A systematic method to identify nonlinear dynamics of BWR by using the reactor noise

For the identification of the dynamics of the Vermont Yankee BWR with the reactor noise, different parametric models have been tested. The widely used ARMA model is unable to identify the nonlinearity in the noise data. A systematic method by using the NARMA model, which takes advantage of both the ANN and ARMA, is developed. Comparisons are made between the identification results with ARMA and NARMA model. The advantages of identification with NARMA model over ARMA model are demonstrated. The linear-kernels of the identified NARMA models are extracted so that the natural frequency, damping ratio and time constants of the BWR are obtained. The values of those characteristics are well corresponded with the eigenvalues calculated by the differential equations of the Vermont Yankee BWR. The damping ratio with negative value is found to be a criterion for the existence of limit-cycle, which can be seen from the impulse response on the (X_t, X_t−1) plane, in stable nonlinear system.     

Propagation noise calculations in VVER-type reactor core

Neutron noise induced by propagating disturbances in VVER-type reactor core is addressed in this paper. The spatial discretization of the governing equations is based on the box-scheme finite difference method for triangular-z geometry. Using the derived equations, a 3-D 2-group neutron noise simulator (called TRIDYN-3) is developed for hexagonal-structured reactor core, by which the discrete form of both the forward and adjoint reactor dynamic transfer functions (in the frequency domain) can be calculated. In addition, both types of noise sources, namely point-like and traveling perturbations, can be modeled by TRIDYN-3. The results are then benchmarked in different cases. Considering the noise source as propagating perturbations of the macroscopic absorption cross sections, the induced neutron noise is calculated throughout the reactor core. For the first time, adjoint approach is applied and examined for modeling moving noise sources. Moreover, the space- and frequency-dependence of the propagation noise are investigated in this paper.     

反应堆堆内部件振动中子噪声物理模型研究

中子噪声分析对反应堆堆内部件振动监测有重要意义。本文采用微扰理论 (系统方程和扰动源项 )、控制理论 (传递函数 )、反应堆动力学方程 (点堆动力学方程 )建立了堆内部件振动中子噪声物理模型 ,并且用它来解释实验 ,较好地解释了实验测量得到的功率谱密度。在理论模型中通过引入一低频噪声项 ,较好地描述了实验测量功率谱密度低频端的抬高。另外对于吊篮梁式振动 ,则采用四个堆外探测器来实现监测。通过这种方法 ,可以较好地监测吊篮梁式振动和进行计算机仿真模拟。     

Description of void effects in a large BWR by means of a stochastic theory and simulation in a zero-power reactor

We first summarize the stochastic point model developed in previous papers to describe void effects in a large BWR and also summarize our results. The most important of these is the existence of a resonance frequency in the auto-power spectral density of the neutron noise (APSD), the position of which depends on the reactor characteristics, such as power or void coefficient.

In order to check the validity of this model, we made experiments simulating heat transfer and steam fluctuations in a BWR. A stochastic interpretation of the experiments is developed, and results are found to be similar to those obtained with the BWR model. In particular, the zero-power reactor with the simulation device exhibits a resonance frequency showing an identical behaviour to the one predicted for a BWR.

The APSD resulting from experiments in the zero-power reactor CROCUS, at a given power level, is fitted on the theoretical curves by means of the least square method, which provides the resonance frequency. The behaviour of this frequency as a function of the power level agrees fairly well with the theoretical prediction.

If we suppose that feedback mechanisms are the same in a large BWR, we can also admit that the stochastic model gives correctly the resonance frequency.     

An analytically solvable,axially non-homogeneous reactor model—I. General solutions

In conceptual and model studies of neutron noise and diagnostics, and even in most practical applications, simple homogeneous reactor models have been used so far, in which closed form analytical solutions are possible. In this paper a less trivial, axially non-homogeneous reactor model is used. In this model both the static flux and the noise equations can still be solved by analytical methods. The model consists of a 2-D homogeneous rectangular core in which a δ-function (Feinberg-Galanin) control rod is inserted partially. Solution of both the static flux and the dynamic equations (the latter corresponding to a rod manoeuvring experiment in which the rod is moved up and down periodically) can be given by two different analytical methodologies. Both are calculated and discussed in the paper. This model will be used later in concrete diagnostic applications as well as in studies of reactor kinetic approximations.     

