Proceedings of the 2017 25th International Conference on Nuclear Engineering ICONE25 July 2-6, 2017, Shanghai, China ICONE25-67936 MIXED OXIDE LWR ASSEMBLY DESIGN OPTIMIZATION USING DIFFERENTIAL EVOLUTION ALGORITHMS Alan J. Charles University of Cambridge Department of Engineering Cambridge, United Kingdom ajc289@cam.ac.uk Geoffrey T. Parks University of Cambridge Department of Engineering Cambridge, United Kingdom gtp10@cam.ac.uk Keywords: fuel assembly design, optimization, differential evolution, genetic algorithms. a Genetic Algorithm (GA), a more conventional algorithm often used as a benchmark. GAs [1] are one of the more popular methods used in nuclear engineering design optimization. These evolutionary algorithms mimic the processes of natural selection and reproduction in an attempt to create successive populations of solutions which converge on an optimal solution. For this work, the Multi-Objective Alliance Algorithm (MOAA) [2] was chosen as it has previously demonstrated its effectiveness in nuclear engineering fuel assembly design problems, producing solutions which are superior to previous ‘expert designs’ and performing better than other GAs [3], thus making it a good algorithm against which to compare performance. In contrast, DE algorithms [4] are a relatively newer type of evolutionary algorithm that work in a similar fashion to GAs but feature key differences in the way the new population is generated. In addition, the selection process is generally more stringent than with GAs (where inferior solutions have a probability of remaining in the population). This allows DE algorithms to offer potentially faster convergence (which is useful for computationally expensive problems, such as those faced in nuclear engineering) at the risk of premature convergence on a non-optimal solution [5]. DE algorithms have previously been successfully applied to core design optimization problems [6]. However, they do not yet appear to have been applied to nuclear fuel assembly optimization problems, thus making this investigation both novel and a useful step in examining DE’s applicability to solving such problems. For this work, new multi-objective forms of the DE algorithms JADE [7] and μJADE [8] are developed. JADE was chosen as it had been shown in the literature to exhibit superior performance over classic DE algorithms through its use of a variation on the classic DE mutation strategy, and the inclusion ABSTRACT Two new multi-objective differential evolution (DE) algorithms are used to optimize heterogeneous low-enriched uranium + mixed oxide fuel assemblies for use in a pressurized water reactor. The objectives were to maximize plutonium content and minimize the power peaking factor. A performance comparison to a genetic algorithm is used to evaluate the applicability of DE algorithms to nuclear fuel assembly design optimization problems. Results show that DE performs highly competitively against a more established algorithm and can arguably better represent the trade-off between both objectives through greater variety in the number of different pin arrangements explored and a higher reliability in finding the ‘true’ Pareto-front. INTRODUCTION In mixed oxide (MOX) fuel assemblies, there is a clear trade-off between in-core fuel performance (achieved through variation of fuel properties in the assembly) and fabrication cost (due to the increased complexity in the fuel). A capability to explore this trade-off rigorously and systematically would be a helpful aid to decision-making, as the non-linear interaction of variables means design optimization becomes too difficult to achieve through conventional engineering judgement alone. Whilst there are a large variety of optimization methods available, few performance comparisons have been made to determine which methods are most suitable for these problems, and a lack of suitable optimization methods has a negative impact on the design process. This study was conducted to determine the suitability of a relatively new class of algorithms called Differential Evolution (DE) in optimizing a typical nuclear fuel assembly design problem, with performance evaluated by comparing results with 1 Copyright © 2017 ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/25/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use All three algorithms used the reactor physics