REMARKS TO A NOVEL NONLINEAR BWR STABILITY ANALYSIS APPROACH (RAM-ROM METHODOLOGY)

Transcription

1 Advanes in Nulear Fuel Management IV (ANFM 2009) Hilton Head Island, South Carolina, USA, April 12-15, 2009, on CD-ROM, Amerian Nulear Soiety, LaGrange Park, IL (2009) REMARKS TO A NOVEL NONLINEAR BWR STABILITY ANALYSIS APPROACH (RAM-ROM METHODOLOGY) Carsten Lange, Dieter Hennig, Antonio Hurtado Tehnishe Universität Dresden, Institute of Power Engineering, Chair of Hydrogen and Nulear Energy, Dresden, Germany Carsten.lange@tu-dresden.de Keywords: nonlinear BWR stability analysis, bifuration analysis, system ode, redued order model, stable and unstable limit yles ABSTRACT A new approah for nonlinear stability analysis is presented. In the framework of this approah, integrated BWR (system) odes and redued order models (ROM s) are used as omplementary tools to examine the stability harateristi of fixed points and periodi solutions of the nonlinear differential equations desribing the stability behaviour of a BWR loop. Hene the methodology demonstrated in this paper is a novel one in a speifi sense: We analyse the highly nonlinear proesses of the BWR dynamis by appliation of system odes (in whih numerial diffusion properties are examined in advane so far as possible) and by appliation of some sophistiated methods of the nonlinear dynamis (bifuration analysis). We laim, the omplementary appliation of independent methodologies to examine the nonlinear stability behaviour an inrease the reliability of BWR stability analysis. This work is a ontinuation of the previous work at the Paul Sherrer Institute (PSI, Switzerland) and at the University of Illinois (USA) on this field. The urrent ROM was extended by adding the reirulation loop model. The neessity of onsideration of the effet of ooled boiling in an approximated manner was disussed. Furthermore, a new alulation methodology for the feedbak reativity was implemented. The modified ROM is oupled with the ode BIFDD whih performs semi-analytial bifuration analysis. In addition to the ROM extensions, a new approah for alulation of the ROM input data was developed. The new approah for nonlinear BWR stability analysis is presented for NPP Leibstadt. This investigation is arried out for an operational point for whih an out-of-phase power osillation has been observed during a stability test at the beginning of yle 7 (KKL yle 7 reord #4). The modified ROM and the new approah for the alulation of the ROM input data are qualified for BWR stability analysis in the framework of the approah (RAM-ROM methodology) demonstrated in this paper. 1. OJECTIVE In general, the dynamis of a BWR an be desribed by a system of oupled nonlinear partial differential equations. From the nonlinear dynamis point of view, it is well known that suh systems show, under speifi onditions, a very omplex temporal behaviour whih is refleted in the solution manifold of the orresponding equation system. Consequently, to understand the nonlinear stability behaviour of a BWR and to get an overview over types of instabilities, the solution manifold of the differential ANS 2009, Topial Meeting ANFM 2009, p. 1/20

2 equation systems must be examined. In partiular, with regard to the existene of operational points where stable and unstable power osillations are observed, stable or unstable fixed points and stable or unstable osillatory solutions are of paramount interest [1-4]. Notie, stable or unstable osillatory (periodial) solutions orrespond to stable or unstable limit yles. Saddle-node bifuration of yles (turning points or fold bifurations), period doubling and other nonlinear phenomena whih are important from the reator safety point of view are also of interest in this work[5-8]. It is worthy to mention that a linear stability analysis is not able to examine the existene of stable and unstable limit yles, e.g. if the unstable limit yle is born in a ritial Hopf bifuration point [7], stable fixed points and unstable limit yles will oexist in the linear stable region (lose to the bifuration point) in whih the asymptoti deay ratio is less than one ( DR < 1). This example show that oneivably unstable onditions (from the nonlinear point of view) are not reognized and the operational safety limits ould be violated. Hene, in order to reveal this kind of phenomena, nonlinear BWR stability analysis like bifuration analysis is neessary. 2. METHODOLOGY In the framework of the approah applied in this work, integrated BWR (system) odes (RAMONA5, Studsvik/Sandpower) and simplified BWR models (redued order model, ROM) are used as omplementary tools to examine the stability harateristi of fixed points and periodi solutions [1,4]. This new approah is alled RAM-ROM method, where RAM is a synonym for system ode. The intention is, firstly, to identify the stability properties of ertain operational points by performing ROM analysis and, seondly, to apply the system ode for a detailed stability investigation in the neighbourhood of these operational points. Some (onstitutive) essential harateristis of system odes and ROMs are summarized in Fig. 1. Fig. 1 Overview over the methodology applied for the nonlinear BWR stability analyses where RAMONA5 and ROM are used as omplementary tools. ANS 2009, Topial Meeting ANFM 2009, p. 2/20

