Development and veriﬁcation of a methodology for neutron noise response to fuel assembly vibrations Annals of Nuclear Energy

Nuclear reactors are stochastic systems, in which several parameters ﬂuctuate even during steady-state conditions. In addition, structural components like fuel assemblies, vibrate due to strong hydraulic forces. This stochasticity is the cause of the neutron population ﬂuctuating behavior; a phenomenon referred as neutron noise. This paper systematically analyzes the capabilities of the Studsvik’s codes to model the fuel assembly vibration and to assess the neutron noise response. To this aim, the CASMO-5 code is uti- lized for generating cross sections by parametrizing the water-gap thickness of fuel segments. The fuel assembly vibration modelling is approximated by the dynamic modiﬁcation of the water-gap widths between adjacent assemblies using the transient code SIMULATE-3K. The cross sections dependency on the water-gap width is analyzed and successfully compared against Serpent-2 reference results. Last, the utilized codes capability is veriﬁed qualitatively or/and quantitatively through a series of cases, at both lattice and nodal level. (cid:1)


Introduction
Noise is a term, commonly regarded as an undesired disturbance that deteriorates the quality of an analyzed process. However, as in many fields of science and engineering, in nuclear reactor physics the systematic study of the noise has a substantial role since it could be used for monitoring and surveilling the reactor operation and more importantly for detecting possible reactor malfunctions and anomalies. Neutron noise corresponds to the stochastic fluctuation of the neutron population, inside the reactor core, as detected by in-core and ex-core neutron detectors. This stochastic behavior of neutron population arises from the continuous fluctuation of neutronic (e.g. cross sections, number of emitted neutrons per fission, etc.) and thermal-hydraulic (e.g. core coolant flow and temperature, boron concentration, etc.) parameters, as well as from the vibration of various mechanical and structural components of the reactor system (e.g. fuel assemblies, core barrel, control rods, etc.).
Nuclear utilities and research centers have developed over the years advanced signal analysis techniques, in both the time and the frequency domains, to study the neutron noise phenomenon and to identify the characteristic behavior, called also ''signature", of specific types of reactors. The identification of the vibrational characteristics of core internals and the estimation of thermal-hydraulic properties for both pressurized and boiling water reactors (PWRs/BWRs) have been already demonstrated in the past using raw plant signals from neutron detectors and process gauges, even without utilizing computational codes (Thie, 1979;Pazsit, 1998;Sweeney, 1985;Bastl and Bauernfeind, 1975). The surveillance of reactor operation using the systematic study of the neutron noise characteristics assisted repeatedly in the past on the identification of malfunction sources, preventing the normal and economically efficient operation of the reactors (J. Thie, Power reactor noise. American Nuclear Society, 1981). The improvements in computational resources have made possible the development of powerful simulation tools that have the capability to analyze neutron noise phenomena and to reproduce the measured characteristics of neutron noise. For instance, the CORE SIM code is the first dedicated neutron noise solver in the frequency domain, being extensively https://doi.org/10.1016/j.anucene.2020.107669 0306-4549/Ó 2020 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). used for modelling core internals vibrations and neutronic and thermal-hydraulic stochastic fluctuations (Demazière, 2011;Demazière et al., 2015;Mylonakis et al., 2020). In addition, the reactor dynamics time-domain deterministic codes SIMULATE-3K and DYN3D have been used for modelling the impact of inlet coolant temperature and flow fluctuations on the neutron noise behavior (Bermejo et al., 2017;Viebach et al., 2018;Rohde et al., 2018). Recently, the capabilities of the neutron diffusion solver PARCS have been extended to model stationary perturbations (Olmo-Juan et al., 2019). Furthermore, the neutron noise transport equation in the frequency domain has been proposed to be solved using probabilistic algorithms (Yamamoto, 2020;Rouchon et al., 2017).
Since two decades, many European PWRs of the KWU pre-Konvoi design have reported a trend with increasing neutron noise levels measured over the operational cycles (Seidl et al., 2015). This unexpected neutron noise increase does not pose any safety issue, however, it causes costly operational perturbations by forcing the utilities to operate at lower core power levels in order to avoid the activation of the power limitation system triggered by neutron flux peaks. One of the hypotheses for this behavior was assumed to be the more vibrational prone behavior of a specific fuel design, which started to be used in the affected reactors at the same time with the increasing neutron noise trend.
In recent versions of the nodal reactor dynamics code SIMULATE-3K, the so-called assembly vibration model has been implemented, allowing the modeling of the fuel assembly vibrational behavior and the assessment of the associated impact on neutron noise. This model simulates the time-dependent lateral movement of selected fuel assemblies by dynamically modifying the water gap widths between adjacent fuel assemblies. To this aim, an additional parameterization of the macroscopic twogroup cross section library needs to be performed by including a branch calculation for the water gap width of all the analyzed segments of the core during the 2D lattice calculations, in CASMO-5. The original developed model has been used for the first time in (Seidl et al., 2015) to study the neutron noise phenomenon in the German KWU pre-Konvoi PWRs. The study carried out in (Seidl et al., 2015) has indicated, among others, that a random perturbation of the fuel assembly results in a white noise in the neutron detectors, as expected. However, the original model was preliminary and did not ensure the conservation of the gap widths on either sides of the vibrating bundle. Following this exploratory study, the Paul Scherrer Institute (PSI) started to utilize the fuel assembly vibration model in SIMULATE-3K in order to estimate the code's capabilities on simulating relevant noise sources and identifying neutron noise characteristics by comparing against real plant data (Chionis, et al., 2017;. This work focuses on the systematic verification and qualification of the capabilities of the Studsvik Scandpower (SSP) CMS codes (i.e. CASMO-5 (Ferrer, 2015), CMS-LINK5 (Bahadir, 1999), SIMULATE-3 (Cronin et al., 1995), and SIMULATE-3K (Grandi, 2011) to model fuel assembly vibrational movements. Through the current paper, the fuel assembly vibration modelling limitations of the CMS codes have been identified. Consequently, further modelling improvements have been developed and implemented in the SIMULATE-3K code to enhance the functionality of the fuel assembly vibrational methodology. The modelling capabilities of the CMS are assessed through a comparative study against reference Monte-Carlo results at a lattice level. In addition, a series of application studies, performed at core level, assist on the deeper understanding of the fuel assembly vibration impact on neutron noise behavior. This paper is structured as follows. First, the utilized simulation codes are briefly introduced, in Section 2. The fuel vibration methodology, based on the preparation of the macroscopic two-group cross sections and the time-dependent modification of the water gap widths at a nodal level, is described in Section 3. The utilized methodology is verified at a lattice level in Section 4 by comparing the cross sections at various water gap widths generated with CASMO-5 against reference results from the probabilistic solver Serpent-2. Then a detailed qualification study to justify the proper modelling of the investigated neutron noise phenomenon at a nodal level is given in Section 5. Finally, the conclusions of this paper and proposal for further investigations are discussed in the last section, Section 6.