Power Noise Spectra of a Water Reactor in Low Frequency Region

The power noise measurements were carried out on the Kyoto University Reactor at various reactor power levels under natural convection for two kinds of core configuration with different temperature coefficients of reactivity. Analysis of the results revealed strong noise in the low frequency region at higher power levels, even with a core configuration of essentially zero temperature coefficient of total reactivity.

A conventional theoretical model assuming a white noise input in reactivity and a reactor transfer function with temperature-reactivity feedbacks was adopted for comparison with the observed data, but it resulted in considerable disagreement at higher power levels.

As a result, it is concluded that there must exist other power effects not yet clarified in the reactivity feedbacks or in the noise inputs to the reactor, which are responsible for the experimental noise spectrum distortion in the low frequency region.     

Problems of diagnostics of physical characteristics of RBMK reactors according to neutron noise

Conclusion Studies of the neutron noise of the RBMK-1000 can be made on the basis of the existing regular CSS equipment. In studying the neutron noise it was revealed that in the frequency range 0.021–0.022 Hz the spectral characteristic displays a resonance whose amplitude depends essentially on the steam coefficient of reactivity; this makes it possible to use analysis of neutron noise to monitor and diagnose changes in the steam coefficient of reactivity of reactors of the RBMK type during normal operation. At the same time, it was established that for analysis of neutron noise in the RBMK-1000 in the frequency range indicated one can use a simplified model of the dynamics of the reactor plant.Translated from Atomnaya Énergiya, Vol. 48, No. 3, pp. 145–148, March, 1980.     

Calculating reactor transfer functions by Padé approximation via Lanczos algorithm

An effective method of solving the one-speed transport equation in frequency domain is demonstrated in this paper, the so-called Padé approximation via Lanczos algorithm (PVL). The advantage of the PVL method is that implementing the calculation process over a considerably reduced model yields a pseudo-analytical expression of the transfer function over a fairly large range of frequency. As a particular application, the dynamic transfer function of a reactor, i.e. the neutron noise induced by a localised perturbation is calculated in one-speed transport theory. The problem is essentially the same as that of the “detector-field-of-view”; studied by other authors. The PVL algorithm is demonstrated through the solution of the problem and its advantages are described. The quantitative results show that although one-speed theory was used, a local component was found, and thus the local-global decomposition could be reconstructed. This shows that unlike in diffusion theory where at least two-group theory is necessary, the local behaviour can be described already by a one-speed equation in transport theory.     

Transfer Function Measurement of Ionization Chambers with the Use of Uncorrelated Reactor Noise

The frequency response of an ionization chamber was measured by utilizing the uncorrelated reactor noise, varying the voltages as parameters. The results obtained made it clear that the break frequency of the ionization chamber is determined by the transit time of positive ions and the applied voltage, and can further be represented by a simple first order time lag.     

Gabor变换用于核压力容器裂纹楞边超声检测

核压力容器不锈钢堆焊层的材料噪声降低了在役检测中超声信号的信噪比。本文应用Ｇａｂｏｒ变换时频分析技术，根据裂纹楞边的频偏特性，提取了裂纹楞边超声回波。该技术可用于定量地检测核压力容器的裂纹状况。     

Application of On-Line Digital Noise Analysis to Reactor Diagnosis in JMTR

The reactor noise analysis technique is particularly useful in reactor diagnosis for on-line monitoring if the raw noise signals can be processed in almost real time.

An on-line reactor noise analysis system has been developed with use made of the mini-computer HITAC-10. This system utilizes functions for calculating the power spectral density in almost real time, plots the output by digital incremental plotter, and displays the results by means of color graphic display equipment, in order to detect anomalous reactor conditions with the statistical technique.