3 System odes are omputer programs whih inlude suffiiently detailed physial models of all nulear power plant omponents whih are signifiant for a partiular transient analysis (inluding 3D ore model). Therefore, suh detailed BWR models should be able to represent the stability harateristis of a BWR lose to the physial reality. Nonlinear BWR stability analysis using large system odes is urrently ommon pratie in many laboratories [1]. A partiular requirement is the integration of a 3D neutron kineti model, whih permits the analysis of regional or higher mode stability behaviour [3]. A detailed investigation of the omplete solution manifold of the nonlinear equations desribing the BWR stability behaviour by applying system odes needs omprehensive parameter variation studies whih require large omputational effort. Hene system odes are inappropriate to reveal the omplete nonlinear stability harateristis (with other words: the whole solution manifold of the DE system) of a BWR. Furthermore, user of system odes must pay attention to the stability behaviour of the algorithms employed. In partiular, physial and numerial effets regarding power osillations and the behaviour of numerial damping of the algorithms should be known in detail. Numerial diffusion, for example, an orrupt the results of system odes signifiantly. Therefore, redued order analytial models ould be helpful to get a first overview over the stability landsape to be expeted. The objetive of the ROM development is to develop a model as simple as possible from the mathematial and numerial point of view while preserving the physis of the BWR stability behaviour [4]. The resulting equation system is haraterized by a minimum number of system equations whih is realized by the redution of the geometrial omplexity. One demand on our ROM is, beause the ROM -models should be as lose as possible to the -models used in RAMONA, that the solution manifold of the RAMONA model should be as lose as possible to the solution manifold of the ROM. E.g., both neutron kineti models (ROM and RAMONA) are based on the two neutron energy group diffusion equations. Both thermal-hydrauli two phase flow models are represented by models whih onsider the mehanial non-equilibrium (different veloities of the phases of the fluid). The main advantage of employing ROM s is the possible oupling with odes inluding methods of nonlinear dynamis like bifuration analysis. Bifuration analysis of a BWR system, for example, leads to an overview over types of instabilities in seleted parameter spaes. The existene of stable and unstable periodial solutions (orresponds to limit yles) an be examined reliably. Thus, the stability behaviour of global and regional power osillation states an be investigated in detail and an be interpreted in physial terms. In the sope of the present ROM analyses two independent tehniques are employed. These are the semi-analytial bifuration analysis with the bifuration ode BIFDD [7] and the numerial integration of the ROM differential equation system [1]. Bifuration analysis with BIFDD determines the stability properties of fixed points and periodial solutions (orrespond to limit yle) [7,8]. For independent onfirmation of these results, the ROM system will be solved diretly by numerial integration for seleted parameters [1,4,8]. ANS 2009, Topial Meeting ANFM 2009, p. 3/20

4 The proedure of the nonlinear BWR stability analysis applying the RAM-ROM method is depited in Fig. 2. Fig. 2 This figure depits the new approah for nonlinear BWR stability analyses using RAMONA5 and ROM as omplementary tools. The goal is to simulate the stability behavior of the power plant with the ROM as lose as possible to that one alulated by RAMONA5 in the neighborhood of a seleted operational point. Hene, at first, the referene OP has to be seleted for whih the nonlinear BWR stability analysis will be performed. Seondly, the new proedure for the ROM input alulation is applied. Thereby, all ROM input data are alulated from the speifi RAMONA5 model and its steady state solution orresponding to the referene point. We demand: the ROM should provide the orret steady state values in the referene operational point. Thereby the most essential values (for the BWR stability behaviour) are the mode feedbak reativity oeffiients, the ore inlet mass flow, the axial void profile and the hannel pressure drops over the reator vessel omponents along the losed flow path. E.g. the ooling number and the pressure loss oeffiients annot be alulated diretly from the RAMONA5 model (and its steady state output) ANS 2009, Topial Meeting ANFM 2009, p. 4/20

