Luận án Study on fuel loading pattern optimization for vver-1000 nuclear reactor

than that of the reference core by about 1580 pcm. Whereas, the PPF is 89 smaller than the reference value by about 2.4%, and the flatness values are approximate. 90 Chapter 4 Conclusions and future work 4.1 Conclusions In-core fuel management is a complicated multi-objective problem with a very large search space. Many methods have been investigated and applied to this problem, but none of them could definitely quarantine a global optimal solution. In this dissertation, the studies aim at developing advanced metaheuristics methods, i.e. ESA and SHADE, and applying to the ICFM problem of the VVER-1000 reactor. The three following studies have been carried out: (1) Development of a new core physics tool, LPO-V code, has been conducted to calculate the neutronic characteristics of VVER reactors with fast computational speed and acceptable accuracy. This tool is coupled with the newly developed optimization methods for solving the ICFM problem of VVER-1000 reactor. The PhD student developed the LPO-V based on the finite dif- ference method for solving multi-group diffusion equations in hexagonal 91 systems to calculate the neutronic characteristics of the VVER reactor core. Verification calculations of the LPO-V code have been performed based on a VVER-1000 MOX core benchmark and compared with MCNP calculations. Four-group cross-section set of the VVER-1000 MOX core was prepared using the PIJ module of the SRAC-2006 code system for the use in the LPO-V code. The results shown that the maximum deviation of keff is 102 pcm and the maximum deviation of the power distribution is 3.5%. Calculation speed of the LOP-V code was also tested and compared with the speed of the CITATION module of the SRAC-2006 code under the same conditions. The LPO-V code needs about 340 seconds to calculate 2000 LPs, while the CITATION needs 1040 seconds. It means that the LPO-V code can performs the core calculations for a large number of LPs (10000 to 100000) with a sufficient accuracy and reasonable computational time. (2) Development of advanced optimization methods, ESA and SHADE, has been conducted and applied successfully to the LP optimization prob- lem of VVER-1000 reactor. The ESA is an improved version of the original SA method, that was proposed by the PhD student. Instead of using binary/ternary exchange operator to generate a new trial LP as in the SA, the ESA used a crossover of two base LPs for generating a new trial LP similar to that used in GA. Numerical calculations shown that this new improvement can enhance the performance of the SA method. In this study, the SHADE method, one of the current highest effi- ciency optimal search methods, was firstly applied to the ICFM problem. The SHADE method is an advanced version of the DE method with the 92 use of success-history based adaptation to determine the control param- eters F and CR automatically. The SHADE method is applied to opti- mization problems of continuous spaces, while the search space in the LP optimization problem is discrete. Therefore, the relative position indexing approach was deployed to convert real vectors to integer vectors in the discrete SHADE method. (3) Numerical calculations were performed for optimizing the LP of the reference VVER-1000 MOX core using the ESA and SHADE methods in comparison with other methods. The fitness function has been constructed based on the neutronic characteristics of the reactor core, i.e. keff , PPF and the flatness of power distribution. The fitness function was used to evaluate the optimization methods and find the optimal LP of the VVER-1000 reactor. A Mann- Whitney U Test was also introduced to evaluate statistical differences be- tween the optimization methods. Calculations for the VVER-1000 MOX core using the ESA method have been carried out in comparison with the original SA and ASA meth- ods. The results show that average fitness values and objective parameters obtained with ESA are better than those of SA and ASA. Whereas, the number of calculated LPs of ESA is smaller than that of SA and ASA by about 5–10%. Mann-Whitney U Test was also applied to evaluated statistical differences between the methods. The results show that ESA is advantageous over SA and ASA in the problem of fuel LP optimization. In case of the SHADE method, calculations based on the VVRE-1000 MOX core were performed and compared with the ESA and DE methods. The comparison shows that the three methods have comparable performance. 93 However, the advantage of SHADE is that the adaptive mechanism sim- plifies significantly the determination of the control parameters compared to DE. The optimal core LPs selected from the SHADE and ESA meth- ods were similar. This LP have a significant improvement of the keff value, which is greater than that of the reference core by about 1580 pcm. Whereas, the PPF is smaller than the reference value by about 2.4%, and the flatness values are approximate. The results demonstrate that the ESA and the discrete SHADE methods have been successfully developed and applied to the fuel loading optimization of the VVER-1000 reactor. The results also show that the efficiency of the two methods are compa- rable in the problem of this study. Nevertheless this is the first time the two methods have been applied to the ICFM problem, therefore further improvement and extensive application of the methods in the problem of fuel loading optimization are being continuously investigated. Comparison of the performance of the ESA and SHADE methods with other methods in the problem of fuel LP optimization will also be planned. 4.2 Future works (1) The current version of the LPO-V code can only handle 2D reactor core with triangular mesh and can not perform burn-up calculation. This code is being continuously upgraded to simulate 3D reactor with triangular and rectangular meshes and perform burn-up calculation. (2) Further investigation of the ESA and SHADE methods are being planned. Besides, new advanced methods will also be considered to apply 94 to the LP optimization problem. The extension of this study to multi-cycle optimization is also taken into account. (3) Extension of the application of these methods to the LP op- timization problem and core design for other reactor types is also being planned. 