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Detecting long-range correlation with detrended fluctuation analysis: Application to BWR stability
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NUCLEAR ENERGY Annals of Nuclear Energy 33 (2006) 1309–1314 www.elsevier.com/locate/anucene
Detecting long-range correlation with detrended fluctuation analysis: Application to BWR stability ´ lvarez-Ramı´rez, Alejandro Va´zquez Gilberto Espinosa-Paredes *, Jose´ A Departamento de Ingenierı´a de Procesos e Hidra´ulica, Universidad Auto´noma Metropolitana-Iztapalapa, Apartado Postal 55-534, Me´xico, DF 09340, Mexico Received 13 July 2006; accepted 8 September 2006
Abstract The aim of this paper is to explore the application of detrended fluctuation analysis (DFA) to study boiling water reactor stability. DFA is a scaling method commonly used for detecting long-range correlations in non-stationary time series. This method is based on the random walk theory and was applied to neutronic power signal of Forsmark stability benchmark. Our results shows that the scaling properties breakdown during unstable oscillations. 2006 Elsevier Ltd. All rights reserved.
1. Introduction Instability analysis in boiling water reactor (BWR) plays a central role in the understanding of the physical mechanism that induces the observed power oscillations, which are unstable and occur at low-core flow and relatively high-power conditions (March-Leuba and Rey, 1993). For instance, large-amplitude power oscillations have been observed in Caorso (occurred in 1984), La Salle (occurred in 1988) and Laguna Verde (occurred in 1995) plants. Such instability phenomena have been the object of a great deal of study. Diverse classical signal processing techniques such as noise techniques, auto regressive moving average (ARMA) and fast Fourier transform (FFT) have been widely applied to stability monitoring (system identification). However, these techniques are limited in the sense that they are not applicable during transient states and are less sensitive to the noise in the signal to determine the fundamental frequency. Fourier transform in any of its forms (short-time Fourier transform, STFT) provides the frequency content of an analyzed signal. Thus, neither *
Corresponding author. Tel.: +52 55 5804 4645/102; fax: +52 55 5804 4900. E-mail address: [email protected] (G. Espinosa-Paredes). 0306-4549/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.anucene.2006.09.004
of them can detect transients; however it does not mean that it cannot be applied to a signal that contains a transient. Available analytical works are based on the Laplace transform for the dynamic analysis in the frequency domain to determine reactor stability (Van Bragt and Van Der Hagen, 1997). For example, March-Leuba and Blakeman (1991) studied the instabilities of the subcritical modes, and obtained stability bounds for the corewide and out-of-phase mode on the power-flow map using the code LAPUR in frequency-domain for BWR analysis. LAPUR provides numeric transfer functions for nuclear and thermo-hydraulics phenomena. In some approaches, the stability properties of the signal are obtained from invariant quantities as the Lyapunov exponents spectrum (Verdu and Ginestar, 2001). The studies above mentioned are widely applied to instabilities analysis and to develop stability monitoring systems in nuclear power plants, the mechanism that generate the oscillations are not easily uncovered and, in general, not applicable during transient phenomena. In this paper, we explore the use of detrended fluctuation analysis (DFA) (Peng et al., 1994) to detect long-range correlations in temporal patterns of neutronic power generation. Specifically, the case 5 of Forsmark stability benchmark is studied.
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In recent years, the DFA invented by Peng et al. (1994) has been established as an important tool for the detection of long-range correlations in time series with non-stationarities. It is successfully been applied to such diverse fields of interest as DNA (Buldyrev et al., 1998), heart rate dynamics (Peng et al., 1998; Absil et al., 1999), neuron spiking (Blesic et al., 1999), human gait (Peng et al., 1998; Hausdorff et al., 1997), long-time weather records (Talkner and Weber, 2000), cloud structure (Ivanova and Ausloos, 1999), economical time series (Ausloos and Ivanova, 2000), and even solid state physics (Kantelhardt et al., 1999), among others. While the Fourier transform and the recently developed wavelet transform modulus maxima (WTMM) method (Muzy et al., 1993) analyze the time series directly, the DFA is based on the random walk theory (Shlesinger et al., 1987). 2. Neutron monitoring in the BWR The neutronic flux in the reactor is monitored by detectors located inside the reactor core (BWR/6, 1975). Two types of neutron monitors are used: source range neutron monitor (SRNM) and average power range monitor (APRM) which are obtained from the local neutron flux monitored by fixed in-core ion chambers (LPRM), which are arranged in a uniform pattern throughout the core. These chambers cover a range from 1% to 125% of rated power in a linear scale. When a control rod or a group of control rods is selected for movement, readings from the adjacent detectors are displayed together on the control switchboard with a display of the position of the rod or group of rods. Each detector assembly contains four fission chambers and a travelling calibration guide tube (TIP). The chambers are uniformly spaced in an axial direction and lie in four horizontal planes. The average power level is measured by six average power range monitoring (APRM). Each APRM measures bulk power in the core by averaging signals from chains of 24 LPRM distributed throughout the core. Output signals from these monitors are displayed and are also used to operate trips in the reactor protection system. 3. Preliminaries We consider a record (yi) of i = 1, . . . , N equidistant measurements. In most applications, the index i will correspond to the time of the measurements. We are interested in the correlation of the values yi and for different time lags, i.e., correlations over different time scales n. In order to get rid of a constant offset in the data, the mean Æyæ, which is given by N 1 X hyi ¼ y N i¼1 i
ð1Þ
stands for the average of yi, i = 1, . . . ,N, is usually subtracted,
y i y i hyi
ð2Þ
Quantitatively, corresponds between y-values separated by n steps are defined by the auto-correlation function: GðnÞ ¼ hy i y iþn i ¼
