# Localizing Perturbations in Pressurized Water Reactors Using One-Dimensional Deep Convolutional Neural Networks

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. The VVER-1000 Nuclear Reactor at Temelín

#### 3.1. Reactor Description

#### 3.2. Measurement Methodology

#### 3.3. Detectors Used for Structural Health Monitoring

## 4. The Simulated Data

#### 4.1. The FEMFFUSION Diffusion Code

#### 4.2. Neutron Noise Diffusion Equation

- Fluctuations are assumed to be small.$$|\delta X(\overrightarrow{r},t)|\ll {X}_{0}\left(\overrightarrow{r}\right),\phantom{\rule{1.em}{0ex}}\forall (\overrightarrow{r},t).$$
- The transient is assumed to be stationary.$$\u2329\delta X(\overrightarrow{r},t)\u232a=0,\phantom{\rule{1.em}{0ex}}\forall (\overrightarrow{r},t).$$
- Second-order terms are neglected.$$\delta X(\overrightarrow{r},t)\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\delta Y(\overrightarrow{r},t)\approx 0.$$

#### 4.3. Generic Absorber of Variable Strength

## 5. The Machine Learning Model

#### 5.1. Model Architecture

## 6. Results and Discussion

#### 6.1. Results

#### 6.1.1. The Efficiency of the Trained Models

#### 6.1.2. Prediction on the Plant Measurements

#### 6.2. Discussion

#### 6.2.1. The Other Available Measurements in the Core

#### 6.2.2. Some Considerations on the Frequency Resolution

#### 6.2.3. Spectra Observation of the Other Available Measurements to Validate the Prediction

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ANN | Artificial Neural Network |