95 Papers published during the dissertation 1. Viet-Phu Tran, Giang T.T. Phan, Van-Khanh Hoang, Haidang Phan, Nhat-Duc Hoang, Hoai-Nam Tran; Success-history based adaptive differential evolution method for optimizing fuel loading pattern of VVER-1000 reactor; Nuclear Engineering and Design 377 (2021) 111125 2. Viet-Phu Tran, Giang T.T. Phan, Van-Khanh Hoang, Pham Nhu Viet Ha, Akio Yamamoto, Hoai-Nam Tran; Evolutionary simulated annealing for fuel loading optimization of VVER-1000 reactor; Annals of Nuclear Energy 151 (2021) 107938 3. Viet-Phu Tran, Hoai-Nam Tran, Akio Yamamoto, Tomohiro Endo; Automated Generation of Burnup Chain for Reactor Analysis Appli- cations; Kerntechnik, ISSN 0932-3902, 82 (2017 ) 2 196-205. 4. Viet-Phu Tran, Hoai-Nam Tran, Van Khanh Hoang; Application of Evolutionary Simulated Annealing Method to Design a Small 200 MWt Reactor Core; Nuclear Science and Technology, ISSN 1810-5408, Vol. 10, No. 4 (2020), pp. 16-23 5. Nguyen Huu Tiep, Nguyen Thi Dung, Tran Viet Phu, Tran Vinh Thanh and Pham Nhu Viet Ha; Burnup calculation of the OECD VVER-1000 LEU benchmark assembly using MCNP6 and SRAC2006; 96 Nuclear Science and Technology, ISSN 1810-5408, Vol. 8, No. 4 (2018), pp. 10-19 6. Tran Vinh Thanh, Tran Viet Phu, Nguyen Thi Dung; A study on the core loading pattern of the VVER-1200/V491; Nuclear Science and Technology, ISSN 1810-5408, Vol. 7, No. 1 (2017), pp. 21-27. 7. Tran Viet Phu, Tran Hoai Nam; Discrete SHADE method for in-core fuel management of VVER-1000 reactor; 45th Vietnam Conference on Theoretical Physics (VCTP-45), 2020 (Poster) 8. Viet-Phu Tran, Hoai-Nam Tran, Van Khanh Hoang; Application of Evolutionary Simulated Annealing Method to Design a Small 200 MWt Reactor Core; 6th Conference on Nuclear Science and Technol- ogy for young researcher, 08-09/10/2020 97 References [1] W. N. Association, Outline history of nuclear energy(accessed on November 2020). URL https://world-nuclear.org/information-library/ current-and-future-generation/outline-history-of-nuclear-energy. aspx [2] W. N. Association, World Nuclear Performance Report 2020, 2020. [3] N. E. Agency, the economics of the nuclear fuel cycle, 1994. [4] P. J. Turinsky, Core Isotopic Depletion and Fuel Management, Springer US, Boston, MA, 2010, pp. 1241–1312. doi:10.1007/ 978-0-387-98149-9_10. [5] E. Gomin, M. Kalugin, D. Oleynik, VVER-1000 MOX Core Com- putational Benchmark, Specification and Results, Vol. NEA/NSC/- DOC(2005)17, 2006. [6] A. Yamamoto, A quantitative comparison of loading pattern op- timization methods for in-core fuel management of pwr, Journal of Nuclear Science and Technology 34 (4) (1997) 339–347. doi: 10.1080/18811248.1997.9733673. 98 [7] Y. P. Mahlers, Core loading pattern optimization based on simulated annealing and successive linear programming, Annals of Nuclear En- ergy 22 (1) (1995) 29–37. doi:10.1016/0306-4549(94)00031-9. [8] J. G. Stevens, K. S. Smith, K. R. Rempe, T. J. Downar, Optimization of pressurized water reactor shuffling by simulated annealing with heuristics, Nuclear Science and Engineering 121 (1) (1995) 67–88. doi:10.13182/NSE121-67. [9] H. C. Lee, H. J. Shim, C. H. Kim, Parallel computing adaptive simulated annealing scheme for fuel assembly loading pattern op- timization in pwrs, Nuclear Technology 135 (1) (2001) 39–50. doi: 10.13182/NT01-A3204. [10] T. Rogers, J. Ragusa, S. Schultz, R. S. Clair, Optimization of pwr fuel assembly radial enrichment and burnable poison location based on adaptive simulated annealing, Nuclear Engineering and Design 239 (6) (2009) 1019–1029. doi:10.1016/j.nucengdes.2009.02. 005. [11] T. K. Park, H. G. Joo, C. H. Kim, H. C. Lee, Multiobjective loading pattern optimization by simulated annealing employing discontinu- ous penalty function and screening technique, Nuclear Science and Engineering 162 (2) (2009) 134–147. doi:10.13182/nse162-134. URL https://doi.org/10.13182/NSE162-134 [12] A. Zameer, S. M. Mirza, N. M. Mirza, Core loading pattern op- timization of a typical two-loop 300 mwe pwr using simulated an- nealing (sa), novel crossover genetic algorithms (ga) and hybrid ga(sa) schemes, Annals of Nuclear Energy 65 (2014) 122–131. doi: 10.1016/j.anucene.2013.10.024. 99 [13] V.-P. Tran, G. T. T. Phan, V.-K. Hoang, P. N. V. Ha, A. Yamamoto, H.-N. Tran, Evolutionary simulated annealing for fuel loading op- timization of VVER-1000 reactor, submitted to Annals of Nuclear Energy. [14] W. Zeng, J. Li, T. Hui, J. Xie, T. Yu, Lqg/ltr controller with simu- lated annealing algorithm for ciads core power control, Annals of Nu- clear Energy 142 (2020) 107422. doi:https://doi.org/10.1016/ j.anucene.2020.107422. [15] S. Tang, M. Peng, G. Xia, G. Wang, C. Zhou, Optimization design for supercritical carbon dioxide compressor based on simulated annealing algorithm, Annals of Nuclear Energy 140 (2020) 107107. doi:https: //doi.org/10.1016/j.anucene.2019.107107. [16] C. Lin, J.-I. Yang, K.-J. Lin, Z.-D. Wang, Pressurized water reactor loading pattern design using the simple tabu search, Nuclear Science and Engineering 129 (1) (1998) 61–71. doi:10.13182/NSE98-A1963. [17] N. J. Hill, G. T. Parks, Pressurized water reactor in-core nuclear fuel management by tabu search, Annals of Nuclear Energy 75 (2015) 64–71. doi:10.1016/j.anucene.2014.07.051. [18] J. Francois, C. Martin-del Campo, R. Francois, L. B. Morales, A practical optimization procedure for radial bwr fuel lattice design using tabu search with a multiobjective function, Annals of Nuclear Energy 30 (12) (2003) 1213–1229. doi:https://doi.org/10.1016/ S0306-4549(03)00055-0. 100 [19] A. Castillo, G. Alonso, L. B. Morales, C. Martin del Campo, J. Fran- cois, E. del Valle, Bwr fuel reloads design using a tabu search tech- nique, Annals of Nuclear Energy 31 (2) (2004) 151–161. doi:https: //doi.org/10.1016/S0306-4549(03)00214-7. [20] J. A. Castillo, J. J. Ortiz, G. Alonso, L. B. Morales, E. del Valle, Bwr control rod design using tabu search, Annals of Nuclear Energy 32 (7) (2005) 741–754. doi:https://doi.org/10.1016/j.anucene.2004. 