energy 35 (2008) 1968–1973
Contents lists available at ScienceDirect
Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene
Technical note
LONSA as a tool for loading pattern optimization for VVER-1000 using synergy of a neural network and simulated annealing A.H. Fadaei *, S. Setayeshi Faculty of Nuclear Engineering and Physics, Amirkabir University of Technology (Tehran Polytechnique), Hafez Street, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 12 December 2007 Received in revised form 3 May 2008 Accepted 9 May 2008 Available online 25 June 2008
a b s t r a c t This paper presents a new method for loading pattern optimization in VVER-1000 reactor core. Because of the immensity of search space in fuel management optimization problems, finding the optimum solution requires a huge amount of calculations in the classical method, while neural network models, with massively parallel structures, accompanied by simulated annealing method are powerful enough to find the best solution in a reasonable time. Hopfield neural network operates as a local minimum searching algorithm; and for improving the obtained result from neural network, simulated annealing is used. Simulated annealing, because of its stochastic nature, can provide for the escape of the result of Hopfield neural network from a local minimum and guide it to the global minimum. In this study, minimization of radial power peaking factor inside the reactor core of Bushehr NPP is considered as the objective. The result is the optimum configuration, which is in agreement with the pattern proposed by the designer. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Fuel management is a branch of nuclear engineering, which calculates a reactor’s first and subsequent cores for producing full power within adequate safety margins (Levine, 1986). Considering the core of a VVER-1000 with 163 fuel assemblies, it is necessary to compute 163! possible configurations to find the best one. In this paper, LONSA (loading optimization by neural network and simulated annealing) program has been introduced. The method of continuous Hopfield neural network accompanied by simulated annealing (SA) is applied for solving fuel management problem of a typical VVER, Bushehr NPP, based on LONSA program. Compared with the classical methods, the proposed method is much simpler to implement and the results are more likely to be optimum. 2. Nuclear fuel management Nuclear fuel management (Turinsky and Parks, 1999; Turinsky et al., 2005) involves making the following decisions: the quantity and attributes of the fresh fuel assemblies that will be purchased, the partially burnt fuel assemblies that will be reinserted, the locations of both the fresh and partially burnt fuel assemblies within the core, i.e., core loading pattern (LP), and for a boiling water reactors the control rod program/core flow (CRP/CF) strategy. These
* Corresponding author. Tel.: +98 21 88630644; fax: +98 21 88417576. E-mail addresses: [email protected] (A.H. Fadaei), [email protected] (S. Setayeshi). 0306-4549/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.anucene.2008.05.001
decisions need to be made for each reload cycle. The objective of nuclear fuel management is to minimize the nuclear fuel cycle cost while satisfying the cycle energy requirement. This must be done such that all safety and operational constraints are satisfied with sufficient margin. To get a grasp of the magnitude of the decision space, consider the following decisions that need to be made in each reload cycle: fresh fuel lattice designs (pin-by-pin radial position fuel enrichment and burnable poison loading), fresh fuel bundle designs (axial span of different lattice designs), number of fresh fuel bundle designs of each type, partially burnt fuel assemblies to be reinserted, location of fresh and partially burnt fuel assemblies within the core, and for BWRs the CRP/CF strategy as a function of cycle exposure. Constraining these decisions are various operational and safety limits as now indicated: reactivity limits (coolant/moderator density reactivity coefficient, shutdown margin, ejected or dropped rod worth, hot excess reactivity), thermal margins (total and radial power peaking factors, MFLCPR, MAPRAT, MFLPD), and mechanical/material limits (pin, assembly and region average discharge burnups, vessel fluence, excore detector count rate, and restrictions on fuel assembly placement {control cell core}). In the past and to this day, the ingenuity of the reload core design engineer has been used to make these decisions, finding feasible and near-optimum core designs. To assist the reload core design engineer various computer aides have been developed, such as computer code linkage buffer codes, automated design calculational sequences (determination of moderator temperature coefficient versus moderator temperature, cycle burnup and soluble boron concentration), graphical interfaces to setup input (core loading pattern) and interpret output, and the application of mathematical optimization capabilities (Oyarzun et al., 2003). It is this
A.H. Fadaei, S. Setayeshi / Annals of Nuclear Energy 35 (2008) 1968–1973
last item, the application of mathematical optimization capabilities that this paper will focus on. From the above discussion, we see that the nuclear fuel management decision-making problem is highly constrained and has a decision space that is very large (can approach 10100). Further complicating the application of mathematical optimization capabilities are that to evaluate the objective functions and core response constraints requires considerable computational effort (solution of the few-group neutron diffusion equation as a function of rods configuration over the cycle); that the problem is nonlinear; that there is a lack of derivative information with respect to objective function and constraint values dependencies on decision variables; that the feasible space is disjointed; that both continuous and integer decision variables enter; and that multi-cycles should be evaluated, since decisions made for the current cycle impact subsequent cycle decisions, and since to evaluate fuel cycle costs, fuel must be tracked from initial fabrication to final disposition. Two prominent mathematical optimization methodologies have evolved that appear to have various degrees of applicability to the nuclear fuel management decision-making problem. More classical mathematical optimization methods, such as linear, quadratic and dynamic programming, so far have met with limited success with a few exceptions (Lam et al., 2003), since they are likely not appropriate due to the attributes noted above. In VVERs, the fuel-loading problem has a significant effect on safety and economics. Loading pattern for the first cycle should be satisfying safety constraints, such as local power peaking factor Pmax, maximum burnup BUmax, and to produce maximum energy during the cycle (Kim et al., 1993). Many fuel-shuffling methods have been studied to satisfy these objectives, that is, to obtain the optimal loading patterns. Some of these methods are as follows: linear programming, dynamic programming, variational OCT, perturbation theory, direct search, heuristic/artificial intelligence (Discroll et al., 1990). None of these optimization approaches ensure the global optimum solution because of the limitations of their search algorithms; they can only find near-optimum solutions (Kim et al., 1993). To resolve these limitations, we suggest using a combination of neural network modeling with a highly parallel structure such as a Hopfield network, which is suitable for optimization ( Hopfield, 1982, 1984; Hopfield and Tank, 1985, 1986), and simulated annealing method (Van Laarhoveen and Aarts, 1987) which is used as a local minimum escape algorithm (Freeman and Skapura, 1991).