APSD | Auto-Power Spectral Density |

BOC | Beginning of Cycle |

CNN | Convolutional Neural Network |

CPSD | Cross-Power Spectral Density |

DC | Direct Current component |

DMTS | Distributed Measuring Test System |

FA | Fuel Assembly |

IAEA | International Atomic Energy Agency |

IRI | Incompatible Rod Insertion |

JTFS | Joint Time–Frequency Spectrogram |

LSTM | Long Short-Term Memory unit |

ML | Machine Learning |

NPP | Nuclear Power Plant |

PWR | Pressurized Water Reactor |

RNN | Recurrent Neural Network |

RVMS | Reactor Vibration Monitoring System |

SPND | Self Power Neutron Detector |

SÚJB | State Office for Nuclear Safety (Czech Republic) |

UJV | Nuclear power engineering in Czech Republic (ÚJV Řež) |

## References

- Ribeiro, F.D.S.; Calivá, F.; Chionis, D.; Dokhane, A.; Mylonakis, A.; Demazière, C.; Leontidis, G.; Kollias, S. Towards a Deep Unified Framework for Nuclear Reactor Perturbation Analysis. In Proceedings of the 2018 IEEE Symposium Series on Computational Intelligence (SSCI), Bangalore, India, 18–21 November 2018; pp. 120–127. [Google Scholar] [CrossRef][Green Version]
- Noise-Based Core Monitoring and Diagnostics—Overview of the Cortex Project; The CORTEX project received funding from the Euratom Research and Training Programme 2014–2018 under grant agreement No. 754316; Zenodo: Geneva, Switzerland, 2017; Available online: https://zenodo.org/record/1230346#.YcVBxmBBxPY (accessed on 15 November 2021).
- Tambouratzis, T.; Giannatsis, J.; Kyriazis, A.; Siotropos, P. Applying the Computational Intelligence Paradigm to Nuclear Power Plant Operation: A Review (1990–2015). Int. J. Energy Optim. Eng.
**2020**, 9, 27–109. [Google Scholar] [CrossRef][Green Version] - Pázsit, I.; Glöckler, O. On the Neutron Noise Diagnostics of Pressurized Water Reactor Control Rod Vibrations. III. Application at a Power Plant. Nucl. Sci. Eng.
**1988**, 99, 313–328. [Google Scholar] [CrossRef] - Pázsit, I.; Garis, N.S.; Glöckler, O. On the Neutron Noise Diagnostics of Pressurized Water Reactor Control Rod Vibrations—IV: Application of Neural Networks. Nucl. Sci. Eng.
**1996**, 124, 167–177. [Google Scholar] [CrossRef] - Karlsson, J.H.; Pázsit, I. Localisation of a channel instability in the Forsmark-1 boiling water reactor. Ann. Nucl. Energy
**1999**, 26, 1183–1204. [Google Scholar] [CrossRef] - Demazière, C.; Mylonakis, A.; Vinai, P.; Durrant, A.; De Sousa Ribeiro, F.; Wingate, J.; Leontidis, G.; Kollias, S. Neutron noise-based anomaly classification and localization using machine learning. EPJ Web Conf.
**2021**, 247, 21004. [Google Scholar] [CrossRef] - Durrant, A.; Leontidis, G.; Kollias, S. 3D convolutional and recurrent neural networks for reactor perturbation unfolding and anomaly detection. EPJ Nucl. Sci. Technol.
**2019**, 5, 20. [Google Scholar] [CrossRef] - Ioannou, G.; Tagaris, T.; Alexandridis, G.; Stafylopatis, A. Intelligent techniques for anomaly detection IN nuclear reactors. EPJ Web Conf.
**2021**, 247, 21011. [Google Scholar] [CrossRef] - Tagaris, T.; Ioannou, G.; Sdraka, M.; Alexandridis, G.; Stafylopatis, A. Putting Together Wavelet-Based Scaleograms and Convolutional Neural Networks for Anomaly Detection in Nuclear Reactors. In Proceedings of the 2019 3rd International Conference on Advances in Artificial Intelligence (ICAAI 2019), Istanbul, Turkey, 26–28 October 2019; Association for Computing Machinery: New York, NY, USA, 2019; pp. 237–243. [Google Scholar] [CrossRef][Green Version]
- Tasakos, T.; Ioannou, G.; Verma, V.; Alexandridis, G.; Dokhane, A.; Stafylopatis, A. Deep Learning-Based Anomaly Detection in Nuclear Reactor Cores. In Proceedings of the International Conference on Mathematics & Computational Methods Applied to Nuclear Science & Engineering (M&C 2021), Online, 3–7 October 2021; pp. 2026–2037. [Google Scholar] [CrossRef]