12.004. [21] A. Castillo, J. J. Ortiz, J. L. Montes, R. Perusquia, Fuel loading and control rod patterns optimization in a bwr using tabu search, Annals of Nuclear Energy 34 (3) (2007) 207–212. doi:https://doi.org/ 10.1016/j.anucene.2006.12.006. [22] C. Lin, Y.-H. Chen, The max min ant system and tabu search for pressurized water reactor loading pattern design, Annals of Nu- clear Energy 71 (2014) 388–398. doi:https://doi.org/10.1016/ j.anucene.2014.04.020. [23] J. Barnes, V. Wiley, J. Moore, D. Ryer, Solving the aerial fleet refueling problem using group theoretic tabu search, Mathemati- cal and Computer Modelling 39 (6) (2004) 617–640, defense trans- portation: Algorithms, models, and applications for the 21st century. doi:https://doi.org/10.1016/S0895-7177(04)90544-4. [24] J. Crino, J. Moore, J. Barnes, W. Nanry, Solving the theater dis- tribution vehicle routing and scheduling problem using group the- oretic tabu search, Mathematical and Computer Modelling 39 (6) (2004) 599–616, defense transportation: Algorithms, models, and 101 applications for the 21st century. doi:https://doi.org/10.1016/ S0895-7177(04)90543-2. [25] M. D. DeChaine, M. A. Feltus, Nuclear fuel management optimiza- tion using genetic algorithms, Nuclear Technology 111 (1) (1995) 109–114. doi:10.13182/NT95-A35149. [26] G. T. Parks, Multiobjective pressurized water reactor reload core de- sign by nondominated genetic algorithm search, Nuclear Science and Engineering 124 (1) (1996) 178–187. doi:10.13182/NSE96-A24233. [27] J. Wan, F. Zhao, Optimization of ap1000 power control system set- points using genetic algorithm, Progress in Nuclear Energy 95 (2017) 23–32. doi:https://doi.org/10.1016/j.pnucene.2016.11.009. [28] J. Washington, J. King, Optimization of plutonium and minor ac- tinide transmutation in an ap1000 fuel assembly via a genetic search algorithm, Nuclear Engineering and Design 311 (2017) 199–212. doi:https://doi.org/10.1016/j.nucengdes.2016.11.030. [29] M. Turkmen, S. Ergun, U. Colak, A new method in beam shaping: Multi-objective genetic algorithm method coupled with a monte- carlo based reactor physics code, Progress in Nuclear Energy 99 (2017) 165–176. doi:https://doi.org/10.1016/j.pnucene.2017. 05.008. [30] E. Israeli, E. Gilad, Novel genetic algorithm for loading pattern opti- mization based on core physics heuristics, Annals of Nuclear Energy 118 (2018) 35–48. doi:https://doi.org/10.1016/j.anucene. 2018.03.042. 102 [31] P. Wrigley, P. Wood, P. Stewart, R. Hall, D. Robertson, Mod- ule layout optimization using a genetic algorithm in light water modular nuclear reactor power plants, Nuclear Engineering and Design 341 (2019) 100–111. doi:https://doi.org/10.1016/j. nucengdes.2018.10.023. [32] R. Kianpour, G. Ansarifar, M. Fathi, Optimal design of a vver- 1000 nuclear reactor core with dual cooled annular fuel based on the reactivity temperature coefficients using thermal hydraulic and neutronic analysis by implementing the genetic algorithms, Annals of Nuclear Energy 148 (2020) 107682. doi:https://doi.org/10. 1016/j.anucene.2020.107682. [33] T. Hui, W. Zeng, T. Yu, Core power control of the ads based on genetic algorithm tuning pid controller, Nuclear Engineering and Design 370 (2020) 110835. doi:https://doi.org/10.1016/j. nucengdes.2020.110835. [34] Z. Li, J. Huang, M. Ding, Comparison and analysis of different selection strategies of genetic algorithms for fuel reloading opti- mization of thorium-based htgrs, Nuclear Engineering and Design 373 (2021) 110969. doi:https://doi.org/10.1016/j.nucengdes. 2020.110969. [35] J. K. Axmann, Parallel adaptive evolutionary algorithms for pressur- ized water reactor reload pattern optimizations, Nuclear Technology 119 (3) (1997) 276–291. doi:10.13182/NT97-A35403. [36] A. Mishra, A. Patwardhan, A. Verma, Safety management in npps using an evolutionary algorithm technique, Nuclear Engineering and Design 237 (12) (2007) 1445–1451, 18th International Conference on 103 Structural Mechanics in Nuclear Engineering. doi:https://doi. org/10.1016/j.nucengdes.2006.10.003. [37] G. H. F. Caldas, R. Schirru, Parameterless evolutionary algorithm applied to the nuclear reload problem, Annals of Nuclear Energy 35 (4) (2008) 583–590. doi:https://doi.org/10.1016/j.anucene. 2007.08.014. [38] S. Mishra, R. Modak, S. Ganesan, Optimization of depleted uranium bundle loading in fresh core of indian phwr by evolutionary algorithm, Annals of Nuclear Energy 37 (2) (2010) 208–217. doi:https://doi. org/10.1016/j.anucene.2009.11.001. [39] S. Mishra, R. Modak, S. Ganesan, Selection of fuel channels for thermal power measurement in 700mwe indian phwr by evolution- ary algorithm, Nuclear Engineering and Design 247 (2012) 116–122. doi:https://doi.org/10.1016/j.nucengdes.2012.03.003. [40] S.-C. Wu, T.-H. Chan, M.-S. Hsieh, C. Lin, Quantum evolution- ary algorithm and tabu search in pressurized water reactor load- ing pattern design, Annals of Nuclear Energy 94 (2016) 773–782. doi:https://doi.org/10.1016/j.anucene.2016.04.039. [41] M.-S. Hsieh, S.-C. Wu, Modified quantum evolutionary algorithm and self-regulated learning for reactor loading pattern design, Annals of Nuclear Energy 127 (2019) 268–277. doi:https://doi.org/10. 1016/j.anucene.2018.12.018. 104 [42] Z. wu Yu, J. feng Mao, A stochastic dynamic model of train- track-bridge coupled system based on probability density evolu- tion method, Applied Mathematical Modelling 59 (2018) 205–232. doi:https://doi.org/10.1016/j.apm.2018.01.038. [43] Y. Zhou, B. Xu, R. Pang, D. Zou, X. Kong, Stochastic seismic response and stability reliability analysis of a vertical retaining wall in front of the pumping station of a nuclear power plant us- ing the probability density evolution method, Nuclear Engineering and Design 334 (2018) 110–120. doi:https://doi.org/10.1016/ j.nucengdes.2018.05.008. [44] D. Babazadeh, M. Boroushaki, C. Lucas, Optimization of fuel core loading pattern design in a vver nuclear power reactors using particle swarm optimization (pso), Annals of Nuclear Energy 36 (7) (2009) 923–930. doi:10.1016/j.anucene.2009.03.007. [45] R. D. Yadav, H. P. Gupta, Optimization studies of fuel loading pattern for a typical pressurized water reactor (pwr) using particle swarm method, Annals of Nuclear Energy 38 (9) (2011) 2086–2095. doi:10.1016/j.anucene.2011.05.019. [46] M. Jamalipour, R. Sayareh, M. Gharib, F. Khoshahval, M. R. Karimi, Quantum behaved particle swarm optimization with differ- ential mutation operator applied to wwer-1000 in-core fuel manage- ment optimization, Annals of Nuclear Energy 54 (2013) 134–140. doi:10.1016/j.anucene.2012.11.008. 