N n 1 X y i y iþn N n i¼1
ð3Þ
If yi are correlated, G(n) is zero for n > 0. Short-range correlations of the yi are described by G(n) declining exponentially, GðtÞ exp ðn=nx Þ
ð4Þ
with a decay time nx. For the so-called long-range correlations, G(n) declines as a power-law: GðnÞ nr
ð5Þ
with an exponent 0 < r < 1. A direct calculation of G(n) is usually not appropriate due to noise superimposed on the collected data yi an due to underlying trends of unknown origin (e.g. the average Æyæ might be different for the first and the second half of the record, if the data are strongly long-range correlated). This makes the definitions of G(n) problematic. Then, the correlation exponent r is determined as is presented in Section 4. 4. DFA algorithm In the DFA method, basically a time series is integrated to obtain a type of random-walk profile, i.e., the method is a modification of root-mean-square analysis of a random walk for non-stationary data (Walleczek, 2000). Then this sequenced is divided into no overlapping boxes of equal sizes n. In each box the local trend is estimated by an m-degree polynomial fitting. In a subsequent step, the root-mean-square fluctuation, denoted by Fq(n) (where a generalized q-norm is used to compute a fluctuation function), of the difference between integrated sequence and the polynomial fit is calculated for each n. Finally, assuming that each fluctuation meets a power-law with respect to the box size, a scaling factor a is computed as the slope of the following fluctuation plot: def I ¼ logðnÞ versus log ðF q ðnÞÞ ð6Þ The DFA method used to analyze the neutronic power dynamics consists of five steps (Peng et al., 1994; Kantelhard et al., 2001): In the first step, we determine the profile xðiÞ ¼
i X
½y k hyi;
i ¼ 1; . . . ; N
ð7Þ
k¼1
of the record (yi) of length N. The subtraction of the mean Æyæ is not compulsory, since it would be eliminated by the later detrending in the third step anyway. In the second step, we cut the profile x(i) into Nn ” [N/n] non-overlapping segments of equal length n.Since the record length N needs not be a multiple of the considered
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time scale n, a short part at the end of the profile will remain in most cases. In order not to disregard this part of the record, the same procedure is repeated starting from the other end of the record. Thus, 2Nn segments are obtained altogether. In the third step, we calculate the local trend for each segment v by a least-squares fit of the data. Then we define the detrended time series for segment duration n, denoted by xn(i), as the difference between the original time series and the fits: xn ðiÞ ¼ xi pv ðiÞ
ð8Þ
where pv(i) is the fitting polynomial in the vth segment. In the fourth step, we calculate for each of the 2Nn segments the variance: n 1X x2 ½ðv 1Þn þ i ð9Þ F 2n ðvÞ ¼ hx2n ðiÞi ¼ n i¼1 n of detrended time series xn(i) by averaging over all data points i in the vth segment. Finally, we average over all segments and take the square root to obtain the DFA fluctuation function: "
#1=2 2N n 1 X 2 F ðnÞ ¼ F ðvÞ 2N n v¼1 n
ð10Þ
The above procedure is repeated for a broad range of segment lengths n. According to the recommendations made by Peng et al. (1994), nmin . 5 and nmax . N/4. If the data (yi) are long-range power-law correlated (Eq. (5)), the fluctuation function F(n) increase by a power-law: F ðnÞ na
ð11Þ
where a is called the scaling exponent, a self-affinity parameter representing the long-range power-law correlation properties of the signal. In the case of short-range (or not correlations at all) the detrended walk displays properties of a standard random walk (e.g., white noise) and a = 0.5. On the other hand, if a < 0.5 the correlation in the signal is anti-persistent (an increment is very likely to be followed by a decrement, and vice versa), and if a > 0.5 the correlation in the signal is persistent (an increment is very likely to be followed by an increment, and vice versa). Notice that integration/differentiation of the signal will increase/ decrease the exponent a by one. It may happen, as we shall see later, that in two neighbouring intervals of n the function F(n) displays two different power-law behaviours. The crossover region is defined by the values of n where the function F(n) changes its behaviour. The DFA algorithm described above can be extended along several directions. For instance, (i) overlapping segments of size n and overlapping size l can be used in Step 2 to reduce possible spurious correlations introduced by discontinuous trending, (ii) instead of the liner trending in Step 3, smoother fluctuation function behaviour can be obtained by computing mth-order polynomial fit. In this case, the mth-order DFA method, denoted by DFA-m, is
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obtained, and (iii) a generalised q-norm can be used to compute a fluctuation function as follow: " #1=q 2N n 1 X F q ðnÞ ¼ F 2 ðvÞ ð12Þ 2N n v¼1 n This approach has been proposed to study multi fractality in non-stationary signals (Kantelhardt et al., 2002). In this case, if the series y(i) is long-range power-law correlated, Fq(n) in creases as a power-law: F q ðnÞ naðqÞ
ð13Þ
where a(q) is the generalised scaling exponent. Notice that a(2) correspond to the standard scaling exponent a. It should be stressed that, in the sequel, we will use the basic DFA algorithm, as described in Steps 1 to 6, to detect the correlation structure of Formark stability benchmark neutronic power signal. 5. Results and discussions The DFA method was applied to neutronic signal of Forsmark stability benchmark, specifically to the case ‘‘c5.aprm’’ (Verdu et al., 2001). The signal corresponds to average power range monitor (APRM) registered during instability event, which is shown in Fig. 1. This event corresponds to a situation where the neutronic power from the reactor displays abnormal, apparently unstable, oscillations. That is, it seems that a medium-frequency unstable oscillation dominates over the stable high-frequency oscillations found during a normal operation. The slower frequency (about 0.05 Hz) found during normal and abnormal operation is related to the neutronic processes. For easiness in the discussion, we will use time-scale s instead of segment size n, which are related by s = nDt, where Dt is the sampling period (here, Dt = 0.08 s). In this form, the fluctuation function is F(s) sa. Additionally, a
Fig. 1. APRM signal, case c5-aprm1. Forsmark stability benchmark.