code WIMS10a [10] to solve the neutron transport equation and obtain the PPF for each tested fuel assembly design, using an arbitrary power level. Assemblies were modelled using a standard CORAIL assembly layout of 264 fuel pins, which can be divided using octant symmetry to give 39 unique fuel pin positions. These pins are labelled as fuel types 1, 2, 3 (MOX type 1, MOX type 2, and LEU, respectively). The quantities N1, N2, and N3 are therefore the total number of each respective pin type, where N1 + N2 + N3 = 39. Some pins are weighted by 0.5 due to octant symmetry. LEU enrichment is fixed at 5% U235 and reactor grade Pu composition is assumed. Two %Pu MOX pin types are allowed (W1, W2). Constraints affecting this problem include a requirement that the minimum number of LEU pins must be half the total, and a maximum amount of Pu within the MOX pins (20%). This gives a total of 39 integer variables, 2 continuous variables, with the mentioned constraints of N3 ≥ 16.5 (264/8) and 0 ≤ W1, W2 ≤ 20. The Pu content of an assembly is therefore MOXT = W1 × N1 + W2 × N2. From this, both objectives to be minimized are therefore PPF and –MOXT. Algorithms were run on the ‘Ray’ computer cluster used by the University of Cambridge’s Department of Engineering, with specifications shown in Table 3. of an archive and self-adapting control parameters. Similarly, μJADE was chosen as it was shown to be effective when working on multimodal problems, but uses a significantly smaller population size. These features should help the algorithm perform well in a multi-objective environment, where diversity and convergence rates are important. The problem investigated concerns optimization of a ‘CORAIL’ assembly [7] containing both low-enriched uranium (LEU) and plutonium MOX pins, with the objectives of minimizing the power peaking factor (PPF) and maximizing plutonium content. Plutonium and LEU fuel have different reaction probabilities and produce a wide range of neutron energies, potentially resulting in an uneven neutron flux and subsequent fuel temperature problems. Optimization of plutonium distribution by varying the position of MOX pins and the amount of plutonium contained within them can result in improved thermal margins, whilst increasing the overall plutonium content above that of the ‘CORAIL’ design. METHODOLOGY Multi-Objective JADE (MOJADE) and μJADE (MOμJADE) were created using C++ and are based on the JADE and μJADE algorithms as described by the originators in [8,9], with the following modifications implemented to allow them to operate in a multi-objective environment. First, selection and ranking are no longer done based on one objective, and the ‘best’ solutions are now a list of nondominated solutions which represents the current Pareto-front. These are determined from the current population. Secondly, the archive was changed to accept solutions from the population that have been dominated by new solutions, and an additional archive was added to accept new solutions that are Paretoequivalent to the existing population. The pseudocode for MOJADE and MOμJADE can be seen in Annex A. Control parameters used for MOAA and MOJADE / MOμJADE are given below in Table 1 and Table 2, respectively. TABLE 3. RAY COMPUTER CLUSTER SPECIFICATIONS Processor RAM Video Card Hard Drive RESULTS Each algorithm was run 30 times, with a limit of 1600 solution evaluations per run. Analysis of the results involved comparing Pareto-fronts found by the three algorithms using a number of indicators and statistical tests to determine the performance of each algorithm. The indicators used were the epsilon and hypervolume indicators. The epsilon indicator [11] represents the minimum distance required to translate all the points of the found Pareto-front from a given algorithm to weakly dominate the reference set (which is the overall Paretofront constructed from all solutions found by all algorithms). The hypervolume indicator [12] is the difference between the hypervolume of the objective space dominated by the Paretofront found by a particular algorithm and the hypervolume of the objective space dominated by the reference set, using the least-optimal solution found from all three algorithms as a reference point. In both cases smaller values indicate better performance. The Kruskal-Wallis test [13] was then used to determine the statistical