1. APPLICATION OF CLUSTER ANALYSIS AND AUTOREGRESSIVE NEURAL NETWORKS FOR THE NOISE DIAGNOSTICS OF THE IBR-2M REACTOR Yu. N. Pepelyshev, Ts. Tsogtsaikhan, G. A. Ososkov XXV International Symposium on Nuclear Electronics and Computing - NEC'2015, On 28 September – 02 October, 2015, Montenegro (Budva) Joint Institute for Nuclear Research Dubna, Russian Federation

2. Contents Introduction Objective Cluster analysis of the pulse energy noise −Determining the number of clusters −Clustering procedure −Clustering results & Conclusion Autoregressive neural network prediction of coolant flow rate −Experimental data & modeling choice −Method −Implementation −Prediction results & Conclusion Summary 2

3. IBR-2M PARAMETERS The IBR–2M reactor operates with a design power of 2 MW in Dubna, Moscow region, Russia 3 Moving reflector Core Stationary reflector Water moderator Principal diagram of the reactor IBR-2 Parameters of the IBR – 2M reactor: Nominal power – 2 MW Peak power pulse – 1830 MW Pulse half-width – 200 µs Frequency – 5 Hz Type of the fuel – PuO 2 Coolant – Na

4. Cluster analysis of the pulse energy noise 4 Objective is to investigate the dynamics of the pulse energy noise during a reactor cycle Variation in the power spectrum of the IBR–2M pulse energy fluctuations. The measurement duration is 10.5 days; 552 spectra are presented

5. Data for clustering Each successive power spectrum of the pulse energy fluctuations reflecting the reactor noise state in the time interval of ~28 min was represented by a point in the 256-D Euclidean space. The dimension of the space is equal to the number of points in the spectrum. The Euclidean distance between the ith and kth points is given by 5

6. Criteria to evaluate the number of the clusters Three criteria are used for determining the number of clusters Silhouette values Dunn index Physical meaning of the clustering results Additional criteria Adjusted Rand Krzanowski-Lai Hartigan R-Squared indexes 6 may not effectively find the correct number of clusters

7. Determining the number of clusters Variation in the silhouette with the number of clusters in some of the cluster analysis methods: ( ■ ) weighted pair-group average, ( ▲ ) k-means, ( ● ) unweighted center of mass distance (complete) 7

8. Clustering procedure The hierarchical cluster algorithm allowing a detailed study of the structure 8 The scheme of the hierarchical clustering algorithm C=С 0 no yes

9. Clustering results 1 Variation in the relative root-mean-square deviation of the total pulse energy fluctuations (a), and of those due to the axial vibrations of the moving reflectors (b). The line shows the transition of the reactor noise state from one cluster to another. The numbers are cluster numbers. 9

10. Clustering results 2 10 Cluster structures compressed from the 256-dimensional to the two - dimensional space for the total pulse energy fluctuations (left) and the fluctuations due to the axial vibrations of the moving reflectors (right)

11. Clustering results 3 11 Cluster structures compressed from the 256-dimensional to the three - dimensional space for the total pulse energy fluctuations (left) and the fluctuations due to axial vibrations of the moving reflector (right).

12. Clustering results 4 12 Cluster structures compressed from the 256-dimensional to the two-dimensional space for the total pulse energy fluctuations (a) and the fluctuations due to axial vibrations of the moving reflectors (b) vs the reactor operating time

13. Clustering results 5 13 Power spectral densities of the total pulse energy fluctuations corresponding to the centers of four clusters. Dashed lines show the spread of the spectra in the clusters

14. Conclusion of cluster analysis The result of a cluster analysis shown as the power noise is successively divided into four stable structures, The first two clusters are observed between 0 to 14 h after the maximum power has been reached, Then, 1.2 days later, the transition region ends, and the reactor noise state is stabilized, It is shown that the transition region of the reactor noise is caused by a change in the vibration state of the moving reflectors. 14

15. Prediction of liquid sodium flow rate through the core of the IBR-2M using autoregressive neural networks 15

16. The core cooling system of the IBR-2M reactor 16

17. Objective 17 To predict the liquid sodium flow rate through the core during a reactor operation Variation in the temperature and flow rate of liquid sodium through the core affects the reactivity fluctuation and power Liquid sodium flow rate

18. Experimental data & modeling choice 18 The original time series is recorded within the four days of reactor operation cycle. The sampling frequency is 0.1 s. The best results were achieved when the feedback was included into neural network scheme and the nonlinear autoregressive neural network with feedback connection (NAR) was used as a predictor

19. Training 19 Structure of NAR network for training (without feedback connection) Network error E to be minimized during training Weights updating The change in weight which is added to the old weight

20. Prediction Open loop network is closed after training and feedback connection is added Previous 2 days of data is used to predict for the next 2 days 20 feedback connection

21. Prediction results 1 Comparison between measured liquid sodium flow rate and predicted data by the model NAR (left) Prediction error (relative difference) between experimental and predicted data (right) 21

22. Prediction results 2 Experimental results versus theoretical predicted data 22

23. Prediction results 3 Power spectral density of experimental results and predicted data 23 Prediction Experiment f, h -1

24. Conclusion on prediction 24 The study was fulfilled to choose the best predictor for the liquid sodium flow rate during the IBR-2M operation. The nonlinear autoregressive neural network (NAR) has allowed more accurate and long-term predicting than standard methods. The prediction results are more accurate when compared with the common feed-forward networks or multilayer perceptron The NAR model allows to predict the slow changes of liquid sodium flow rate up to two days within the prediction error less than 5%.

25. Summary 25 Application of hierarchical cluster analysis and nonlinear autoregressive neural network for the noise analysis of the IBR- 2M has allowed to get more detailed information about the reactor noise state in a task of the reactor diagnostics. It is very important for safety operation of the reactor

26. 26 Thank you for your attention!