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Fig. 2. Fluctuation function F(s) for Forsmark benchmark (c5.aprm1).
crossover will be denoted by scr. The results obtained with DFA for the case ‘‘c5.aprm’’ of Forsmark stability benchmark can be described as follows. Fig. 2 shows the presence of a plateau between the crossover scr,1 = 2.35 s and scr,2 = 12.45 s. Above and below the plateau the scaling exponent a, is around 1.323 indicating that time-scales smaller than 2.35 s, the correlation in the signal is persistent showing that neutronic power fluctuations occur out of the time-scale range [scr,1, scr,2] behaves like a fractional Brownian motion. In the plateau, the slope of the log–log plot is close to zero, reflection the lack of a power-law scaling behaviour. Similar plateau structure has been observed in other systems when a dominant periodic trend becomes superimposed on an otherwise non-periodic fluctuation signal (Viswanathan et al., 1997). For time scales bigger than the crossover scr,2, dominant periodic fluctuation has been observed, indicating stability with a scaling exponent about 0.7, i.e., the ‘‘dissipation’’ regions were disconnected. Summarizing, the DFA of the neutronic power signals led to the following main results: during the instable oscillation, such a scaling pattern is broken. Here, no scaling pattern is defined for a wide range (from scr,1 to scr,2) of time scales. The physics generating the scaling patterns detected for the case ‘‘c5.aprm’’ of Forsmark stability benchmark involves very complex nuclear processes, two-phase flow, heat transfer, and feedback control mechanism. In the following, we will discuss the detected scaling patterns in relation to the physical phenomena in a BWR nuclear reactor. An interesting issue is to find a physical interpretation of the observed crossover and the scaling exponents. In this direction, let us recall some features of BWR operation in normal conditions, i.e., operating under stable and stationary conditions. Nuclear activity generates a neutronic flow across the reactor core. The fission neutrons are born at very high energies in the MeV range. Then, the neutrons slow down via scattering coalitions in the water (neutron moderator), and after the fission process the energy is subsequently converted into thermal energy via conductive and convective heat transfer mechanism. At this point, vapour bubbles are generated leading to a very complex
two-phase flow pattern. Bubbles are of different sizes and their diameter can range from 0.01 to 1.0 cm. Energy transfer, hence neutronic power level, is reduced in flow regions with large amounts of bubbles, i.e., large void fractions. In this way, stationary neutronic power fluctuations could be generated by the non-homogeneous nature of the twophase flow through the BWR core. In principle, bubbles are formed almost independent of each other, and only secondary factors such as bubble collisions and shocks have some effects on the flow-pattern variations. As shown by Zebende et al. (2004), the values of scaling exponent a water for liquid–vapour-phase transitions ranges from 0.6 to 1.1. Pressure and wave oscillations of the order of 0.1– 1 Hz have been observed, both experimentally and numerically, in bubbling two-phase flows (Boure, 1978; Ranshaw and Trapp, 1978; Cheng et al., 1985), which introduces a dynamical effect in the two-phase flow pattern. These phenomena can contribute to the observed long-range correlation for time scales after of the crossover scr,2 = 12.45 s. That is, it seems that the large bubble formation and the recirculation flow dynamics inducing pressure and void fraction waves become the main contribution on the correlated fluctuations after of the observed crossover scr,2 = 12.45 s. On the other hand, after of the crossover one finds high-frequency fluctuations with anti-persistency nature, i.e. displays long-range correlation. Such anti-persistent scaling could be attributed, e.g., to high-frequency phenomena associated to the interaction between the nuclear reaction and the generation and subsequent growth of small bubbles, which yield a sort of sub-percolation neutronic diffusion across the reactor core. In other words, such a phenomena yield a neutron-scattering process, which is reflected as noise-like high-frequency neutronic power fluctuations (of the order of 0.1 Hz). Overall, the fact that neutronic power fluctuations are stable, stationary and correlated for relatively small frequencies (considered as the inverse of time scales) reflect the presence of an effective control structure that is able to damp neutronic power fluctuations. In principle, such control structure composed of intrinsic dissipative mechanism (e.g., the amount of generated neutrons equals the amount of lost neutrons) and feedback control mechanism (e.g., Doppler effect, cooling processes: void fraction and moderator temperature, and control rod). Fig. 2 suggests a possible periodic trend in the signal with a frequency about 0.5 Hz (n = 25), which dominates the neutronic power dynamics just after the situation where the neutronic power from the reactor displays normal, apparently stable behaviour (about 200 s of time elapsed, as is observed in Fig. 1). During unstable behaviour the physical phenomena in a BWR nuclear reactor is analyzed. This phenomenon can be caused by the lag introduced into thermal-hydraulic system by the finite speed of perturbation of density perturbation (Lahey and Podowski, 1989). At high-core void fractions, the feedback becomes so strong that it induces oscillations at frequency about 0.5 Hz. When this feedback increases,
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the oscillation become more pronounced, and oscillatory instability is reached. This can be observed in Fig. 1. On the other hand, both the interaction the single-phase and the two-phase pressure drop components in the coolant channels and the negative feedback of the void content to the reactor power are factors for instable behaviour. Along the line of the discussion for this event, the breakdown of the scaling properties during the unstable behaviour could be reflecting a fault in the intrinsic and extrinsic (feedback) control mechanism, as suggested by diverse analysis with different techniques (March-Leuba, 1985; Lahey and Podowski, 1989; Verdu et al., 2001; Blazquez and Ruiz, 2003). In this form, the unstable behaviour is displayed as an accumulation of energy fluctuations at low and high times scales, i.e., high and low frequencies, while fluctuations in the time-scale range [2.35, 12.45] are absent. The observed scaling structure at high time-scales (i.e., high-frequency) could be induced by variations in neutronic flux, while the corresponding scaling structure at low time-scale (i.e., low-frequency) can be attributed to recirculation water flow-rate disturbances. In this way, an interesting issue is to look at the origin of the scaling properties breakdown during event. Previous studies (March-Leuba, 1985; Lahey and Podowski, 1989) have shown that the unstable oscillation phenomenon was probably introduced by dominant density wave phenomena. Physically, density waves are intrinsically related to the two-phase hydraulic phenomena. In fact, it has been discussed that the large neutronic power oscillations observed were introduced by a sort of resonant dynamics between the void fraction formation and with pressure waves. In this way, fast pressure waves were slowed down in the region where bubbles start to appear. In turn, such a phenomenon leads to a significant fluctuation in the void fraction. By effects of neutron scattering, which is strongly effected by the void fraction, the above phenomenon was detected as the large neutronic power oscillations. Summing up, it seems that extreme changes in the two-phase flow pattern induced the large neutronic power fluctuations (low- and high-frequencies). Fig. 3 shows the evolution of the fluctuation function F(n) for three different sequential. Notice that, in the initial stages the nuclear reactor undergoes an unstable behaviour with high amplitude oscillations, the fluctuation function F(n) retain partially the scaling properties displayed in Fig. 2. However, as time increases the nuclear reactor undergoes a stable behaviour. This behaviour is captured by the progressive evolution of F(n), which converges to the structure displayed during normal operation. Additional evidence can be found when a short-time Fourier transform (STFT)-based power spectrum is carried out on the neutronic power dynamics of the event. In BWR instability regime characterization via neutron noise analysis, the fundamental frequency identification is important in order to know the possible origin of the instability. The results are shown in Fig. 4 where the oscillation is damped after 150 s. The fundamental frequency of 0.54 Hz and second harmonic about 1 Hz are present since
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Fig. 3. Evolution of the fluctuation function F(s).
