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.     

BWR linear stability analysis

A state-of-the-art analytical model has been developed for investigating the linear stability of boiling water nuclear reactors (BWR). This model has been implemented into a computer code named NUFREQ-NP. The NUFREQ-NP code permits an investigation of the stability of a single heated channel, as well as the multi-channel core of a BWR. It accommodates various neutronics models, including point kinetics, one-dimensional, two-dimensional, and three-dimensional neutron kinetics. It is capable of evaluating system stability in terms of any of the following perturbed state variables: inlet flow rate, external reactivity, and system pressure.This paper presents the modeling principles used in NUFREQ-NP to describe the transient two-phase flow and heat transfer phenomena. Special emphasis is placed on those aspects of the model development which are related to variable system pressure effects. Also, the derivation of various transfer functions is given. The results of NUFREQ-NP testing and verification include a parametric study of the effects of various modeling assumptions, and comparisons with both out-of-core and in-core-experimental data. These comparisons indicate very good agreement between the calculated and measured system transfer functions.     

BWR transient analysis payoffs from LOCA research

The extensive reactor safety programs directed toward Loss of Coolant Accident (LOCA) responses have provided both advanced technology and successful programmatic approaches that have significant applicability to current reactor transient studies. Extension of the benchmark TRAC-BWR computer program is proving to be highly successful for the full range of BWR transient phenomena. Work in progress will extend this technology to accurate Engineering Analyzers and Training Simulators that operate on low cost desktop workstations. The payoffs of these developments for improved reactor design, analysis and operation are summarized.     

A simplified model of the BWR depressurization transient

A simplified model is proposed to evaluate the BWR plant depressurization transient subsequent to the activation of a single safety relief valve. The model is validated with a series of RELAP5 Mod 3 calculations yielding reasonable comparisons. The depressurization behavior of the 26 US BWR plants is analyzed with the model and the relevant parameters controlling the process are identified. Of these, four are plant-specific and their effect on the final outcome of the depressurization transient is quantified. The minimum vessel inventory at the activation of the low pressure emergency core coolant injection is selected as the figure of merit. Core power is the only parameter that significantly affects the outcome of the transient. The role of the low pressure injection and of the discharge port cross-sectional area is discussed in detail. The effect of operator actions on the control rod drive (CRD) injection flow is also discussed.     

Application of a reduced order model to BWR corewide stability analysis

The determination of system stability parameters from power readings is a problem usually solved by time series techniques such as autoregressive modeling. These techniques are capable of determining the system stability, but ignore the physics of the process and focus on the determination of a nth order linear model. A nonlinear reduced order system is used in conjunction with estimation techniques to present a different approach for stability determination. The simulation of the reduced order model shows the importance of the feedback reactivity imposed by the thermal-hydraulics; the dominant contribution to this feedback is provided by the void reactivity, being a function of power, burnup, power distribution, and in general of the operating conditions of the system. The feedback reactivity is estimated from power measurements and used in conjunction with a reduced order model to determine the system stability properties in terms of the decay ratio.     

Time dependence of linear stability parameters of a BWR

In this work we present several tools to study the time dependence of the linear stability parameters of a BWR using neutron noise analysis. Particularly, we have studied the variation in time of the fundamental frequency of a signal using the short-time Fourier transform and we have compared this method with the calculation of a time dependent Power Spectral Density (PSD) function. The temporal variation of the decay ratio is analysed using a method based on an autoregressive model to fit the different blocks of the signal. The performance of the tools presented is compared analysing analytic signals and a real signal of Forsmark 1&2 Stability Benchmark.     

