Reduced Order Modeling of Once-through Steam Generator Based on Dynamic Mode Decomposition
-
摘要: 直流蒸汽发生器（OTSG）的运行特性对反应堆的安全有重要影响。大尺度、精细化的仿真模型为OTSG热工水力特性和安全分析提供高保真的模拟结果，但也对计算资源提出了挑战。带控制的动态模式分解（DMDc）是一种数据驱动的模型降阶（MOR）方法，其在动态模式分解（DMD）的基础上能够为含控制输入的系统建立低维的、准确的输入输出模型以替代高保真模型进行快速计算。考虑到实际运行中如蒸汽压力等OTSG热工参数受反应堆控制系统影响，利用RELAP5程序建立的全阶模型获取了OTSG在快速降负荷和快速升负荷工况下的主要热工参数的高保真模拟结果，基于DMDc建立了OTSG的降阶模型（ROM）。结果表明DMDc能够提取OTSG在变负荷工况下的动态特性，ROM的计算结果与全阶模型之间的最大相对误差小于2%。此外，对比了DMDc与DMD方法对OTSG降阶建模的效果，证明了DMDc方法的优越性。
-
关键词:
- 直流蒸汽发生器（OTSG） /
- 动态模式分解（DMD） /
- 带控制的动态模式分解（DMDc） /
- 模型降阶（MOR）
Abstract: The operation characteristics of once-through steam generator (OTSG) have an important influence on the safety of the reactor. The large-scale and refined simulation model provides high fidelity simulation results for the thermal hydraulic characteristics and safety analysis of OTSG, but it also challenges the computing resources. Dynamic Mode Decomposition with Control (DMDc) is a data-driven model order reduction (MOR) method, which can establish a low-dimensional and accurate input-output model for the system with control inputs on the basis of dynamic mode decomposition (DMD) to replace the high-fidelity model for fast calculation. Considering that the thermal parameters of OTSG, such as steam pressure, are affected by the reactor control system in actual operation, the full-order model established by RELAP5 is used to obtain the high-fidelity simulation results of the main thermal parameters of OTSG under the conditions of rapid load reduction and rapid load increase, and the reduced-order model (ROM) of OTSG is established based on DMDc. The results show that DMDc can extract the dynamic characteristics of OTSG under variable load conditions, and the maximum relative error between the calculation results of reduced-order model and the full-order model is less than 2%. In addition, the effects of DMDc and DMD methods on OTSG reduced-order modeling are compared, which proves the superiority of DMDc method. -
图 1 RELAP5模型计算节点图
A—环管；P—管道；B—分支；V—阀门；T—时间相关控制体；TDJ—时间相关接管；S—单一控制体
Figure 1. Nodalization of RELAP5 Model
图 3 快速降负荷工况下的DMDc结果与全阶模型结果对比
图例中n1~n30表示第1~第30个节点的对应参数值
Figure 3. Comparison of DMDc Results and Full-order Model Results under Rapid Load Reduction Conditions
图 4 DMDc重构误差
1—最大相对误差；2—L2范数误差；3—给水温度误差；4—含气率误差；5—换热系数误差；6—冷却剂温度误差；7—压力误差
Figure 4. Reconstruction Errors of DMDc
图 5 DMDc主要模态对应的特征值在复平面上的分布
Figure 5. Distribution of Eigenvalues Corresponding to Main Modes of DMDc on Complex Plane
图 7 DMD与DMDc重构误差对比
1—最大误差；2—总体误差；3—给水温度误差；4—含气率误差；5—换热系数误差；6—冷却剂温度误差；7—压力误差
Figure 7. Comparison of Reconstruction Errors between DMD and DMDc
表 1 IP200 主要热工设计参数
Table 1. Main Thermal Design Parameters of IP200
参数 数值 反应堆额定功率/MW 220.0 堆芯进口温度/K 557.15 堆芯出口温度/K 589.15 一回路冷却剂平均温度/K 573.15 稳压器压力/MPa 15.5 主冷却剂流量/(kg·s−1) 1200.0 给水流量/(kg·s−1) 90.0 主给水压力/MPa 4.5 主蒸汽压力/MPa 3.0 主给水温度/K 373.15 主蒸汽温度/K 543.15 
下载: 导出CSV
表 2 DMD与DMDc方法的计算步骤
Table 2. Computational Steps of DMD and DMDc Method
