Analysis of a BWR Stability Behavior using Generalized Discrete Shannon Functions. Application to Ringhals and Forsmark Benchmarks Data

In this paper we have introduced a new methodology to on-line signal conditioning and monitoring to determine the stability parameters of the BWR NPP, that is, the determination of the effective Decay Ratio DR and the frequency of the main oscillation causing instability events. This method is based on the generalized discrete Shannon function convolution, which removes the noise and filters the signal in a specified frequency band. We have focused our attention in noise signals, first on analytic ones to check how the algorithm works, and then we have tested it with some real neutron signals. The algorithm works very well with dirty real signals providing good results, even in the case of short time series. Main attemption has been focused on decomposing signals to detect when a global and/or a regional oscillations are taking place in a BWR. This methodology can be implemented in on-line monitors to determine the stability parameters of the BWR reactors, that is, the determination of the effective decay ratio (DR) and the frequency of the main oscillation causing instability events.     

Analyses of pressure perturbation events in boiling water reactor

In order to design more stable and safer core configurations, experimental and theoretical studies about BWR (Boiling Water Reactor) instability have been performed to characterize the phenomenon and to predict the conditions for its occurrence. The instabilities can be caused by interdependencies between thermal-hydraulic and reactivity feedback parameters such as the void-coefficient, for example, during a pressure perturbation event. In this work, the RELAP5-MOD3.3 thermal-hydraulic system code and the PARCS-2.4 3D neutron kinetic code were coupled to simulate BWR transients. The pressure perturbation is considered in order to study in detail this type of transient. Two different algorithms developed at the University of Pisa were used to calculate the Decay Ratio (DR) and the natural frequency (NF) from the power oscillation signals obtained from the transient calculations. The validation of a code model set up for the Peach Bottom-2 BWR plant is performed against Low-Flow Stability Tests (LFST). The four series of Stability Tests were performed at Peach Bottom Unit 2 in 1977 at the end of cycle 2 in order to measure the reactor core stability margins at the limiting conditions used in design and safety analysis.     

Global physical and numerical stability of a nuclear reactor core

Low order models are used to investigate the influence of integration methods on observed power oscillations of some nuclear reactor simulators. The zero-power point reactor kinetics with six-delayed neutron precursor groups are time discretized using explicit, implicit and Crank–Nicholson methods, and the stability limit of the time mesh spacing is exactly obtained by locating their characteristic poles in the z-transform plane. These poles are the s to z mappings of the inhour equation roots and, except for one of them, they show little or no dependence on the integration method. Conditions for stable power oscillations can be also obtained by tracking when steady state output signals resulting from reactivity oscillations in the s-Laplace plane cross the imaginary axis. The dynamics of a BWR core operating at power conditions is represented by a reduced order model obtained by adding three ordinary differential equations, which can model void and Doppler reactivity feedback effects on power, and collapsing all delayed neutron precursors in one group. Void dynamics are modeled as a second order system and fuel heat transfer as a first order system. This model shows rich characteristics in terms of indicating the relative importance of different core parameters and conditions on both numerical and physical oscillations observed by large computer code simulations. A brief discussion of the influence of actual core and coolant conditions on the reduced order model is presented.     

A semiempirical prediction of the decay ratio for the boiling water reactors start-up process

During the start-up of a commercial boiling water reactor (BWR), the power and the coolant flow are continuously monitored. In order to prevent power instability events, the decay ratio (DR) could also be monitored. The process can be made safer if the operator could anticipate the DR too. DR depends on the power, the flow and many other quantities such as axial and radial neutron flux distribution, feed water temperature, void fraction, etc. A simple relationship for DR is derived. Three independent variables seem to be enough: the power, the flow and a single parameter standing for all other quantities which affect the DR. The relationship is validated with data from commercial BWR start-ups. A practical procedure for the start-up of a BWR is designed; it could help preventing instability events.     