Assessment of the Prony's method for BWR stability analysis

It is known that Boiling Water Reactors are susceptible to present power oscillations in regions of high power and low coolant flow, in the power-flow operational map. It is possible to fall in one of such instability regions during reactor startup, since both power and coolant flow are being increased but not proportionally. One other possibility for falling into those areas is the occurrence of a trip of recirculation pumps. Stability monitoring in such cases can be difficult, because the amount or quality of power signal data required for calculation of the stability key parameters may not be enough to provide reliable results in an adequate time range. In this work, the Prony's Method is presented as one complementary alternative to determine the degree of stability of a BWR, through time series data. This analysis method can provide information about decay ratio and oscillation frequency from power signals obtained during transient events. However, so far not many applications in Boiling Water Reactors operation have been reported and supported to establish the scope of using such analysis for actual transient events. This work presents first a comparison of decay ratio and frequency oscillation results obtained by Prony's method and those results obtained by the participants of the Forsmark 1 & 2 Boiling Water Reactor Stability Benchmark using diverse techniques. Then, a comparison of decay ratio and frequency oscillation results is performed for four real BWR transient event data, using Prony's method and two other techniques based on an autoregressive modeling. The four different transient signals correspond to BWR conditions from quasi-steady to power oscillations. Power signals from such transients present a challenge for stability analysis, either because of the low number of data points or need of much iteration, and thus reducing their capability for real time analysis. The results show that Prony's method can be a complementary reliable tool in determining BWR's stability degree.     

Gap conductance analysis for BWR rods relocation model

Pellet relocation is estimated as a function of irradiation history.Fuel fracture at BOL causes an ‘initial relocation’, whereby the fragments are displaced towards the canning. If columnar grain growth does not take place, the pellet remains cracked into pie shaped pieces without circumferential fractures; pellet diameter is stably increased by a term independent on linear power.If columnar grain growth occurs, cracks are filled in the restructured zone and, during a shut-down, circumferential cracks appear. The internal void volume is partially filled during up-ramps so that the relocation is an inverse function of linear power. Cracks healing, causing a void volume to be transferred inwards, produces a ‘time dependent relocation’, related to reactor cycling; as exposure proceeds, the pellet approaches a limiting diameter.The semiempirical model was derived from in-pile temperature measurements at low burn-up and BWR Garigliano rods PIE at high burn-up.A proper choice of literature data was also considered in comparing the parameters with experimental points.     

Parameter estimation during a transient – application to BWR stability

The estimation of system parameters is of obvious practical interest. During transient operation, these parameters are expected to change, whereby the system is rendered time-varying and classical signal processing techniques are not applicable. A novel methodology is proposed here, which combines wavelet multi-resolution analysis and selective wavelet coefficient removal with classical signal processing techniques in order to provide short-term estimates of the system parameters of interest. The use of highly overlapping time-windows further monitors the gradual changes in system parameter values. The potential of the proposed methodology is demonstrated with numerical experiments for the problem of stability evaluation of boiling water reactors during a transient.     

A macroscopic cross-section model for BWR pin-by-pin core analysis

A macroscopic cross-section model used in boiling water reactor (BWR) pin-by-pin core analysis is studied. In the pin-by-pin core calculation method, pin-cell averaged cross sections are calculated for many combinations of core state and depletion history variables and are tabulated prior to core calculations. Variations of cross sections in a core simulator are caused by two different phenomena (i.e. instantaneous and history effects). We treat them through the core state variables and the exposure-averaged core state variables, respectively. Furthermore, the cross-term effect among the core state and the depletion history variables is considered. In order to confirm the calculation accuracy and discuss the treatment of the cross-term effect, the k-infinity and the pin-by-pin fission rate distributions in a single fuel assembly geometry are compared. Some cross-term effects could be negligible since the impacts of them are sufficiently small. However, the cross-term effects among the control rod history (or the void history) and other variables have large impacts; thus, the consideration of them is crucial. The present macroscopic cross-section model, which considers such dominant cross-term effects, well reproduces the reference results and can be a candidate in practical applications for BWR pin-by-pin core analysis on the normal operations.     