3. LONSA (loading optimization by neural network and simulated annealing) In this study, a LONSA as a new program for fuel management has been developed. This program can obtain an optimized loading pattern for VVER-1000 reactor core. The LONSA (loading optimization by neural network and simulated annealing) program uses the FORTRAN programming language as the base medium and has the following three subroutines: neural network, Hopfield; simulated annealing; and neutronic calculations.
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the optimization problem if it is close to a corner of the hypercube (i.e., if the activation value of each unit is close to zero or one). The continuous model is superior to the discrete model in terms of the local minima problem because of its smoother energy surface (Peng et al., 1996). More information about Hopfield neural network for optimization procedure is published by Hopfield and Tank (1985) and Sadighi et al. (2002). In the neural network subroutine, at first, a neuron is selected, and then selected neuron is updated by using equations of motion based on energy function. Neutronic calculation is performed by using of neutronic calculation’ subroutine for calculating energy function. 3.2. Simulated annealing To resolve the problem in this study, we needed to investigate other algorithms to escape local minimum and ensure reaching global minimum (Kirkpatrick et al., 1983). In this study, simulated annealing (SA) method is used as a local minimum escape algorithm. The SA subroutine is performing the algorithm, which provides for the escape of the network from local minimum and guide it to global minimum. SA is probably the most developed of the stochastic optimization methods (Van Laarhoveen and Aarts, 1987; Aarts and Korst, 1989). Although it may appear to be a very simple method to implement; however, the reality is that to obtain both robustness (so that it consistently locates the family of nearoptimum decisions) and efficiency (so that it minimizes the number of histories that must be examined to locate the family of near-optimum decisions), some thought must be given. SA is based upon the analogy of a solid slowly cooling to its lowest energy state, i.e., annealing. More information about simulated annealing algorithm for optimization procedures can be found in Van Laarhoveen and Aarts (1987), Aarts and Korst (1989), and Sadighi et al. (2002). Neural network and simulated annealing works in tandem, whereby at first the neural network calculates a local minima, and then LONSA improves this result by applying simulated annealing for escaping from local minima and reaching global minima. In other words, the initial point of SA is the Hopfield neural network result. 3.3. Neutronic calculations The neutronic simulation of reactor core is performed by using of WIMS (for cell calculation) and CITATION (for core calculation) codes. CITATION is a three-dimensional, multi group diffusion code (Oak Ridge National Laboratory, 1972), and is used for core calculations. The macroscopic cross-sections are prepared by using WIMS D-4 code (Winfrith, 1985). These two codes were linked with FORTRAN software for simulating the reactor core. Unlike PWR reactors, which have square lattice fuel assemblies, the VVER-1000 reactor core uses a hexagonal lattice. A detailed discussion about the simulation of VVER-1000 reactor core can be found in Faghihi et al. (2007).
3.1. Hopfield neural network
4. Mapping of the fuel management problem on the Hopfield network
Neural network subroutine is performed network calculation and update of each neuron based on energy function that was obtained from neutronic calculation and network parameters. Hopfield and Tank extended the original model to a fully interconnected network of nonlinear analog units, where the activation level of each unit is a value in the interval [0,1]. This network operates as a local minima-searching algorithm. Obviously, the final configuration of the network can be decoded into a solution of
One of the main objective functions in the fuel management optimization is power peaking factor minimization. The fuel assemblies should thus be arranged in the core in such a manner that the neutron flux is flattened as high possible (Discroll et al., 1990). In view of the fact that each fuel should be assigned to one position and each position should be occupied by one fuel, the constraints of the problem are characterized. Objective function,
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A.H. Fadaei, S. Setayeshi / Annals of Nuclear Energy 35 (2008) 1968–1973
minimization of power peaking factor, in this case should be calculated in each neuron transition using the neutronic codes. We design a Hopfield neural network with N N = N2 neurons (N rows and N columns), where N is the number of fuel assemblies in 1/6th of the reactor core. In this way Vi,j represents the output of neuron in row ‘i’ and column ‘j’, which corresponds to the weight of placing fuel ‘j’ in position ‘i’ in the core. The energy function for this case is as follows (Hopfield and Tank, 1985):
E ¼ E1 þ E2 þ E3 þ E4 8 !2 9
ð1Þ ð2Þ
ð3Þ ð4Þ
i
E4 ¼ D=2ðF R 1Þ2
ð5Þ
where, FR is the radial power peaking factor, defined as:
R þH=2 FR ¼
1 p R2
PðRHC ; zÞdz H=2 R R R þH=2 Pðr; zÞdzdr 0 H=2
ð6Þ
where, P(r, z) is the power of reactor in radius r and height z and RHC is the radius at which that hot channel has occurred. The parameter E1 is the energy of the first constraint (if there is one ‘1’ in each row this term will be zero), E2 is the energy of the second constraint (if there is one ‘1’ in each column, this term will be zero), and E3 is the energy that corresponds to the total value of a neuron’s output. The parameter E4 is the energy of the objective function. The method outlined in this paper could be applied to the other objective functions, which are not considered here. Parameters A, B, and D are constants that will be determined by trial and error method. In this form of the energy function, the network will converge if the activity of each neuron changes with time according to the differential equations previously published by Sadighi et al. (2002). The network is converged when terms E1 and E2 (constraint terms) are zero. The quality of the obtained solution depends on the E4 value; a lower value of E4 is an indication of a better fuel assembly configuration. After finding one acceptable solution by the Hopfield network, it will be improved by SA application.