DMD DMDc 步骤一：收集并构造快照${\boldsymbol{X}}$和${{{\boldsymbol{X}}^{^{{\prime}}}}}$ 步骤一：收集并构造系统状态快照和控制输入快照 步骤二：对输出快照矩阵${{{\boldsymbol{X}}^{^{{\prime}}}}}$奇异值分解 步骤二：对全部快照矩阵奇异值分解 步骤三：计算DMD算子A的近似 步骤三：对输出快照矩阵${{{\boldsymbol{X}}^{^{{\prime}}}}}$奇异值分解 步骤四：对算子A的近似进行特征值分解 步骤四：计算DMDc算子A和算子B的近似 步骤五：计算DMD模态 步骤五：对算子A的近似进行特征值分解 步骤六：计算DMDc模态
下载: 导出CSV
表 3 快照包含的输入变量和控制变量
Table 3. Input Parameters and Control Variables for Snapshot Collection
名称 快照类别 传热管一次侧冷却剂温度 输入变量 传热管二次侧流体温度 输入变量 传热管冷却剂进出口压力 输入变量 传热管换热系数 输入变量 传热管二次侧流体含气率 输入变量 主蒸汽压力 输入变量 主给水压力 输入变量 反应堆功率 控制变量 反应性 控制变量 给水流量 控制变量 蒸汽压力偏差 控制变量
下载: 导出CSV
表 4 不同的采样时间步长下的DMDc误差对比
Table 4. Comparison of Reconstruction Errors under DifferentSampling Time Steps
采样时间
步长/s总体误差 传热管一
次侧冷却
剂温度误差传热管二
次侧温
度误差压力重
构误差传热管
换热系
数误差传热管
含气率
误差0.1 0.0015 0.0046 0.0005 0.0113 0.0018 0.0013 0.2 0.0015 0.0047 0.0005 0.0114 0.0018 0.0013 0.3 0.0015 0.0047 0.0005 0.0115 0.0018 0.0014
下载: 导出CSV
-
[1] XIA G L, PENG M J, DU X. Research of flow instability in OTSG under low load conditions[J]. Annals of Nuclear Energy, 2015, 75: 421-427. doi: 10.1016/j.anucene.2014.08.059 [2] AVRAMOVA M N, IVANOV K N. Verification, validation and uncertainty quantification in multi-physics modeling for nuclear reactor design and safety analysis[J]. Progress in Nuclear Energy, 2010, 52(7): 601-614. doi: 10.1016/j.pnucene.2010.03.009 [3] IVANOV K, AVRAMOVA M. Progress and challenges in the development and qualification of multi-level multi-physics coupled methodologies for reactor analysis[C]//ICAPP 2007 - International Congress on Advances in Nuclear Power Plants. Paris, France: SFEN, 2007. [4] FICK L, MADAY Y, PATERA A T, et al. A stabilized POD model for turbulent flows over a range of Reynolds numbers: optimal parameter sampling and constrained projection[J]. Journal of Computational Physics, 2018, 371: 214-243. doi: 10.1016/j.jcp.2018.05.027 [5] MANTHEY R, KNOSPE A, LANGE C, et al. Reduced order modeling of a natural circulation system by proper orthogonal decomposition[J]. Progress in Nuclear Energy, 2019, 114: 191-200. doi: 10.1016/j.pnucene.2019.03.010 [6] SMITH T R, MOEHLIS J, HOLMES P. Low-dimensional modelling of turbulence using the proper orthogonal decomposition: a tutorial[J]. Nonlinear Dynamics, 2005, 41(1-3): 275-307. doi: 10.1007/s11071-005-2823-y [7] LORENZI S, CAMMI A, LUZZI L, et al. A reduced order model for investigating the dynamics of the Gen-IV LFR coolant pool[J]. Applied Mathematical Modelling, 2017, 46: 263-284. doi: 10.1016/j.apm.2017.01.066 [8] GERMAN P, TANO M, FIORINA C, et al. GeN-ROM—An OpenFOAM®-based multiphysics reduced-order modeling framework for the analysis of Molten Salt Reactors[J]. Progress in Nuclear Energy, 2022, 146: 104148. doi: 10.1016/j.pnucene.2022.104148 [9] ZAREI M. On a reduced order modeling of the nuclear reactor dynamics[J]. Applied Mathematics and Computation, 2021, 393: 125819. doi: 10.1016/j.amc.2020.125819 [10] SUN Y, YANG J H, WANG Y H, et al. A POD reduced-order model for resolving the neutron transport problems of nuclear reactor[J]. Annals of Nuclear Energy, 2020, 149: 107799. doi: 10.1016/j.anucene.2020.107799 [11] VALOCCHI G, TOMMASI J, RAVETTO P. Reduced order models in reactor kinetics: a comparison between point kinetics and multipoint kinetics[J]. Annals of Nuclear Energy, 2020, 147: 107702. doi: 