Towards a Stability Monitor in Laguna Verde Nuclear Power Plant Based on the Empirical Mode Decomposition

The th International Topical Meeting on Nuclear Thermal-Hydraulics, Operation and Safety (NUTHOS-) Okinawa, Japan, December 4-8, 24 NUTHOS-2 Towards a Stability Monitor in Laguna Verde Nuclear Power Plant Based on the Empirical Mode Decomposition Alfonso Prieto-Guerrero and Gilberto Espinosa-Paredes División de Ciencias Básicas e Ingeniería, Universidad Autónoma Metropolitana-Iztapalapa Av. San Rafael Atlixco 86 Col. Vicentina, México, D.F., 934 MEXICO apg@xanum.uam.mx, gepe@xanum.uam.mx ABSTRACT In this paper a new method based on the empirical mode decomposition (EMD) to estimate a parameter associated with instability in boiling water reactors (BWR), is presented. The proposed method allows to decompose the analyzed signal in different levels or intrinsic mode functions (IMF). One or more of these different modes can be associated to the instability problem in BWRs. By tracking the instantaneous frequency (IF) and the autocorrelation function of the intrinsic mode functions associated to the instability of the BWR, the estimation of the proposed instability parameter can be achieved. The proposed methodology was applied on a real instability event occurred in Laguna Verde Nuclear Power Plant (LV). The instability event happened during a Cycle 4 power ascension without fuel damage. Hence this is a very interesting case to validate our algorithm. Indeed, the LV signal exhibits very clear trends: at the beginning the LV signal is increasing very slowly, in the middle the amplitude increases much faster and in the end the amplitude decreases rapidly, due to reactor scram. From results, we can observe that the estimated decay ratio (DR) behaves as follows: first increasing and then decreasing around. Around s, the estimated DR stays over until around 2 s when decreases for two consecutive analyzed segment. After these two consecutive segments, again the estimated DR increases over the value and then it stayed over this threshold until the final analyzed segment. Also it can be observed that the IF behaves as expected: around the.5 Hz at the beginning and perfectly reinforced in the middle and final of the event. The mean estimated value for this resonant frequency is.54 Hz with a standard deviation of.3 Hz, obtained fitting the Gaussian distribution showed by this frequency. It is clearly shown that the proposed algorithm permits to tracking the IF associated to the instability event and gives a perfectly estimation of the DR. Thinking for a real-time stability monitor, with the estimated classical DR only is possible to say that the reactor is unstable between 3 and 4 s. With our proposed method, around 5 s it is already possible to launch the alarm after confirmation of several short analyzed segments of 5 s. KEYWORDS Empirical Mode Decomposition, Boiling Water Reactors, Stability Monitor, Intrisic Mode Functions, Hilbert-Huang transform.. INTRODUCTION In BWR instability events, two kinds of instabilities are found: in-phase (global, core-wide) oscillations, and out-of-phase (regional) oscillations. In-phase oscillations are caused by the lag / 2

The th International Topical Meeting on Nuclear Thermal-Hydraulics, Operation and Safety (NUTHOS-) Okinawa, Japan, December 4-8, 24 NUTHOS-2 introduced into the thermal-hydraulic system by the finite speed of propagation of density perturbation []. At high-core void fractions and low flow conditions, the feedback becomes so strong that it induces oscillations at frequency about.5 Hz. When this feedback increases, the oscillation becomes more pronounced, and oscillatory instability is reached. The term out-of-phase oscillation is applied to those instabilities in which different core zones show a considerable phase shift (8 o ) in neutron flux oscillation. It has been shown that stability depends on several variables such as control rod patterns, void fraction, burnup, inlet mass flow, among others. In the last few decades, research has been devoted to the study of power oscillations and the mechanisms that generate them (e.g. [-6]). Several approaches have been taken to address the