Codes and simulation methods
This section briefly describes the computational tools that have been used throughout this research. The lattice code CASMO-5, employing the delta gap model, has been used for the preparation of the macroscopic cross section library. The CASMO-5 cross sections, for various water gap width sizes, will be compared to those obtained by Serpent-2 in Section 4.2. The post-processing of CASMO-5 library is performed employing the CMS-LINK5 code. The steady-state and transient nodal calculations are carried out using the SIMULATE-3 and SIMULATE-3K codes, respectively.

CMSYS methodology
The CMSYS platform, developed at PSI within the STARS program, comprises reference steady-state core models, which are continuously developed and validated for all Swiss operating reactors and cycles. The CMSYS includes the state-of-the-art codes CASMO-5, SIMULATE-3 and SIMULATE-3K. The fuel assembly vibration can be modelled within the CMSYS platform by utilizing the so-called delta gap model in CASMO-5 and the fuel vibration model in SIMULATE-3K.

CASMO-5
CASMO-5 is a multi-group two-dimensional lattice and depletion simulator, based on the method of characteristics transport calculation, being extensively used in both research and industry for modelling light water reactor (LWR) fuel assemblies. CASMO-5 uses up to 586 neutron groups (in this work, the default group structure of 19 neutron groups was used) and 18 gamma energy groups cross section library, based on the ENDF/B-VII.1 data library. The two-group homogenized nuclear data (i.e. cross sections, discontinuity factors etc.) are generated for every 2D fuel segment 1 at various state points (e.g. moderator temperature, boron concentration, control rod positions, etc.). CASMO-5 typically prepares a defined case matrix of branch calculations for the nodal simulator SIMULATE-3. For special applications, related to the study of fuel assembly bowing or fuel assembly vibration, CASMO-5 offers the capability to extend the default case matrix by including a delta water gap branch calculation by activating the delta gap model. The lateral fuel assembly vibration is modelled by assuming a variation of the moderator thickness surrounding the fuel rods. To this aim, the delta gap model allows the modification of the water gap widths at any side of the fuel assembly by an increment amount d. Therefore, it is possible to generate two-group homogenized cross sections that depend on gap sizes and can be utilized, then, by the nodal simulators, i.e. SIMULATE-3 and SIMULATE-3K, to model static bowing and/or dynamic lateral oscillation of fuel assemblies. 1 A fuel bundle is discretized in axial zones, called segments, for taking into account the fuel composition (e.g. enrichment levels, gadolinium rods, etc.) and fuel design (e.g. part length rods, water rods/channel variation, etc.) characteristics.

CMS-LINK5
All data generated during the lattice depletion calculation with CASMO-5 (such as two-group macroscopic cross sections and discontinuity factors, kinetics data, pin power reconstruction data, etc.) are collected by the CMS-LINK5 code (Bahadir, 1999) and post-processed into a binary formatted nuclear data library for use in SIMULATE-3 and SIMULATE-3K.