5 beause the models desribing the axial power profile and the pressure drop along the losed flow path are different in both odes. Hene a speial alulation proedure for the pressure loss oeffiients and the ore inlet ooling of the ROM is developed and applied. Finally, after the alulation of the ROM input parameters, nonlinear BWR analysis is performed by using ROM and RAMONA5 as omplementary tools. To summarize and roughly speaking, the ROM analysis provides an overview about the stability harateristis of fixed points and limit yles in seleted parameter spaes. Besides, the ourrene of further types of nonlinear phenomenon suh as saddle-node bifuration of yles an be revealed [8]. Hene, the system ode analysis an be performed with more pre-information (is made more reliable). 2.1 Loal bifuration analysis using BIFDD In the framework of the bifuration analysis with BIFDD, the so-alled Poinarè- Andronov-Hopf bifuration (PAH-B) theorem plays a dominant role [1,4,5,7,8]. This theorem guarantees the existene of stable and unstable periodi solutions of nonlinear differential equations if ertain onditions are satisfied. A mathematial desription is given in [1,7-9] and thus is not repeated here. In order to get information about the stability property of the periodi solution, the (linear) Floquet theory is applied (and several tehniques suh as Lindstedt-Poinarè asymptoti expansion, entre manifold redution, transformation into the Poinarè normal form presented in [7-10]) where the so-alled Floquet exponent (Floquet parameter) β appears [1,7-9] whih determine the stability of the periodi solution. If β < 0, the periodi solution is stable (superritial bifuration) while if β > 0, the periodi solution is unstable (ritial bifuration). Roughly speaking, the Floquet parameter an be interpreted to be a stability indiator for limit yles and is a result of a speial tehnique from nonlinear dynamis. Notie, the existene of a Hopf bifuration is the (mathematial) reason for the sudden appearane of periodi osillations (limit yles). These periodi solutions will be observed in the BWR reator dynamis as global (in-phase) or regional (e.g. out-of-phase or azimuthal mode) power osillations. The bifuration analysis is arried out with the bifuration ode BIFDD developed by Hassard [7].The user of BIFDD has to provide the input parameter vetor, a set of nonlinear ODEs, the orresponding Jaobian matrix and the initial guess for the phase spae variables. The bifuration analysis starts with seletion of the so alled iteration and bifuration parameter. Thereby the iteration parameter will be varied in the interval defined by the user. For eah iteration step BIFDD omputes the ritial value γ k, of the bifuration parameter, the amplitude ε of the osillation, the Floquet parameter β β2 and further ertain expansion parameters defined in [1,7-10]. As a result of the bifuration analysis using BIFDD, a set of fixed points where the Hopf onditions are fulfilled will be obtained in the two dimensional parameter spae, whih is spanned by the iteration and bifuration parameter (left diagram of Fig. 3 b)) [4]. This set of fixed points is alled linear stability boundary. In eah of these fixed points a periodial solution is born whose stability property is determined by the Floquet ANS 2009, Topial Meeting ANFM 2009, p. 5/20

6 exponent [1,4]. The right diagram of Fig. 3 b) shows the Floquet exponent β β2 for eah iteration step. Fig. 3 Stability boundary in the two dimensional parameter spae whih is spanned by the iteration and bifuration parameter and the orresponding bifuration harateristi. In Fig. 3b is shown a seleted stability map (inremental parameter, e.g. N, vs. ontrol or bifuration parameter, γ k, left hand side) and the orresponding bifuration harateristi ( β 2 vs. inremental parameter, right hand side). The bifuration harateristi predits unstable limit yles oexisting with stable fixed points (see Fig. 3a) and stable limit yles oexisting with unstable fixed points (see Fig. 3) in the environment of the stability line. Notie, an unstable limit yle born in a ritial Hopf bifuration separates a set of trajetories (in phase spae) whih spiral into the steady state solution (singular fixed point) from a set of trajetories whih spiral away ad infinitum of the phase spae (the state variables diverges in an osillatory manner; the system behaves unstable). 2.2 Numerial integration Semi-analytial bifuration analysis is only valid in the viinity of the ritial bifuration parameter (SB) in a small neighbourhood of the singular fixed point. In order ANS 2009, Topial Meeting ANFM 2009, p. 6/20

7 to get information of the stability behaviour beyond the loal bifuration findings numerial integration (in the time domain) of the set of the ODEs is neessary. Besides, the preditions of the semi-analytial bifuration analysis an be onfirmed independently [1,4,8]. 3. THE ROM The urrent BWR redued order model onsists of three oupled -models. These are a neutron kineti model, a fuel heat ondution model and a two-hannel thermal-hydrauli model (presented in [1, 2]). Fig. 4 depits a shemati sketh of the ROM. In this paper, only the most important ROM assumptions are presented. Fig. 4 Shemati sketh of the ROM 3.1 Neutron kineti model The neutron kinetis model is based on (an effetive) two energy groups (thermal and fast neutrons). The spatial mode expansion approah of the neutron flux in terms of lambda modes ( λ -modes) [12] is used (thereby it is assumed that the amplitude funtion is independent on the energy). Only the first two modes (fundamental and the first mode) and one effetive one group of delayed neutron preursors are onsidered. The ontribution of the delayed neutron preursors to the feedbak reativity is negleted. A new alulation methodology for the mode feedbak reativities is implemented (lose oupling to the system ode RAMONA: the mode feedbak reativities are alulated with the aid of the 3D power distributions alulated by the LAMBDA ode [12]; in turn, RAMONA provides the marosopi ross setions). Taking into aount these assumptions, four mode kineti equations ould be developed, oupled with the equations of the heat ondution and the thermal-hydrauli via the feedbak reativity terms (void and Doppler feedbak reativities). ANS 2009, Topial Meeting ANFM 2009, p. 7/20