stability of BWRs, March-Leuba [7] pioneered the study of reduced-order models for coupled thermal-neutron dynamics, followed by Turso et al. [8] and Muñoz-Cobo et al. [9], among others. Some of those works were developed to gain insights about the BWRs dynamics, while others were focused on a more complete description of the heat transfer process (e.g. [-2]). The models presented in the mentioned works were able to describe, in a qualitative way, low-frequency oscillations and even instabilities, but neither sustained oscillations of relatively high frequency nor highly non-stationary behavior could be described accurately by such models [3]. Recently, the wavelet theory has been used to explore new alternatives for transient instability analysis [4-5]. A key point is that in general, BWR signals are non-stationary, therefore traditional methods such as the Fourier transform, might lead to biased stability parameters. Sunde and Pázsit [6] proposed an original work using the wavelet transform in combination with the autocorrelation function (ACF) to estimate the DR. Prieto-Guerrero and Espinosa-Paredes [7] propose the application of wavelet ridges to track the instantaneous frequency and determine the DR. Also recently in [8-2], the idea of an instantaneous DR was introduced. In [2], Prieto-Guerrero and Espinosa-Paredes propose a new method based on empirical mode decomposition (EMD) to estimate a parameter associated to instability in BWRs. This instability parameter is not exactly the classical DR, but it will be associated with this. The methodology is based on the implementation of the empirical mode decomposition algorithm that allows the decomposition of the analyzed signal in different levels or intrinsic mode functions (IMF). One or more of these different modes can be associated to the instability problem in BWRs. Based on the Hilbert-Huang transform it is possible to get the instantaneous frequency (IF) associated to each IMF. By tracking this instantaneous frequency and the autocorrelation function of the IMF associated to the instability of the BWR, the estimation of the proposed instability parameter can be achieved. Based on this research, in this work the methodology was applied on a real instability event occurred in Laguna Verde Power Plant (LV): on January 24, 995 a power instability event occurred in the Unit, which is a BWR-5 and is operated since 99 at a rated power of 93 MWt. The instability event happened during a Cycle 4 power ascension without fuel damage. Considering the fact that an accurate prediction for the onset of BWR instability is indispensable for the safety of the BWR core, this is a very interesting case to validate our algorithm; permitting to establish the bases for a future real-time stability monitor. The rest of this paper is organized as follows: in Section 2 the basic background to understand our methodology is presented. In Section 3, the methodology to estimate the instantaneous frequency and the proposed DR is discussed. Then in Section 4, the validation of the 2 / 2

The th International Topical Meeting on Nuclear Thermal-Hydraulics, Operation and Safety (NUTHOS-) Okinawa, Japan, December 4-8, 24 NUTHOS-2 methodology presented in this paper is performed doing experiments with a real instability event from the LV. Last, in Section 5, our conclusions are presented. 2. PRELIMINARIES 2.. Empirical Mode Decomposition and Intrinsic Mode Functions The empirical mode decomposition (EMD) algorithm was proposed by Huang et al. [22] in order to analyze non-stationary signals from non-linear processes. EMD extracts intrinsic oscillatory modes defined by the time scales of oscillation. The components that result from the EMD algorithm are called Intrinsic Mode Functions (IMFs). These obtained IMFs result in a composed AM-FM (Amplitude Modulation-Frequency Modulation) signal. The fundamental step of the empirical mode decomposition (EMD) is the next iterative sifting process:. Consider a signal xt with M maxima and L minima. The sifting process starts with identifying the extrema of the signal, xt, given by the sets Emax xmax t, xmax t2, xmax t3,, xmax tm, and Emin xmin t, xmin t2, xmin t3,, xmin tl. We