Science and Technology of Nuclear Installations

Volume 2018, Article ID 9852925, 13 pages

https://doi.org/10.1155/2018/9852925

## A Novel Nonlinear BWR Stability Indicator Based on the Sample Entropy

División de Ciencias Básicas e Ingeniería, Universidad Autónoma Metropolitana-Iztapalapa, Ciudad de México 09340, Mexico

Correspondence should be addressed to Alfonso Prieto-Guerrero; xm.mau.munax@gpa

Received 15 June 2018; Accepted 9 October 2018; Published 1 November 2018

Academic Editor: Alejandro Clausse

Copyright © 2018 Omar Alejandro Olvera-Guerrero et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

BWRs are thus far the simplest energy systems to transform fission energy into electrical power. However, there are still many aspects in their operation that, under certain conditions, may induce BWR unstable behavior. The default indicator to study BWR unstable behavior is the Decay Ratio (DR). However, due to the fact that BWRs show very complex responses under instability and responses that may even be chaotic, the DR might not be a suitable choice to rely on to accommodate for such intricate behavior. In this work a novel methodology based on the Sample entropy (SampEn) and the noise-assisted multivariate empirical mode decomposition (NA-MEMD) is introduced. Such methodology was developed thinking for a real time-implementation of a stability monitor. The proposed methodology was tested with a set of signals that stem from several nuclear power plants in operation today that have experienced in the past unstable events, each one of a different nature.

#### 1. Introduction

The issue of BWR instability has been a topic of concern of many scientific and technological works for more than forty years. BWR unstable events are quite rare and may happen during BWR start up or during transients that may interfere with the conventional operating region of the reactor. BWR instability phenomena are strongly linked with the interaction between thermal hydraulic and neutron kinetics processes such interaction under certain conditions and may induce undesirable BWR behavior that could jeopardize BWR operation. Thus, a reliable prediction for the onset of BWR instability is of utter importance for BWR core safety [1].

D’Auria [2] presented all BWR unstable events recorded up to 1999, which were due to various incidents (e.g., [3, 4]), while others were intentionally induced for experimental purposes (e.g., [5, 6]). Unstable oscillations in the neutron flux were recorded during these unstable events through the use of the electronic instrumentation available in the reactor. After the first events of instability were observed, the respective authorities fostered the growth of research projects to study the physical phenomena involved in BWR instability.

The most common instability type for commercial BWRs is the density-wave instability [7–9]. This type of instability can be described as follows: given a flow perturbation, a* wave *of voids travels upwards through the channel producing a pressure drop (, is approximately constant, where is the pressure drop in the single phase in the liquid region and is the pressure drop in the two-phase flow region) which is delayed with respect to the original perturbation. An increase in flow might induce an increase in pressure drop and a negative feedback that reduces the flow perturbation. The density wave phenomena delays this feedback and, at some frequency, the delay is equivalent to a phase lag; so at this frequency the pressure drop feedback is positive and if the gain is large enough, the channel flow becomes unstable and oscillates at that frequency highly concentrated around 0.5 Hz, where the associated resonance frequency is related with the fluid transit time along the heated channel, which could differ from 2 seconds, i.e., approximately 1 Hz.

In practice, the most commonly used indicator to study BWR stability due to wave oscillation is the decay ratio (DR), which is computed from the impulse response function obtained from a model fitting a linear second order system, accommodating the core behavior of a BWR. The DR is the current output indicator of most, if not all, of the stability monitors implemented in NPPs today [10]. The DR as a feasible BWR stability measure has been widely accepted but it has been observed that a BWR working at an operating point with a small DR can be close to instability [11]. Sometimes, the DR jumps from the stable to the far unstable region [12]. According to these works, the DR might not be a reliable option, under certain conditions. Besides, the need for linear and stationary signals might be a handicap for DR estimation. Therefore, it is necessary to explore new methodologies and indicators adapted to accommodate at their theoretical framework nonlinear and nonstationary behavior in order to study BWR unstable phenomena with as much realism as possible.

We are trying to convey in this work this idea, for this we explore the Sample Entropy (SampEn) linked to the noise-assisted multivariate empirical mode decomposition (NA-MEMD) to infer whether BWR signals are stable or not. The SampEn [13] is a measure that provides an index of signal* complexity* or* irregularity* of a time series. In this way, the SampEn might work as a possible nonlinear BWR stability indicator. SampEn was in principle developed almost exclusively to analyze physiological time series [13] but its utility has expanded to other domains. For instance, SampEn has been tested with daily weather temperature to measure climate complexity [14]. It has been used to measure the complexity of the dynamic reconfiguration of the brain [15] to infer its association with normal aging.