significance of these values. For this work, the Kruskal-Wallis test results represent the probability that the given indicator values are not a true representation of the algorithm’s relative performance against another, and are instead the result of random chance. TABLE 1. MOAA CONTROL PARAMETERS Number of tribes Probability 1 for the creation of tribes Probability 2 for the creation of tribes Initial standard deviation Final standard deviation Probability 3 for the creation of alliances Alliance standard deviation Total number of Pareto-optimal solutions Factor for evaluation neighbourhood 6 0.5 0.2 0.3 0.01 2 / variables 0.1 100 10 TABLE 2. MOJADE AND MOμJADE CONTROL PARAMETERS Parameter Rate of parameter adaptation Greediness of mutation strategy Population Generations MOJADE 0.1 0.05 32 50 Dual Intel Xeon Processor E5-2650 (8-core hyper threading, 2.6 GHz Turbo with 20 MB cache 64 GB (8×8 GB) 1866 MHz DDR3 512 MB AMD FirePro 2270 1TB 7,200 rpm MOμJADE 0.05 3 / population 8 200 2 Copyright © 2017 ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/25/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use FIGURE 1. RESULTS OF MOAA, MOJADE AND MOµJADE OPTIMIZATION OF MOX FUEL ASSEMBLIES FIGURE 2. COMPARISON OF NON-DOMINATED SOLUTIONS FOUND USING MOAA, MOJADE AND MOµJADE ALGORITHMS TO OPTIMIZE MOX FUEL ASSEMBLIES 3 Copyright © 2017 ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/25/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use A graph showing the results from all 30 runs for each of the three algorithms can be seen in Figure 1. Filtering these to show just the Pareto-Optimal (PO) solutions for each algorithm produces Figure 2. Both figures plot solutions in PPF against (–MOXT) space, so that both objectives are to be minimized, and thus the bottom-left corner represents an ideal solution. It can be seen from these two graphs that MOJADE and MOμJADE are clearly performing comparably to MOAA, and both significantly contribute to the highlighted overall Pareto-front shown in Figure 2. MOAA appears to dominate the Pareto-front at both the extremes of plutonium content, with MOJADE and MOμJADE being most effective around the middle of the Pareto-front. A small amount of clustering is also present in the PO solutions of MOAA, with clear gaps in the Pareto-front that are populated by MOJADE and MOμJADE solutions. The clustering of MOAA results can be explained by looking at the patterns themselves. Each MOAA run tends to converge on a single pin pattern with one or two changes in the pins, and the non-dominated solutions therefore show the effect of increasing or decreasing the values of W1 and/or W2 within the same pin pattern. This results in a number of solutions that have very similar values for MOXT and PPF. In contrast, both MOJADE and MOμJADE produce many different patterns within each run and thus arguably better explore the search space of different pin arrangements. Results from each of the 30 runs for each algorithm were used to compute the mean and standard deviation of the hypervolume and epsilon indicators along with their corresponding p-values from the Kruskal-Wallis test. These can be seen in Tables 4 – 6 respectively. Table 4 indicates that MOJADE is most consistent at producing results which dominate the entirety of the known search space. MOμJADE performs slightly worse and also has a large standard deviation. This is likely due to the small population of MOμJADE, making its performance highly influenced by the number of times the algorithm is run (more so than the other algorithms), in order to provide more data. Table 5, however, suggests that whilst MOμJADE results do not give as much data as MOJADE as to the size of the search space, they are more likely to be closer to the ‘true’ Pareto-front. Table 6 gives the p-value results of the Kruskal-Wallis test for the hypervolume and epsilon indicators for both DE algorithms versus the GA, as well as against each other. Against MOAA, hypervolume test results show that MOJADE performs better due to methodology and not by chance. However, for the epsilon indicator, the result shows a statistical likelihood that differences in epsilon performance between MOAA and MOJADE are by chance. For MOμJADE, the results suggest that the DE algorithm outperforms the GA on both indicators due to the methodology. Finally, when comparing