Decay ratio estimation based on time–frequency representations

A novel method based on bilinear time–frequency representations (TFRs) is proposed to determine the time evolution of the linear stability parameters of a boiling water reactor (BWR) using neutronic noise signals. TFRs allow us to track the instantaneous frequencies contained in a signal to estimate an instantaneous decay ratio (IDR) that closely follows the signal envelope changes in time, making the IDR a measure of local stability. In order to account for long term changes in BWR stability, the ACDR measure is introduced as the accumulated product of the local IDRs. As it is shown in this paper, the ACDR measure clearly reflects major long term changes in BWR stability. Last to validate our method, synthetic and real neutronic signals were used. The methodology was tested on the Laguna Verde Unit 1, two events were reported in the Forsmark stability benchmark.     

Monitoring the stability of BWR oscillation by nonlinear time series modeling

Monitoring the dynamics evolution of BWR oscillation has great importance in evaluating safety of the nuclear systems. Time series analysis methodology has been widely accepted as a powerful tool for this subject. BWR stability has been so far evaluated by decaying ratio (DR) calculated from the impulse response function of autoregressive (AR) model. To explore much more reliable method for detecting BWR instability, this paper introduces a nonlinear time series analysis approach namely exponential autoregressive (ExpAR) modeling. The ExpAR model is available for revealing types of nonlinear dynamics such as fixed point, limit cycle, and even chaos. Furthermore, the model is real-time estimated so that it is suitable for the purpose of on-line BWR instability detection. Empirical analysis of typical benchmark neutronic signal shows the effectiveness of this proposal.     

Comments on the application of bifurcation analysis in BWR stability analysis

Currently, BWR stability analysis is most often performed by the application of system codes which provide the time evolution of the neutron flux or thermal power at a defined operational point (OP) after imposing a system parameter perturbation. However, in general it is impossible to understand the real stability state of the BWR at a specific OP by the application of system code analysis alone. Hence, we are exploring methods developed in the nonlinear dynamics field in order to reveal the nature of the BWR stability states when power oscillations are observed. A powerful method is bifurcation analysis. In order to motivate this “nonlinear thinking” versus “linear thinking”, in this paper we will demonstrate some examples of phenomena which can only be understood in nonlinear terms by application of bifurcation theory and where linear interpretation leads to incorrect conclusions.     

An advanced reduced order model for BWR stability analysis

An advanced reduced order model was developed and qualified in the framework of a novel approach for nonlinear stability analysis of boiling water nuclear reactors (BWRs). This approach is called the RAM-ROM method where RAM is a synonym for system code and ROM stands for reduced order model. In the framework of the RAM-ROM method, integrated BWR (system) codes and reduced order models are used as complementary tools to examine the stability characteristics of fixed points and periodic solutions of the nonlinear differential equations describing the stability behaviour of a BWR loop. This methodology is a novel one in a specific sense: we analyse the highly nonlinear processes of BWR dynamics by applying validated system codes and by the sophisticated methods of nonlinear dynamics, e.g. bifurcation analysis. We claim and we will show that the combined application of independent methodologies to examine nonlinear stability behaviour can increase the reliability of BWR stability analysis.This work is a continuation of previous work at the Paul Scherrer Institute (PSI, Switzerland) of the second author and at the University of Illinois (USA) in this field. In the scope of a PhD work at the Technical University Dresden (Germany), the current ROM was extended to an advanced ROM by adding a recirculation loop model, a quantitative assessment of the necessity for consideration of the effect of sub-cooled boiling and a new calculation methodology for feedback reactivity. A crucial point of ROM qualification is a new calculation procedure for ROM input data based on steady-state RAM (ONA) results. The modified ROM is coupled with the BIFDD bifurcation code which performs a semi-analytical bifurcation analysis (see Appendix C). In this paper, the advanced ROM (TU Dresden ROM, TUD-ROM) is briefly described and the results of a nonlinear BWR stability analysis based on the RAM-ROM method are summarised for NPP Leibstadt, NPP Ringhals and NPP Brunsbüttel. The results show that the TUD-ROM including the new approach for ROM input data calculation is qualified for BWR stability analysis in the framework of the RAM-ROM method.     