8 3.2 Fuel rod heat ondutions The heat ondution model assumptions (this -model is ompletely adopted from Karve (1998) [2]): (1) Two axial regions, orresponding to the single and two-phase regions, are onsidered; (2) three distint radial regions, the fuel pellet, the gap and the lad are modeled in eah of the two axial regions; (3) azimuthal symmetry for heat ondution in the radial diretion is assumed; (4) heat ondution in the z- diretion is negleted; (5) time-dependent, spatially uniform volumetri heat generation is assumed. These assumptions result in a one-dimensional (radial) time dependent partial differential equation (PDE). By assuming a two-pieewise quadrati spatial approximation for the fuel rod temperature, the PDE an be redued to a system of ODEs by applying the variation priniple approah. A detailed derivation is presented in [1,2]. 3.3 Thermal-hydrauli model The thermal-hydrauli behavior of the BWR is represented by two heated hannels oupled by the neutron kinetis and by the reirulation loop. This model is based on the following assumptions: (1) the heated hannel, whih has a onstant flow ross setion, is divided into two axial regions, the single and the twophase region; (2) all thermal-hydrauli values are averaged over the flow ross setion; (3) the dynamial behavior of the two-phase region is presented by a drift flux model (DFM, Rizwan-uddin, 1981]) where mehanial non equilibrium (differene between the two phase veloities, and a radial non-uniform void distribution is onsidered) is assumed (the DFM represents the stability behavior of the two-phase more aurately than a homogeneous equilibrium model, in partiular for high void ontent); (4) the two phases are assumed to be in thermodynami equilibrium; (5) the system pressure is onsidered to be onstant; (6) the fluid in both axial regions and the downomer is assumed to be inompressible; (7) the following terms are negleted in the energy balane: the kineti energy, potential energy, pressure gradient, frition dissipation; (8) the PDEs (three-dimensional mass, momentum and energy balane equation) are onverted into the final ODEs by applying the weighted residual method in whih spatial approximations (spatially quadrati but time-dependent profiles) for the single phase enthalpy [2] and the two-phase quality are used (is equivalent to a oarse grained axial disretization). The thermal-hydrauli model is extended by a reirulation loop model based on the following assumptions: (9) the downomer (onstant flow ross setion) region is onsidered to be a single phase region; (10) all physial proesses whih lead to energy inrease and energy derease are negleted in the downomer; (11) the pump head due to the reirulation pumps is onsidered to be onstant ( Phead = onst); (12) the effet of the ooled boiling phenomenon on the BWR stability behaviour is assessed. (Profile-Fit model [Levy, 1967]) To summarize, the dynamial system of the redued BWR model onsists of 22 ODEs, four from the neutron kineti model, eight to desribe the fuel rod heat ondution (two equations for eah phase, in eah hannel) and ten that desribe the thermalhydrauli model (five for eah hannel) [1]. ANS 2009, Topial Meeting ANFM 2009, p. 8/20

9 4. NONLINEAR STABILITY ANALYSIS FOR NPP LEIBSTADT (KLL7_re4) In order to gather stability data for yle 7 of the NPP Leibstadt (KKL) a stability test was performed on September 6 th 1990 [11]. Comprehensive post-alulation stability analyses using system odes were onduted for KKL measurement reord #4 and #5 (KKL7_re4-OP and KKL7_re5-OP). The paper ontains a short presentation of results of the nonlinear stability analyses for KKL7_re4-OP [12] where inreasing regional power osillations ourred at approximately 60% power and 37% ore mass flow [12]. Fig. 5 and Fig. 6 show the time evolutions of LPRM signals of the measurement and the RAMONA5 alulation. Relative Amplitude NPP Leibstadt yle 7 reord 4 70 Measured signals of LPRM 9 and LPRM 31 of KKL7_re4 68 LPRM 9 LPRM NF = 0.58 Hz Time 700 s 800 Fig. 5 Time evolution of the LPRM signals, measured in KKL7_re4-OP. RAMONA5 analysis for NPP Leibstadt yle 7 reord 4 Time evolution of the LPRM signals (fourth level) Relative Amplitude Time s Time s 160 LPRM 9 LPRM 26 NF = Hz s Time Fig. 6 RAMONA5 result for the referene OP. The relative amplitudes of signals are shown for LPRM 9 and LPRM 26. Both LPRM signals have a phase shift of π. ANS 2009, Topial Meeting ANFM 2009, p. 9/20