SIMULATE-3
SIMULATE-3 is a three-dimensional full core nodal code that performs coupled neutronics and thermal-hydraulics simulations to estimate the nodal power and thermal-hydraulic parameter distributions of both PWR and BWR cores. Every fuel assembly is discretized in Z axial nodes of equal size, and every node can be further discretized in 2x2 sub-nodes, in order to increase the precision of the calculation. SIMULATE-3 solves the two-group three-dimensional diffusion equation for every node utilizing the interpolated cross section at the local conditions from CASMO-5. The spatial flux is represented as a fourth-order polynomial with a quadratic transverse leakage and the intra assembly exposure is described by a quadratic polynomial in two directions (Cronin et al., 1995). In addition, every fuel assembly is modelled separately in the thermal-hydraulic model, by solving the total mass, energy and momentum equations and by determining the void fraction by a drift flux model. SIMULATE-3 analyzes the steadystate behavior of a LWR during an operating cycle by depleting the core and performing the so-called core follow calculations. In addition, SIMULATE-3 generates restart files at specific exposure points and operating conditions of interest, from which a transient calculation can be initiated using the SIMULATE-3K code.

SIMULATE-3K
SIMULATE-3K is a full core transient nodal code including coupled neutronic and thermal-hydraulic capabilities (Grandi, 2011). The steady-state neutronics models are the same in SIMULATE-3 and SIMULATE-3K. The transient solution is based on the timedependent two-group diffusion equation, using six-group delayed neutron groups. A five-equation hydraulic model is utilized in SIMULATE-3K by solving two continuity equations, two energy equations and one mixture momentum equation. Among other capabilities, SIMULATE-3K mimics fuel assembly lateral oscillations by dynamically modifying the water gap width sizes between assemblies. The fuel vibration model in SIMULATE-3K requires homogenized cross sections generated by CASMO-5, using the delta gap model.

Serpent-2
Serpent-2 is a three-dimensional Monte-Carlo particle transport code, developed at the VTT Technical Research Center of Finland, which has been extensively used since 2009 in a large range of reactor physics applications (Leppänen et al., 2015). Serpent-2 estimates the kinetic parameters and cross sections based on the neutron flux distribution, evaluated by tracking neutron paths within an analyzed geometry. Serpent-2 is a continuous energy code providing more accurate neutron flux distribution, in contrast to deterministic lattice codes, like CASMO-5, using energy discretization techniques for estimating the parameters of interest. In this work, a typical PWR lattice is modelled in both CASMO-5 and Serpent-2. The results of Serpent-2 are used as a reference for verifying the homogenized two-group cross sections generated by CASMO-5 when the water gap width of the analyzed lattice is increased, as presented in Section 4.2.

Methodology for simulated neutron noise based on fuel assemblies' vibration
This section describes the modelling of fuel assembly vibration using the CMSYS platform which involves three main processes; first, the generation of homogenized two-group cross section in CASMO-5, using the delta gap model; second, the actual modelling of lateral assembly vibrations using the fuel vibration model in SIMULATE-3K. Last, the time-dependent treatment of the bundle's lateral movement is specified as a user input, based on an automatized process which is fully developed in the MATLAB environment within this work.