- Demazière, C. CORE SIM: A multi-purpose neutronic tool for research and education. Ann. Nucl. Energy
**2011**, 38, 2698–2718. [Google Scholar] [CrossRef] - Mylonakis, A.; Vinai, P.; Demazière, C. CORE SIM+: A flexible diffusion-based solver for neutron noise simulations. Ann. Nucl. Energy
**2021**, 155, 108149. [Google Scholar] [CrossRef] - Ioannou, G.; Tasakos, T.; Mylonakis, A.; Alexandridis, G.; Demaziere, C.; Vinai, P.; Stafylopatis, A. Feature Extraction and Identification Techniques for the Alignment of Perturbation Simulations with Power Plant Measurements. In Proceedings of the International Conference on Mathematics & Computational Methods Applied to Nuclear Science & Engineering (M&C 2021), Online, 3–7 October 2021; pp. 2048–2059. [Google Scholar] [CrossRef]
- Studsvik. SIMULATE-3K. Available online: https://www.studsvik.com/what-we-do/products/simulate3-k/ (accessed on 15 November 2021).
- Vidal-Ferràndiz, A.; Carreño, A.; Ginestar, D.; Verdú, G. FEMFFUSION: A Finite Element Code for Nuclear Reactor Modelling. 2020. Available online: https://www.femffusion.imm.upv.es/ (accessed on 15 November 2021).
- Stulik, P.; Bem, M.; Tschiesche, J.; Machek, J. CORTEX WP4 Progress Report on subtask T4.2.3. In Progress Report ver.01, Core Monitoring Techniques and Experimental Validation and Demonstration (CORTEX), Horizon 2020 EU Framework Programm (No. 754316); European Commission: Brussels, Belgium, 2020. [Google Scholar]
- Stulik, P.; Torres, L.; Montalvo, C.; García-Berrocal, A.; Salazar, C.; Alexandridis, G.; Tabouratzis, T.; Machek, J.; Pantera, L.; Bem, M. CORTEX Deliverable 3.3: Development of advanced signal processing techniques and evaluation results. In Deliverable D3.3, Core Monitoring Techniques and Experimental Validation and Demonstration (CORTEX), Horizon 2020 EU Framework Programm (No. 754316); European Commission: Brussels, Belgium, 2019. [Google Scholar]
- Dokhane, A.; Mylonakis, A. CORTEX Deliverable 3.2: Description of simulated data. In Deliverable D3.2, Core Monitoring Techniques and Experimental Validation and Demonstration (CORTEX), Horizon 2020 EU Framework Programm (No. 754316); European Commission: Brussels, Belgium, 2019. [Google Scholar]
- Vidal-Ferràndiz, A.; Ginestar, D.; Carreño, A.; Verdú, G.; Demazière, C. A finite element method for neutron noise analysis in hexagonal reactors. EPJ Web Conf.
**2021**, 247, 21007. [Google Scholar] [CrossRef] - Vidal-Ferrandiz, A.; Fayez, R.; Ginestar, D.; Verdú, G. Solution of the Lambda modes problem of a nuclear power reactor using an h–p finite element method. Ann. Nucl. Energy
**2014**, 72, 338–349. [Google Scholar] [CrossRef][Green Version] - Vidal-Ferràndiz, A.; Carreño, A.; Ginestar, D.; Verdú, G. A block Arnoldi method for the SPN equations. Int. J. Comput. Math.
**2020**, 97, 341–357. [Google Scholar] [CrossRef] - Stacey, P.W.M. Nuclear Reactor Physics, 2nd ed.; WILEY-VCH Verlag GmbH & Co.KGaA: Weinheim, Germany, 2007. [Google Scholar] [CrossRef]
- Pázsit, I.; Demazière, C. Noise Techniques in Nuclear Systems. In Handbook of Nuclear Engineering; Cacuci, D.G., Ed.; Springer: Boston, MA, USA, 2010; pp. 1629–1737. [Google Scholar] [CrossRef]
- Duncan, W.J. Galerkin’s Method in Mechanics and Differential Equations; Defense Technical Information Center: Ft. Belvoir, VA, USA, 1937. [Google Scholar]
- Van der Vorst, H.A. Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems. SIAM J. Sci. Stat. Comput.
**1992**, 13, 631–644. [Google Scholar] [CrossRef] - Schwarzenberg-Czerny, A. On matrix factorization and efficient least squares solution. Astron. Astrophys. Suppl. Ser.
**1995**, 110, 405. [Google Scholar] - Ba, J.L.; Kiros, J.R.; Hinton, G.E. Layer Normalization. arXiv
**2016**, arXiv:1607.06450. [Google Scholar] - Kingma, D.P.; Ba, J. Adam: A Method for Stochastic Optimization. arXiv
**2017**, arXiv:1412.6980. [Google Scholar]