Using this system, reactor noise signals have been measured and analyzed under various operational conditions in the JMTR. The variance of the power spectral density is found to fit a logarithmic probability density function. This function is independent of the frequency, but is dependent on the number of sampling functions.

A logical procedure for anomaly detection based on statistical characteristics has been developed. It is applied to a case where it is supposed that the PWR operating mode in the OWL-1 is the normal process and that the BWR mode is the anomalous. It is demonstrated as a result, that this procedure can successfully detect anomalous processes.     

Identification of reactor internals'' vibration modes of a Korean standard PWR using structural modeling and neutron noise analysis

The vibration characteristics of a Korean standard PWR reactor internals have been estimated through a three-dimensional finite element analyses and verified by using the mode separated power spectral density functions obtained from the ex-core neutron noise signals. Also the natural vibration modes of the fuel assembly have been identified measuring both the ex-core and the in-core neutron noise signals which are close to each other. As a result, the fundamental bending mode frequency of the reactor internal structure is found to be around 8 Hz and the fundamental shell mode frequency 14.5 Hz, respectively. It is also shown that the fundamental bending mode frequency of the fuel assembly is 2.3 Hz and the 2^nd bending mode frequency 5.8 Hz, respectively. These results can be used for the supplements of the Korean standard PWR's CVAP (Comprehensive Vibration Assessment Program) data.     

Identification and localization of absorbers of variable strength in nuclear reactors

This paper investigates the possibility of localising a noise source of the type “absorber of variable strength” (or reactor oscillator) from as few as five neutron detectors evenly distributed throughout the core of a commercial nuclear reactor. The novelty of this investigation lies with the fact that the calculations are performed for a realistic 2-D heterogeneous reactor in the 2-group diffusion approximation, via the prior determination of the corresponding reactor transfer function. It is first demonstrated that the response of such a reactor to a localized perturbation deviates significantly from point-kinetics. The space-dependence of the induced neutron noise thus carries enough information about the location of the noise source, which makes it possible to determine its position from a few detector readings. The identification of the type of noise source is easily performed from the in-phase behaviour of the induced neutron noise. Different unfolding techniques are finally tested. All these techniques rely on the use of the reactor transfer function. One of these techniques is based on the comparison between the actual measured neutron noise and the neutron noise calculated for every possible location of the noise source. This technique is very reliable and almost insensitive to the contamination of the detector signals by background noise, but also extremely CPU consuming. Another technique, based on the piece-wise inversion of the reactor transfer function and requiring little CPU effort, was developed. Although this technique is much less reliable when background noise is present, this technique is useful to indicate a region of the reactor where a noise source is likely to be located.     

Analytical Study of Operating Reactor Noise on a Single Channel and a Single Phase Flow Model, (I)

Basic formulas to describe the propagation of a periodically modulated driving source are derived on the distributed model of at-power reactors.

Disturbances in the neutron field are regarded as being promptly spread over the whole core in the frequency domain of usual interest for noise and frequency response analyses. On the basis of this approximation, a coupled set of kinetic equations for neutrons, fuel and coolant temperature is derived and solved for the case where a driving source of unit strength is imparted. The relations between the solutions obtained and the formulas for analyzing experiments are established. Comparisons are made qualitatively between the results and those of the lumped model. The existence of a sink and a resonance frequency is examined. The power spectral density of neutron fluctuations in a water reactor under forced circulated cooling shows a sharply varying configuration when the inlet coolant temperature fluctuations are the major noise sources. When the reactor is under natural convective cooling or under liquid metal cooling, no such characteristic features appear.     