10.1016/j.anucene.2020.107702 [12] ZHANG C Y, CHEN G. Fast solution of neutron transport SP3 equation by reduced basis finite element method[J]. Annals of Nuclear Energy, 2018, 120: 707-714. doi: 10.1016/j.anucene.2018.06.042 [13] BUCHAN A G, CALLOO A A, GOFFIN M G, et al. A POD reduced order model for resolving angular direction in neutron/photon transport problems[J]. Journal of Computational Physics, 2015, 296: 138-157. doi: 10.1016/j.jcp.2015.04.043 [14] CHATTERJEE A. An introduction to the proper orthogonal decomposition[J]. Current Science, 2000, 78(7): 808-817. [15] BENNER P, GRIVET-TALOCIA S, QUARTERONI A, et al. Model order reduction: volume 2: snapshot-based methods and algorithms[M]. Boston: De Gruyter, 2020: 307-308. [16] SCHMID P J. Dynamic mode decomposition of numerical and experimental data[J]. Journal of Fluid Mechanics, 2010, 656: 5-28. doi: 10.1017/S0022112010001217 [17] NATHAN K J, STEVEN L B, BINGNI W B, et al. Dynamic mode decomposition: data-driven modeling of complex systems[M]. SIAM, 2016: 1. [18] BRUNTON S L, KUTZ J N. Data-driven science and engineering: machine learning, dynamical systems, and control[M]. 2nd ed. Cambridge: Cambridge University Press, 2022: 281-283. [19] TU J H. Dynamic mode decomposition: theory and applications[D]. Princeton: Princeton University, 2013. [20] DI RONCO A, CAMMI A, GIACOBBO F, et al. Application of the dynamic mode decomposition approach to the dispersion of radioactive contaminants in air[C]//Proceedings of the 27th International Conference Nuclear Energy for New Europe (NENE 2018). Ljubljana. Nuclear Society of Slovenia. 2018: 1-8. [21] DI RONCO A, INTROINI C, CERVI E, et al. Dynamic mode decomposition for the stability analysis of the Molten Salt Fast Reactor core[J]. Nuclear Engineering and Design, 2020, 362: 110529. doi: 10.1016/j.nucengdes.2020.110529 [22] CHEN K K, TU J H, ROWLEY C W. Variants of dynamic mode decomposition: boundary condition, koopman, and fourier analyses[J]. Journal of Nonlinear Science, 2012, 22(6): 887-915. doi: 10.1007/s00332-012-9130-9 [23] SCHMID P J. Dynamic mode decomposition and its variants[J]. Annual Review of Fluid Mechanics, 2022, 54: 225-254. doi: 10.1146/annurev-fluid-030121-015835 [24] PROCTOR J L, BRUNTON S L, KUTZ J N. Dynamic mode decomposition with control[J]. SIAM Journal on Applied Dynamical Systems, 2016, 15(1): 142-161. doi: 10.1137/15M1013857 [25] XIA G L, PENG M J, DU X. Calculation analysis on the natural circulation of a passive residual heat removal system for IPWR[J]. Annals of Nuclear Energy, 2014, 72: 189-197. doi: 10.1016/j.anucene.2014.02.018 [26] 刘建阁,彭敏俊,张志俭,等. 套管式直流蒸汽发生器负荷跟随动态特性分析[J]. 原子能科学技术,2010, 44(2): 175-182. [27] GENGLEI X, MINJUN P, XUE D. Analysis of load-following characteristics for an integrated pressurized water reactor[J]. International Journal of Energy Research, 2014, 38(3): 380-390. doi: 10.1002/er.3053 [28] GAVISH M, DONOHO D L. The optimal hard threshold for singular values is $ {4 / \sqrt{3}} $[J]. IEEE Transactions on Information Theory, 2014, 60(8): 5040-5053. doi: 10.1109/TIT.2014.2323359 -
计量
- 文章访问数: 102
- HTML全文浏览量: 19
- PDF下载量: 58
- 被引次数: 0