5. Simulation results We have applied the continuous Hopfield model accompanied by SA to find the optimum-loading pattern for the first cycle of Bushehr NPP and compare it with the pattern proposed by the designer (the standard method). In this study, the positions of seven fuel assemblies in the periphery of the core and the central assembly are fixed; therefore should determine the best positions for remaining 20 assemblies. It means that the states of 20 20 = 400 neurons should be obtained which will be finally 0 or 1. Figs. 1–5 show the evolution of the Hopfield network from the initial to the final states where it is converged to an acceptable solution. As expected, the solution found by Hopfield network is a local minimum of energy function. Most of the times these solutions have good quality. In this study with the well-tuned parameters of the Hopfield network (A = 2000, B = 500, D = 1000, a = 0.05, and dt = 0.5), the network energy normally falls between 60 and 53,000. By comparing the energy of this example, Fig. 4 (E = 61.32), with the energy of the configuration proposed by the VVER Russian Designer (Fig. 5), which is 76.05, we can conclude that the solution found by the Hopfield network is near optimum. As the Hopfield solution in our study is near optimum, LONSA start the SA subroutine with this initial point at a low temperature (Tin = 100 K) and reduces it gradually in each step (annealing coefficient = 0.95). Note that Tin and the annealing coefficient parameters are selected by investigating the SA method’s performance, i.e., velocity and quality of convergence, by analyzing the results for different seeds. In this way reaching to the global minimum in an acceptable time is assured. Figs. 6–8 show the process of reaching the global minimum. As the power peaking factor is reduced, the neutron flux tends towards a flat distribution. In Table 1, we have calculated FR by using the CITATION code’s output and Eq. (6) for the three core configurations that have been discussed. As noted, the calculated results by LONSA for loading optimization has lower power peaking factor in comparison with the configuration proposed by VVER-1000. The LONSA code thus has good reliability for finding the optimum core configuration. 6. Concluding remarks In this study, we introduce the LONSA (loading optimization by neural network and simulated annealing) program for solving the fuel-loading pattern optimization. We applied the continuous
Fig. 1. Evolution of neurons in the Hopfield network – first step.
A.H. Fadaei, S. Setayeshi / Annals of Nuclear Energy 35 (2008) 1968–1973
Fig. 2. Evolution of neurons in the Hopfield network – 67th step.
Fig. 3. Evolution of neurons in the Hopfield network – 222nd step.
Fig. 4. Evolution of neurons in the Hopfield network – 376th step (network is converged) (E = 61.32).
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A.H. Fadaei, S. Setayeshi / Annals of Nuclear Energy 35 (2008) 1968–1973
Fig. 5. Configuration that is proposed by VVER for first loading of Bushehr NPP – (E = 76.05).
Fig. 8. Simulated annealing process for first loading of Bushehr NPP – last iteration (T = 0.246 101 and E = 22.79).
Table 1 Comparison of the results Method
Network energy E4
FR
Number of iterations
Hopfield VVER-1000(Standard) Hopfield + SA
61.32 76.05 22.79
1.35 1.39 1.21
376 – 539
design proposed by the designer (BUSHER NPP, 1998). Since the solution that is provided by the Hopfield network is a local minimum, we improved the solution of this network by using the simulated annealing (SA) method. The final result that is obtained by this method is closer to the optimum solution. Appendix 1. Hopfield neural network
Fig. 6. Simulated annealing process for first loading of Bushehr NPP – first iteration (T = 100 °K and E = 64.24).
Hopfield and Tank extended the original model to a fully interconnected network of nonlinear analog units, where the activation level of each unit is a value in the interval [0,1]. The Hopfield model has two versions: continuous and discrete. For the continuous version, neurons have continuous values within (0,1) and the dynamics of the network are governed by a set of non-linear differential equations. For the discrete version, the neurons have only two possible values, 0 or 1, and the dynamics of the network are characterized by a set of non-linear difference equations (Hopfield, 1982, 1984). The space of possible configurations {Vi}i = 1,...,L in continuous model is rather than discrete, and is bounded by the L-dimensional hypercube defined by Vi = 0 or 1. The evolution of the units over time is now characterized by the following differential equations (usually called ‘‘equations of motion”)
dU i =dt ¼
X
T V j ij j
þ Ii U i ;
i ¼ 1; :::; L
ðA1Þ
where Ui, Ii, and Vi are the input, input bias, and activation level of unit i, respectively. The activation level of unit i is a function of its input, namely Fig. 7. Evolution of energy in the simulated annealing procedure for first loading of Bushehr NPP.
Hopfield network to the fuel management optimization problem of Bushehr NPP, and the obtained solution was compared with the
V i ¼ gðU i Þ ¼ 1=2 ð1 þ tanh aU i Þ ¼ 1=ð1 þ e2aUi Þ
ðA2Þ
The activation function g is the well-known sigmoidal function, which always returns a value between 0 and 1. The parameter a is used to modify the slope of the function. In Fig. 1, for example, the a value is lower for curve (A2) than for curve (A1).
A.H. Fadaei, S. Setayeshi / Annals of Nuclear Energy 35 (2008) 1968–1973
Since 1985, when the analog (continuous) Hopfield network was first used for the traveling salesman problem (TSP), its application has been extended too many areas (Hopfield and Tank, 1985). Hopfield network operates as a local minima-searching algorithm. Obviously, the final configuration of the network can be decoded into a solution of the optimization problem if it is close to a corner of the hypercube (i.e., if the activation value of each unit is close to zero or one). The continuous model is superior to the discrete in terms of the local minima problem because of its smoother energy surface (Peng et al., 1996). The energy function for the continuous Hopfield–Tank model is:
E ¼ 1=2
XX i
T VV j ij i j
X
VI þ i i i
Z
Vi
g 1 ðxÞdx
ðA3Þ
0
Note in particular that dUi/dt = dE/dVi. Accordingly, when the units obey the dynamics of the equations of motion, the network is performing a gradient descent in the network’s configuration space with respect to that energy function, and stabilizes at a local minimum. At that point, dUi/dt = 0 and the input to any given unit i is the weighted sum of the activation levels of all the other units plus the bias, that is
Ui ¼
X
T V j ij j
þ Ii
ðA4Þ
To resolve problem in this study, we should investigate other algorithms to escape local minimum and ensure reaching global minimum (Kirkpatrick et al., 1983; Kirkpatrick et al., 1984). In this study, SA method is used as a local minimum escape algorithm. Appendix 2. Simulated annealing (SA) Likely the most developed of the stochastic optimization methods is the SA method (Van Laarhoveen and Aarts, 1987; Aarts and Korst, 1989). Although it appears to be very simple method to implement, in the reality to obtain both robustness and efficiency, careful consideration is required. SA is based upon the analogy of a solid slowly cooling to its lowest energy state, i.e., annealing. Given a minimization objective, let F denote the value of the objective function for the currently accepted decision variables, and F denote the value of the objective function after some perturbations are made to the currently accepted decision variables. Whether the perturbed decision variables are accepted, implying they now become the currently accepted decision variables, is determined by the following rules.