Fig. 4. STFT-based power spectrum of the neutronic power dynamics along Forsmark stability benchmark (c5.aprm.1).
the beginning of the event. After 180 s, the second harmonic disappears and the third harmonic around 1.6 Hz is present between 100 s and 180 s. The instability is due to density wave phenomena, where the fundamental frequency, first and second harmonics are presents. As it can be seen the fundamental frequency and the second harmonic decrease when the oscillation is damped, but the second harmonic is only present when the energy of the oscillation is higher. This harmonic is observed for Forsmark case beginning of the event. Therefore, the second harmonic could be a good parameter together with DFA for indicating the evolution in time of the oscillation event. 6. Conclusions In this work we present a methodology based on DFA techniques for instability analysis in BWR. The method
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was applied to neutronic power signal of Forsmark stability benchmark. Our results suggest that DFA can help in the understanding and monitoring of the dynamics of power generation in nuclear plants with BWR. A feature that can be exploited to detect failures in neutronic power dynamics is that the scaling properties breakdown during unstable oscillations. Fig. 2 shows the breakdown of scaling properties for the full time series. The STFT method allows the detection of the fundamental frequency and a second harmonic only when the energy of the oscillations is high, so this harmonic together with DFA method can be used in on-line for surveillance monitoring of BWR nuclear power plant. References Absil, P.-A., Sepulchre, R., Bilge, A., Gerard, P., 1999. Nonlinear analysis of cardiac rhythm fluctuations using DFA method. Physica A 272, 235–244. Ausloos, M., Ivanova, K., 2000. Introducing false EUR and false EUR exchange rates. Physica A 286, 353–366. Blazquez, J., Ruiz, J., 2003. The Laguna Verde BWR/5 instability event. Prog. Nucl. Energ. 43, 195–200. Blesic, S., Milosevic, S., Stratimirovic, D., Ljubisavljevic, M., 1999. Detrended fluctuation analysis of time series of a firing fusimotor neuron. Physica A 268, 275–282. Boure, J.A., 1978. In: Ginoux, J.J. (Ed.), Two-Phase Flows and Heat Transfer with Application to Nuclear Reactor Design Problems. Hemisphere, Washington, DC. Buldyrev, S.V., Dokholyan, N.V., Goldberger, A.L., Havlin, S., Peng, C.K., Stanley, G.M., Viswanathan, G.M., 1998. Analysis of DNA sequences using methods of statistical physics. Physica A 249, 430–438. BWR/6, 1975. BWR/6 general description of a boiling water reactor, Nuclear Energy Division, General Electric Company. Cheng, L.Y., Drew, D.A., Lahey, R.T., 1985. An analysis of wave propagation in bubble two-component two-phase flows. J. Heat Transfer 107, 402–408. Hausdorff, J.M., Mitchell, S.L., Firtion, R., Peng, C.-K., Cudkowicz, M.E., Wei, J.Y., Goldberger, A.L., 1997. Altered fractal dynamics of gait: reduced stride-interval correlations with aging and Huntington’s disease. J. Appl. Physiol. 82, 262–269. Ivanova, K., Ausloos, M., 1999. Application of the detrended fluctuation analysis (DFA) method for describing cloud breaking. Physica A 274, 349–354. Kantelhardt, J.W., Berkovits, R., Havlin, S., Bunde, A., 1999. Are the phases in the Anderson model long-range correlated? Physica A 266, 461–464. Kantelhard, J.W., Koscielny-Bunde, E., Rego, H.H.A., Havlin, S., Bunde, A., 2001. Detecting long-range correlations with detrended fluctuation analysis. Physica A 295, 441–454.