10 The transient behaviour presented in Fig. 6 was initiated by introduing a 2 node sinusoidal ontrol rod movement resulting in a perturbation of the state variables of the BWR system. The signals of the LPRM 9 and 26 of the fourth level are loated in different ore half s (RAMONA predits a fixed symmetry line for the present ase []). As an be seen, an inreasing out of phase power osillation is ourring in the referene * 1 OP. RAMONA5 predits a natural frequeny of NF = 0.537s while the measurement * 1 gives NF = 0.58s. All RAMONA5 investigations for the referene OP and its lose neighbourhood have shown that the out of phase power osillation will not disharge into a stable limit yle. The existene of a stable limit yle annot be verified by the RAMONA5 and measurement results but it must not be exluded. For the ROM analyses, KKL7_re4-OP is defined to be the referene OP. All design parameters of the ROM are alulated from the speifi KKL-RAMONA5 model. The operating parameters are estimated from the steady state solution provided by RAMONA5 for KKL7_re4-OP where the new alulation proedure for the ROMinput was applied. The values of the heated hannel pressure drops simulated by the ROM are lose to the referene values provided by RAMONA5. In addition to that the axial void profiles alulated by RAMONA5 and ROM are in good agreement. Thereby the deviation of the total volumetri void fration is less than 1%. These steady state results are presented in []. The results of the numerial integration of the ROM equation system in the referene OP onfirm the transient behavior predited by RAMONA5. The time evolution of the fundamental mode n 0 () t, first azimuthal mode n 1 () t and the hannel inlet veloities v 1, inlet ( t ) and v 2, inlet ( t ) are shown in Fig. 7. n 0 -n 0,steady Time evolutions of the fundamental and first azimuthal mode and the hannel inlet veloities n 0 -n 0,steady n 1 v inlet, i /v 0, inlet n 1 NF = Hz v inlet,1 v inlet, s Time Fig. 7 Time evolutions of the fundamental n 0 () t and first azimuthal mode n 1 () t and the hannel inlet veloities v 1, inlet ( t ) and v 2, inlet ( t ). ANS 2009, Topial Meeting ANFM 2009, p. 10/20

11 It an be seen that the amplitudes of the fundamental mode osillation are deaying for the first 250 s and then inreasing while the first azimuthal mode osillation is inreasing ontinuously, after the perturbation (an in-phase osillation was triggered) was imposed on the system. This means, an inreasing out of phase power osillation is ourring in the referene OP. Thus, the predition of the RAMONA5 investigation in the referene OP ould be verified by the ROM. The osillation frequeny predited by the ROM is * 1 NF = 0.457s. The behaviour of the fundamental mode osillation (deaying for the first 250 s and then inreasing) an be explained by the solution of a linearized system where eah omponent of the solution depends on eah eigenvalue of the Jaobian matrix. This means, if there is at least one pair of omplex onjugated eigenvalues with a real part larger then zero, all omponents of the solution will diverge asymptotially in an osillatory manner. This was shown in detail in [1]. 4.1 Loal bifuration analysis with BIFDD The results of the semi-analytial bifuration analysis of the ROM equation system are presented in the N - DPext -operating plane ( N and DP ext are defined in Nomenlature at the end of this paper). Notie, a variation of DP ext (steady state external pressure drop) orresponds to a movement on the rod-line whih rosses the referene OP while the 3D-distributions will not be affeted. Thus the stability properties of operational points along a fixed rod-line (fixed ontrol rod onfiguration) and its lose neighborhood are analyzed. The stability boundary (SB) and the bifuration harateristi are shown in Fig. 8. The SB is defined as the set of fixed points for whih the Hopf onditions are fulfilled. Roughly speaking, this means, in eah of these fixed points a limit yle is born and exists either in the (linear) stable or (linear) unstable region. The stability harateristi of the limit yle is determined by the Floquet parameter β 2 [7-9]. In Fig. 9 is shown the SB transformed into the power flow map. N Stability boundary unstable region KKL referene-op: 1.0 N = DP ext = stable region DP ext stable region Bifuration harateristi β 2 >0 (ritial) β 2 <0 (superritial) β 2 Fig. 8 Stability boundary and the bifuration harateristi (nature of the Poinarè-Andronov-Hopf bifuration) for the referene OP. ANS 2009, Topial Meeting ANFM 2009, p. 11/20

12 80 Power Flow Map for KKL 70 Thermal Power unstable region stable region KKL7 reord4 OP: Thermal Power (59.5%) Core Flow (36.5%) SB (ritial PAH-B) SB (superritial PAH-B) 112% Rod Line exlusion region Core Flow Fig. 9 Stability boundary transformed into the power flow map. Unstable periodial solutions (unstable limit yle) lose to the KKL7_re4-OP are predited by the semi-analytial bifuration analysis (for 0.53 < N < 1.38, see Fig. 8). These solutions are loated in the linear stable region lose to the stability boundary. This means, in this region oexist stable fixed points and unstable limit yles (see Fig. 3a). Notie, the asymptoti deay ratio (linear stability indiator) is less than 1 ( DR < 1) in this region. A linear stability analysis is not apable to examine the stability properties of limit yles. 4.2 Numerial Integration: Loal Consideration For independent onfirmation of the results whih are predited by the bifuration analyses, numerial integration of the ROM equation system (in the time domain) has been arried out for seleted parameters. The ROM equations are solved in an operational point that is loated in the (linear) stable region (see Fig. 10) lose to the SB. In this region unstable limit yles are predited by the bifuration analysis N numerial integration: N = DP ext = Stability boundary referene OP unstable region stable region SB (ritial PAH-Bifuration) DP ext Fig. 10 SB and the point for whih the unstable limit yle will be verified by numerial integration. ANS 2009, Topial Meeting ANFM 2009, p. 12/20