also set x t s t. 2. The set points of Emax are interpolated to form the upper envelope of the signal, xˆu t. Similarly, the set points of E min are interpolated to form the minimum envelope, xˆl t. xˆu t xˆl t 3. The average envelope xˆ t, is computed. A 4. This average envelope is subtracted from the original signal k intermediate signal: d t x t xˆ t A 2 xt resulting in a residue with k indicating the iteration number. The k k iteration on k is continued until the scalar product d t, d t and the number of k extreme (maxima and minima) and the number of zero-crossings of d k more than one. This sifting process produces the first IMF given by IMF t d t j obtained at the th k iteration. 5. Following this, the function with s t s t IMF t process is repeated (steps to 4), resulting in the second IMF, i.e., t may differ by no j with is created, and the sifting IMF2 j t. Considering this procedure, the other IMFs are generated until the residue r t x t IMF t accomplished. The functions IMFj t, j,2,..., N orthogonal to one another. decompose N is j xt and are nearly j 3 / 2

The th International Topical Meeting on Nuclear Thermal-Hydraulics, Operation and Safety (NUTHOS-) Okinawa, Japan, December 4-8, 24 NUTHOS-2 The sifting process essentially extracts scales of the signal. Further, the IMF have symmetric envelopes and a zero mean value. Due to these characteristics, the IMF is referred to as being monocomponent. Since the residue is computed by successively subtracting the sifted functions from the original signal, the EMD algorithm is data driven and adaptive, i.e. the basis functions are derived from the signal itself in contrast to the traditional methods where the basis functions are fixed. Furthermore, interpolation is an inexact procedure. The performance of the EMD algorithm is sensitive to the interpolation procedure. The EMD has been shown to be effective in extracting relevant components in a variety of applications involving non-stationary signals. 2.2 Hilbert-Huang Transform Consider the analytic representation of a signal given by: where At and In this equation j t Z t x t jx t A t e () x ˆ t are the envelope and the instantaneous phase of the signal, respectively. ˆx t is the Hilbert transform of the signal xˆ t xt, which is defined by [23]: xt d, (2) This complex or analytic signal is completely characterized by their amplitude phase At and their t with values in the interval [, 2π), forming the so-called canonical pair. Considering that we can observe that this canonical representation of x t Re Z x t A t cos t (3) xt corresponds to a signal varying simultaneously in amplitude and phase all the time (exactly like each IMF from the EMD). Based on this representation, Ville [24] proposed to calculate the instantaneous frequency associated to the instantaneous phase t. From Eq. (), the instantaneous frequency is defined by: It is clear that the analytic signal frequency range with t d inst t (4) dt Zx t associated with xt, it also comprises the phase information of the original signal xt has the same amplitude and xt, which can be used to compute the instantaneous amplitude and to analyze the frequency performance of xt. Based on this fact, we can construct the analytic signal corresponding to each IMF using the Hilbert transform, defined in Eq. (2). The combination of the EMD applied 4 / 2

IF [Hz] IF [Hz] The th International Topical Meeting on Nuclear Thermal-Hydraulics, Operation and Safety (NUTHOS-) Okinawa, Japan, December 4-8, 24 NUTHOS-2 to xt to generate IMFs, and the Hilbert transform of each IMF is called the Hilbert-Huang Transform (HHT). The different IMFs resulting from the EMD algorithm are orthogonal to each other. The IMFs, thus represent different time-scales of oscillations, which form a set of bases functions. This implies that there is no redundancy in the information contained in the different IMFs. Using this property, a distribution can be constructed from the instantaneous frequencies of each of the IMFs. This distribution is called the Hilbert-Huang Spectrum (HHS) [22]. Original signal: Forsmark Case 6.2 LPRM 5 42 4 42 38 5 5 2 analyzed segment 25 3 35 4 42 4 38 5 38 96 98 2 4 5 Time [s] 6 8.5 4 IMF 