**Figure 2.**SPND positions in the core of the VVER-1000/320 reactor with data from the U1C09 cycle. (

**a**) Radial detector positions with marked fa1, fa3, fa4 and fa6 sets. (

**b**) Vertical detector positions in the TVSA-T fuel assembly.

**Figure 3.**JTFS from the $N205$ SPND (selected from the fa1 configuration set) during the U1C09 cycle.

**Figure 7.**Prediction on plant measurements when no filter is applied (sampling rate 100 $\mathrm{Hz}$). (

**a**) fa1. (

**b**) fa3. (

**c**) fa4. (

**d**) fa6.

**Figure 8.**Prediction on plant measurements for filtered bandwidth $[0\phantom{\rule{0.166667em}{0ex}};1.1]\phantom{\rule{0.166667em}{0ex}}\mathrm{Hz}$ and sampling rate of 100 $\mathrm{Hz}$ (Section 6.2.2). (

**a**) fa1. (

**b**) fa3. (

**c**) fa4. (

**d**) fa6.

**Figure 9.**Prediction on plant measurements for filtered bandwidth $[0\phantom{\rule{0.166667em}{0ex}};2]\phantom{\rule{0.166667em}{0ex}}\mathrm{Hz}$ and sampling rate of 100 $\mathrm{Hz}$. (

**a**) fa1. (

**b**) fa3. (

**c**) fa4. (

**d**) fa6.

**Figure 10.**Prediction on plant measurements for filtered bandwidth $[9\phantom{\rule{0.166667em}{0ex}};11]\phantom{\rule{0.166667em}{0ex}}\mathrm{Hz}$ and sampling rate of 100 $\mathrm{Hz}$. (

**a**) fa1. (

**b**) fa3. (

**c**) fa4. (

**d**) fa6.

**Figure 11.**Available measurements for the U1C09 cycle, used to verify the machine learning performance.

**Table 1.**Detector signal groups (letter N designating internal detectors, the first two digits their radial position and the last digit the axial position).

Group Name | Detectors |
---|---|

fa1 | $N221$, $N223$, $N225$, $N227$, $N241$, $N243$, $N245$, $N247$, $N271$, $N273$, |

$N275$, $N277$, $N201$, $N203$, $N205$, $N207$ | |

fa3 | $N411$, $N413$, $N415$, $N417$, $N381$, $N383$, $N385$, $N387$, $N471$, $N473$, |

$N475$, $N477$, $N401$, $N403$, $N405$, $N407$ | |

fa4 | $N151$, $N153$, $N155$, $N157$, $N081$, $N083$, $N085$, $N087$, $N061$, $N063$, |

$N065$, $N067$, $N111$, $N113$, $N115$, $N117$ | |

fa6 | $N571$, $N573$, $N575$, $N577$, $N541$, $N543$, $N545$, $N547$, $N511$, $N513$, |

$N515$, $N517$, $N561$, $N563$, $N565$, $N567$ |

Group Name | fa1 | fa3 | fa4 | fa6 |
---|---|---|---|---|

Accuracy | $93.25\%$ | $93.95\%$ | $93.74\%$ | $93.26\%$ |

Model Characteristic | 0.1 Hz | 1 Hz | 10 Hz |
---|---|---|---|

Prediction accuracy of the radial location | low | medium | high |

Spectral resolution | not well described | partially described | well described |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Pantera, L.; Stulík, P.; Vidal-Ferràndiz, A.; Carreño, A.; Ginestar, D.; Ioannou, G.; Tasakos, T.; Alexandridis, G.; Stafylopatis, A. Localizing Perturbations in Pressurized Water Reactors Using One-Dimensional Deep Convolutional Neural Networks. *Sensors* **2022**, *22*, 113.
https://doi.org/10.3390/s22010113

**AMA Style**

Pantera L, Stulík P, Vidal-Ferràndiz A, Carreño A, Ginestar D, Ioannou G, Tasakos T, Alexandridis G, Stafylopatis A. Localizing Perturbations in Pressurized Water Reactors Using One-Dimensional Deep Convolutional Neural Networks. *Sensors*. 2022; 22(1):113.
https://doi.org/10.3390/s22010113

**Chicago/Turabian Style**

Pantera, Laurent, Petr Stulík, Antoni Vidal-Ferràndiz, Amanda Carreño, Damián Ginestar, George Ioannou, Thanos Tasakos, Georgios Alexandridis, and Andreas Stafylopatis. 2022. "Localizing Perturbations in Pressurized Water Reactors Using One-Dimensional Deep Convolutional Neural Networks" *Sensors* 22, no. 1: 113.
https://doi.org/10.3390/s22010113