MOμJADE against MOJADE, the Kruskal-Wallis test results indicate that the lower mean hypervolume indicator for MOJADE and the lower mean epsilon indicator for MOμJADE are due to differences in their methodology and not by chance. TABLE 4. HYPERVOLUME INDICATOR VALUES Algorithm MOAA MOJADE MOμJADE Mean 1.6664 0.7672 1.1267 Standard Deviation 0.5169 0.1047 0.7723 TABLE 5. EPSILON INDICATOR VALUES Algorithm MOAA MOJADE MOμJADE Mean 0.3897 0.3941 0.3320 Standard Deviation 0.1478 0.1204 0.1081 TABLE 6. KRUSKAL-WALLIS TEST RESULTS Algorithm comparison MOJADE vs MOAA MOμJADE vs MOAA MOμJADE vs MOJADE Hypervolume 3.879E-11 8.513E-07 9.497E-05 Epsilon 9.528E-01 7.363E-02 5.650E-02 CONCLUSIONS These results suggest that DE is able to find solutions comparable in quality to those found by MOAA and arguably better examines the search space of pin patterns. Both DE algorithms show good performance in this exploratory optimization problem, despite the algorithms originally being designed for single-objective optimization. Although further optimization of algorithm control parameters, as well as determining the robustness and scalability of this method, are still required, this study demonstrates that DE has been shown to be an effective method alongside a proven GA when looking at the optimization of nuclear fuel assembly designs. NOMENCLATURE DE Differential Evolution GA Genetic Algorithm LEU Low-Enriched Uranium MOAA Multi-Objective Alliance Algorithm MOJADE Multi-Objective JADE MOμJADE Multi-Objective μJADE MOX Mixed Oxide PO Pareto-Optimal PPF Power Peaking Factor REFERENCES [1] Holland, J.H., 1992, Adaptation in Natural and Artificial Systems, MIT Press, Cambridge, Massachusetts. [2] Lattarulo, V., and Parks, G.T., 2012, “A Preliminary Study of a new Multi-objective Optimization Algorithm”, Proc. IEEE Congress on Evolutionary Computation, IEEE, Brisbane, pp. 1–8. [3] Lattarulo, V., Lindley, B.A., and Parks, G.T., 2014, “Application of the MOAA for the Optimization of CORAIL Assemblies for Nuclear Reactors”, Proc. IEEE Congress on Evolutionary Computation, IEEE, Beijing, pp. 1413–1420. 4 Copyright © 2017 ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/25/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use [4] Storn, R., and Price, K., 1997, “Differential Evolution – A Simple and Efficient Heuristic for Global Optimization over Continuous Spaces”, J. Global Optim., 11, (4), pp. 341–359. [5] Zio, E., and Viadana, G., 2011, “Optimization of the Inspection Intervals of a Safety System in a Nuclear Power Plant by Multi-Objective Differential Evolution (MODE)”, Reliab. Eng. Syst. Safe., 96, (11), pp. 1552–1563. [6] Sacco, W.F., Henderson, N., Rios-Coelho, A.C., Ali, M.M., and Pereira, C.M.N.A., 2009, “Differential Evolution Algorithms Applied to Nuclear Reactor Core Design”, Ann. Nucl. Energy, 36, (8), 1093–1099. [7] Youinou, G., Zaetta, A., Vasile, A., Delpech, M., Rohart, M., and Guillet, J.L., 2001, “Heterogeneous assembly for plutonium multi recycling in PWRs: The CORAIL concept”, Proc. International Conference on Back-End of the Fuel Cycle: From Research to Solutions (GLOBAL 2001), Paris, pp. 1–5. [8] Zhang, J., and Sanderson, A.C., 2009, “JADE: Adaptive Differential Evolution with Optional External Archive”, IEEE Trans. Evol. Comput., 13, (5), pp. 945–958. [9] Brown, C., Jin, Y., Leach, M., Hodgson, M., 2013, “μJADE: Adaptive Differential Evolution with a Small Population”. Soft Comput., 1, pp. 1–10. [10] Lindley, B.A., Newton, T.D., Hosking, J.G., Smith, P.N., Powney, D.J., Tollit, B., and Smith, P.J., 2015, “Release of WIMS10: A Versatile Reactor Physics Code for Thermal and Fast Systems”, Proc. International Congress on Advances in Nuclear Power Plants (ICAPP 2015), ANS, Nice, pp. 1793– 1801. [11] Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C.M., and Grunert Da Fonseca, V., 2003, “Performance Assessment of Multiobjective Optimizers: An Analysis and Review”, IEEE Trans. Evol. Comput., 7, (2), pp. 1–22. [12] Knowles, J., Thiele, L., Zitzler, E., 2006, “A Tutorial on the Performance Assessment of Stochastic Multiobjective Optimizers”, TIK Report 214, Swiss Federal Institute of Technology (ETHZ), Zurich. [13] Kruskal, W.H., Wallis, W.H., 1952, “Use of Ranks in One-criterion Variance Analysis”, J. Am. Stat. Assoc., 47, (260), 583–621. 