Delta gap model (lattice level)
CASMO-5 generates few group macroscopic cross sections (R x;G ) based on Eq. (1), following a homogenization and condensation procedure in order to integrate the fine energy cross sections over the analyzed volume (including both the fuel and moderator regions) and then condenses the fine energy group structure into two groups (Jo et al., 2018). In Eq. (1); x corresponds to a reaction rate, while g and G are the fine and few energy group indices, respectively, and / g is the flux in energy group g. In this work, emphasis is given to the absorption (R a;G ), capture (R c;G ), scattering (R s;G ) and v-weighted fission (vR f ;G ) cross sections, in the fast (G = 1) and thermal (G = 2) energy groups. v is the average number of emitted neutrons per fission.
In addition, CASMO-5 uses the fundamental buckling mode method for the critical spectrum calculations in order to account for neutron leakage effects when reflective boundary conditions are utilized. The diffusion coefficients are estimated based on Eq.
(2) using the in-scatter correction method for the estimation of the transport cross section (R trÁin;G ).
The transport cross section is evaluated based on Eq.
(3), where R t;G is the total cross section, R 1 s;g 0 !g is the P 1 scattering macroscopic cross section from group g 0 to g, and / 1 g 0 is the P 1 flux moment (Ferrer, 2015;Jo et al., 2018).
The capability of the assembly bow model for both PWR and BWR applications has been already introduced in CASMO-4 (Bahadir, 2010). The delta gap model allows the user to increase the water gap thickness by an increment amount d in any of the four surfaces (i.e. west, east, south and north) and consequently to simulate the lateral deformation/bow of a selected lattice. The delta gap model in CASMO-5 is the first important step for the modelling of fuel assembly vibration downstream in 3D full core simulations, and gives the possibility to modify the water gap thicknesses on any side of the analyzed assembly and to calculate the associated cross sections, discontinuity factors, and pin powers during a branch lattice calculation. To do so, both the increment amount d of the lattice water width and the lattice symmetry have to be defined in the CASMO-5 input file by the user. It is recalled that, typical LWRs bundles have square geometry, and therefore, CASMO-5 can perform lattice calculations not only in a full geometry but also in half, quadrant, or even octant symmetry. This significantly decreases both the computational memory and time. The lattice symmetry is of key importance for the nuclear data generation, since for a user-defined water width change d on a lattice side, the delta gap model in CASMO-5 automatically imposes the same water width increase on the symmetric side.
This modelling approach can be better understood with the following example (Fig. 1), in which the nuclear data are generated for a given square lattice using the CASMO-5 delta gap model. The water width at the east side is increased by an increment amount d (dg E = d), corresponding to a lattice lateral displacement towards the west direction. However, the input definition of the water width increase depends on the lattice symmetry, as specified by the user. Fig. 1a shows that, the requested lattice calculation (i.e. increase of the east side water width by d) is correctly performed only for a full lattice geometry. On the contrary, the same lattice calculation in a half, quarter or octant symmetry automatically imposes larger water width increase than the requested one (i.e. Fig. 1b-d). In a half or quarter lattice symmetry, the delta gap model imposes an additional water gap increase on the west side of the lattice (i.e. dg E =dg W =d; Fig. 1b, c), while, in an octant symmetry the water gap increase d is applied in all four sides of the lattice (i.e. Fig. 1d). This clearly shows that, the moderator content of the lattice in the half, quarter and octant symmetric calculations is larger compared to the user-defined one. Undoubtedly, the additional moderator content affects the homogenized twogroup cross sections, however, this is corrected during the nuclear data post-processing and nuclear library preparation steps using CMS-LINK5, as described in Section 4.1.
The generation of nuclear data library using the delta gap branches is a crucial step, since these neutronic data will be further used in downstream core-wide calculations for simulating fuel assembly vibrations. The modelling of fuel assembly vibration in SIMULATE-3K is based on the delta gap model of CASMO-5, as the modification of water gap thicknesses between neighboring assemblies is employed for mimicking the dynamic lateral assembly movement, as it is described in more details in the next subsection. In other words, this modelling approach assumes that the lateral movement of a fuel assembly corresponds to an increase of the water content within the analyzed lattice. In reality, however, during the lateral movement of a fuel assembly only the fuel pins are laterally affected from their initial positions without affecting the water content of an analyzed lattice. Nevertheless, the proposed approximation of modifying the water content between a vibrating bundle and its adjacent ones, is a clever and pragmatic approach since, in reality, the water content in the local area between the selected bundles is also modified (increases or decreases). Therefore, since SIMULATE-3K is a nodal solver, the only way to model this behavior is to modify the water content at a nodal level for the bundles involved in the process, instead of modifying locally the water content only in the area between the adjacent fuel assemblies. The validity of this modelling assumption is further assessed in Section 4 by comparing CASMO-5 against reference results based on Serpent-2. Last, it is important to state that, so far, SIMULATE-3K can simulate the fuel assembly vibration modelling if and only if the nuclear data are generated using the CASMO-5 delta gap model in an octant or quarter lattice calculation (i.e. Fig. 1c, d). Any other lattice modelling approach produces nuclear data which are not suitable for the fuel assembly vibration in SIMULATE-3K.

Fuel vibration model (nodal level)
One of the latest versions of the SIMULATE-3K code (version 2.06.00) offers the capability to simulate time-dependent lateral movement of one or more fuel assemblies in the x-or/and ydirection by automatically modifying the water gap width sizes of both the vibrating fuel assembly and its first neighbors (Bahadir, 2010). The so-called ''assembly vibration model" in SIMULATE-3K is based on the cross section library branch generated by CASMO-5 using the delta gap model, as described in the previous sub-section. The modelling process of the assembly vibration model in SIMULATE-3K is described with the help of the following simplified example.
A fuel assembly FA i , located between two neighboring assemblies FA iÀ1 and FA iþ1 , is modelled here to laterally vibrate in one direction (Fig. 2). Let's assume that, the central fuel assembly FA i moves towards the assembly FA iÀ1 at a time step t. Therefore, the distance between FA i and FA iÀ1 is decreased by an increment amount d, and consequently, the distance between the assemblies FA i and FA iþ1 will be increased by the same increment amount d, in order to preserve the geometry of the analyzed assemblies. SIMULATE-3K uses a fixed computational mesh, and therefore, the code mimics the fuel assembly movement by utilizing the homogenized cross sections that correspond to a change of the water gap width size by d=2 for each one of the eight sub-nodes (each FA is divided into 4 sub-nodes), involved in this process. In other words, the two right sub-nodes of the central assembly FA i , and the two left sub-nodes of FA iþ1 will use the corresponding cross sections based on an increased water gap width of d=2, whereas, the two left sub-nodes of FA i , and the two right sub-nodes of FA iÀ1 will use the corresponding cross sections based on a decreased water gap width of d=2 2 . This approach ensures the preservation of the fuel assemblies' geometry. The lateral vibration of the fuel is achieved by repeating the above process at every time step and by using the corresponding water gap widths as imposed by the user.
The fuel assembly vibration model allows the user to impose a vibration of specific amplitude and frequency based on the fuel design type. Therefore, all the fuel assemblies of the same fuel design type will undergo a perturbation of the same characteristics (i.e. amplitude and frequency). However, the systematic study of the fuel assembly vibration model in SIMULATE-3K should start by analyzing simplified scenarios, which allow a first assessment of the model's capabilities. Such simplified scenarios include the modelling of the lateral vibration of a single fuel assembly (independently from any design type) or even the vibration of a single node, following vibration characteristics, explicitly imposed by the user. Therefore, further code enhancements and modifications were performed, allowing the user to dynamically modify the water gap thicknesses of any selected node in the core. In addition to the code enhancements, an additional input file must be prepared by the user to explicitly define the water gap widths between all the fuel assemblies and at every time step. The preparation of such input file is a rather easy task for a simple scenario, like a dynamical lateral movement of a single node in only one direction (i.e. the x-direction). However, the complexity of the preparation of such input files significantly increases by modelling the simultaneous vibration of multiple fuel assemblies in two directions and with different vibration characteristics (i.e. amplitude of displacement, frequency of vibration, type of movement; random or following a sine function).