105 [47] A. Zameer, M. Muneeb, S. M. Mirza, M. A. Z. Raja, Fractional- order particle swarm based multi-objective PWR core loading pat- tern optimization, Annals of Nuclear Energy 135 (2020) 106982. doi:https://doi.org/10.1016/j.anucene.2019.106982. [48] B. Tavron, E. Shwageraus, Pebble bed reactor fuel cycle optimiza- tion using particle swarm algorithm, Nuclear Engineering and Design 307 (2016) 96–105. doi:https://doi.org/10.1016/j.nucengdes. 2016.06.033. [49] T.-Y. Lin, J.-T. Yeh, W.-S. Kuo, Using particle swarm optimiza- tion algorithm to search for a power ascension path of boiling water reactors, Annals of Nuclear Energy 102 (2017) 37–46. doi:https: //doi.org/10.1016/j.anucene.2016.12.019. [50] Y. Ling, Q. Yue, C. Chai, Q. Shan, D. Hei, W. Jia, Nuclear accident source term estimation using kernel principal component analysis, particle swarm optimization, and backpropagation neural networks, Annals of Nuclear Energy 136 (2020) 107031. doi:https://doi. org/10.1016/j.anucene.2019.107031. [51] S. M. H. Mousakazemi, Computational effort comparison of genetic algorithm and particle swarm optimization algorithms for the pro- portional integral derivative controller tuning of a pressurized wa- ter nuclear reactor, Annals of Nuclear Energy 136 (2020) 107019. doi:https://doi.org/10.1016/j.anucene.2019.107019. [52] W. Zeng, W. Zhu, T. Hui, L. Chen, J. Xie, T. Yu, An imc-pid controller with particle swarm optimization algorithm for msbr core power control, Nuclear Engineering and Design 360 (2020) 110513. doi:https://doi.org/10.1016/j.nucengdes.2020.110513. 106 [53] H. Wang, M. jun Peng, A. Ayodeji, H. Xia, X. kun Wang, Z. kang Li, Advanced fault diagnosis method for nuclear power plant based on convolutional gated recurrent network and enhanced particle swarm optimization, Annals of Nuclear Energy 151 (2021) 107934. doi: https://doi.org/10.1016/j.anucene.2020.107934. [54] A. M. de Lima, R. Schirru, F. C. da Silva, J. A. C. C. Medeiros, A nuclear reactor core fuel reload optimization using artificial ant colony connective networks, Annals of Nuclear Energy 35 (9) (2008) 1606 – 1612. doi:https://doi.org/10.1016/j.anucene.2008.03. 002. [55] L. Machado, R. Schirru, The Ant-Q algorithm applied to the nuclear reload problem, Annals of Nuclear Energy 29 (2002) 1455–1470. doi: 10.1016/S0306-4549(01)00118-9. [56] A. M. de Lima, R. Schirru, F. C. da Silva, J. A. C. C. Medeiros, A nuclear reactor core fuel reload optimization using artificial ant colony connective networks, Annals of Nuclear Energy 35 (9) (2008) 1606–1612. doi:https://doi.org/10.1016/j.anucene.2008.03. 002. [57] C.-D. Wang, C. Lin, Automatic boiling water reactor loading pattern design using ant colony optimization algorithm, Annals of Nuclear Energy 36 (8) (2009) 1151–1158. doi:https://doi.org/10.1016/ j.anucene.2009.04.004. [58] R. Prasad, M. Ali, P. Kwan, H. Khan, Designing a multi-stage mul- tivariate empirical mode decomposition coupled with ant colony op- timization and random forest model to forecast monthly solar radi- ation, Applied Energy 236 (2019) 778–792. doi:https://doi.org/ 107 10.1016/j.apenergy.2018.12.034. [59] F. Ezazi, M. H. Mallah, J. Karimi Sabet, A. Norouzi, A. Mah- moudian, Investigation on the net cascade using ant colony opti- mization algorithm, Progress in Nuclear Energy 119 (2020) 103169. doi:https://doi.org/10.1016/j.pnucene.2019.103169. [60] B. Jiang, J. Zhou, X. Huang, P. Wang, Prediction of critical heat flux using gaussian process regression and ant colony optimization, Annals of Nuclear Energy 149 (2020) 107765. doi:https://doi. org/10.1016/j.anucene.2020.107765. [61] W. F. Sacco, A. A. de Moura Meneses, N. Henderson, Some studies on differential evolution variants for application to nuclear reactor core design, Progress in Nuclear Energy 63 (2013) 49–56. doi:https: //doi.org/10.1016/j.pnucene.2012.10.003. [62] A. Charles, G. Parks, Application of differential evolution algo- rithms to multi-objective optimization problems in mixed-oxide fuel assembly design, Annals of Nuclear Energy 127 (2019) 165–177. doi:https://doi.org/10.1016/j.anucene.2018.12.002. [63] G. T. T. Phan, Q. B. Do, Q. H. Ngo, T. A. Tran, H.-N. Tran, Appli- cation of differential evolution algorithm for fuel loading optimiza- tion of the dnrr research reactor, Nuclear Engineering and Design 362 (2020) 110582. doi:https://doi.org/10.1016/j.nucengdes. 2020.110582. 108 [64] W. F. Sacco, N. Henderson, A. C. Rios-Coelho, Topographical clear- ing differential evolution: A new method to solve multimodal opti- mization problems, Progress in Nuclear Energy 71 (2014) 269–278. doi:https://doi.org/10.1016/j.pnucene.2013.12.011. [65] A. Glotic, A. Zamuda, Short-term combined economic and emission hydrothermal optimization by surrogate differential evolution, Ap- plied Energy 141 (2015) 42–56. doi:https://doi.org/10.1016/j. apenergy.2014.12.020. [66] W. Sacco, N. Henderson, Balancing exploration and exploitation in differential evolution via variable scaling factors: An application to practical problems, Progress in Nuclear Energy 83 (2015) 365–373. doi:https://doi.org/10.1016/j.pnucene.2015.04.014. [67] J. Zhang, Y. Wu, Y. Guo, B. Wang, H. Wang, H. Liu, A hybrid harmony search algorithm with differential evolution for day-ahead scheduling problem of a microgrid with consideration of power flow constraints, Applied Energy 183 (2016) 791–804. doi:https://doi. org/10.1016/j.apenergy.2016.09.035. [68] F. D. Maio, S. Baronchelli, M. Vagnoli, E. Zio, Determination of prime implicants by differential evolution for the dynamic reliability analysis of non-coherent nuclear systems, Annals of Nuclear Energy 102 (2017) 91–105. doi:https://doi.org/10.1016/j.anucene. 