5 2 -.5 5 5.5 -.5 5 5-5 5.2 -.2 5 5.2 -.2 5 5 Time [s] IMF 2 IMF 3 IMF 4 IMF 5 IF [Hz] IF [Hz] IF [Hz] 5 5.5 5 5 Fig. IMFs and IFs of a specific LPRM from the Forsmark benchmark, case 6. Figure plots the five IMFs of the EMD decomposition of a real signal obtained from a BWR during 5 s. We observe that the IMFs have a zero mean, symmetric envelopes (AM-FM signals), and the number of extreme is at most one more than the number of zero crossings. The frequency of an IMF is directly proportional to the number of zero crossings in one time duration of IMFs. Also we can observe perfectly the frequencies tracking instantaneous frequency (IF) associated to each IMF. Both IMFs and IFs, determine the mode or modes associated to the instability phenomena. In the presented signal, this instability problem can be clearly observed at the IMF 3 of Figure. 3. DECAY RATIO ESTIMATION BASED ON THE EMD-HHT.5 5 5.5 5 5.5 5 5 Time [s] Instantaneous Frequency (IF) 2 3 4 5 6 7.2.4.6.8 In this work, decay ratio (DR) estimation based on the EMD-HHT technique and applied to the BWR signals, was done following the next methodology proposed in [2]: 4 2 5.2.4.6.8 5.2.4.6.8 4 2.2.4.6.8 Frequency [Hz] IF Histrograms 5 / 2

The th International Topical Meeting on Nuclear Thermal-Hydraulics, Operation and Safety (NUTHOS-) Okinawa, Japan, December 4-8, 24 NUTHOS-2. The considered signal (APRM or LPRM) obtained from the BWR is segmented in time windows of 5 s of duration. This short time period was chosen because it has demonstrated in [2] that it is possible to estimate perfectly the IF associated to the instability mode in 5 s. Furthermore, the estimated ACF (see step 5) decays to zero in this same time period. 2. Each segmented signal (APRM or LPRM) is analyzed (decomposed) using the EMD algorithm, described in Section 2., obtaining in this way the corresponding IMFs. It is worth mentioning that the APRM or LPRM signals are not being processed before. For instance, to remove the signal trend, it is not necessary to filter the BWR signal before, due that this information is contained in the last scale computed from the EMD. It is also worth mentioning that the trend does not represent an IMF. 3. The Hilbert transform for each IMF is computed in order to get the instantaneous frequencies contained in each IMF. 4. When tracking these frequencies, it is possible to get the mode or modes associated to instability processes. In this way, only the mode (IMF) or modes associated to the BWRs instability are considered for further processing. 5. The autocorrelation function (AFC) of the considered IMF is calculated. 6. Based on these ACF values, an instability parameter is estimated. This parameter is determined as the Decay Ratio (DR) but unlike it, is calculated considering two consecutive maxima from the ACF magnitude of the analyzed IMF. This consideration (DR estimation) was inferred based on the analysis of a synthetic signal and the discussion on BWR real signals. 7. Finally, the mean and variance of the DR are calculated averaging the estimated DR obtained in each window of 5 s along time. 4. EXPERIMENTS AND RESULTS The methodology described in the prior section, was validated first using a synthetic signal and then with BWR s real signals in a previous article [2]. In [2] two complete cases from the Forsmark stability benchmark corresponding to cases 4 and 6 were presented and deeply analyzed. In this work, the LV instability event is added to the discussion about the potential of this proposed method to predict an instability event and to implement a real-time stability monitor in a BWR. In our implementation, a modified version of the EMD code provided in [25] was used. 4.. Case of the Instability Event of Laguna Verde Power Plant The proposed methodology was applied in a real instability event. On January 24, 995 a power instability event occurred in Laguna Verde Unit, which is a BWR-5 and is operated since 99 at a rated power of 93 MWt. The instability event happened during a Cycle 4 power ascension without fuel damage. When the thermal power reached 37% of the rated power, the recirculation pumps were running at low speed driving 37.8% of the total core flow. The