Modelling platform
In the above context and in the framework of the current research, an in-house script has been developed and prepared in 2 CASMO-5 cannot perform lattice calculation for a configuration with a negative water gap width for PWR fuel assemblies. However, SIMULATE-3K models the decrease of water gap width by an increment (or a decrease) amount d between two fuel assemblies (for mimicking the approaching of an assembly to a neighbor) by extrapolating below zero the corresponding cross section set. the MATLAB environment in order to automatize the preparation of such input files. Fig. 3 summarizes the capabilities of the developed MATLAB script. First, the user defines the basic simulation parameters, such as the time step, the duration of simulation, and the desired maximum amplitude of the fuel assembly lateral displacement (step 1). Then, the user selects the total number and the location of the vibrating fuel assemblies (step 2). In step 3, the direction of vibration (in x-and/or in y-direction) of the fuel assemblies is selected by the user. Afterwards, the type of vibration (i.e. random or a sine function at a specific nominal frequency) is given (step 4). Finally, the user specifies if all the selected fuel assemblies vibrate identically (synchronized) or randomly between each other (step 5). This development extends the flexibility and capabilities of the fuel vibration model in SIMULATE-3K and allows the user to impose explicitly any kind of lateral fuel assembly vibration and to bypass the predefined and the limited simulation options.