To properly estimate the SampEn from real BWR signals, the NA-MEMD [16] was explored. The NA-MEMD is an algorithm that decomposes non-stationary signals that stem from nonlinear sources. This technique also mitigates the* mode mixing *phenomena [17] of the default EMD [18] technique. The EMD is the root technique that inspired the development of the NA-MEMD. The NA-MEMD produces a local and fully data-driven decomposition of a studied signal into its fast and slow oscillations. At the end of the procedure, the studied signal can be expressed as a sum of amplitude and frequency modulated (AM-FM) functions called intrinsic mode functions (IMFs), or simply called* modes*, plus a residue (the trend) of the decomposition. The combination of the NA-MEMD and the Hilbert transform is known as the Hilbert-Huang transform (HHT). The methodology we introduce here is based on the HHT and it computes an indicator linked to BWR stability, in this case the aforementioned SampEn. The NA-MEMD decomposes the analyzed BWR signal into IMFs. One or more of these extracted modes can be associated to the instability problem in BWRs. Through HHT it is possible to get the instantaneous frequency (IF) linked to each IMF. By tracking this IF and the SampEn of the IMF linked to instability, the estimation of the SampEn-based stability indicator is accomplished. The methodology proposed in here is a continuation of two previous works [19, 20], in which a Shannon Entropy estimator [21] was used in conjunction with other members of the EMD family to study the stability of artificial and real BWR signals. In the present work the Shannon Entropy estimator was replaced by the SampEn practical formula to infer whether BWR signals are stable or not.

This work is organized as follows: in Section 2, a full review of the SampEn and of the NA-MEMD techniques is given. In Section 3, the methodology to compute the instantaneous frequency and the proposed SampEn novel stability indicator is described. In Section 4, the validation of the methodology presented in this paper is performed doing experiments with real signals taken from the Forsmark [5] and Ringhals [6] stability benchmarks and from a typical BWR power plant during stable and unstable operation. Our major observations regarding our novel methodology based on SampEn and the NA-MEMD are thoroughly discussed in Section 5.

#### 2. Preliminaries

##### 2.1. The Sample Entropy

The procedure for computing SampEn was introduced by Richman and Moorman [13]. In this work, we grant a brief summary of their findings. SampEn is a measure of complexity. Let be a time series of length . To compute the complexity of this time series via SampEn, follow the next steps:(1)Build a vector with consecutive data points taken from where is the length of sequences to be compared, also called the* embedding dimension*.(2)For each define where varies in the interval . In here, is the tolerance for accepting matches, , where is a scaling parameter and std(*x*) is the standard deviation of . is the Heaviside function: and is the Chebyshev distance, defined as(3) represents the proportion of whose distances to are less than . Now, for each we also define where represents the proportion corresponding to the dimension of . and have the same mold, but embedding vectors in both cases are defined in different spaces.(4)Average across all embedding vectors, to obtain and(5)The SampEn is computed as

SampEn is the negative natural logarithm of the conditional probability that two sequences similar for points remain similar at the next point, where self matches are not included in calculating the probability. Thus, a* lower* value of SampEn indicates more self-similarity (i.e., high order) of the studied time series whereas a* higher* value of SampEn points to higher complexity of the time series.

The calculation of SampEn requires* a priori* determination of two unknown parameters, and (the length of data is up to the user). The suggested values of are located in the range of 0.1-0.2 times the standard deviation of the studied signal . In this work, we looked for the value that grants the global SampEn maximum because this maximum value leads to the correct interpretation of signals complexity [22]. So, this of interest is fixed in our simulations at . The value of can be computed via the estimation of false nearest neighbor [23]. In our particular BWR stability discipline, that was found through false nearest neighbors for our signals was most of the time equal to 2. Therefore, is fixed at for all of our computations. SampEn is, theoretically speaking, a fraction in the interval . However, the next two formulas can be used to find the lower bound and the upper bound of SampEn for fixed values of and .

The lower bound is computed as

The upper bound is computed asFor all of our simulations* m*,* r* and are fixed at , and data points. Thus the lower bound is practically whereas the upper bound is close to 11. However, the upper bound value was never attained for any simulation.

##### 2.2. The Noise-Assisted Multivariate Empirical Mode Decomposition

The multivariate empirical mode decomposition [16], commonly referred to as the MEMD, is a nonlinear filter to make the classic EMD [18] suitable for processing of multichannel signals. The behavior of the MEMD was analyzed in the presence of white Gaussian noise [20, 24] and it was found that the MEMD in essence acts as a dyadic filter bank on each channel of the multivariate input signal; such MEMD property is illustrated in Figure 1 and its steps are given below.