flow control valves were set to their minimum closed position in order to operate the recirculation pumps at a high speed. The drop in drive flow resulted in a core flow reduction of 32% and, a power reduction also of 32%. Two control rods were also partially withdrawn during valve closure. The new low flow operating conditions resulted in growing power oscillations [26]. Figure 2 shows the register of the averaged power range monitor (APRM) obtained via the Integral Information Process System (IIPS) with a sampling period of.2 s for the whole time series. This figure shows that the power signal is non-stationary and the oscillations reached up 6 / 2

Power (%) The th International Topical Meeting on Nuclear Thermal-Hydraulics, Operation and Safety (NUTHOS-) Okinawa, Japan, December 4-8, 24 NUTHOS-2 to.5% of the maximum peak to peak amplitude. In order to get a clear interpretation of results, each signal is depicted as a function of time instead of indexing it using samples. For the sampling period of.2 s, the time axis for this signal varies from to 7 s. According to Gonzalez [26] and Farawila [27] the channel A of the APRM trace shows no unstable behavior at 3:28: h. The valve closure was initiated at 3:28:2 h. A small core flow reduction was noticeable 4 seconds later, and the APRM-A trace depicts signs of instability although the variations in the magnitude of the signal remained within the noise level. As the valve continued to close, the APRM-A trace shows clear unstable behavior starting at 3:3:3 h. The valve reached the minimum position at 3:3:3 h, and the oscillations continued without a significant increase in their growth rate. The operator attempted to stabilize the power level by increasing the core flow opening the valves at 3:33:2 h. As a result of increasing the core flow, the oscillation started to decay at 3:34:4 h. At 3:35:2 h the oscillation reached 3% of amplitude, when the reactor was manually scrammed. 4 38 36 34 32 3 28 2 3 4 5 6 7 8 Time [s] Fig. 2 Laguna Verde APRM-A signal. Hence this is a very interesting case to validate our algorithm. Indeed, the LV signal exhibits very clear trends (see the three red marked segments in Figure 2). At the beginning the LV signal is increasing very slowly, in the middle the amplitude increases much faster and in the end the amplitude decreases rapidly, since manual operator intervention was required to avoid a potential dangerous emergency at the LV reactor. From Figure 3, we observe that the estimated DR behaves as follows: first increasing and then decreasing around. Around s, the estimated DR stays over until around 2 s when decreases for two consecutive analyzed segment. After these two consecutive segments, again the estimated DR increases over the value and then it stayed over this threshold until the final analyzed segment. Also in Figure 3 can be observed that the IF behaves as expected: around the.5 Hz at the beginning and perfectly reinforced in the middle and final of the event. The mean 7 / 2

IF [[Hz] IF [Hz] Power [%] Power [%] Power [%] Power [%] The th International Topical Meeting on Nuclear Thermal-Hydraulics, Operation and Safety (NUTHOS-) Okinawa, Japan, December 4-8, 24 NUTHOS-2 estimated value for this resonant frequency is.54 Hz with a standard deviation of.3 Hz, obtained fitting the Gaussian distribution plotted in Figure 4. In Figure 3 it is clearly shown that the proposed algorithm permits to tracking the IF associated to the instability event and gives a perfectly estimation of the DR. 4 LV signal LV signal 35 3 2 3 4 5 6 7 LV signal: LV analyzed segment 4 35 3 7 7 72 73 74 75 76 77 78 79 7 Instantaneous Frequency frequency (IF).5.5 2 3 4 5 6 7 ACF magnitude ACF for magnitude the considered for the IMF from the the analyzed analyzed segment segment 2 2 3 4 5 6 7 8 9 Estimated DR along time time.5.5 2 3 4 5 6 7 Time [s] Fig. 3 Laguna Verde case. Instantaneous frequency (IF) and DR estimations. 3 25 2 5 5.4.45.5.55.6.65 Instantaneous Frequency (IF) [Hz].7 Fig. 4 Laguna Verde case. Histogram of the instantaneous frequency (IF). 8 / 2