Kantelhardt, J.W., Zschiegner, S.A., Koscielny-Bunde, E., Havlin, Sh., Bunde, A., Stanley, H.E., 2002. Multifractal detrended fluctuation analysis of nonstationary time series. Physica A 316, 87–114. Lahey, R.T., Podowski, M.Z., 1989. On the analysis of various instabilities in two-phase flow. Multiphase Sci. Technol. 4, 183–370. March-Leuba, J., 1985. A reduced-order model of boiling water reactor linear dynamics. Nucl. Technol. 75, 15–22. March-Leuba, J., Blakeman, E.D., 1991. A mechanism for out-of-phase power instabilities in boiling water reactor. Nucl. Sci. Eng. 107, 173– 179. March-Leuba, J., Rey, J.M., 1993. Coupled thermal-hydraulics neutronic stabilities in boiling water reactor: a review of the stat of the art. Nucl. Eng. Des. 145, 97–111. Muzy, J.F., Bacry, E., Arneodo, A., 1993. Multifractal formalism for fractal signals: The structure–function approach versus the wavelettransform modulus-maxima method. Phys. Rev. E 47, 875–884. Peng, C.-K., Buldyrev, S.V., Havlin, S., Simons, M., Stanley, H.E., Goldberger, AL., 1994. Mosaic organization of DNA nucleotides. Phys. Rev. E 49, 1685–1689. Peng, C.-K., Hausdorff, J.M., Havlin, S., Mietus, J.E., Stanley, H.E., Goldberger, A.L., 1998. Multiple-time scales analysis of physiological time series under neural control. Physica A 249, 491–500. Ranshaw, J.D., Trapp, J.A., 1978. Characteristic, stability, and shortwavelength phenomena in two-phase flow equations systems. Nucl. Sci. Eng. 66, 93–102. Shlesinger, M.F., West, B.J., Klafter, J., 1987. Le´vy dynamics of enhanced diffusion: application to turbulence. Phys. Rev. Lett. 58, 1100–1103. Talkner, P., Weber, R.O., 2000. Power spectrum and detrended fluctuation analysis: application to daily temperatures. Phys. Rev. E 62, 150– 160. Van Bragt, D.D.B., Van Der Hagen, H.J.J., 1997. Stability of natural circulation boiling water reactors: Part I-description stability model and theoretical analysis in terms of dimensional groups. Nucl. Technol. 121, 40–51. Verdu, G., Ginestar, D., 2001. Neutronic signal conditioning using a singular system analysis. Ann. Nucl. Sci. 28, 565–583. Verdu, G., Ginestar, D., Mun˜oz-Cobo, J.L., Navarro-Esbri, J., Palomo, M.J., Lansaker, P., Conde, J.M., Recio , M., Sartori, E., 2001. Forsmark 1&2 Stability Benchmark. Time series analysis methods for oscillations during BWR operation. Final Report, NEA/NSC/DOC (2001) 2. Viswanathan, G.M., Buldyrev, S.V., Havlin, S., Stanley, H.E., 1997. Quantification of DNA patchiness using long-range correlation measures. Biophys. J. 72, 866–875. Walleczek, J. (Ed.), 2000. Fractal mechanism in neural control: human heartbeat and gait dynamics in health and disease. Self-Organized Biological Dynamics and Nonlinear Control. Cambridge University Press. Zebende, G.F., da Silva, M.V.S., Rosa, A.C.P., Alves, A.S., de Jesu´s, J.C.O., Moret, M.A., 2004. Studying long-range correlations in a liquid–vapor-phase transition. Physica A 342, 322–328.
NUCLEAR ENERGY Annals of Nuclear Energy 33 (2006) 1309–1314 www.elsevier.com/locate/anucene
Detecting long-range correlation with detrended fluctuation analysis: Application to BWR stability ´ lvarez-Ramı´rez, Alejandro Va´zquez Gilberto Espinosa-Paredes *, Jose´ A Departamento de Ingenierı´a de Procesos e Hidra´ulica, Universidad Auto´noma Metropolitana-Iztapalapa, Apartado Postal 55-534, Me´xico, DF 09340, Mexico Received 13 July 2006; accepted 8 September 2006
Abstract The aim of this paper is to explore the application of detrended fluctuation analysis (DFA) to study boiling water reactor stability. DFA is a scaling method commonly used for detecting long-range correlations in non-stationary time series. This method is based on the random walk theory and was applied to neutronic power signal of Forsmark stability benchmark. Our results shows that the scaling properties breakdown during unstable oscillations. 2006 Elsevier Ltd. All rights reserved.
1. Introduction Instability analysis in boiling water reactor (BWR) plays a central role in the understanding of the physical mechanism that induces the observed power oscillations, which are unstable and occur at low-core flow and relatively high-power conditions (March-Leuba and Rey, 1993). For instance, large-amplitude power oscillations have been observed in Caorso (occurred in 1984), La Salle (occurred in 1988) and Laguna Verde (occurred in 1995) plants. Such instability phenomena have been the object of a great deal of study. Diverse classical signal processing techniques such as noise techniques, auto regressive moving average (ARMA) and fast Fourier transform (FFT) have been widely applied to stability monitoring (system identification). However, these techniques are limited in the sense that they are not applicable during transient states and are less sensitive to the noise in the signal to determine the fundamental frequency. Fourier transform in any of its forms (short-time Fourier transform, STFT) provides the frequency content of an analyzed signal. Thus, neither *
Corresponding author. Tel.: +52 55 5804 4645/102; fax: +52 55 5804 4900. E-mail address: [email protected] (G. Espinosa-Paredes). 0306-4549/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.anucene.2006.09.004
of them can detect transients; however it does not mean that it cannot be applied to a signal that contains a transient. Available analytical works are based on the Laplace transform for the dynamic analysis in the frequency domain to determine reactor stability (Van Bragt and Van Der Hagen, 1997). For example, March-Leuba and Blakeman (1991) studied the instabilities of the subcritical modes, and obtained stability bounds for the corewide and out-of-phase mode on the power-flow map using the code LAPUR in frequency-domain for BWR analysis. LAPUR provides numeric transfer functions for nuclear and thermo-hydraulics phenomena. In some approaches, the stability properties of the signal are obtained from invariant quantities as the Lyapunov exponents spectrum (Verdu and Ginestar, 2001). The studies above mentioned are widely applied to instabilities analysis and to develop stability monitoring systems in nuclear power plants, the mechanism that generate the oscillations are not easily uncovered and, in general, not applicable during transient phenomena. In this paper, we explore the use of detrended fluctuation analysis (DFA) (Peng et al., 1994) to detect long-range correlations in temporal patterns of neutronic power generation. Specifically, the case 5 of Forsmark stability benchmark is studied.