2016.12.018. [69] S. Ikeda, R. Ooka, Application of differential evolution-based constrained optimization methods to district energy optimization and comparison with dynamic programming, Applied Energy 254 109 (2019) 113670. doi:https://doi.org/10.1016/j.apenergy.2019. 113670. [70] T. Smuc, D. Pevec, B. Petrovic, Annealing strategies for loading pattern optimization, Annals of Nuclear Energy 21 (6) (1994) 325– 336. doi:10.1016/0306-4549(94)90028-0. [71] D. J. Kropaczek, P. J. Turinsky, G. T. Parks, G. I. Maldonado, The efficiency and fidelity of the in-core nuclear fuel management code formosa-p, in: Report of the International conference on reactor physics and reactor computations, Tel Aviv (Israel), no. 25044744, 1994, pp. 572–579. [72] S. M. Mirza, A. Abrar, N. M. Mirza, Adaptive simulated annealing methodology for fuel loading pattern optimization, Nuclear Science Journal 37 (5) (2000) 340–350. [73] R. Storn, K. Price, Differential evolution - a simple and effi- cient heuristic for global optimization over continuous spaces, Jour- nal of Global Optimization 11 (1997) 341–359. doi:10.1023/A: 1008202821328. [74] K. V. Price, R. M. Storn, J. A. Lampinen, Differential Evolution-A Practical Approach to Global Optimization, Vol. 141, 2005. doi: 10.1007/3-540-31306-0. [75] S. Das, P. Suganthan, Differential evolution: A survey of the state-of- the-art, IEEE Transactions on Evolutionary Computation 15 (2011) 4 – 31. doi:10.1109/TEVC.2010.2059031. [76] J. Vesterstrom, R. Thomsen, A comparative study of differential evo- lution, particle swarm optimization, and evolutionary algorithms on 110 numerical benchmark problems, in: Proceedings of the 2004 Congress on Evolutionary Computation (IEEE Cat. No.04TH8753), Vol. 2, 2004, pp. 1980–1987 Vol.2. [77] T. Eltaeib, A. Mahmood, Differential evolution: A survey and anal- ysis, Applied Sciences 8 (2018) 1945. doi:10.3390/app8101945. [78] Bilal, M. Pant, H. Zaheer, L. Garcia-Hernandez, A. Abraham, Dif- ferential evolution: A review of more than two decades of research, Engineering Applications of Artificial Intelligence 90 (2020) 103479. URL S095219762030004X [79] R. Tanabe, A. Fukunaga, Success-history based parameter adapta- tion for differential evolution, in: 2013 IEEE Congress on Evolution- ary Computation, 2013, pp. 71–78. [80] W. F. Sacco, N. Henderson, A. C. Rios-Coelho, M. M. Ali, C. M. N. A. Pereira, Differential evolution algorithms applied to nuclear reactor core design, Annals of Nuclear Energy 36 (8) (2009) 1093– 1099. doi:https://doi.org/10.1016/j.anucene.2009.05.007. [81] W. F. Sacco, N. Henderson, Differential evolution with topographi- cal mutation applied to nuclear reactor core design, Progress in Nu- clear Energy 70 (2014) 140–148. doi:https://doi.org/10.1016/ j.pnucene.2013.09.012. [82] H. Ding, G. Sun, L. Hao, B. Wu, Y. Wu, A loading pattern op- timization method based on discrete differential evolution, Annals of Nuclear Energy 137 (2020) 107057. doi:https://doi.org/10. 1016/j.anucene.2019.107057. 111 [83] G. Phan, H.-N. Tran, K.-C. Nguyen, V.-P. Tran, V.-K. Hoang, P. N. V. Ha, H. A. T. Kiet, Comparative analysis of the Dalat nuclear research reactor with HEU fuel using SRAC and MCNP5, Science and Technology of Nuclear Installations Volume 2017, Arti- cle ID 2615409 (Article ID 2615409) (2017) 10 pages. doi:https: //doi.org/10.1155/2017/2615409. [84] Q. B. Do, G. T. T. Phan, K.-C. Nguyen, Q. H. Ngo, H.-N. Tran, Criticality and rod worth analysis of the DNRR research reac- tor using the SRAC and MCNP5 codes, Nuclear Engineering and Design 343 (2019) 197–209. doi:https://doi.org/10.1016/j. nucengdes.2019.01.011. [85] Nuclear Power Reactors in the World, no. 2 in Reference Data Series, INTERNATIONAL ATOMIC ENERGY AGENCY, Vienna, 2020. URL https://www.iaea.org/publications/14756/ nuclear-power-reactors-in-the-world [86] S. B. Ryzhov, V. A. Mokhov, M. P. Nikitenko, G. G. Bessalov, A. K. Podshibyakin, D. A. Anufriev, J. Gadó, U. Rohde, VVER-Type Re- actors of Russian Design, Springer US, Boston, MA, 2010, pp. 2249– 2320. doi:10.1007/978-0-387-98149-9_20. [87] T. S. A. E. C. ROSATOM, The VVER today: Evolution, Design, Safety, 2018. URL 0be1220af25741375138ecd1afb18743.pdf [88] S. Marco, G. Aleksander, P. Ghislain, Z. Jiri, D. Jiri, B. David, V. Ladislav, I. Melnikov, M. Valery, F. Fichot, P. Matejovi, G. Pavlin, 112 E. Alexandre, B. Sophie, B. Nikolai, J. J. Mathieu, C. L. Guennic, M. Buck, R. Krasen, I. N. Ivanov, K. Petar, N. Anna, In-Vessel Melt Retention (IVMR) Analysis of a VVER-1000 NPP, Publications Office of the European Union, 2016. doi:doi/10.2790/438713. [89] Z. Hozer, K. Trambauer, J. Duspiva, VVER-specific features regard- ing core degradation - Status Report, Nuclear Energy Agency of the OECD, 2018. URL https://inis.iaea.org/collection/NCLCollectionStore/_Public/ 41/041/41041076.pdf?r=1 [90] Status report 108 - VVER-1200 (V-491) (VVER-1200 (V-491)), In- ternational Atomic Energy Agency, Vienna, 2011. URL https://aris.iaea.org/PDF/VVER-1200(V-491).pdf [91] W. N. Association, Nuclear power in russia, (accessed on April 2021). URL https://world-nuclear.org/information-library/ country-profiles/countries-o-s/russia-nuclear-power.aspx# ECSArticleLink6 [92] M. Dwiddar, A. Badawi, H. Abou Gabal, I. El-Osery, From vver- 1000 to vver-1200: Investigation of the effect of the changes in core, 2014. doi:10.13140/2.1.4876.1281. [93] A. Glebov, A. Klushin, Development of supercritical-water cooled reactors in russia and abroad, Atomic Energy 116 (2014) 320 – 329. doi:https://doi.org/10.1007/s10512-014-9860-x. 113 [94] A. Glebov, A. Klushin, Y. Baranaev, Prospects of vver-skd reactor in a closed fuel cycle, Nuclear Energy and Technology 1 (1) (2015) 60–67. doi:https://doi.org/10.1016/j.nucet.2015.11.013. [95] J. Bajgl, M. Lehmann, New code for vver-440 loading pattern design, in: Proceedings of the ninth Symposium of Atomic Energy Research, Hungary: Kiadja and KFKI Atomenergia Kutato Intezet, 1999, p. 780. [96] J. Prehradny, K. Katovsky, R. Cada, Wwer 440 fuel cycles optimiza- tion by the athena code, in: Proceedings of the twenty-first sympo- sium of atomic energy research on WWER physics and reactor safety, 2011. [97] Y. Rahmani, A. Pazirandeh, M. Ghofrani, M. Sedighi, Using a com- bination of weighting factors, genetic algorithm and ant colony meth- ods to speed up the reloading pattern optimization of vver-1000 re- actors, 2013. [98] A. Fadaei, S. Setayeshi, Lonsa as a tool for loading pattern optimiza- tion for vver-1000 using synergy of a neural network and simulated annealing, Annals of Nuclear Energy 35. doi:10.1016/j.anucene. 