Boiling water reactor stability revisited: The effects of flashing

There are numerous investigations of two-phase flow stability with particular emphasis to BWR stability; these have become increasingly sophisticated and complete over the years. The basic features of a new development and frequency-domain code capable of considering all the channels (bundles) in a BWR, flashing of the coolant at low pressure, full coupling with 3D, two-group neutronics, etc. are described. The basic thermal-hydraulic model is used to study the effects of flashing on stability in a BWR-like channel. The behavior of the channel is highly dynamic. Contrary to what could have been intuitively guessed, the effect of flashing is stabilizing; the reasons and mechanisms leading to this are discussed.     

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.     

The influence of direct energy deposition on BWR stability

Direct energy deposition into the coolant of a BWR via high-energy γs and neutrons is always neglected in reduced-order BWR dynamics models. The correctness of such a simple approach is investigated by comparing reactor stability predictions of the March–Leuba–Cacuci–Pérez reduced-order model, with a model where direct heating has been included. It is shown that neglecting direct energy deposition leads to a strong underestimation of the stability of the reactor.     

Theory of stochastic bifurcation in BWRs and applications

We have found that noise destroys the Hopf bifurcation in BWR so that the Lyapunov exponent takes negative values in any case near the onset of power oscillation. Consequently, the values of decay ratio (DR) always take a value less than 1. Theoretical prediction based on the theory of stochastic bifurcation is verified with the use of practical time series data. A new method of the stochastic dynamics of phase-difference is also proposed, and its powerfulness is demonstrated.     

Two novel procedures for aggregating randomized model ensemble outcomes for robust signal reconstruction in nuclear power plants monitoring systems

Detecting anomalies in sensors and reconstructing the correct values of the measured signals is of paramount importance for the safe and reliable operation of nuclear power plants. Auto-associative regression models can be used for the signal reconstruction task but in real applications the number of sensors signals may be too large to be handled effectively by one single model. In these cases, one may resort to an ensemble of reconstruction models, each one handling a small group of sensor signals; the outcomes of the individual models are then combined to produce the final reconstruction. In this work, three methods for aggregating the outcomes of a feature-randomized ensemble of Principal Components Analysis (PCA)-based regression models are analyzed and applied to two case studies concerning the reconstruction of 215 signals monitored at a Finnish nuclear Pressurized Water Reactor (PWR) and 920 simulated signals of the Swedish Forsmark-3 Boiling Water Reactor (BWR). Based on the insights gained, two novel aggregation procedures are developed for optimal signal reconstruction.     

Limitation of Burn Time due to Ash Accumulation in a Moving Ring Reactor

A new method for evaluating the decay ratios in a boiling water reactor (BWR) using the singular value decomposition (SVD) method had been proposed. In this method, a signal component closely related to the BWR stability can be extracted from independent components of the neutron noise signal decomposed by the SVD method. However, real-time stability monitoring by the SVD method requires an efficient procedure for screening such compoments. For efficient screening, an artificial neural network (ANN) with three layers was adopted. The trained ANN was actually applied to decomposed components of local power range monitor (LPRM) signals that were measured in stability experiments conducted in the Ringhals-1 BWR. In each LPRM signal, multiple candidates were screened from the decomposed components. However, decay ratios could be estimated by introducing appropriate criterions for selecting the most suitable component among the candidates. The estimated decay ratios are almost identical to those evaluated by visual screening in a previous study. The selected components commonly have the largest singular value, the largest decay ratio and the least squared fitting error among the candidates. By virtue of excellent screening performance of the trained ANN, the real-time stability monitoring by the SVD method can be applied in practice.     