Development of Reactor Noise Monitor

A reactor noise monitor has been developed to provide a tool that serves the real time routine function of detecting anomalous states of operation in power reactors. In this monitor, RMS (root mean square) values and average frequencies of noise signals are calculated on a real time basis. The average frequency is defined as the frequency averaged by weighting with PSD (power spectral density) of the noise signal and proved to be equivalent to the ratio of an RMS value of the differentiated noise signal to an RMS value of the noise signal itself. Thus, the average frequency can be obtained on a real time basis by using a differentiator, an RMS calculator and a divider. The frequency components of 12 kinds of signals ranging of 0.01–36 Hz are divided into two or three frequency regions by band-pass filters and the RMS value and the average frequency of each frequency region are obtained by the digital calculation method. The performance of this monitor was tested by using a simulated anomalous signal with a peak on its PSD. From the test, it is proved that this monitor can detect more than changes of the RMS value and average frequency. The functions of this monitor were also confirmed at the actual BWR power plant.

In addition, a method to detect peak location of PSD is developed and the accuracy of the peak frequency measurement of the proposed method is investigated to be satisfactory.     

Analysis of pool-type reactor noise

A theoretical and experimental analysis is presented of pool-type reactor noise caused by thermal hydraulic and mechanical phenomena. The thermal hydraulic model is based on fluctuations of coolant inlet temperature and coolant velocity as noise sources. Resonance peaks in the reactor noise spectrum are explained by simple mechanical models of the reactor structure and the instrument tubes.     

Power spectral analysis for a thermal subcritical reactor system driven by a pulsed 14 MeV neutron source

A series of power spectral analyses for a thermal subcritical reactor system driven by a pulsed 14 MeV neutron source was carried out at Kyoto University Critical Assembly (KUCA), to determine the prompt-neutron decay constant of the accelerator-driven system (ADS). The cross-power spectral density between time-sequence signal data of two neutron detectors was composed of a familiar continuous reactor noise component and many delta-function-like peaks at the integral multiple of pulse repetition frequency. The prompt-neutron decay constant inferred from the reactor noise component of the cross-power spectral density was consistent with that obtained by a pulsed neutron experiment. However, the reactor noise component of the auto-power spectral density of each detector was hidden by a white chamber noise in the higher-frequency range and this feature resulted in a considerable underestimation of the decay constant. For several runs with a low pulse-repetition frequency, furthermore, we attempted to infer the decay constant from point data of the delta-function-like peaks. The analysis for a run under a slightly subcritical state resulted in the consistent decay constant; however, those for other runs under significantly subcritical states underestimated the decay constant. Considering the contribution of a spatially higher mode to the point data, the above underestimation was solved to obtain the consistent decay constant. While the Feynman-α formula for a pulsed neutron source is too complicated to be fitted directly to variance-to-mean ratio data, the present analysis on frequency domain is much simpler and the conventional formula based on the first-order reactor transfer function is available for fitting to power spectral density data.     

Finite Element Analysis of Vibration of Thin Elastic Plates by Simplified Hybrid Displacement Method

Temperature noise, measured by thermocouples mounted at each core fuel subassembly, is considered to be the most useful signal for detecting and locating local cooling anomalies in an LMFBR core. However, the core outlet temperature noise contains background noise due to fluctuations in the operating parameters including reactor power. It is therefore necessary to reduce this background noise for highly sensitive anomaly detection by subtracting predictable components from the measured signal. In the present study, both a physical model and an autoregressive model were applied to noise data measured in the experimental fast reactor JOYO. The results indicate that the autoregressive model has a higher precision than the physical model in background noise prediction. Based on these results, an “autoregressive model modification method” is proposed, in which a temporary autoregressive model is generated by interpolation or extrapolation of reference models identified under a small number of different operating conditions. The generated autoregressive model has shown sufficient precision over a wide range of reactor power in applications to artificial noise data produced by an LMFBR noise simulator even when the coolant flow rate was changed to keep a constant power-to-flow ratio.     

Negative Energy Level and Temperature Coefficient of Thermal Neutron Absorption Rate of Th-232

A review is presented on the recent research activities in Japan in the field of reactor noise analysis. It covers six selected areas : nonlinear stochastic theory and identification of nonlinearity in nuclear reactor dynamics, structural identification via reactor noise analysis, autoregressive model and contraction of information, knowledge-based support system for reactor noise analysis and diagnosis, reactor noise analysis benchmark tests and application of noise analysis to BWR plant diagnosis.