2008.05.001. [99] C. Guler, S. Levine, K. Ivanov, J. Svarny, V. Krysl, P. Mikolas, J. Sustek, Development of the vver core loading optimization sys- tem, Annals of Nuclear Energy 31 (2004) 747–772. doi:10.1016/j. anucene.2003.09.005. [100] M. R. Karahroudi, S. A. M. Shirazi, K. Sepanloo, Optimization of designing the core fuel loading pattern in a vver-1000 nuclear power 114 reactor using the genetic algorithm, Annals of Nuclear Energy 57 (2013) 142–150. doi:10.1016/j.anucene.2013.01.051. [101] M. A. Moghanloo, S. Mahmoudi, A novel multi objective load- ing pattern optimization by gravitational search algorithm (gsa) for wwer1000 core, Progress in Nuclear Energy 93 (2016) 1–11. doi:10.1016/j.pnucene.2016.07.014. [102] T. V. Phu, et al., Calculation of neutronic characteristics of the VVER-1000 reactor core, Ministerial project 2014-2015, 2014. [103] T. D. Long, et al., Research and build interface software combining MCNP5 and COBRA-EN for fuel channel of VVER-1000 reactor, Elementary project, 2015. [104] N. H. Tiep, et al., Research on the effect of neutron current on the VVER reactor vessel, Elementary project, 2017. [105] T. V. Phu, et al., Research on loading pattern optimization of VVER reactor, Ministerial project 2015-2016, 2015. [106] J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, third edition, Prentice-Hall, 2001. [107] Y. Oka, Nuclear Reactor Design, Springer, Tokyo, 2014. doi:https: //doi.org/10.1007/978-4-431-54898-0. [108] J. David M Young, Iterative methods for solving partial difference equations of elliptic type, Ph.D. thesis, Havard University (1950). [109] R. J. Stamm’ler, M. J. Abbate, Methods of steady-state reactor physics in nuclear design, Academic Press INC. (London, 1983. 115 [110] K. Okumura, T. Kugo, K. Kaneko, K. Tsuchihashi, SRAC2006: A comprehensive neutronics calculation code system, Tech. rep., JAEA- Data/Code 2007-004. Japan Atomic Energy Agency, Tokai, Japan (2007). [111] M. B. Chadwick, P. Oblozinsky, M. Herman, N. M. Greene, R. D. McKnight, D. L. Smith, et al., Endf/b-vii.0: Next generation evalu- ated nuclear data library for nuclear science and technology, Nuclear Data Sheets 107 (12) (2006) 2931–3060. doi:https://doi.org/10. 1016/j.nds.2006.11.001. [112] J. Zhang, A. C. Sanderson, Jade: Adaptive differential evolution with optional external archive, IEEE Transactions on Evolutionary Computation 13 (5) (2009) 945–958. [113] G. C. Onwubolu, D. Davendra, Differential Evolution: A Hand- book for Global Permutation-Based Combinatorial Optimization, Vol. 175, Springer, Berlin, Heidelberg, 2009. doi:10.1007/ 978-3-540-92151-6. [114] N. Poursalehi, A. Zolfaghari, A. Minuchehr, Differential harmony search algorithm to optimize pwrs loading pattern, Nuclear Engineer- ing and Design 257 (2013) 161–174. doi:10.1016/j.nucengdes. 2013.01.020. [115] N. Poursalehi, A. Zolfaghari, A. Minuchehr, Pwr loading pattern op- timization using harmony search algorithm, Annals of Nuclear En- ergy 53 (2013) 288–298. doi:10.1016/j.anucene.2012.06.037. [116] R. Gharari, N. Poursalehi, M. Abbasi, M. Aghaie, Implementation of strength pareto evolutionary algorithm ii in the multiobjective 116 burnable poison placement optimization of kwu pressurized water reactor, Nuclear Engineering and Technology 48 (5) (2016) 1126– 1139. doi:10.1016/j.net.2016.04.004. [117] G. W. Corder, D. I. Foreman, Nonparametric Statistics for Non- Statisticians, John Wiley and Sons, Inc., 2009. [118] R. R. Swain, T. Dash, P. M. Khilar, A complete diagnosis of faulty sensor modules in a wireless sensor network, Ad Hoc Networks 93 (2019) 101924. doi:https://doi.org/10.1016/j.adhoc.2019. 101924. [119] M. E. H. Pedersen, Good Parameters for Differential Evolution, Tech- nical Report no. HL1002, Hvass Laboratories, 2010. 117 Appendix A VVER-1000 MOX core Benchmark specification Table A.1: Dimension of the cell zones [5]. Cell name Zone in the cell Radius (cm) Fuel cell Fuel pellet radius 0.386 Cladding outer radius 0.455 Central tube cell/ guide tube cell Tube inner radius 0.55 Tube outer radius 0.63 Guide tube with absorber rod cell Absorber pellet radius 0.35 Absorber cladding outer radius 0.41 Tube inner radius 0.55 Tube outer radius 0.63 Appendix B Cross sections of materials 118 Table A.2: Material names used in the fuel assemblies [5]. Assembly type Material name Material description UO2 assemblies type 1 U_4.2 Uranium fuel with 235U enrichment 4.2% wt. TVEG_5 Uranium-gadolinium fuel with enrichment 3.3% wt. on 235U and 5% wt. on Gd2O3 U_3.7 UO2 fuel with 235U enrichment 3.7% wt. MOX assemblies type 2 PU_3.6 MOX fuel with fissile plutonium isotopes enrichment 3.62% wt. TVEG_4 Uranium-gadolinium fuel with enrichment 3.6% wt. on 235U and 4% wt. on Gd2O3 PU_2.7 MOX fuel with fissile plutonium isotopes enrichment 2.69% wt. PU_2.4 MOX fuel with fissile plutonium isotopes enrichment 2.42% wt. Table A.3: Isotopic composition of fuel U_4.2, atoms/barn ∗ cm2 [5]. U_4.2 Burnup, MWd/kg 0 15 32 40 U235 9.0411E-04 5.8139E-04 3.2990E-04 2.4314E-04 U236 5.7700E-05 9.7452E-05 1.0890E-04 U238 2.0362E-02 2.0161E-02 1.9905E-02 1.9773E-02 NP37 3.7658E-06 1.0351E-05 1.3577E-05 PU38 4.5135E-07 2.7546E-06 4.6336E-06 PU39 9.6584E-05 1.2788E-04 1.3134E-04 PU40 1.7820E-05 4.4214E-05 5.4717E-05 PU41 7.8291E-06 2.4361E-05 3.0501E-05 PU42 8.5918E-07 6.9401E-06 1.1811E-05 AM41 9.6169E-08 5.7445E-07 8.2251E-07 O 4.2532E-02 4.2532E-02 4.2532E-02 4.2532E-02 SM49 9.1807E-08 8.9565E-08 8.5421E-08 