The th International Topical Meeting on Nuclear Thermal-Hydraulics, Operation and Safety (NUTHOS-) Okinawa, Japan, December 4-8, 24 NUTHOS-2 A final discussion for this event involves the decision about to considerate the DR as the two first consecutive peaks obtained from the ACF magnitude calculated from interested IMF. The first reason was that in certain cases the ACF vanishes quickly (i.e. when the processed IMF is quite small in amplitude). The second reason is to give a conservative prediction for an instability event. In order to show this case we plotted in Figure 5 both estimated DR: in red (dashed line) the classical DR estimation (two consecutive maxima of the ACF), in blue (solid line) the proposed DR (two consecutive maxima of the ACF magnitude). For both cases the method permits to follows the instability event. However, thinking a real-time stability monitor, with the estimated classical DR only is possible to say that the reactor is unstable between 3 and 4 s. With our proposed method, around 5 s it is already possible to launch the alarm after confirmation of several analyzed segments..5.5 Proposed DR Classical DR 2 3 4 5 6 7 8 Time [s] Fig. 5 Comparison of DR between classical estimation and proposed method in this work. It is clear that to implement this method as a real-time stability monitor; we need to introduce a decision rule about when an alarm is launched. This escapes from the scope of this work. However we can observe the great potential of this method. It is worth mentioning this method was implemented using the MATLAB software in a PC desktop with a processor Intel Core Duo at 2.4 GHz and 4 GBytes in RAM being the complete processing time (including displayed figures) around 2 s for each segment of 5 s. Indeed, this proposed method does not represent high computational complexity and the code can be optimized and accelerated using tools like DSPs (Digital Signal Processors). 5. CONCLUSIONS In this work, we propose a novel approach to stability analysis of boiling water reactors (BWRs) based on empirical mode decomposition (EMD) and the Hilbert-Huang transform 9 / 2

The th International Topical Meeting on Nuclear Thermal-Hydraulics, Operation and Safety (NUTHOS-) Okinawa, Japan, December 4-8, 24 NUTHOS-2 (HHT). The method consists of obtaining the different intrinsic modes functions (IMF) associated to the EMD. Applying the HHT it is possible to get from each IMF, the associated instantaneous frequency (IF) along time. This instantaneous frequency allows to decide which IMF will be associated to a possible instability event. In this manner, it is possible to track this instability situation. An estimated decay ratio (DR) is obtained along time, considering the magnitude of the autocorrelation function (ACF) of followed IMF. Some important conclusions can be delivered. First, the method is relatively simple to implement and it does not represent a high computational complexity. Second, results obtained from the implementation of this method are promising and we can conclude that the proposed method clearly contributes on the fact to detect events of instability along time. Indeed, with the proposed estimation for the DR, this method allows to predict the instability event through time before the oscillation going to a critical state. REFERENCES. R. T. Lahey, M. Z. Podoswski, On the analysis of various instabilities in two-phase flow, Multiphase Science and Technology, 4, pp. 83-37 (989). 2. P. Saha, N. Zuber, An analytical study of the thermally induced two-phase flow instabilities including the effects of thermal non-equilibrium, Int. J. Heat Mass Transfer, 2, pp. 45-426 (978). 3. S. J. Peng, M. Z. Podowski, R. T. Lahey, M. Becker, NUFREQ-NP: a computer code for the instability analysis of boiling water nuclear reactors, Nuclear Science Engineering, 88, pp. 44-4 (984). 4. J. March-Leuba, LAPUR benchmark against in-phase and out-of-phase stability test, NUREG/CR-565, ORNL/TM-62 (99). 5. J. March-Leuba, E. D. Blakeman, A mechanism for out-of-phase power instabilities in Boiling Water Reactors, Nuclear Science and Engineering, 7, pp. 73-79 (99). 6. J. March-Leuba, J. M. Rey, Coupled thermal-hydraulic neutronic stabilities, in boiling water reactor a review of the state of the art, Nuclear Engineering and Design, 45, pp. 97- (993). 