Modelling assumption
In reality, the fuel pins of a vibrating fuel assembly are laterally displaced over their reference positions. A precise modelling of such a mechanism using neutronic solvers would require the dynamic modification of the computational mesh, resulting into a huge computational cost. In order to overcome this limitation, the CASMO-5 delta gap model uses a simplified approximation in which the water gap thickness of the lattices' sides is modified by an increment amount d to mimic the lattice lateral displacement, as introduced in the previous section. It is recalled that, SIMULATE-3K can read only nuclear data generated with the delta gap model in a quarter or octant lattice symmetry to simulate the fuel assembly vibration.
The current section studies the CASMO-5 delta gap modelling simplification and assesses the impact on the generated nuclear data. To this aim, the fuel pins of a lattice are assumed to be displaced from their initial position in the x-direction. This behavior can be modelled in CASMO-5 using the fuel displacement model (Rhodes, 2015). Using this option, it is possible to statically displace a fuel pin within its own pin cell, and more importantly, without increasing the water content of the lattice as the delta gap model does. In other words, the fuel displacement model corresponds to a realistic modelling of the lattice lateral displacement, and therefore, it is compared against the performance of the delta gap model. To do so, the homogenized two-group cross sections are generated using either the CASMO-5 fuel displacement or the delta gap models for a simplified lattice with 8x8 fuel pins layout using reflective boundary conditions (Fig. 4). The impact of the water gap thickness increase by d = 0:1cm on the homogenized two-group cross sections using the default fuel pins displacement model and the delta gap approach and its variations is studied with the help of five cases (Fig. 4): Case a -unperturbed case: An unperturbed lattice in which all the fuel pins are positioned in their reference locations is modelled. Neither the fuel pin displacement nor the delta gap model is used in the unperturbed case (Fig. 4a). This calculation allows to estimate the two-group homogenized cross sections under unperturbed conditions. Case b: represents the realistic approach in CASMO-5 for modelling the lateral displacement of a lattice. All fuel pins are displaced to the west direction by d=0:1cm, using the fuel pin displacement model in CASMO-5. Due to the reflective boundary conditions, this scenario corresponds to the movement of the assembly towards its west neighbor (Fig. 4b). This calculation allows the estimation of the cross sections change as a result of a realistic modelling of a fuel lattice displacement. However, this approach cannot generate nuclear data used for the fuel assembly vibration modelling in SIMULATE-3K. Case c: the east side water gap is increased by d=0:1cm, using the delta gap model, in full assembly geometry calculation in CASMO-5 (Fig. 4c). The cross sections are generated over the homogenized region which includes both the original geometry of the studied lattice and an additional moderator layer of a 0:1cm width in its east side. The water gap increase is not imposed to any other face of the lattice, since the lattice calculation is performed in full geometry. However, nuclear data generated using this approach cannot be read by SIMULATE-3K for the fuel assembly vibration. Case d: the east side water gap is increased by d=0:1cm, using the delta gap model, in quarter assembly symmetry calculation in CASMO-5. Automatically, CASMO-5 imposes the same moderator layer increase on the symmetric face of the lattice (i.e. west side) due to the quarter symmetry (Fig. 4d). Cross sections Fig. 1. Impact on water gap width sizes on the four sides of a lattice, based on the assembly symmetry (a: full, b: half, c: quarter, d: octant), when the water gap thickness of the east side is requested to increase by d in CASMO-5 using the delta gap model. generated using this variation of the delta gap model are compatible with downstream 3D full core calculation of fuel assembly vibration in SIMULATE-3K. Case e: the east side water gap is increased by d=0:1cm, using the delta gap model, in octant assembly symmetry calculation in CASMO-5. In this case, CASMO-5 will automatically impose a water gap increase by an incremental amount d in all four faces of the lattice due to the octant symmetry (Fig. 4e). This delta gap model variation is the second available option for generating cross section used for simulating fuel assembly vibration in SIMULATE-3K.
The macroscopic absorption (R a;G ) and nu-fission (vR f ;G ) cross sections and the diffusion coefficients (D G ) in fast and thermal energy groups are computed for every case and the results are summarized in Table 1. First, the results are compared with respect to the reference case a, in order to assess the impact of the different modelling approaches on the macroscopic cross sections. Clearly, every modelling approach (case b, c, d, or e) results to slightly different cross sections compared to the unperturbed case. The displacement of fuel pins, in case b, results in a relative change of cross sections from 0:13% up to 0:68%, whereas, the increase of the water gap thickness, in cases c, d and e, modifies the cross sections up to 1:70%, compared to the reference case a. The discrepancy between case b and cases d and e is expected, as the nuclear data generated in the latter cases are based on larger moderator content compared to the former case. It should be noted that, the thermal diffusion coefficient is the parameter that is affected the most from the lattice modification, in all the analyzed cases. In addition, the cross section results, based on the delta gap model (i.e. cases c, d, and e), are compared with respect to case b (fuel pin displacement model), representing the most realistic approach of fuel assembly displacement (Table 1). Case c (i.e. delta gap model in full geometry) indicate slightly higher values for almost all cross sections, up to 0:29% compared to those of fuel pin displacement model, whereas, case e (i.e. delta gap model in octant assembly symmetry calculation) shows smaller values for almost all cross sections, up to 1:07%. The quarter symmetric lattice calculation (case d) shows the lowest discrepancy, from the fuel displacement model, compared to cases b and e. Once again, the deviation of cases d and e compared to the realistic modelling case b is attributed to the fact that, the CASMO-5 delta gap model in quarter and octant symmetry assumes a larger moderator content compared to the fuel displacement model. This inconsistency is, though, corrected during the nuclear data postprocessing at the CMS-LINK5 level, in which the cross sections differences between the unperturbed lattice and the branch delta gap model calculation in quarter or octant symmetry are divided by two (Bahadir, 2010). As a result, the nuclear data library used for downstream 3D simulation of the fuel assembly vibration contain the correctly treated cross sections. Therefore, it is concluded that the delta gap model generates similar cross section results with the fuel pin displacement model in CASMO-5, even if the modelling approach of the former does not entirely correspond to the reality. Consequently, it can be stated that, the delta gap model can be properly used in downstream core-wide calculations for mimicking fuel assembly vibrations with SIMULATE-3K.