SM51 3.8524E-07 4.9458E-07 5.3181E-07 TC99 1.9865E-05 4.0006E-05 4.8545E-05 RH03 1.1241E-05 2.2541E-05 2.6925E-05 CS33 2.1660E-05 4.3134E-05 5.2008E-05 ND43 1.7145E-05 3.0721E-05 3.5092E-05 ND45 1.2231E-05 2.3856E-05 2.8579E-05 PM47 5.1135E-06 7.0579E-06 7.2585E-06 SM52 2.0001E-06 4.0165E-06 4.7868E-06 119 Table A.4: Isotopic composition of fuel TVEG_5, atoms/barn ∗ cm2 [5]. TVEG_5 Burnup, MWd/kg 0 15 32 40 U235 6.6163E-04 4.8382E-04 2.6776E-04 1.9449E-04 U236 3.5164E-05 7.0094E-05 8.0032E-05 U238 1.9143E-02 1.8968E-02 1.8728E-02 1.8603E-02 NP37 2.5863E-06 7.7190E-06 1.0347E-05 PU38 2.9960E-07 2.0282E-06 3.5111E-06 PU39 8.8781E-05 1.1559E-04 1.1801E-04 PU40 1.5353E-05 4.1294E-05 5.1119E-05 PU41 6.2524E-06 2.2283E-05 2.8043E-05 PU42 6.0179E-07 6.3481E-06 1.1103E-05 AM41 6.9561E-08 4.9976E-07 7.2654E-07 O 4.1938E-02 4.1938E-02 4.1938E-02 4.1938E-02 GD52 3.2142E-06 2.2242E-06 1.1130E-06 7.6549E-07 GD54 3.4579E-05 3.1600E-05 2.7446E-05 2.5465E-05 GD55 2.3321E-04 3.2385E-07 1.5769E-07 1.4181E-07 GD56 3.2053E-04 5.4422E-04 5.3058E-04 5.2333E-04 GD57 2.4346E-04 1.8330E-07 1.6774E-07 1.5847E-07 GD58 3.8403E-04 6.3019E-04 6.3494E-04 6.3726E-04 GD60 3.3373E-04 3.3229E-04 3.3037E-04 3.2938E-04 SM49 8.0821E-08 8.0584E-08 7.7330E-08 SM51 3.1290E-07 4.0538E-07 4.4104E-07 TC99 1.2031E-05 3.0625E-05 3.8586E-05 RH03 7.4977E-06 1.8625E-05 2.2940E-05 CS33 1.3175E-05 3.3172E-05 4.1526E-05 ND43 1.0380E-05 2.3428E-05 2.7732E-05 ND45 7.3069E-06 1.8017E-05 2.2419E-05 PM47 3.3064E-06 5.7735E-06 6.1251E-06 SM52 1.2690E-06 3.2490E-06 3.9837E-06 120 Table A.5: Isotopic composition of fuel U_3.7, atoms/barn ∗ cm2 [5]. U_3.7 Burnup, MWd/kg 0 15 32 40 U235 7.9649E-04 4.8884E-04 2.6496E-04 1.9092E-04 U236 5.4042E-05 8.8494E-05 9.7726E-05 U238 2.0469E-02 2.0262E-02 2.0000E-02 1.9864E-02 NP37 3.7015E-06 9.9287E-06 1.2901E-05 PU38 4.6522E-07 2.7516E-06 4.5708E-06 PU39 9.5675E-05 1.2378E-04 1.2659E-04 PU40 1.9149E-05 4.5926E-05 5.6139E-05 PU41 8.3590E-06 2.4695E-05 3.0494E-05 PU42 1.0147E-06 7.7274E-06 1.2925E-05 AM41 1.0325E-07 5.7739E-07 8.0933E-07 O 4.2530E-02 4.2530E-02 4.2530E-02 4.2530E-02 SM49 8.3005E-08 8.2706E-08 7.9571E-08 SM51 3.4935E-07 4.5427E-07 4.9115E-07 TC99 1.9485E-05 3.8798E-05 4.6961E-05 RH03 1.1189E-05 2.2235E-05 2.6478E-05 CS33 2.1245E-05 4.1820E-05 5.0289E-05 ND43 1.6565E-05 2.9007E-05 3.2889E-05 ND45 1.1935E-05 2.2961E-05 2.7419E-05 PM47 4.9515E-06 6.7031E-06 6.8639E-06 SM52 2.0058E-06 3.9584E-06 4.6994E-06 121 Table A.6: Isotopic composition of fuel PU_3.6, atoms/barn ∗ cm2 [5]. PU_3.6 Burnup, MWd/kg 0 17 33 U235 4.3057E-05 3.0534E-05 2.0186E-05 U236 2.5385E-06 4.2696E-06 U238 2.0386E-02 2.0144E-02 1.9894E-02 NP37 2.4045E-06 4.2797E-06 PU38 1.0841E-06 1.3292E-06 2.7311E-06 PU39 7.5661E-04 4.7406E-04 2.9852E-04 PU40 5.3794E-05 1.4795E-04 1.7846E-04 PU41 9.5720E-06 5.7132E-05 8.3282E-05 PU42 3.5119E-06 9.8236E-06 2.5860E-05 AM41 1.3594E-06 2.9942E-06 O 4.2506E-02 4.2506E-02 4.2506E-02 SM49 1.4783E-07 1.2062E-07 SM51 7.7056E-07 7.6447E-07 TC99 2.2348E-05 4.0862E-05 RH03 2.2129E-05 3.6232E-05 CS33 2.4904E-05 4.4872E-05 ND43 1.5770E-05 2.7603E-05 ND45 1.1215E-05 2.0679E-05 PM47 5.0355E-06 6.6403E-06 SM52 3.2199E-06 5.2658E-06 122 Table A.7: Isotopic composition of fuel TVEG_4, atoms/barn ∗ cm2 [5]. TVEG_4 Burnup, MWd/kg 0 17 33 U235 7.3225E-04 5.4783E-04 3.4998E-04 U236 3.9989E-05 7.3537E-05 U238 1.9360E-02 1.9139E-02 1.8901E-02 NP37 3.7973E-06 9.6340E-06 PU38 4.9031E-07 2.5298E-06 PU39 1.2109E-04 1.5184E-04 PU40 1.7377E-05 4.4993E-05 PU41 7.0208E-06 2.4072E-05 PU42 5.4476E-07 5.0595E-06 AM41 8.8029E-08 5.5623E-07 O 4.2056E-02 4.2055E-02 4.2055E-02 GD52 2.5815E-06 1.8321E-06 1.0821E-06 GD54 2.7772E-05 2.5080E-05 2.2136E-05 GD55 1.8730E-04 1.0215E-06 1.6393E-07 GD56 2.5743E-04 4.3334E-04 4.2171E-04 GD57 1.9553E-04 2.5976E-07 2.0690E-07 GD58 3.0843E-04 5.0718E-04 5.1162E-04 GD60 2.6804E-04 2.6656E-04 2.6501E-04 SM49 1.0735E-07 1.0224E-07 SM51 4.0519E-07 5.0972E-07 TC99 1.2602E-05 2.9713E-05 RH03 7.9971E-06 1.8557E-05 CS33 1.3779E-05 3.2177E-05 ND43 1.0973E-05 2.3818E-05 ND45 7.6416E-06 1.7521E-05 PM47 3.2794E-06 5.5753E-06 SM52 1.2631E-06 3.0576E-06 123 Table A.8: Isotopic composition of fuel PU_2.7, atoms/barn ∗ cm2 [5]. PU_2.7 Burnup, MWd/kg 0 17 33 U235 4.3057E-05 2.8612E-05 1.7606E-05 U236 2.7944E-06 4.5372E-06 U238 2.0598E-02 2.0347E-02 2.0087E-02 NP37 2.4008E-06 4.2361E-06 PU38 8.0774E-07 1.0993E-06 2.5148E-06 PU39 5.6222E-04 3.3450E-04 2.1853E-04 PU40 3.9987E-05 1.2215E-04 1.4136E-04 PU41 7.1160E-06 4.9442E-05 6.9024E-05 PU42 2.6131E-06 9.5258E-06 2.5765E-05 AM41 1.1240E-06 2.3584E-06 O 4.2508E-02 4.2508E-02 4.2508E-02 SM49 1.1354E-07 9.8177E-08 SM51 5.8719E-07 6.0027E-07 TC99 2.0699E-05 3.7403E-05 RH03 2.0188E-05 3.2327E-05 CS33 2.3035E-05 4.0983E-05 ND43 1.4448E-05 2.4621E-05 ND45 1.0429E-05 1.9007E-05 PM47 4.6128E-06 5.9544E-06 SM52 3.0318E-06 4.7872E-06 124 Table A.9: Isotopic composition of fuel PU_2.4, atoms/barn ∗ cm2 [5]. PU_2.4 Burnup, MWd/kg 0 17 33 U235 4.3057E-05 2.7777E-05 1.6391E-05 U236 2.9076E-06 4.6650E-06 U238 2.0660E-02 2.0405E-02 2.0140E-02 NP37 2.3897E-06 4.1976E-06 PU38 7.2271E-07 1.0335E-06 2.4576E-06 PU39 5.0579E-04 2.9332E-04 1.9329E-04 PU40 3.5961E-05 1.1497E-04 1.3072E-04 PU41 6.4023E-06 4.6985E-05 6.4083E-05 PU42 2.3413E-06 9.6174E-06 2.6319E-05 AM41 1.0464E-06 2.1281E-06 O 4.2508E-02 4.2508E-02 4.2508E-02 SM49 1.0281E-07 9.0532E-08 SM51 5.3010E-07 5.4672E-07 TC99 2.0326E-05 3.6699E-05 RH03 1.9698E-05 3.1334E-05 CS33 2.2608E-05 4.0174E-05 ND43 1.4108E-05 2.3798E-05 ND45 1.0254E-05 1.8670E-05 PM47 4.5137E-06 5.7880E-06 SM52 2.9973E-06 4.6882E-06 Table A.10: Isotopic composition of the structural material, atoms/barn ∗ cm2 [5]. Material name Material zone Material isotopic composition Zirconium alloy Fuel cladding Zr 4.26E-02 Central tube Nb 4.22E-04 Guide tube Hf 6.59E-06 Steel Absorber cladding Fe 5.93E-02 Steel buffer Cr 1.69E-02 Steel barrel Ni 8.48E-03 Steel vessel Ti 9.90E-04 C 4.74E-04 B4C 80% enrichment of B10 Absorber rod B10 6.57E-02 B11 1.64E-02 C 2.05E-02 125 Table A.11: Moderator and water in reflector materials, atoms/barn ∗ cm2 [5]. Material name Material description Material