7. J. March-Leuba J., A reduced-order model of Boiling Water Reactor linear dynamics, Nuclear Technology, 75, pp. 5-22 (986). 8. J. A. Turso, J. March-Leuba, M. Edwards, A modal-based reduced-order model of BWR out-of-phase instabilities, Annals of Nuclear Energy, 2, pp. 92-934 (997). 9. J. L. Muñoz-Cobo, S. Chiva, A. Sekhri, A reduced order model of BWR dynamics with subcooled boiling and modal kinetics: application to out of phase oscillations, Annals of Nuclear Energy, 3, pp. 35-62 (24).. M. Uehiro, Y. F. Rao, K. Fukuda, Linear stability analysis on instabilities of in-phase and out-of-phase modes in Boiling Water Reactors, J. Nuclear Science and Technology, 33, pp. 628-635 (996). / 2

The th International Topical Meeting on Nuclear Thermal-Hydraulics, Operation and Safety (NUTHOS-) Okinawa, Japan, December 4-8, 24 NUTHOS-2. G. Guido, J. Converti, A. Clause, Density wave oscillations in parallel channels, an analytical approach, Nuclear Engineering and Design, 25, pp. 2-39 (997). 2. M. Z. Podowski, M. Pinheiro, Modeling and numerical simulation of oscillatory two phase flow with application to boiling water nuclear reactor, Nuclear Engineering and Design, 77, pp. 79-88 (997). 3. G. Verdu, D. Ginestar, J. L. Muñoz-Cobo, J. Navarro-Esbri, M. J. Palomo, P. Lansaker, J. M. Conde, M. Recio, E. Sartori, Forsmark &2 Stability Benchmark. Time Series Analysis Methods for Oscillations During BWR Operation Final Report, NEA/NSC/DOC 2 (2). 4. G. Espinosa-Paredes, A. Prieto-Guerrero, A. Núñez-Carrera, R. Amador-García, Wavelet-based method for instability analysis in boiling water reactors, Nuclear Technology, 5, pp. 25-26 (25). 5. G. Espinosa-Paredes, A. Nuñez-Carrera, A. Prieto-Guerrero, M. Ceceñas, Wavelet approach for analysis of neutronic power using data of Ringhals stability benchmark, Nuclear Engineering and Design, 237, pp. 9-5 (27). 6. C. Sunde, I. Pázsit, Wavelet techniques for the determination of the decay ratio in boiling water reactors, Kerntechnik, 72, pp. 7-9 (27). 7. A. Prieto-Guerrero, G. Espinosa-Paredes, Decay Ratio Estimation of BWR Signals based on Wavelet Ridges, Nuclear Science and Engineering, 6, pp. 32-37 (28). 8. J. E. Torres-Fernández, A. Prieto-Guerrero, G. Espinosa-Paredes, Decay ratio estimation using the instantaneous autocorrelation function, Nuclear Engineering and Design, 24, pp. 3559-357 (2). 9. J. E. Torres-Fernández, A. Prieto-Guerrero, G. Espinosa-Paredes, Decay ratio estimation based on time-frequency representations, Annals of Nuclear Energy, 37, pp. 93-2 (2). 2. J. E. Torres-Fernández, G. Espinosa-Paredes, A. Prieto-Guerrero, Decay ratio estimation in BWRs using the exponential time-scale representation, Annals of Nuclear Energy, 49, pp. 43-53 (22). 2. A. Prieto-Guerrero, G. Espinosa-Paredes, Decay Ratio estimation in boiling water reactors based on the empirical mode decomposition and the Hilbert-Huang transform, Progress in Nuclear Energy, 7, pp. 22-33 (24). 22. N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N. C. Yen, C. C. Tung, H. H. Liu, The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis, Mathematical Physical and Engineering Sciences, 454, pp. 679-699 (998). / 2

The th International Topical Meeting on Nuclear Thermal-Hydraulics, Operation and Safety (NUTHOS-) Okinawa, Japan, December 4-8, 24 NUTHOS-2 23. N. Delprat, B. Escudie, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, B. Torresani, Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies, IEEE Transactions on Information Theory, 38, pp. 644-664 (992). 24. J. Ville, Theorie et application de la notion de signal analytique, Cables Trans., A(), pp. 6-74 (948). 25. G. Rilling, P. Goncalves, EMD Toolbox for Matlab., http://perso.ens-lyon.fr/patrick.flandrin/emd.html, (28). 26. V. M. Gonzalez, R. Amador, R. Castillo, Análisis del evento de oscilaciones de potencia en la CNLV: Informe Preliminar, CNSNS-TR-3, REVISION, Comisión Nacional de Seguridad Nuclear y Salvaguardias, México (995). 27. Y. M. Farawila, D. W. Pruitt, P. E. Smith, L. Sanchez, L. P. Fuentes, Analysis of the Laguna Verde instability event, Proceedings of National Heat Transfer Conference, Houston, Texas, August 3-6, Vol. 9, pp. 98-22 (996). 2 / 2