Application of TRACE/PARCS to BWR stability analysis

The work described here is the validation of TRACE/PARCS for Boiling Water Reactor stability analysis. A stability methodology was previously developed, verified, and validated using data from the OECD Ringhals stability benchmark. The work performed here describes the application of TRACE/PARCS to all the stability test points from cycle 14 of the Ringhals benchmark. The benchmark points from cycle 14 were performed using a half-core symmetric, 325 channel TRACE model. Several parametric studies are performed on test point 10 of cycle 14. Two temporal difference methods, Semi-Implicit method (SI) and Stability Enhanced Two Step (SETS) method are applied to three different mesh sizes in heated channels with series of time step sizes. The results show that the SI method has a smaller numerical damping than the SETS method. When applying the SI method with adjusted mesh and Courant time step sizes (the largest time step size under the Courant limit), the numerical damping is minimized, and the predicted Decay Ratio (DR) agrees well with the reference values which were obtained from the measured noise signal. The SI method with adjusted mesh and Courant time step size is then applied to all test points of cycle 14 with three types of initiating perturbations, control rod (CR), pressure perturbation, and noise simulation (NS). There is good agreement between the decay ratios and frequencies predicted by TRACE/PARCS and those from the plant measurements. Sensitivities were also performed to investigate the impact on the decay ratio and natural frequency of the heat conductivity of the gap between fuel and clad, as well as the impact of the pressure loss coefficient of spacers.     

Spatial and model-order based reactor signal analysis methodology for BWR core stability evaluation

A new methodology for the boiling water reactor core stability evaluation from measured noise signals has been recently developed and adopted at the Paul Scherrer Institut (PSI). This methodology consists in a general reactor noise analysis where as much as possible information recorded during the tests is investigated prior to determining core representative stability parameters, i.e. the decay ratio (DR) and the resonance frequency, along with an associated estimate of the uncertainty range. A central part in this approach is that the evaluation of the core stability parameters is performed not only for a few but for ALL recorded neutron flux signals, allowing thereby the assessment of signal-related uncertainties. In addition, for each signal, three different model-order optimization methods are systematically employed to take into account the sensitivity upon the model-order.     

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.     

Kuosheng BWR/6 stability analysis with LAPUR5 code

Unstable power/flow oscillation of a nuclear power reactor core is one of the main reasons that cause minor core damage. Stability analysis to determine system’s decay ratio needs to be performed at each core reload design to prevent core instability events from happening. Making use of LAPUR5 and SIMULATE-3 codes, we have established a methodology to conduct such analysis. Comparisons made with vendor’s STAIF results indicated close agreements, within acceptable ±0.2 in decay ratios, for Kuosheng NPP Unit2 Cycle 17 reloads design. Sensitivity studies have shown that density reactivity coefficient, delayed-neutron fractions (β) and decay constants (λ), total core flow, and core power axial shape are the most important parameters that might affect the accuracy of decay ratios. We have also found that core conditions at EOC result in larger decay ratios than those at BOC.     

Bayesian inference of BWR model parameters by Markov chain Monte Carlo

In this paper, the Markov chain Monte Carlo approach to Bayesian inference is applied for estimating the parameters of a reduced-order model of the dynamics of a boiling water reactor system. A Bayesian updating strategy is devised to progressively refine the estimates, as newly measured data become available. Finally, the technique is used for detecting parameter changes during the system lifetime, e.g. due to component degradation.     

BWR core-wide stability analysis with respect to dominant feedback mechanisms and characteristic poles

BWR core-wide stability is studied from the viewpoint of linear dynamic stability treated via poles of a closed-loop transfer function. The quantitative study is performed using a BWR noise model describing neutronic and thermal-hydraulic core dynamics. Transfer functions of neutron power to reactivity and core inlet flow are derived in explicit forms and their poles are evaluated both numerically and analytically. It is shown that the characteristic poles may be classed into three groups relating to neutronic process, fuel heat transfer and core void dynamics. In particular, the poles for the void dynamics take complex values and hence give rise to core-wide damped oscillation of neutron power. Furthermore, the study of characteristic poles serves for the stability analysis of the Ringhals-1 benchmark test data. It is shown and clarified that two stability indexes, decay ratio and resonance frequency, have clear dependence on reactor power and core inlet flow.     