isotopic composition M575B1.3 Moderator with boron content 1300 ppm, Tm = 575K, ρ = 0.7241g/cm3 H 4.8410E-02 O16 2.4205E-02 B10 1.0381E-05 B11 4.2049E-05 M575B0 Moderator without boron, Tm = 575K, ρ = 0.7241g/cm3 H 4.8410E-02 O16 2.4205E-02 B10 0.0 B11 0.0 M560B1.3 Moderator with boron content 1300 ppm, Tm = 560K, ρ = 0.7533g/cm3 H 5.0362E-02 O16 2.5181E-02 B10 1.0800E-05 B11 4.3744E-05 M560B0.6 Moderator with boron content 600 ppm, Tm = 560K, ρ = 0.7533g/cm3 H 5.0362E-02 O16 2.5181E-02 B10 4.9845E-06 B11 2.0190E-05 M560B0 Moderator without boron, Tm = 560K, ρ = 0.7533g/cm3 H 5.0362E-02 O16 2.5181E-02 B10 0.0 B11 0.0 M553B0 Moderator without boron, Tm = 553K, ρ = 0.7657g/cm3 H 5.1192E-02 O16 2.5596E-02 B10 0.0 B11 0.0 M300B2.8 Moderator with boron content 2800 ppm, Tm = 300K, ρ = 1.0033g/cm3 H 6.7076E-02 O16 3.3538E-02 B10 3.0981E-05 B11 1.2549E-04 Table B.1: Four groups structure with three fast groups and one thermal group. Group Energy (eV) Upper Lower 1 1.0000E+07 2.4788E+04 2 2.4788E+04 2.0347E+03 3 2.0347E+03 1.8554E+00 4 1.8554E+00 1.0000E-05 126 Table B.2: Four groups cross sections of fuel assemblies. Group PRODUCTION FISSION CAPTURE ABSORPTION FISS.SPCTR DIFFUSION1 g->1 g->2 g->3 g->4 * * A1B1A010 * * * * * * * * 1 4.14E-03 1.52E-03 1.01E-03 2.53E-03 9.98E-01 1.81E+00 1.43E-01 3.56E-02 3.07E-03 0.00E+00 2 2.31E-03 9.48E-04 5.42E-03 6.37E-03 1.48E-03 8.42E-01 0.00E+00 1.91E-01 1.99E-01 0.00E+00 3 1.53E-02 6.29E-03 2.31E-02 2.93E-02 3.32E-05 7.99E-01 0.00E+00 0.00E+00 3.17E-01 7.12E-02 4 1.36E-01 5.59E-02 2.68E-02 8.27E-02 0.00E+00 4.29E-01 0.00E+00 0.00E+00 2.62E-04 6.93E-01 * * A1B2A010 * * * * * * * * 1 4.05E-03 1.46E-03 1.00E-03 2.46E-03 9.98E-01 1.82E+00 1.42E-01 3.54E-02 3.05E-03 0.00E+00 2 1.70E-03 6.85E-04 5.36E-03 6.05E-03 1.51E-03 8.44E-01 0.00E+00 1.90E-01 1.99E-01 0.00E+00 3 1.27E-02 5.06E-03 2.40E-02 2.90E-02 3.37E-05 7.99E-01 0.00E+00 0.00E+00 3.17E-01 7.10E-02 4 1.40E-01 5.43E-02 3.69E-02 9.11E-02 0.00E+00 4.19E-01 0.00E+00 0.00E+00 2.88E-04 7.05E-01 * * A1B3A010 * * * * * * * * 1 3.90E-03 1.40E-03 9.90E-04 2.39E-03 9.98E-01 1.84E+00 1.41E-01 3.53E-02 3.04E-03 0.00E+00 2 1.19E-03 4.66E-04 5.30E-03 5.76E-03 1.53E-03 8.46E-01 0.00E+00 1.89E-01 1.99E-01 0.00E+00 3 1.01E-02 3.86E-03 2.48E-02 2.86E-02 3.43E-05 8.00E-01 0.00E+00 0.00E+00 3.17E-01 7.08E-02 4 1.27E-01 4.72E-02 4.42E-02 9.13E-02 0.00E+00 4.11E-01 0.00E+00 0.00E+00 2.87E-04 7.19E-01 * * A1B4A010 * * * * * * * * 1 3.84E-03 1.37E-03 9.84E-04 2.36E-03 9.98E-01 1.84E+00 1.41E-01 3.52E-02 3.04E-03 0.00E+00 2 1.01E-03 3.87E-04 5.27E-03 5.66E-03 1.54E-03 8.47E-01 0.00E+00 1.88E-01 1.99E-01 0.00E+00 3 9.00E-03 3.39E-03 2.51E-02 2.84E-02 3.45E-05 8.00E-01 0.00E+00 0.00E+00 3.17E-01 7.08E-02 4 1.19E-01 4.36E-02 4.60E-02 8.96E-02 0.00E+00 4.09E-01 0.00E+00 0.00E+00 2.80E-04 7.25E-01 * * A2B1A010 * * * * * * * * 1 4.61E-03 1.61E-03 9.99E-04 2.61E-03 9.98E-01 1.83E+00 1.41E-01 3.51E-02 3.02E-03 0.00E+00 2 1.61E-03 5.69E-04 5.49E-03 6.05E-03 1.52E-03 8.42E-01 0.00E+00 1.91E-01 1.99E-01 0.00E+00 3 1.64E-02 5.81E-03 2.41E-02 2.99E-02 3.39E-05 7.97E-01 0.00E+00 0.00E+00 3.17E-01 7.08E-02 4 2.36E-01 8.26E-02 8.04E-02 1.63E-01 0.00E+00 4.27E-01 0.00E+00 0.00E+00 5.13E-04 6.17E-01 * * A2B2A010 * * * * * * * * 1 4.30E-03 1.51E-03 9.93E-04 2.50E-03 9.98E-01 1.84E+00 1.41E-01 3.51E-02 3.03E-03 0.00E+00 2 1.24E-03 4.37E-04 5.38E-03 5.82E-03 1.54E-03 8.44E-01 0.00E+00 1.90E-01 1.99E-01 0.00E+00 3 1.34E-02 4.72E-03 2.45E-02 2.92E-02 3.45E-05 7.98E-01 0.00E+00 0.00E+00 3.18E-01 7.07E-02 4 1.96E-01 6.86E-02 7.28E-02 1.41E-01 0.00E+00 4.10E-01 0.00E+00 0.00E+00 4.43E-04 6.71E-01 * * A2B3A010 * * * * * * * * 1 4.07E-03 1.43E-03 9.87E-04 2.42E-03 9.98E-01 1.84E+00 1.40E-01 3.52E-02 3.03E-03 0.00E+00 2 9.84E-04 3.45E-04 5.31E-03 5.65E-03 1.57E-03 8.46E-01 0.00E+00 1.89E-01 1.99E-01 0.00E+00 3 1.11E-02 3.88E-03 2.52E-02 2.91E-02 3.51E-05 7.99E-01 0.00E+00 0.00E+00 3.18E-01 7.05E-02 4 1.60E-01 5.57E-02 6.82E-02 1.24E-01 0.00E+00 4.05E-01 0.00E+00 0.00E+00 3.86E-04 6.99E-01 127 Table B.3: Four groups cross sections of non fuel materials. Group PRODUCTION FISSION CAPTURE ABSORPTION FISS.SPCTR DIFFUSION1 g->1 g->2 g->3 g->4 * * STB1A080 * * * * * * * * 1 0.00E+00 0.00E+00 8.48E-04 8.48E-04 0.00E+00 1.40E+00 1.97E-01 3.69E-02 3.10E-03 0.00E+00 2 0.00E+00 0.00E+00 9.24E-04 9.24E-04 0.00E+00 5.14E-01 0.00E+00 5.28E-01 1.20E-01 0.00E+00 3 0.00E+00 0.00E+00 6.28E-03 6.28E-03 0.00E+00 4.71E-01 0.00E+00 0.00E+00 6.53E-01 4.87E-02 4 0.00E+00 0.00E+00 1.14E-01 1.14E-01 0.00E+00 3.02E-01 0.00E+00 0.00E+00 4.15E-04 9.89E-01 * * STB2A090 * * * * * * * * 1 0.00E+00 0.00E+00 2.15E-04 2.15E-04 0.00E+00 1.91E+00 5.35E-02 1.11E-01 9.74E-03 0.00E+00 2 0.00E+00 0.00E+00 8.83E-05 8.83E-05 0.00E+00 8.77E-01 0.00E+00 2.60E-02 3.54E-01 0.00E+00 3 0.00E+00 0.00E+00 9.13E-04 9.13E-04 0.00E+00 7.50E-01 0.00E+00 0.00E+00 3.00E-01 1.43E-01 4 0.00E+00 0.00E+00 1.72E-02 1.72E-02 0.00E+00 3.03E-01 0.00E+00 0.00E+00 1.65E-04 1.08E+00 * * STB3A0A0 * * * * * * * * 1 0.00E+00 0.00E+00 5.43E-04 5.43E-04 0.00E+00 1.58E+00 1.31E-01 7.33E-02 6.37E-03 0.00E+00 2 0.00E+00 0.00E+00 5.18E-04 5.18E-04 0.00E+00 6.36E-01 0.00E+00 2.89E-01 2.35E-01 0.00E+00 3 0.00E+00 0.00E+00 3.71E-03 3.71E-03 0.00E+00 5.74E-01 0.00E+00 0.00E+00 4.82E-01 9.51E-02 4 0.00E+00 0.00E+00 6.67E-02 6.67E-02 0.00E+00 2.95E-01 0.00E+00 0.00E+00 2.92E-04 1.06E+00 * * H2ORA0H0 * * * * * * * * 1 0.00E+00 0.00E+00 1.71E-04 1.71E-04 0.00E+00 1.97E+00 4.27E-02 1.16E-01 1.02E-02 0.00E+00 2 0.00E+00 0.00E+00 3.26E-05 3.26E-05 0.00E+00 9.24E-01 0.00E+00 -8.83E-03 3.69E-01 0.00E+00 3 0.00E+00 0.00E+00 5.41E-04 5.41E-04 0.00E+00 7.82E-01 0.00E+00 0.00E+00 2.76E-01 1.49E-01 4 0.00E+00 0.00E+00 1.08E-02 1.08E-02 0.00E+00 3.04E-01 0.00E+00 0.00E+00 1.49E-04 1.08E+00 * * STEAA0L0 * * * * * * * * 1 0.00E+00 0.00E+00 1.13E-03 1.13E-03 0.00E+00 1.32E+00 2.48E-01 2.29E-03 1.06E-06 0.00E+00 2 0.00E+00 0.00E+00 1.30E-03 1.30E-03 0.00E+00 4.40E-01 0.00E+00 7.47E-01 1.00E-02 0.00E+00 3 0.00E+00 0.00E+00 8.66E-03 8.66E-03 0.00E+00 3.99E-01 0.00E+00 0.00E+00 8.22E-01 4.53E-03 4 0.00E+00 0.00E+00 1.60E-01 1.60E-01 0.00E+00 3.10E-01 0.00E+00 0.00E+00 5.32E-04 9.15E-01 128