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In recent years, the DFA invented by Peng et al. (1994) has been established as an important tool for the detection of long-range correlations in time series with non-stationarities. It is successfully been applied to such diverse fields of interest as DNA (Buldyrev et al., 1998), heart rate dynamics (Peng et al., 1998; Absil et al., 1999), neuron spiking (Blesic et al., 1999), human gait (Peng et al., 1998; Hausdorff et al., 1997), long-time weather records (Talkner and Weber, 2000), cloud structure (Ivanova and Ausloos, 1999), economical time series (Ausloos and Ivanova, 2000), and even solid state physics (Kantelhardt et al., 1999), among others. While the Fourier transform and the recently developed wavelet transform modulus maxima (WTMM) method (Muzy et al., 1993) analyze the time series directly, the DFA is based on the random walk theory (Shlesinger et al., 1987). 2. Neutron monitoring in the BWR The neutronic flux in the reactor is monitored by detectors located inside the reactor core (BWR/6, 1975). Two types of neutron monitors are used: source range neutron monitor (SRNM) and average power range monitor (APRM) which are obtained from the local neutron flux monitored by fixed in-core ion chambers (LPRM), which are arranged in a uniform pattern throughout the core. These chambers cover a range from 1% to 125% of rated power in a linear scale. When a control rod or a group of control rods is selected for movement, readings from the adjacent detectors are displayed together on the control switchboard with a display of the position of the rod or group of rods. Each detector assembly contains four fission chambers and a travelling calibration guide tube (TIP). The chambers are uniformly spaced in an axial direction and lie in four horizontal planes. The average power level is measured by six average power range monitoring (APRM). Each APRM measures bulk power in the core by averaging signals from chains of 24 LPRM distributed throughout the core. Output signals from these monitors are displayed and are also used to operate trips in the reactor protection system. 3. Preliminaries We consider a record (yi) of i = 1, . . . , N equidistant measurements. In most applications, the index i will correspond to the time of the measurements. We are interested in the correlation of the values yi and for different time lags, i.e., correlations over different time scales n. In order to get rid of a constant offset in the data, the mean Æyæ, which is given by N 1 X hyi ¼ y N i¼1 i
ð1Þ
stands for the average of yi, i = 1, . . . ,N, is usually subtracted,
y i y i hyi
ð2Þ
Quantitatively, corresponds between y-values separated by n steps are defined by the auto-correlation function: GðnÞ ¼ hy i y iþn i ¼
N n 1 X y i y iþn N n i¼1
ð3Þ
If yi are correlated, G(n) is zero for n > 0. Short-range correlations of the yi are described by G(n) declining exponentially, GðtÞ exp ðn=nx Þ
ð4Þ
with a decay time nx. For the so-called long-range correlations, G(n) declines as a power-law: GðnÞ nr
ð5Þ
with an exponent 0 < r < 1. A direct calculation of G(n) is usually not appropriate due to noise superimposed on the collected data yi an due to underlying trends of unknown origin (e.g. the average Æyæ might be different for the first and the second half of the record, if the data are strongly long-range correlated). This makes the definitions of G(n) problematic. Then, the correlation exponent r is determined as is presented in Section 4. 4. DFA algorithm In the DFA method, basically a time series is integrated to obtain a type of random-walk profile, i.e., the method is a modification of root-mean-square analysis of a random walk for non-stationary data (Walleczek, 2000). Then this sequenced is divided into no overlapping boxes of equal sizes n. In each box the local trend is estimated by an m-degree polynomial fitting. In a subsequent step, the root-mean-square fluctuation, denoted by Fq(n) (where a generalized q-norm is used to compute a fluctuation function), of the difference between integrated sequence and the polynomial fit is calculated for each n. Finally, assuming that each fluctuation meets a power-law with respect to the box size, a scaling factor a is computed as the slope of the following fluctuation plot: def I ¼ logðnÞ versus log ðF q ðnÞÞ ð6Þ The DFA method used to analyze the neutronic power dynamics consists of five steps (Peng et al., 1994; Kantelhard et al., 2001): In the first step, we determine the profile xðiÞ ¼
i X
½y k hyi;
i ¼ 1; . . . ; N
ð7Þ
k¼1
of the record (yi) of length N. The subtraction of the mean Æyæ is not compulsory, since it would be eliminated by the later detrending in the third step anyway. In the second step, we cut the profile x(i) into Nn ” [N/n] non-overlapping segments of equal length n.Since the record length N needs not be a multiple of the considered
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time scale n, a short part at the end of the profile will remain in most cases. In order not to disregard this part of the record, the same procedure is repeated starting from the other end of the record. Thus, 2Nn segments are obtained altogether. In the third step, we calculate the local trend for each segment v by a least-squares fit of the data. Then we define the detrended time series for segment duration n, denoted by xn(i), as the difference between the original time series and the fits: xn ðiÞ ¼ xi pv ðiÞ
ð8Þ
where pv(i) is the fitting polynomial in the vth segment. In the fourth step, we calculate for each of the 2Nn segments the variance: n 1X x2 ½ðv 1Þn þ i ð9Þ F 2n ðvÞ ¼ hx2n ðiÞi ¼ n i¼1 n of detrended time series xn(i) by averaging over all data points i in the vth segment. Finally, we average over all segments and take the square root to obtain the DFA fluctuation function: "
#1=2 2N n 1 X 2 F ðnÞ ¼ F ðvÞ 2N n v¼1 n
ð10Þ
The above procedure is repeated for a broad range of segment lengths n. According to the recommendations made by Peng et al. (1994), nmin . 5 and nmax . N/4. If the data (yi) are long-range power-law correlated (Eq. (5)), the fluctuation function F(n) increase by a power-law: F ðnÞ na
ð11Þ
where a is called the scaling exponent, a self-affinity parameter representing the long-range power-law correlation properties of the signal. In the case of short-range (or not correlations at all) the detrended walk displays properties of a standard random walk (e.g., white noise) and a = 0.5. On the other hand, if a < 0.5 the correlation in the signal is anti-persistent (an increment is very likely to be followed by a decrement, and vice versa), and if a > 0.5 the correlation in the signal is persistent (an increment is very likely to be followed by an increment, and vice versa). Notice that integration/differentiation of the signal will increase/ decrease the exponent a by one. It may happen, as we shall see later, that in two neighbouring intervals of n the function F(n) displays two different power-law behaviours. The crossover region is defined by the values of n where the function F(n) changes its behaviour. The DFA algorithm described above can be extended along several directions. For instance, (i) overlapping segments of size n and overlapping size l can be used in Step 2 to reduce possible spurious correlations introduced by discontinuous trending, (ii) instead of the liner trending in Step 3, smoother fluctuation function behaviour can be obtained by computing mth-order polynomial fit. In this case, the mth-order DFA method, denoted by DFA-m, is
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obtained, and (iii) a generalised q-norm can be used to compute a fluctuation function as follow: " #1=q 2N n 1 X F q ðnÞ ¼ F 2 ðvÞ ð12Þ 2N n v¼1 n This approach has been proposed to study multi fractality in non-stationary signals (Kantelhardt et al., 2002). In this case, if the series y(i) is long-range power-law correlated, Fq(n) in creases as a power-law: F q ðnÞ naðqÞ
ð13Þ
where a(q) is the generalised scaling exponent. Notice that a(2) correspond to the standard scaling exponent a. It should be stressed that, in the sequel, we will use the basic DFA algorithm, as described in Steps 1 to 6, to detect the correlation structure of Formark stability benchmark neutronic power signal. 5. Results and discussions The DFA method was applied to neutronic signal of Forsmark stability benchmark, specifically to the case ‘‘c5.aprm’’ (Verdu et al., 2001). The signal corresponds to average power range monitor (APRM) registered during instability event, which is shown in Fig. 1. This event corresponds to a situation where the neutronic power from the reactor displays abnormal, apparently unstable, oscillations. That is, it seems that a medium-frequency unstable oscillation dominates over the stable high-frequency oscillations found during a normal operation. The slower frequency (about 0.05 Hz) found during normal and abnormal operation is related to the neutronic processes. For easiness in the discussion, we will use time-scale s instead of segment size n, which are related by s = nDt, where Dt is the sampling period (here, Dt = 0.08 s). In this form, the fluctuation function is F(s) sa. Additionally, a
Fig. 1. APRM signal, case c5-aprm1. Forsmark stability benchmark.