On the possibility of the space-dependence of the stability indicator (decay ratio) of a BWR

A model is proposed for the explanation of the space-dependence of the so-called decay ratio (DR) which is used to quantify the stability properties of boiling water reactors (BWRs). The study was prompted by the observation of a strongly space-dependent decay ratio in an instability event at the Swedish Forsmark-1 BWR. Prior to that event, the space-dependence of the DR was neither observed, nor assumed possible in the theoretical models of instability.The model proposed here is based on a previous suggestion by one of the authors on how to model the estimation of the DR in case of two different types of oscillations (instabilities) being present in the core simultaneously. The model was earlier only used in a space-independent form, but here its applicability is extended such that space-dependence of the oscillations is also accounted for, by using a noise simulator. The investigations show that the DR, as determined by the individual LPRMs (neutron detectors) at different positions, can be strongly space-dependent if at least two different oscillations with differing DR and space-dependence exist in the core simultaneously. The observed space-dependence of the DR in the Forsmark case can be reconstructed by the model.     

BWR stability and bifurcation analysis using reduced order models and system codes: Identification of a subcritical Hopf bifurcation using RAMONA

The system code RAMONA, as well as a recently developed BWR reduced order model (ROM), are employed for the stability analysis of a specific operational point of the Leibstadt nuclear power plant. This has been done in order to assess the ROM’s applicability and limitations in a quantitative manner.     

Simulation of an hypothetical out-of-phase instability case in boiling water reactor by RELAP5/PARCS coupled codes

A very complex type of power instability occurring in boiling water reactor (BWR) consists of out-of-phase regional oscillations, in which normally subcritical neutronic modes are excited by thermal-hydraulic feedback mechanisms. The out-of-phase mode of oscillation is a very challenging type of instability and its study is relevant because of the safety implications related to the capability to promptly detect any such inadvertent occurrence by in-core neutron detectors, thus triggering the necessary countermeasures in terms of selected rod insertion or even reactor shutdown. In this work, simulations of out-of-phase instabilities in a BWR obtained by assuming an hypothetical continuous control rod bank withdrawal are being presented. The RELAP5/Mod3.3 thermal-hydraulic system code coupled with the PARCS/2.4 3D neutron kinetic code has been used to simulate the instability phenomenon. Data from a real BWR nuclear power plant (NPP) have been used as reference conditions and reactor parameters. Simulated neutronic power signals from local power range monitors (LPRM) have been used to detect and study the local power oscillations. The decay ratio (DR) and the natural frequency (NF) of the power oscillations (typical parameters used to evaluate the instabilities) have been used in the analysis. The results are discussed also making use of two-dimensional plots depicting relative core power distribution during the transient, in order to clearly illustrate the out-of-phase behavior.     

Simplified distributed parameters BWR dynamic model for transient and stability analysis

This paper describes a simplified model to perform transient and linear stability analysis for a typical boiling water reactor (BWR). The simplified transient model was based in lumped and distributed parameters approximations, which includes vessel dome and the downcomer, recirculation loops, neutron process, fuel pin temperature distribution, lower and upper plenums reactor core and pressure and level controls. The stability was determined by studying the linearized versions of the equations representing the BWR system in the frequency domain. Numerical examples are used to illustrate the wide application of the simplified BWR model. We concluded that this simplified model describes properly the dynamic of a BWR and can be used for safety analysis or as a first approach in the design of an advanced BWR.     

An Analytical Study of Nonlinear Reactor Dynamics Based on Bifurcation Theory with Reference to BWR Power Oscillation

A nonlinear reactor dynamics model of reduced order is derived and an analytical study on BWR power oscillation is made using this model. It provides some essential features which are not given by numerical studies, such as the explicit expressions of the linear stability condition and the weak stability condition which is related to the periodic motion. In addition, the relation between the reactivity feedback and these conditions is obtained. The application of the analytical results to the qualitative analysis of BWR dynamics is easy and quick in comparison with numerical approaches.     

Development of advanced core noise monitoring system for BWRS

A BWR core noise monitoring system is developed for addressing core anomaly problems in future advanced core operation. In order to monitor in-core status from a limited number of signals, various up-to-date signal processing algorithms are introduced to compensate for a lack of information. These algorithms, such as independent component analysis, factor analysis and model based parameter estimation, are demonstrated to be effective through real plant data analysis to evaluate core and regional stability index, reactivity coefficients and core flow rate. Through these practices, we demonstrate that the core noise monitoring system is an effective general platform for providing a variety of monitoring tools to meet the requirements in future advanced core operation.