If F 6 F or if F > F and random½0; 1 < eðF
FÞ=KT
then F ¼ F otherwise F ¼ F ðB1Þ
So if the perturbed decision variables produce a lower objective function value, then they become the currently accepted decision variables. However, if the perturbed decision variable produces a higher objective function value, they only become the currently accepted decision variables if a random number between zero and one exceeds the expression noted in Eq. (5); otherwise, the currently accepted decision variables are unaltered. The conditional acceptance of inferior solutions allows the search algorithm to escape local minimums, and as we shall see when utilizing penalty functions, traverse the infeasible decision space. The parameter T plays the analog of material temperature in annealing. At high temperatures, many inferior solutions are ac-
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cepted, allowing the search space to be extensively transversed in search of the vicinity of the global minimum. As temperature decreases, the probability of acceptance of inferior solutions decreases, since if the cooling schedule is done appropriately one should now be within the vicinity of the family of global optimum solutions. What the initial temperature should be and how fast cooling should occur determine the robustness and efficiency of the SA implementation. Fortunately, both of these attributes can be determined based upon the specific behavior of the optimization problem that is being solved as the search progresses (Kropaczek et al., 1994). Another item that must be addressed is at what temperature cooling should be ended and the optimization search terminated. This can be triggered by a combination of lower temperature limit, maximum number of histories, and lack of improvement in the objective function value. References Aarts, E., Korst, J., 1989. Simulated Annealing and Boltzmann Machines – A Stochastic Approach to Combinatorial Optimization and Neural Computing. John Wiley & Sons, New York. Discroll, M.J., Downar, T.J., Pilat, E.E., 1990. The Linear Reactivity Model for Nuclear Fuel Management. American Nuclear Society. Freeman, J.A., Skapura, D.M., 1991. Neural Network Algoritms; Application and Programming Techniques. Addison-Wesley Publishing Co. Faghihi, F., Fadaie, A.H., Sayareh, R., 2007. Reactivity coefficients simulation of the Iranian VVER-1000 nuclear reactor using WIMS and CITATION codes. Progress in Nuclear Energy (Elsevier Journal) 49 (1), 68–78. Hopfield, J.J., 1982. Neural networks and physical systems with emergent collective computational properties. Proceedings the National Academy of Science (USA) 79, 2554–2558. Hopfield, J.J., 1984. Neurons with graded response have collective computational properties like those of two-state neurons. Proceedings the National Academy of Science (USA) 81, 3088–3092. Hopfield, J.J., Tank, D.W., 1985. Neural computations of decisions in optimization problems. Biological Cybernetics 52, 141–152. Hopfield, J.J., Tank, D.W., 1986. Computing with neural networks: a model. Science 233, 625–632. Kirkpatrick, S., Gelatt, C.D., Vecchi, M.P., 1983. Optimization by simulated annealing. Science 220 (4598), 671. Kim, H.G., Change, S.H., Lee, B.H., 1993. Optimal fuel loading pattern design using an artificial neural network and a fuzzy rule-based system. Nuclear Science and Engineering 115, 152–163. Kropaczek, D.J., Turinsky, P.J., Maldonado, G.I., Parks, G.T., 1994. The Efficiency and Fidelity of the In-Core Fuel Management Code FROMOSA-P. In: Proceedings of the International Conference on Reactor Physics and Reactor Computations, Tel Aviv, 572, INS/ENS. Levine, S., 1986. In-Core Fuel Management of Four Reactor Types. Handbook of Nuclear Reactor Calculation, vol. II. CRC Press. Lam, H.Q., Sato, D., Little, D.C., Popa, F.D., 2003. LP-FUN Applications to Reload Design. In: Proceedings of Advances in Nuclear Fuel Management III Topical Meeting, Hilton Head, SC. Oak Ridge National Laboratory, 1972. CITATION -LDI2 code. Oyarzun, C.C., Kropaczek, D.J., Sutton, S.B., Russell, W.E., 2003. The Global Nuclear Fuel Optimization System for BWR Fuel Cycle Management. In: Proceedings of Advances in Nuclear Fuel Management III Topical Meeting, Hilton Head, SC. Peng, M., Gupta, N.K., Amitage, A.F., 1996. An investigation into improvement of local minima of the Hopfield network. Neural Network 9 (7), 1241–1253. Sadighi, M., Setayeshi, S., Salehi, A.A., 2002. PWR fuel management optimization using neural networks. Annals of Nuclear Energy 29 (41–45). Turinsky, P.J., Parks, G.T., 1999. Chapter: Advances in Nuclear Fuel Management Optimization for Light Water Reactors. Advances in Nuclear Science and Technology, vol. 37. Kluwer Academic/Plenum Publishers, NYC. pp. 137. Turinsky, P.J., Keller, P.M., Abdel-khalik, H.S., 2005. Evolution of nuclear fuel management and reactor operator aid tools. Nuclear Engineering and Technology 37 (1). Van Laarhoveen, P.J.M., Aarts, E.H.L., 1987. Simulated Annealing: Theory and Applications. D. Reidel Publishing Company, Dordrecht, Holland. Winfrith, 1985. LWR-WIMS, a Computer Code For Light Water Reactor Calculations, UK: AEE, AEEW-R 1498. BUSHER NPP Neutronic calculations of the core. Report RRC KI No 32/1-38-498, 1998.