13 0.01 Time evolution of the first azimuthal mode numerial integration: n "small" perturbation Flow = 36.0 % Power = 58.8 % δv inlet = 0.01 ("small") n "large" perturbation numerial integration: Flow = 36.0 % P ower = 58.8 % δ V inlet = ("large") s 200 Time Fig. 11 Numerial integration is arried out in an operational point in whih an unstable limit yle is predited by the bifuration analysis. The transient was initiated by imposing perturbations of the ore inlet mass flow with (small δ vinlet = 0.01, large δ vinlet = 0.025). In order to verify the existene of the unstable limit yle, perturbations of different amplitudes are imposed on the system. This means, aording to X () t = X 0 + δ X () t, the steady state solution X 0 is perturbed by different perturbation amplitudes δ Xt () and the transient behavior of the system state Xt () is alulated by numerial integration of the ROM equation system. If a suffiient small perturbation is imposed on the system, the state variables will return to the steady state solution. The terminus suffiient small perturbation means that the trajetory starts within the basin of attration of the fixed point. Roughly speaking, the perturbation amplitude is less than the repellor amplitude (see phase spae portrait depited in Fig. 3a). But if a suffiient large perturbation is imposed on the system, the state variables will diverge in an osillatory manner. The terminus suffiient large perturbation means that the perturbation amplitude is larger than the repellor amplitude. In this ase the trajetory will start out of the basin of attration of the fixed point. As shown in Fig. 11, the results of the numerial integration method onfirm loally the predition of the bifuration analysis. 4.3 Numerial Integration: Global Consideration The loal ROM analysis has shown that the bifuration analyses using BIFDD and the numerial integration method provide loally in the origin of the dynamial system (phase or state spae) in the viinity of the ritial parameter value γ k, (parameter spae; index k is ignored in the following disussion) onsistent results. The terminus loally in the origin of the dynamial system means that the lose neighbourhood of the steady state solution X 0 (the singular fixed point) is taken into aount in the phase spae. Further analyses in the referene OP and its neighbourhood, whereby numerial integration is arried out for a time period of 800 s revealed the existene of stable limit yles. The result of the time integration in the referene OP using the numerial integration ode is shown in Fig. 12. ANS 2009, Topial Meeting ANFM 2009, p. 13/20

14 n 0 -n 0,steady 0,00 Time evolutions of the fundamental and first azimuthal mode and the hannel inlet veloities 0,02 n 0 -n 0,steady -0,02 0,3 n 1 (1) n 1 0,0 v inlet, i /v 0, inlet -0,3 1,05 1,02 0,99 0,96 v inlet,1 v inlet,2 Fig. 12: s 800 Time Result of (a long time) numerial integration in the referene OP where a long time integration is arried out (referene OP of KKL7re4). The (ursory) onlusion is: The existene of a stable limit yle in the linear unstable region is inonsistent with the result of the bifuration analysis whih delivers ritial Hopf bifurations. Hene, unstable limit yles are predited in the linear stable region for this analysis ase. In order to understand this behaviour, more in depth onsiderations are neessary. The above analysis reveals that the system behaviour annot be examined only by loal onsiderations suh as semi-analytial bifuration analysis using BIFDD. The oexistene of a ritial bifuration point (where an unstable limit yle is born) and stable limit yles in the linear unstable region ould be an unique indiator for a possible existene of global bifuration. In ontrast to the Hopf bifuration, global bifurations involve large regions of the phase spae rather than just the neighbourhood of a singular fixed point [6]. Thus, in the sope of the present work, the post-bifuration state an only be determined through numerial integration of the ROM equations. For this purpose, the amplitudes of the limit yles vs. the ore inlet ooling will be determined by numerial integration. Thereby, all the other parameters are fixed. The results are plotted in Fig. 13 and Fig. 14. The diagram shown in these figures is also known as bifuration diagram. In partiular, the global behaviour in the lose neighbourhood of the stability boundary will be analysed. The ooling number N is varied between 0.6 and 0.9 and the stable limit yle amplitudes of the first azimuthal mode A( n 1) are determined. Fig. 13, Fig. 14 and Fig. 15 summarises results of this analysis. The results show, that limit yle amplitudes A( n 1) dereases with dereasing ore inlet ooling. Below the ritial value N, (the Hopf onditions are fulfilled at N, ) stable limit yles still exist ANS 2009, Topial Meeting ANFM 2009, p. 14/20