Benchmarking against Serpent-2
SIMULATE-3K mimics the dynamic lateral movement of a fuel assembly by modifying the water gap width between the vibrating fuel assembly and its neighboring assemblies, at every time step. The time-dependent modification of the water gap width corresponds to a time-dependent variation of the cross sections of all the nodes, involved in this process. This section examines how the key homogenized two-group cross sections are affected by the increase of the water gap width of an assembly. To this aim, a simplified lattice is used. The utilized lattice has a 14 Â 14 pins layout consisting of 180 fuel pins with 2:2% U 235 initial enrichment and 16 guide tubes, as presented in Fig. 5. A set of nine calculations are performed with CASMO-5 (using the modelling approach d from Fig. 4. i.e. in quarter assembly), in which the water gap thickness of the analyzed lattice is increased gradually from 0cm up to 0:15cm. The key macroscopic two-group cross sections (i.e. absorption, capture, m-fission, scattering) and the diffusion coefficients in  two-groups, and the multiplication factor k inf , are evaluated for every lattice calculation. In addition, the same lattice is modelled with Serpent-2 and the same set of nine calculations is performed, in which the water gap thickness is gradually increased, as it is done in the CASMO-5 calculations. It should be noted that, all the calculations have been performed at beginning of life conditions and at room temperature conditions (300K).
In order to ensure a consistent comparison between CASMO-5 and Serpent-2, the following procedure was followed. The homogenized two-group cross sections were generated in CASMO-5 without considering any thermal expansion of the fuel pins. In addition, the fundamental mode for the critical buckling correction is omitted in CASMO-5, in order to agree with the default solution from Serpent-2. Moreover, the transport cross section and diffusion coefficient are estimated in Serpent-2 using the cumulative migration method, shown to produce consistent results with the in-scatter method utilized by CASMO-5 (Liu et al., 2016). Last, the same nuclear data library (ENDF/B-VII.1) is used in both codes. Serpent-2 results' uncertainties are reduced by increasing the total number of neutrons (i.e. 10 3 cycles of 10 5 neutrons with 50 inactive cycles). Fig. 6 presents the macroscopic cross sections comparison between CASMO-5 and Serpent-2 for various water gap thicknesses. As can be seen, over all, the agreement between the two codes is satisfactory (relative difference less than 1:5%) for all the analyzed parameters, except for the diffusion coefficient in the thermal group (D 2 ), which exhibits a discrepancy of about 9:5%. In addition, the relative difference in terms of k inf between the two codes is in the acceptable range of 175-215pcm, showing that both CASMO-5 and Serpent-2 predict relevant results and similar trends as the water gap thickness is increased.
The tendencies of the macroscopic cross sections with respect to the water gap increase can be explained as follows. An increase of the water gap in the lattice configuration consequently leads to higher moderator to fuel ratio. Since the moderator content is increased, then the overall absorption, capture and fission rates (which are mainly driven by the fuel properties) will be reduced within the homogenized lattice. Therefore, in Fig. 6, it is observed that the absorption, capture and m-fission cross sections in both energy groups have a decreasing trend as the water gap thickness is increased. In addition, the fast diffusion coefficient (D 1 ) exhibits an increasing trend, since neutrons of higher energy (fast group) will diffuse in larger area by increasing the water gap width. However, as the water gap width is increased the neutrons will have more space to interact with hydrogen atoms (i.e. larger scattering cross section, as depicted in Fig. 6), and consequently, the thermalized neutrons will be absorbed easier in the moderator leading to smaller diffusion lengths in the thermal group (i.e. decrease of the thermal diffusion coefficient, D 2 , as depicted in Fig. 6). Finally, it can be observed that the reactivity of the lattice increases as the water gap width increases too (i.e. increase of the moderator to fuel ratio). Further increase of the water gap width will make the system to be over-moderated, and therefore, the k inf will be expected to start decreasing when the optimal moderation configuration is exceeded, which is not the case here.
In summary, this scoping analysis shows that, both the tendency and the magnitude of change of the homogenized twogroup cross sections are quantitatively and qualitatively very similar between CASMO-5 and Serpent-2, when the water gap width thickness increases. Therefore, it can be concluded that, the cross section library generated by CASMO-5 using the delta gap model is qualified to be used with high confidence by SIMULATE-3K for modelling fuel assembly lateral vibrations. This verification and qualification study is of paramount importance before analyzing full-core transient calculations.

Application studies at nodal level
This section focuses on the qualitative verification of the fuel assembly vibration model, used in SIMULATE-3K being the first commercial code with such capabilities. The fuel assembly vibration model has been developed by SSP in order to study how the lateral oscillations of PWR assemblies are affecting the neutron noise behavior of the core. This model has been first applied on PWR in (Seidl et al., 2015), then preliminary investigations on its capabilities has been demonstrated in (Chionis, et al., 2017;. A series of transient scenarios are presented in the following sub-sections. The cross sections are generated using the delta gap model in quarter lattice symmetry in CASMO-5. The transient scenarios in SIMULATE-3K comprise the random lateral vibrations of a single central node, a single fuel assembly, or a cluster of fuel assemblies. All transient simulations have been performed with the SIMULATE-3K code, with a duration of 35s and a time step of 0:01s, in order to minimize the statistical error of the results in the time and frequency domains. Finally, it should be noted that, all the simulations have been fully converged during the initialization stage of the steady-state solution to ensure the elimination of potential numerical noise. Table 1 Comparison of two-group homogenized cross sections for five different configurations of a simplified assembly (Fig. 4) This section illustrates the impact of different vibration patterns on the neutron noise behavior and the propagation of such perturbations across the core, both axially and radially. Neutron noise is expressed as the magnitude of the neutron flux fluctuations, which is estimated here in terms of the coefficient of variation (CV), defined by Eq. (4): where, r / i;j;z G and / i;j;z G is the standard deviation and the mean value, respectively, of neutron flux in the energy group G in a node located at the core position i; j; z. The time-dependent fast and thermal fluxes are estimated at every node of a typical PWR with a radial size 15x15 and 40 axial nodes of the active zone. Every node is a square lattice with a dimension of 21:56cm and a height of 8:95cm. A single row of radial and axial reflector surrounds the active core in the simulated model, as presented in Fig. 7.