Contents lists available at ScienceDirect
Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene
Technical note
LONSA as a tool for loading pattern optimization for VVER-1000 using synergy of a neural network and simulated annealing A.H. Fadaei *, S. Setayeshi Faculty of Nuclear Engineering and Physics, Amirkabir University of Technology (Tehran Polytechnique), Hafez Street, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 12 December 2007 Received in revised form 3 May 2008 Accepted 9 May 2008 Available online 25 June 2008
a b s t r a c t This paper presents a new method for loading pattern optimization in VVER-1000 reactor core. Because of the immensity of search space in fuel management optimization problems, finding the optimum solution requires a huge amount of calculations in the classical method, while neural network models, with massively parallel structures, accompanied by simulated annealing method are powerful enough to find the best solution in a reasonable time. Hopfield neural network operates as a local minimum searching algorithm; and for improving the obtained result from neural network, simulated annealing is used. Simulated annealing, because of its stochastic nature, can provide for the escape of the result of Hopfield neural network from a local minimum and guide it to the global minimum. In this study, minimization of radial power peaking factor inside the reactor core of Bushehr NPP is considered as the objective. The result is the optimum configuration, which is in agreement with the pattern proposed by the designer. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Fuel management is a branch of nuclear engineering, which calculates a reactor’s first and subsequent cores for producing full power within adequate safety margins (Levine, 1986). Considering the core of a VVER-1000 with 163 fuel assemblies, it is necessary to compute 163! possible configurations to find the best one. In this paper, LONSA (loading optimization by neural network and simulated annealing) program has been introduced. The method of continuous Hopfield neural network accompanied by simulated annealing (SA) is applied for solving fuel management problem of a typical VVER, Bushehr NPP, based on LONSA program. Compared with the classical methods, the proposed method is much simpler to implement and the results are more likely to be optimum. 2. Nuclear fuel management Nuclear fuel management (Turinsky and Parks, 1999; Turinsky et al., 2005) involves making the following decisions: the quantity and attributes of the fresh fuel assemblies that will be purchased, the partially burnt fuel assemblies that will be reinserted, the locations of both the fresh and partially burnt fuel assemblies within the core, i.e., core loading pattern (LP), and for a boiling water reactors the control rod program/core flow (CRP/CF) strategy. These
* Corresponding author. Tel.: +98 21 88630644; fax: +98 21 88417576. E-mail addresses: [email protected] (A.H. Fadaei), [email protected] (S. Setayeshi). 0306-4549/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.anucene.2008.05.001
decisions need to be made for each reload cycle. The objective of nuclear fuel management is to minimize the nuclear fuel cycle cost while satisfying the cycle energy requirement. This must be done such that all safety and operational constraints are satisfied with sufficient margin. To get a grasp of the magnitude of the decision space, consider the following decisions that need to be made in each reload cycle: fresh fuel lattice designs (pin-by-pin radial position fuel enrichment and burnable poison loading), fresh fuel bundle designs (axial span of different lattice designs), number of fresh fuel bundle designs of each type, partially burnt fuel assemblies to be reinserted, location of fresh and partially burnt fuel assemblies within the core, and for BWRs the CRP/CF strategy as a function of cycle exposure. Constraining these decisions are various operational and safety limits as now indicated: reactivity limits (coolant/moderator density reactivity coefficient, shutdown margin, ejected or dropped rod worth, hot excess reactivity), thermal margins (total and radial power peaking factors, MFLCPR, MAPRAT, MFLPD), and mechanical/material limits (pin, assembly and region average discharge burnups, vessel fluence, excore detector count rate, and restrictions on fuel assembly placement {control cell core}). In the past and to this day, the ingenuity of the reload core design engineer has been used to make these decisions, finding feasible and near-optimum core designs. To assist the reload core design engineer various computer aides have been developed, such as computer code linkage buffer codes, automated design calculational sequences (determination of moderator temperature coefficient versus moderator temperature, cycle burnup and soluble boron concentration), graphical interfaces to setup input (core loading pattern) and interpret output, and the application of mathematical optimization capabilities (Oyarzun et al., 2003). It is this
A.H. Fadaei, S. Setayeshi / Annals of Nuclear Energy 35 (2008) 1968–1973
last item, the application of mathematical optimization capabilities that this paper will focus on. From the above discussion, we see that the nuclear fuel management decision-making problem is highly constrained and has a decision space that is very large (can approach 10100). Further complicating the application of mathematical optimization capabilities are that to evaluate the objective functions and core response constraints requires considerable computational effort (solution of the few-group neutron diffusion equation as a function of rods configuration over the cycle); that the problem is nonlinear; that there is a lack of derivative information with respect to objective function and constraint values dependencies on decision variables; that the feasible space is disjointed; that both continuous and integer decision variables enter; and that multi-cycles should be evaluated, since decisions made for the current cycle impact subsequent cycle decisions, and since to evaluate fuel cycle costs, fuel must be tracked from initial fabrication to final disposition. Two prominent mathematical optimization methodologies have evolved that appear to have various degrees of applicability to the nuclear fuel management decision-making problem. More classical mathematical optimization methods, such as linear, quadratic and dynamic programming, so far have met with limited success with a few exceptions (Lam et al., 2003), since they are likely not appropriate due to the attributes noted above. In VVERs, the fuel-loading problem has a significant effect on safety and economics. Loading pattern for the first cycle should be satisfying safety constraints, such as local power peaking factor Pmax, maximum burnup BUmax, and to produce maximum energy during the cycle (Kim et al., 1993). Many fuel-shuffling methods have been studied to satisfy these objectives, that is, to obtain the optimal loading patterns. Some of these methods are as follows: linear programming, dynamic programming, variational OCT, perturbation theory, direct search, heuristic/artificial intelligence (Discroll et al., 1990). None of these optimization approaches ensure the global optimum solution because of the limitations of their search algorithms; they can only find near-optimum solutions (Kim et al., 1993). To resolve these limitations, we suggest using a combination of neural network modeling with a highly parallel structure such as a Hopfield network, which is suitable for optimization ( Hopfield, 1982, 1984; Hopfield and Tank, 1985, 1986), and simulated annealing method (Van Laarhoveen and Aarts, 1987) which is used as a local minimum escape algorithm (Freeman and Skapura, 1991).