Applicability of the Diffusion and Simplified P3 Theories for Pin-by-Pin Geometry of BWR

The pin-by-pin fine-mesh core calculation method is considered as a candidate next-generation core calculation method for BWR. In this study, the diffusion and simplified P3 (SP3) theories are applied to the BWR pin-by-pin fine-mesh calculation. The performances of the diffusion and SP3 theories for cell-homogeneous pin-by-pin fine-mesh calculation for BWR are evaluated through comparison with a cell-heterogeneous detailed transport calculation by the method of characteristics (MOC). Two-dimensional, 2 × 2 multi-assemblies geometry is used to compare the prediction accuracies of the diffusion and SP3 theories. The 2 × 2 multi-assemblies geometry consists of 9 × 9 UO₂ fuel assemblies that have two different enrichment splittings. To minimize the cell-homogenization error, the SPH method is applied for the pin-by-pin fine-mesh calculation. The SPH method is a technique that reproduces a result of heterogeneous calculation using that of homogeneous calculation. The calculation results indicated that the diffusion theory shows a discrepancy larger than that of the SP3 theory on the pin-wise fission rate distribution. In contrast to the diffusion theory, the SP3 theory shows a much better accuracy on the pin-wise fission rate distribution. The computation time using the SP3 theory is about 1.5 times longer than that using the diffusion theory. The BWR core analysis consists of various calculations, e.g., the cross section interpolation, neutron flux calculation, thermal hydraulic calculation, and burn-up calculation. The function of the calculation time for the neutron flux calculation is usually less than half in the typical BWR core analysis. Therefore, the difference in the calculation time between the diffusion and SP3 theories would have no significant impact on the calculation time of the BWR core analysis. For these reasons, the SP3 theory is more suitable than the diffusion theory and is expected to have sufficient accuracy for the 2 × 2 multi-assemblies geometry used in this study, which simulates a typical situation of the actual BWR core.     

Detecting long-range correlation with detrended fluctuation analysis: Application to BWR stability

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.     

Stability analysis of BWR nuclear-coupled thermal-hydraulics using a simple model

A simple mathematical model is developed to describe the dynamics of the nuclear-coupled thermal-hydraulics in a boiling water reactor (BWR) core. The model, which incorporates the essential features of neutron kinetics and single-phase and two-phase thermal-hydraulics, leads to a simple dynamical system comprised of a set of nonlinear ordinary differential equations (ODEs). The stability boundary is determined and plotted in the inlet-subcooling-number (enthalpy)/external-reactivity operating parameter plane. The eigenvalues of the Jacobian matrix of the dynamical system also are calculated at various steady-states (fixed points); the results are consistent with those of the direct stability analysis and indicate that a Hopf bifurcation occurs as the stability boundary in the operating parameter plane is crossed. Numerical simulations of the time-dependent, nonlinear ODEs are carried out for selected points in the operating parameter plane to obtain the actual damped and growing oscillations in the neutron number density, the channel inlet flow velocity, and the other phase variables. These indicate that the Hopf bifurcation is subcritical, hence, density wave oscillations with growing amplitude could result from a finite perturbation of the system even when it is being operated in the parameter region thought to be safe, i.e. where the steady-state is stable. Finally, the power-flow map, frequently used by reactor operators during start-up and shut-down operation of a BWR, is mapped to the inlet-subcooling-number/neutron-density (operating-parameter/phase-variable) plane, and then related to the stability boundaries for different fixed inlet velocities corresponding to selected points on the flow-control line. Also, the stability boundaries for different fixed inlet subcooling numbers corresponding to those selected points, are plotted in the neutron-density/inlet-velocity phase variable plane and then the points on the flow-control line are related to their respective stability boundaries in this plane. The relationship of the operating points on the flow-control line to their respective stability boundaries in these two planes provides insight into the instability observed in BWRs during low-flow/high-power operating conditions. It also shows that the normal operating point of a BWR is very stable in comparison with other possible operating points on the power-flow map.