5 Copyright © 2017 ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/25/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use ANNEX A PSEUDOCODE OF MOJADE AND MOµJADE MOJADE NOMENCLATURE µCR is the adaptive crossover probability µF is the adaptive mutation probability A1 is the archive used for dominated solutions A2 is the archive used for Pareto-equivalent solutions c is the rate of parameter adaptation D is the number of dimensions (variables) G is the number of generations meanA is the arithmetic mean meanL is the Lehmer mean NP is the last member of the population p is the greediness of the mutation strategy P is the population randn is a normal distribution randc is a Cauchy distribution SCR is the set of successful crossover factors SF is the set of successful mutation factors Begin Set µCR = 0.5; µF = 0.5; A1, A2 = 0 Create random initial population {xi, 0|i = 1, 2, …, NP} Evaluate and rank population, determine 100p% best vectors For g = 1 to G SF = 0, SCR = 0 For i = 1 to NP CRi = randni (µCR, 0.1), Fi = randci (µF, 0.1) Randomly choose xp_best from 100p% Randomly choose xr1 =/= xi from P Randomly choose xr2 =/= xr1 =/= xi from P ∪ A1 + A2 vi = xi + Fi · (xp_best - xi) + Fi · (xr1 - xr2) Generate jrand = randint(1, D) For j = 1 to D If j = jrand or rand(0, 1) < CRi ui,j = vi,j Else ui,j = xi,j End If End For If f(ui) dominates f(xi) xi → A1 (replaces random member of A1 if A1 is full) xi = ui CRi → SCR, Fi → SF Else If f(ui) is Pareto-equivalent to f(xi) && f(ui) is NOT dominated by f(A2) Remove members of A2 that are dominated by ui ui → A2 End If End If Rerank 100p% best vectors End For µCR = (1 – c) · µCR + c · meanA(SCR) µF = (1 – c) · µF + c · meanL(SF) End For End vi is the ith test vector following mutation ui is the ith test vector following crossover xi is the ith member of the population 6 Copyright © 2017 ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/25/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use MOµJADE NOMENCLATURE Begin Set µCR = 0.5; µF = 0.5; A1, A2 = 0 Create random initial population {xi, 0|i = 1, 2, …, NP} Evaluate and rank population, determine 100p% best vectors For g = 1 to G SF = 0, SCR = 0 For i = 1 to NP CRi = randni (µCR, 0.1), Fi = randci (µF, 0.1) Randomly choose xa =/= xi from P Randomly choose xb =/= xa =/= xi from P Randomly choose xp_best =/= xa from 100p% Randomly choose xc from P ∪ A1 + A2 vi = xi + Fi · (xp_best - xa) + Fi · (xb - xc) Generate jrand = randint(1, D) For j = 1 to D If j = jrand or rand(0, 1) < CRi ui,j = vi,j , bi,j = 1 Else ui,j = xi,j , bi,j = 0 End If End For For j = 1 to D If rand(0, 1) ≤ 0.005 ui,j = low_lim + rand(0, 1) · (up_lim – low_lim) bi,j = 0 Else ui,j = ui,j, bi,j = bi,j End If End For CRi = ∑ b / D If f(ui) dominates f(xi) xi → A1 (replaces random member of A1 if A1 is full) xi = ui CRi → SCR, Fi → SF Else If f(ui) is Pareto-equivalent to f(xi) && f(ui) is NOT dominated by f(A2) Remove members of A2 that are dominated by ui ui → A2 End If End If Rerank 100p% best vectors If ui ∪ 100p% best vectors BIR = BIR + 1 End If End For If mod(g, max(100, 10D) = 0 µCR = (1 – c) · µCR + c · meanA(SCR) µF = (1 – c) · µF + c · meanL(SF) End If If mod(g, max(1000, 100D) = 0 If BIR== 0 Reinitialize pop, include random member of 100p% BIR = 0 End If End If End For End µCR is the adaptive crossover probability µF is the adaptive mutation probability A1 is the archive used for dominated solutions A2 is the archive used for Pareto-equivalent solutions BIR is a restart variable used if no improvement has been made c is the rate of parameter adaptation D is the number of dimensions (variables) G is the number of generations meanA is the arithmetic mean meanL is the Lehmer mean NP is the last member of the population p is the greediness of the mutation strategy P is the population randn is a normal distribution randc is a Cauchy distribution SCR is the set of successful crossover factors SF is the set of successful mutation factors up_lim / low_lim are the limits set by the variable constraints bi is used to repair the crossover rate after crossover and perturbation vi is the ith test vector following mutation ui is the ith test vector following crossover and perturbation xi is the ith member of the population 7 Copyright © 2017 ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/25/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use

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