3. LONSA (loading optimization by neural network and simulated annealing) In this study, a LONSA as a new program for fuel management has been developed. This program can obtain an optimized loading pattern for VVER-1000 reactor core. The LONSA (loading optimization by neural network and simulated annealing) program uses the FORTRAN programming language as the base medium and has the following three subroutines: neural network, Hopfield; simulated annealing; and neutronic calculations.
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the optimization problem if it is close to a corner of the hypercube (i.e., if the activation value of each unit is close to zero or one). The continuous model is superior to the discrete model in terms of the local minima problem because of its smoother energy surface (Peng et al., 1996). More information about Hopfield neural network for optimization procedure is published by Hopfield and Tank (1985) and Sadighi et al. (2002). In the neural network subroutine, at first, a neuron is selected, and then selected neuron is updated by using equations of motion based on energy function. Neutronic calculation is performed by using of neutronic calculation’ subroutine for calculating energy function. 3.2. Simulated annealing To resolve the problem in this study, we needed to investigate other algorithms to escape local minimum and ensure reaching global minimum (Kirkpatrick et al., 1983). In this study, simulated annealing (SA) method is used as a local minimum escape algorithm. The SA subroutine is performing the algorithm, which provides for the escape of the network from local minimum and guide it to global minimum. SA is probably the most developed of the stochastic optimization methods (Van Laarhoveen and Aarts, 1987; Aarts and Korst, 1989). Although it may appear to be a very simple method to implement; however, the reality is that to obtain both robustness (so that it consistently locates the family of nearoptimum decisions) and efficiency (so that it minimizes the number of histories that must be examined to locate the family of near-optimum decisions), some thought must be given. SA is based upon the analogy of a solid slowly cooling to its lowest energy state, i.e., annealing. More information about simulated annealing algorithm for optimization procedures can be found in Van Laarhoveen and Aarts (1987), Aarts and Korst (1989), and Sadighi et al. (2002). Neural network and simulated annealing works in tandem, whereby at first the neural network calculates a local minima, and then LONSA improves this result by applying simulated annealing for escaping from local minima and reaching global minima. In other words, the initial point of SA is the Hopfield neural network result. 3.3. Neutronic calculations The neutronic simulation of reactor core is performed by using of WIMS (for cell calculation) and CITATION (for core calculation) codes. CITATION is a three-dimensional, multi group diffusion code (Oak Ridge National Laboratory, 1972), and is used for core calculations. The macroscopic cross-sections are prepared by using WIMS D-4 code (Winfrith, 1985). These two codes were linked with FORTRAN software for simulating the reactor core. Unlike PWR reactors, which have square lattice fuel assemblies, the VVER-1000 reactor core uses a hexagonal lattice. A detailed discussion about the simulation of VVER-1000 reactor core can be found in Faghihi et al. (2007).
3.1. Hopfield neural network
4. Mapping of the fuel management problem on the Hopfield network
Neural network subroutine is performed network calculation and update of each neuron based on energy function that was obtained from neutronic calculation and network parameters. Hopfield and Tank extended the original model to a fully interconnected network of nonlinear analog units, where the activation level of each unit is a value in the interval [0,1]. This network operates as a local minima-searching algorithm. Obviously, the final configuration of the network can be decoded into a solution of
One of the main objective functions in the fuel management optimization is power peaking factor minimization. The fuel assemblies should thus be arranged in the core in such a manner that the neutron flux is flattened as high possible (Discroll et al., 1990). In view of the fact that each fuel should be assigned to one position and each position should be occupied by one fuel, the constraints of the problem are characterized. Objective function,
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minimization of power peaking factor, in this case should be calculated in each neuron transition using the neutronic codes. We design a Hopfield neural network with N N = N2 neurons (N rows and N columns), where N is the number of fuel assemblies in 1/6th of the reactor core. In this way Vi,j represents the output of neuron in row ‘i’ and column ‘j’, which corresponds to the weight of placing fuel ‘j’ in position ‘i’ in the core. The energy function for this case is as follows (Hopfield and Tank, 1985):
E ¼ E1 þ E2 þ E3 þ E4 8 !2 9
ð1Þ ð2Þ
ð3Þ ð4Þ
i
E4 ¼ D=2ðF R 1Þ2
ð5Þ
where, FR is the radial power peaking factor, defined as:
R þH=2 FR ¼
1 p R2
PðRHC ; zÞdz H=2 R R R þH=2 Pðr; zÞdzdr 0 H=2
ð6Þ
where, P(r, z) is the power of reactor in radius r and height z and RHC is the radius at which that hot channel has occurred. The parameter E1 is the energy of the first constraint (if there is one ‘1’ in each row this term will be zero), E2 is the energy of the second constraint (if there is one ‘1’ in each column, this term will be zero), and E3 is the energy that corresponds to the total value of a neuron’s output. The parameter E4 is the energy of the objective function. The method outlined in this paper could be applied to the other objective functions, which are not considered here. Parameters A, B, and D are constants that will be determined by trial and error method. In this form of the energy function, the network will converge if the activity of each neuron changes with time according to the differential equations previously published by Sadighi et al. (2002). The network is converged when terms E1 and E2 (constraint terms) are zero. The quality of the obtained solution depends on the E4 value; a lower value of E4 is an indication of a better fuel assembly configuration. After finding one acceptable solution by the Hopfield network, it will be improved by SA application.