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Fig. 2. Fluctuation function F(s) for Forsmark benchmark (c5.aprm1).
crossover will be denoted by scr. The results obtained with DFA for the case ‘‘c5.aprm’’ of Forsmark stability benchmark can be described as follows. Fig. 2 shows the presence of a plateau between the crossover scr,1 = 2.35 s and scr,2 = 12.45 s. Above and below the plateau the scaling exponent a, is around 1.323 indicating that time-scales smaller than 2.35 s, the correlation in the signal is persistent showing that neutronic power fluctuations occur out of the time-scale range [scr,1, scr,2] behaves like a fractional Brownian motion. In the plateau, the slope of the log–log plot is close to zero, reflection the lack of a power-law scaling behaviour. Similar plateau structure has been observed in other systems when a dominant periodic trend becomes superimposed on an otherwise non-periodic fluctuation signal (Viswanathan et al., 1997). For time scales bigger than the crossover scr,2, dominant periodic fluctuation has been observed, indicating stability with a scaling exponent about 0.7, i.e., the ‘‘dissipation’’ regions were disconnected. Summarizing, the DFA of the neutronic power signals led to the following main results: during the instable oscillation, such a scaling pattern is broken. Here, no scaling pattern is defined for a wide range (from scr,1 to scr,2) of time scales. The physics generating the scaling patterns detected for the case ‘‘c5.aprm’’ of Forsmark stability benchmark involves very complex nuclear processes, two-phase flow, heat transfer, and feedback control mechanism. In the following, we will discuss the detected scaling patterns in relation to the physical phenomena in a BWR nuclear reactor. An interesting issue is to find a physical interpretation of the observed crossover and the scaling exponents. In this direction, let us recall some features of BWR operation in normal conditions, i.e., operating under stable and stationary conditions. Nuclear activity generates a neutronic flow across the reactor core. The fission neutrons are born at very high energies in the MeV range. Then, the neutrons slow down via scattering coalitions in the water (neutron moderator), and after the fission process the energy is subsequently converted into thermal energy via conductive and convective heat transfer mechanism. At this point, vapour bubbles are generated leading to a very complex
two-phase flow pattern. Bubbles are of different sizes and their diameter can range from 0.01 to 1.0 cm. Energy transfer, hence neutronic power level, is reduced in flow regions with large amounts of bubbles, i.e., large void fractions. In this way, stationary neutronic power fluctuations could be generated by the non-homogeneous nature of the twophase flow through the BWR core. In principle, bubbles are formed almost independent of each other, and only secondary factors such as bubble collisions and shocks have some effects on the flow-pattern variations. As shown by Zebende et al. (2004), the values of scaling exponent a water for liquid–vapour-phase transitions ranges from 0.6 to 1.1. Pressure and wave oscillations of the order of 0.1– 1 Hz have been observed, both experimentally and numerically, in bubbling two-phase flows (Boure, 1978; Ranshaw and Trapp, 1978; Cheng et al., 1985), which introduces a dynamical effect in the two-phase flow pattern. These phenomena can contribute to the observed long-range correlation for time scales after of the crossover scr,2 = 12.45 s. That is, it seems that the large bubble formation and the recirculation flow dynamics inducing pressure and void fraction waves become the main contribution on the correlated fluctuations after of the observed crossover scr,2 = 12.45 s. On the other hand, after of the crossover one finds high-frequency fluctuations with anti-persistency nature, i.e. displays long-range correlation. Such anti-persistent scaling could be attributed, e.g., to high-frequency phenomena associated to the interaction between the nuclear reaction and the generation and subsequent growth of small bubbles, which yield a sort of sub-percolation neutronic diffusion across the reactor core. In other words, such a phenomena yield a neutron-scattering process, which is reflected as noise-like high-frequency neutronic power fluctuations (of the order of 0.1 Hz). Overall, the fact that neutronic power fluctuations are stable, stationary and correlated for relatively small frequencies (considered as the inverse of time scales) reflect the presence of an effective control structure that is able to damp neutronic power fluctuations. In principle, such control structure composed of intrinsic dissipative mechanism (e.g., the amount of generated neutrons equals the amount of lost neutrons) and feedback control mechanism (e.g., Doppler effect, cooling processes: void fraction and moderator temperature, and control rod). Fig. 2 suggests a possible periodic trend in the signal with a frequency about 0.5 Hz (n = 25), which dominates the neutronic power dynamics just after the situation where the neutronic power from the reactor displays normal, apparently stable behaviour (about 200 s of time elapsed, as is observed in Fig. 1). During unstable behaviour the physical phenomena in a BWR nuclear reactor is analyzed. This phenomenon can be caused by the lag introduced into thermal-hydraulic system by the finite speed of perturbation of density perturbation (Lahey and Podowski, 1989). At high-core void fractions, the feedback becomes so strong that it induces oscillations at frequency about 0.5 Hz. When this feedback increases,