15 (see Fig. 14). This means, stable and unstable limit yles oexist in the linear stable region. The oexistene of stable and unstable limit yles for N < N, is verified by numerial integration for N = by imposing different perturbation amplitudes δ X () t on the system. A suffiient small perturbation leads to a stable behaviour. But when a suffiient large perturbation is imposed on the system, the state variables are attrated by the limit yle. This is presented in Fig Bifuration Diagram Amplitude of the limit yle Amplitude of the unstable limit yle (assessed) ritial value of N (bifuration point) for DP ext,ref A(n 1 ) N,ref saddle node bifuration of the yle (turning point) N Fig. 13 The results of the numerial integration are plotted as bifuration diagram, where N is the bifuration parameter and DPext = DPext, ref. A(n 1 ) Amplitute of the stable limit yle Amplitute of the unstable limit yle (assessed) saddle node bifuration of the yle (turning point) Bifuration Diagram linear stable region linear unstable region ritial value of N (bifuration point) Fig. 14 ( Zoom in of Fig. 13) Notie, the funtion of the amplitudes of the unstable limit yle is an assessment only, not a alulated one. N ANS 2009, Topial Meeting ANFM 2009, p. 15/20

16 0,02 0,01 Time evolution of the first azimuthal mode "small" perturbation (δv inlet = 0.02) n 1 0,00-0,01-0,02 0,05 numerial integration: N = DP ext,ref = n 1 0,00 "large" perturbation (δv inlet = 0.1) -0, s 600 Time Fig. 15 This figure shows the results of numerial integration for N = and DPext = DPext, ref, where a suffiient small δ vinlet = 0.02 and suffiient large δ vinlet = 0.1 perturbation is imposed on the system. When a small perturbation is imposed on the system, the state variables are returning to the steady state solution (origin of the dynamial system). When a large perturbation is imposed on the system, the state variables are attrated by the limit yle. The amplitudes of the unstable limit yle orrespond to the boundary whih separates the basin of attration of the singular fixed point and basin of attration of the stable limit yle. The analysis has shown that stable and unstable limit yles do not exist for ore inlet ooling less than N < Hene, there is a ritial ore inlet ooling N, t, where the two limit yles oalese and annihilate. Due to (numerial) onvergene problems of numerial integration, N, t annot be alulated exatly. The estimated value of N, t is N, t From this result, it an be onluded that in N, t there is a saddle-node bifuration of a yle. This bifuration type belongs to the lass of global bifurations and is also referred to as turning point or fold bifuration [6]. The behaviour of a dynamial system whih undergoes a saddle-node bifuration of a yle is summarized in Fig. 16 (there is depited the bifuration diagram and the orresponding radial phase portraits). In the following Fig. 16 is disussed more in detail. To this end, the ontrol parameter γ in Fig. 16 is assumed to be N ( γ = N ). ANS 2009, Topial Meeting ANFM 2009, p. 16/20

17 Fig. 16 This figure depits a bifuration diagram of a saddle-node bifuration of a yle and the orresponding radial phase portraits. 1. The bifuration at γ t is a saddle-node bifuration of a yle. In this point, a stable and an unstable periodial solution (limit yle) are born ( out the lear blue sky ). As depited in Fig. 16, the phase portrait is hanging signifiantly when passing γ t. 2. In the range γ t < γ < γ, two qualitatively different stable states oexist, namely the origin (fixed point solutions) and the stable limit yle. Both states are separated by an unstable limit yle (see phase portrait in Fig. 16). In other words, due to the saddle-node bifuration at γ = γ t, a stable and an unstable limit yle oexist with stable fixed points within the parameter range γ t < γ < γ. One onsequene is that the origin is stable to small perturbations, but not to large ones. In this sense the origin is loally stable, but not globally stable. 3. From the stability analysis point of view, a saddle-node bifuration is operational safety signifiant, if the amplitude of the stable limit yle is suffiient large. Supposing the system is in steady state (origin) below γ t, and the ontrol parameter γ is slowly inreased. As long as γ < γ the state remains at the origin. But at γ = γ the origin loses stability and thus the slightest nudge will ause the state to jump to the limit yle. In this ase, the state will start to osillate when reahing γ = γ and the osillations are growing as long as they will be attrated by the stable limit yle. With further inrease of γ, the state moves out along the limit yle solution. But if γ is now redued, the state remains on the stable limit yle osillation, even when γ is dereased below γ. The system will return to the origin when the ontrol parameter γ is redued below γ t. This harateristi (alled hysteresis) an be onsidered as a loss of reversibility as the ontrol parameter is varied. ANS 2009, Topial Meeting ANFM 2009, p. 17/20