5. Simulation results We have applied the continuous Hopfield model accompanied by SA to find the optimum-loading pattern for the first cycle of Bushehr NPP and compare it with the pattern proposed by the designer (the standard method). In this study, the positions of seven fuel assemblies in the periphery of the core and the central assembly are fixed; therefore should determine the best positions for remaining 20 assemblies. It means that the states of 20 20 = 400 neurons should be obtained which will be finally 0 or 1. Figs. 1–5 show the evolution of the Hopfield network from the initial to the final states where it is converged to an acceptable solution. As expected, the solution found by Hopfield network is a local minimum of energy function. Most of the times these solutions have good quality. In this study with the well-tuned parameters of the Hopfield network (A = 2000, B = 500, D = 1000, a = 0.05, and dt = 0.5), the network energy normally falls between 60 and 53,000. By comparing the energy of this example, Fig. 4 (E = 61.32), with the energy of the configuration proposed by the VVER Russian Designer (Fig. 5), which is 76.05, we can conclude that the solution found by the Hopfield network is near optimum. As the Hopfield solution in our study is near optimum, LONSA start the SA subroutine with this initial point at a low temperature (Tin = 100 K) and reduces it gradually in each step (annealing coefficient = 0.95). Note that Tin and the annealing coefficient parameters are selected by investigating the SA method’s performance, i.e., velocity and quality of convergence, by analyzing the results for different seeds. In this way reaching to the global minimum in an acceptable time is assured. Figs. 6–8 show the process of reaching the global minimum. As the power peaking factor is reduced, the neutron flux tends towards a flat distribution. In Table 1, we have calculated FR by using the CITATION code’s output and Eq. (6) for the three core configurations that have been discussed. As noted, the calculated results by LONSA for loading optimization has lower power peaking factor in comparison with the configuration proposed by VVER-1000. The LONSA code thus has good reliability for finding the optimum core configuration. 6. Concluding remarks In this study, we introduce the LONSA (loading optimization by neural network and simulated annealing) program for solving the fuel-loading pattern optimization. We applied the continuous
Fig. 1. Evolution of neurons in the Hopfield network – first step.
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Fig. 2. Evolution of neurons in the Hopfield network – 67th step.
Fig. 3. Evolution of neurons in the Hopfield network – 222nd step.
Fig. 4. Evolution of neurons in the Hopfield network – 376th step (network is converged) (E = 61.32).
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Fig. 5. Configuration that is proposed by VVER for first loading of Bushehr NPP – (E = 76.05).
Fig. 8. Simulated annealing process for first loading of Bushehr NPP – last iteration (T = 0.246 101 and E = 22.79).
Table 1 Comparison of the results Method
Network energy E4
FR
Number of iterations
Hopfield VVER-1000(Standard) Hopfield + SA
61.32 76.05 22.79
1.35 1.39 1.21
376 – 539
design proposed by the designer (BUSHER NPP, 1998). Since the solution that is provided by the Hopfield network is a local minimum, we improved the solution of this network by using the simulated annealing (SA) method. The final result that is obtained by this method is closer to the optimum solution. Appendix 1. Hopfield neural network
Fig. 6. Simulated annealing process for first loading of Bushehr NPP – first iteration (T = 100 °K and E = 64.24).
Hopfield and Tank extended the original model to a fully interconnected network of nonlinear analog units, where the activation level of each unit is a value in the interval [0,1]. The Hopfield model has two versions: continuous and discrete. For the continuous version, neurons have continuous values within (0,1) and the dynamics of the network are governed by a set of non-linear differential equations. For the discrete version, the neurons have only two possible values, 0 or 1, and the dynamics of the network are characterized by a set of non-linear difference equations (Hopfield, 1982, 1984). The space of possible configurations {Vi}i = 1,...,L in continuous model is rather than discrete, and is bounded by the L-dimensional hypercube defined by Vi = 0 or 1. The evolution of the units over time is now characterized by the following differential equations (usually called ‘‘equations of motion”)
dU i =dt ¼
X
T V j ij j
þ Ii U i ;
i ¼ 1; :::; L
ðA1Þ
where Ui, Ii, and Vi are the input, input bias, and activation level of unit i, respectively. The activation level of unit i is a function of its input, namely Fig. 7. Evolution of energy in the simulated annealing procedure for first loading of Bushehr NPP.
Hopfield network to the fuel management optimization problem of Bushehr NPP, and the obtained solution was compared with the
V i ¼ gðU i Þ ¼ 1=2 ð1 þ tanh aU i Þ ¼ 1=ð1 þ e2aUi Þ
ðA2Þ
The activation function g is the well-known sigmoidal function, which always returns a value between 0 and 1. The parameter a is used to modify the slope of the function. In Fig. 1, for example, the a value is lower for curve (A2) than for curve (A1).
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Since 1985, when the analog (continuous) Hopfield network was first used for the traveling salesman problem (TSP), its application has been extended too many areas (Hopfield and Tank, 1985). Hopfield network operates as a local minima-searching algorithm. Obviously, the final configuration of the network can be decoded into a solution of the optimization problem if it is close to a corner of the hypercube (i.e., if the activation value of each unit is close to zero or one). The continuous model is superior to the discrete in terms of the local minima problem because of its smoother energy surface (Peng et al., 1996). The energy function for the continuous Hopfield–Tank model is:
E ¼ 1=2
XX i
T VV j ij i j
X
VI þ i i i
Z
Vi
g 1 ðxÞdx
ðA3Þ
0
Note in particular that dUi/dt = dE/dVi. Accordingly, when the units obey the dynamics of the equations of motion, the network is performing a gradient descent in the network’s configuration space with respect to that energy function, and stabilizes at a local minimum. At that point, dUi/dt = 0 and the input to any given unit i is the weighted sum of the activation levels of all the other units plus the bias, that is
Ui ¼
X
T V j ij j
þ Ii
ðA4Þ
To resolve problem in this study, we should investigate other algorithms to escape local minimum and ensure reaching global minimum (Kirkpatrick et al., 1983; Kirkpatrick et al., 1984). In this study, SA method is used as a local minimum escape algorithm. Appendix 2. Simulated annealing (SA) Likely the most developed of the stochastic optimization methods is the SA method (Van Laarhoveen and Aarts, 1987; Aarts and Korst, 1989). Although it appears to be very simple method to implement, in the reality to obtain both robustness and efficiency, careful consideration is required. SA is based upon the analogy of a solid slowly cooling to its lowest energy state, i.e., annealing. Given a minimization objective, let F denote the value of the objective function for the currently accepted decision variables, and F denote the value of the objective function after some perturbations are made to the currently accepted decision variables. Whether the perturbed decision variables are accepted, implying they now become the currently accepted decision variables, is determined by the following rules.