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the oscillation become more pronounced, and oscillatory instability is reached. This can be observed in Fig. 1. On the other hand, both the interaction the single-phase and the two-phase pressure drop components in the coolant channels and the negative feedback of the void content to the reactor power are factors for instable behaviour. Along the line of the discussion for this event, the breakdown of the scaling properties during the unstable behaviour could be reflecting a fault in the intrinsic and extrinsic (feedback) control mechanism, as suggested by diverse analysis with different techniques (March-Leuba, 1985; Lahey and Podowski, 1989; Verdu et al., 2001; Blazquez and Ruiz, 2003). In this form, the unstable behaviour is displayed as an accumulation of energy fluctuations at low and high times scales, i.e., high and low frequencies, while fluctuations in the time-scale range [2.35, 12.45] are absent. The observed scaling structure at high time-scales (i.e., high-frequency) could be induced by variations in neutronic flux, while the corresponding scaling structure at low time-scale (i.e., low-frequency) can be attributed to recirculation water flow-rate disturbances. In this way, an interesting issue is to look at the origin of the scaling properties breakdown during event. Previous studies (March-Leuba, 1985; Lahey and Podowski, 1989) have shown that the unstable oscillation phenomenon was probably introduced by dominant density wave phenomena. Physically, density waves are intrinsically related to the two-phase hydraulic phenomena. In fact, it has been discussed that the large neutronic power oscillations observed were introduced by a sort of resonant dynamics between the void fraction formation and with pressure waves. In this way, fast pressure waves were slowed down in the region where bubbles start to appear. In turn, such a phenomenon leads to a significant fluctuation in the void fraction. By effects of neutron scattering, which is strongly effected by the void fraction, the above phenomenon was detected as the large neutronic power oscillations. Summing up, it seems that extreme changes in the two-phase flow pattern induced the large neutronic power fluctuations (low- and high-frequencies). Fig. 3 shows the evolution of the fluctuation function F(n) for three different sequential. Notice that, in the initial stages the nuclear reactor undergoes an unstable behaviour with high amplitude oscillations, the fluctuation function F(n) retain partially the scaling properties displayed in Fig. 2. However, as time increases the nuclear reactor undergoes a stable behaviour. This behaviour is captured by the progressive evolution of F(n), which converges to the structure displayed during normal operation. Additional evidence can be found when a short-time Fourier transform (STFT)-based power spectrum is carried out on the neutronic power dynamics of the event. In BWR instability regime characterization via neutron noise analysis, the fundamental frequency identification is important in order to know the possible origin of the instability. The results are shown in Fig. 4 where the oscillation is damped after 150 s. The fundamental frequency of 0.54 Hz and second harmonic about 1 Hz are present since
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Fig. 3. Evolution of the fluctuation function F(s).
Fig. 4. STFT-based power spectrum of the neutronic power dynamics along Forsmark stability benchmark (c5.aprm.1).
the beginning of the event. After 180 s, the second harmonic disappears and the third harmonic around 1.6 Hz is present between 100 s and 180 s. The instability is due to density wave phenomena, where the fundamental frequency, first and second harmonics are presents. As it can be seen the fundamental frequency and the second harmonic decrease when the oscillation is damped, but the second harmonic is only present when the energy of the oscillation is higher. This harmonic is observed for Forsmark case beginning of the event. Therefore, the second harmonic could be a good parameter together with DFA for indicating the evolution in time of the oscillation event. 6. Conclusions In this work we present a methodology based on DFA techniques for instability analysis in BWR. The method
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was applied to neutronic power signal of Forsmark stability benchmark. Our results suggest that DFA can help in the understanding and monitoring of the dynamics of power generation in nuclear plants with BWR. A feature that can be exploited to detect failures in neutronic power dynamics is that the scaling properties breakdown during unstable oscillations. Fig. 2 shows the breakdown of scaling properties for the full time series. The STFT method allows the detection of the fundamental frequency and a second harmonic only when the energy of the oscillations is high, so this harmonic together with DFA method can be used in on-line for surveillance monitoring of BWR nuclear power plant. References Absil, P.-A., Sepulchre, R., Bilge, A., Gerard, P., 1999. Nonlinear analysis of cardiac rhythm fluctuations using DFA method. Physica A 272, 235–244. Ausloos, M., Ivanova, K., 2000. Introducing false EUR and false EUR exchange rates. Physica A 286, 353–366. Blazquez, J., Ruiz, J., 2003. The Laguna Verde BWR/5 instability event. Prog. Nucl. Energ. 43, 195–200. Blesic, S., Milosevic, S., Stratimirovic, D., Ljubisavljevic, M., 1999. Detrended fluctuation analysis of time series of a firing fusimotor neuron. Physica A 268, 275–282. Boure, J.A., 1978. In: Ginoux, J.J. (Ed.), Two-Phase Flows and Heat Transfer with Application to Nuclear Reactor Design Problems. Hemisphere, Washington, DC. Buldyrev, S.V., Dokholyan, N.V., Goldberger, A.L., Havlin, S., Peng, C.K., Stanley, G.M., Viswanathan, G.M., 1998. Analysis of DNA sequences using methods of statistical physics. Physica A 249, 430–438. BWR/6, 1975. BWR/6 general description of a boiling water reactor, Nuclear Energy Division, General Electric Company. Cheng, L.Y., Drew, D.A., Lahey, R.T., 1985. An analysis of wave propagation in bubble two-component two-phase flows. J. Heat Transfer 107, 402–408. Hausdorff, J.M., Mitchell, S.L., Firtion, R., Peng, C.-K., Cudkowicz, M.E., Wei, J.Y., Goldberger, A.L., 1997. Altered fractal dynamics of gait: reduced stride-interval correlations with aging and Huntington’s disease. J. Appl. Physiol. 82, 262–269. Ivanova, K., Ausloos, M., 1999. Application of the detrended fluctuation analysis (DFA) method for describing cloud breaking. Physica A 274, 349–354. Kantelhardt, J.W., Berkovits, R., Havlin, S., Bunde, A., 1999. Are the phases in the Anderson model long-range correlated? Physica A 266, 461–464. Kantelhard, J.W., Koscielny-Bunde, E., Rego, H.H.A., Havlin, S., Bunde, A., 2001. Detecting long-range correlations with detrended fluctuation analysis. Physica A 295, 441–454.
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