18 4. The system behaviour near the origin is similar to the behaviour of a system whih undergoes a ritial Hopf bifuration at γ = γ. To summarize, loal ROM analyses for KKL7_re4 have shown that the bifuration analyses and the numerial integration method provide onsistent results only in a small neighbourhood of the Hopf bifuration point γ and in the viinity of the origin (loal onsisteny; Notie, it must be differentiated between the terminus a small neighbourhood in parameter spae and a small neighbourhood in the phase or state spae ; see Fig. 17). In order to study the global harater of the nonlinear system, numerial integration is neessary. For this purpose, numerial integration of the ROM equation system have been arried out, where N was varied in the range [0.62,...,0.9]. The analyses have shown that in the range N, t < N < 0.9 ( N, t = 0.63) stable limit yles exist, even though the bifuration analysis predits only unstable limit yles for N < N, ( N, is the ritial bifuration parameter, for whih the Hopf onditions are fulfilled). Hene, in the referene OP the state variables will also be attrated by the limit yle. In addition to that, for N [ N, t,..., N, ] with N, t < N, stable fixed point solution, unstable periodial solution and stable periodial solution oexist. Below N, only stable fixed point solution exist., t One would think that the preditions of BIFDD and the results of the numerial integration are inonsistent. But notie: BIFDD is based on loal methods. Thus the stability investigation using BIFDD is onentrated only in the origin of the system lose to the ritial parameter γ (see Fig. 17). Fig. 17 Summary of the main harateristis of a system, whih undergoes a saddle-node bifuration of yles. The results of BIFDD are only valid loally in the lose neighbourhood of the origin of the system lose to γ (region Γ ). The global harater of the nonlinear system an only be examined by numerial integration. ANS 2009, Topial Meeting ANFM 2009, p. 18/20

19 5. CONCLUSIONS The nonlinear analysis has shown that the stability behaviour of the referene OP and its lose neighbourhood an be simulated reliably by the new ROM. In this OP, the results of RAMONA5 and ROM are loally onsistent. Under stability related parameter variations the stability behaviour alulated by both ROM and RAM are onsistent. Hene, the new ROM and the new proedure for the alulation of the ROM input data are qualified for BWR stability analysis in the framework of the new approah (RAM- ROM methodology): The ROM analysis provides an overview about types of instability whih have to be expeted (in a seleted BWR power-flow map region). With the RAM (system ode), these phenomena are analyzed in detail (but with preinformation reeived by ROM analysis). Hene the reliability of the stability analysis is improved. A more in depth nonlinear analysis, where global aspets are taken into aount, revealed a system behavior whih an be explained by the existene of a saddle-node bifuration of yles. Methods and results of this work will be the basis for further nonlinear BWR stability analyses. An in-depth understanding of the BWR stability behavior should definitely ontribute to an effiient design of detetion and suppression systems. NOMENCLATURE The ooling number N represents the ore inlet ooling and appears as a boundary ondition in the single phase energy equation. The ooling number and the steady state external pressure drop are written as N ( h h ) ρ DP = = * * * * sat inlet ext DP * * ext * *2 hfg ρg ρ f v0, (1) * where h sat is the saturation enthalpy, * * * * h inlet inlet enthalpy, ρ = ρ f ρg liquid-vapor * * * * * density, hfg = hg hf liquid-vapor enthalpy, v 0 referene veloity and DP ext is the steady n state pressure drop over the vessel high. X() t with X() t is the state vetor of the dynamial system. An asterix on a variable or parameter indiates the original dimensional quantity and any quantity without an asterix is dimensionless. REFERENCES [1] A. Dokhane, BWR Stability and Bifuration Analysis using a Novel Redued Order Model and the System Code RAMONA, Dotoral Thesis, EPFL, Switzerland, [2] Karve, A.A.; Nulear-Coupled Thermal-hydrauli Stability Analysis of Boiling Water Reators ; Ph.D. Dissertation; Virginia University; USA; [3] D.Hennig, A Study of BWR Stability Behavior, Nulear Tehnology, Vol.126, ANS 2009, Topial Meeting ANFM 2009, p. 19/20

20 p.10-31, [4] C.Lange; D.Hennig; A.Hurtado; V.G.Llorens; G.Verdù; In depth analysis of the nonlinear stability behavior of BWR-Systems ; Proeedings of PHYSOR 08; Interlaken; September; Interlaken; Switerland; 2008 [5] J.Guggenheimer; P.Holmes; "Nonlinear Osillation, Dynamial Systems, and Bifuration in Vetor Fields"; Applied Mathematial Sienes 42; Springer Verlag; [6] S. H. Strogatz; Nonlinear dynamis and haos ; Addison-Wesley Publishing Company; [7] D. Hassard, D. Kazarinoff, and Y-H Wan, Theory and Appliations of Hopf Bifuration, Cambridge University Press, New York, [8] Rizwan-uddin, Turning points and - and superritial bifurations in a simple BWR model, to appear in Nulear Engineering and Design; [9] A. Neyfeh and B. Balahandran, Applied nonlinear dynamis: analytial, omputational, and experimental methods John Wiley & Sons, In., New York, [10] A. Neyfeh; Introdution to Perturbation Tehniques; John Wiley & Sons; [11] Blomstrand, J.; The KKL Core Stability Test, Conduted in September 1990 ; ABB-Report; BR91-245; [12] M. Miro; D. Ginestar; D. Hennig; G. Verdu; On the Regional Osillation Phenomenon in BWR s ; Progress in Nulear Energy; Vol. 36; No. 2; pp ; ANS 2009, Topial Meeting ANFM 2009, p. 20/20