If F 6 F or if F > F and random½0; 1 < eðF
FÞ=KT
then F ¼ F otherwise F ¼ F ðB1Þ
So if the perturbed decision variables produce a lower objective function value, then they become the currently accepted decision variables. However, if the perturbed decision variable produces a higher objective function value, they only become the currently accepted decision variables if a random number between zero and one exceeds the expression noted in Eq. (5); otherwise, the currently accepted decision variables are unaltered. The conditional acceptance of inferior solutions allows the search algorithm to escape local minimums, and as we shall see when utilizing penalty functions, traverse the infeasible decision space. The parameter T plays the analog of material temperature in annealing. At high temperatures, many inferior solutions are ac-
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cepted, allowing the search space to be extensively transversed in search of the vicinity of the global minimum. As temperature decreases, the probability of acceptance of inferior solutions decreases, since if the cooling schedule is done appropriately one should now be within the vicinity of the family of global optimum solutions. What the initial temperature should be and how fast cooling should occur determine the robustness and efficiency of the SA implementation. Fortunately, both of these attributes can be determined based upon the specific behavior of the optimization problem that is being solved as the search progresses (Kropaczek et al., 1994). Another item that must be addressed is at what temperature cooling should be ended and the optimization search terminated. This can be triggered by a combination of lower temperature limit, maximum number of histories, and lack of improvement in the objective function value. References Aarts, E., Korst, J., 1989. Simulated Annealing and Boltzmann Machines – A Stochastic Approach to Combinatorial Optimization and Neural Computing. John Wiley & Sons, New York. Discroll, M.J., Downar, T.J., Pilat, E.E., 1990. The Linear Reactivity Model for Nuclear Fuel Management. American Nuclear Society. Freeman, J.A., Skapura, D.M., 1991. Neural Network Algoritms; Application and Programming Techniques. Addison-Wesley Publishing Co. Faghihi, F., Fadaie, A.H., Sayareh, R., 2007. Reactivity coefficients simulation of the Iranian VVER-1000 nuclear reactor using WIMS and CITATION codes. Progress in Nuclear Energy (Elsevier Journal) 49 (1), 68–78. Hopfield, J.J., 1982. Neural networks and physical systems with emergent collective computational properties. Proceedings the National Academy of Science (USA) 79, 2554–2558. Hopfield, J.J., 1984. Neurons with graded response have collective computational properties like those of two-state neurons. Proceedings the National Academy of Science (USA) 81, 3088–3092. Hopfield, J.J., Tank, D.W., 1985. Neural computations of decisions in optimization problems. Biological Cybernetics 52, 141–152. Hopfield, J.J., Tank, D.W., 1986. Computing with neural networks: a model. Science 233, 625–632. Kirkpatrick, S., Gelatt, C.D., Vecchi, M.P., 1983. Optimization by simulated annealing. Science 220 (4598), 671. Kim, H.G., Change, S.H., Lee, B.H., 1993. Optimal fuel loading pattern design using an artificial neural network and a fuzzy rule-based system. Nuclear Science and Engineering 115, 152–163. Kropaczek, D.J., Turinsky, P.J., Maldonado, G.I., Parks, G.T., 1994. The Efficiency and Fidelity of the In-Core Fuel Management Code FROMOSA-P. In: Proceedings of the International Conference on Reactor Physics and Reactor Computations, Tel Aviv, 572, INS/ENS. Levine, S., 1986. In-Core Fuel Management of Four Reactor Types. Handbook of Nuclear Reactor Calculation, vol. II. CRC Press. Lam, H.Q., Sato, D., Little, D.C., Popa, F.D., 2003. LP-FUN Applications to Reload Design. In: Proceedings of Advances in Nuclear Fuel Management III Topical Meeting, Hilton Head, SC. Oak Ridge National Laboratory, 1972. CITATION -LDI2 code. Oyarzun, C.C., Kropaczek, D.J., Sutton, S.B., Russell, W.E., 2003. The Global Nuclear Fuel Optimization System for BWR Fuel Cycle Management. In: Proceedings of Advances in Nuclear Fuel Management III Topical Meeting, Hilton Head, SC. Peng, M., Gupta, N.K., Amitage, A.F., 1996. An investigation into improvement of local minima of the Hopfield network. Neural Network 9 (7), 1241–1253. Sadighi, M., Setayeshi, S., Salehi, A.A., 2002. PWR fuel management optimization using neural networks. Annals of Nuclear Energy 29 (41–45). Turinsky, P.J., Parks, G.T., 1999. Chapter: Advances in Nuclear Fuel Management Optimization for Light Water Reactors. Advances in Nuclear Science and Technology, vol. 37. Kluwer Academic/Plenum Publishers, NYC. pp. 137. Turinsky, P.J., Keller, P.M., Abdel-khalik, H.S., 2005. Evolution of nuclear fuel management and reactor operator aid tools. Nuclear Engineering and Technology 37 (1). Van Laarhoveen, P.J.M., Aarts, E.H.L., 1987. Simulated Annealing: Theory and Applications. D. Reidel Publishing Company, Dordrecht, Holland. Winfrith, 1985. LWR-WIMS, a Computer Code For Light Water Reactor Calculations, UK: AEE, AEEW-R 1498. BUSHER NPP Neutronic calculations of the core. Report RRC KI No 32/1-38-498, 1998.