Single node lateral vibration in one direction
The performance of the fuel assembly vibration model in SIMULATE-3K is first assessed using a simplified scenario. The 20 th node (starting from the core-bottom) of the central fuel assembly at location H8 is modelled to randomly vibrate only in the x-direction, with a maximum amplitude of 0:11cm which Fig. 6. Cross sections dependency on fuel assembly water gap thickness (left axis: circle and square blue marks correspond to CASMO-5 and Serpent-2 results, respectively; right axis: red asterisks provide the relative difference between CASMO-5 and Serpent-2). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) corresponds to the half of a water gap distance between two adjacent fuel bundles in typical PWRs (the location of the vibrating node is illustrated with red color in Fig. 7). To this aim, the water gap widths between the central node (n 20 H8 ) and its neighbors (n 20

H7
and n 20 H9 ) are dynamically modified, at every time step, following the modelling approach, described in section 3.2. For instance, if it is assumed that at time step t, the water gap width between n 20 H8 and n 20 H7 is decreased by d=0:04cm, consequently, the water gap width between n 20 H8 and n 20 H9 is increased byd = 0:04cm. This lateral movement of the central node in the x-direction is translated to a modification of the homogenized cross sections of the central node and its two neighbors. Therefore, the homogenized cross sections of the nodes n 20 H8 and n 20 H7 will be both modified, at time step t, based on a water gap width variation of -d=2=-0:02cm. In parallel, the homogenized cross sections of the nodes n 20 H8 and n 20 H9 will be changed based on a water gap width variation of +d=2=0:02cm, at time step t. As a result, the modifications of the cross sections for the central node n 20 H8 will cancel out, at time step t, and therefore, the lateral movement of n 20 H8 is expected to have an impact only on the neighboring nodes n 20 H7 and n 20 H9 . This dynamic behavior of the localized perturbation of the central node n 20 H8 is illustrated in Fig. 8, in terms of neutron noise amplitude for the fast (left plot) and thermal (right plot) neutron fluxes. The top plots show the neutron noise axial distribution, in which the perturbation location can be easily identified. The bottom plots represent the neutron noise amplitude in a 2D radial representation, at the middle of the core. It is observed clearly that, both fast and thermal neutron noise exhibit the highest values at the nodes n 20 H7 and n 20 H9 , located right next to the perturbation source. In addition, as expected, the neutron noise shows a fully symmetric shape. Hence, this simulation scenario indicates clearly that SIMULATE-3K can properly model the lateral movement of a single node.
In addition to the neutron noise amplitude, the neutron noise phase is equally important for the qualitative assessment of the neutron noise modelling correctness. The central node lateral vibration, in the x-direction, is expected to produce an out-of-phase response between the two core halves. In order to examine this pattern, the neutron noise phase 3 in both energy groups is calculated for each node in the x-direction; i.e. the 15 nodes in the row H at the axial level 20. Fig. 9 shows that, the H7 and H9 nodes, which are the adjacent nodes to the vibrating node H8, exhibit a clear phase difference of 180 , as expected. In addition, the two core halves show a clear out-of-phase behavior too, due to the lateral displacement of the central node. It is to note that, since the same pattern is observed in all the simulation cases of the current chapter, the corresponding noise phase plots are omitted.

Single fuel assembly lateral vibration in one and two directions
In this section, the central fuel assembly located at position H8 is modeled to vibrate randomly, in two scenarios, i.e. first scenario corresponds to x-direction vibration, while the second one is associated to vibration in xy-direction. In both cases, the maximum amplitude of displacement is 0:11cm and all the axial nodes are displaced simultaneously by the same amount, at every time step (Fig. 10).
The fast and thermal components of the neutron noise for the two scenarios are presented in Fig. 11 and Fig. 12, respectively. The vibrating assembly perturbation results in a maximum neutron noise level at the neighboring assemblies, as described in section 5.1. In addition, it is clearly observed that, the neutron noise has a symmetric shape for both fast and thermal components, in both scenarios, as expected. Furthermore, the impact of the direction in which the fuel assembly vibrates on the noise shape can be analyzed by comparing Fig. 11 and Fig. 12. The lateral vibration in one direction (x) results in highest noise in the two neighboring assemblies in the direction of vibration (Fig. 11), whereas, the vibration of the central fuel assembly in two directions (xy-) affects equally the neutron noise amplitude of all four neighboring assemblies (Fig. 12). Moreover, the neutron noise shows an increase of its level axially, leading to highest amplitudes at the core top level. This trend is observed in both scenarios and it is related to the decrease of the coolant density (and the corresponding more negative moderator temperature coefficient) at the core-top due to the increase of coolant temperature, leading to higher noise amplitudes, as it has been already mentioned in (Chionis, et al., 2017).

Vibration of cluster of fuel assemblies
This section studies how the neutron noise phenomenology depends on the number of vibrating fuel assemblies (i.e. collective movement of the fuel structure). To this aim, several simulation scenarios are performed, in which clusters of different sizes of centrally located fuel assemblies are modelled to vibrate in the xdirection or in the xy-directions. The cluster size varies from 1x1 (i.e. vibration of only the central fuel assembly, see Section 5.2) up to 11 Â 11 (i.e. vibration of 121 fuel assemblies, arranged in 11 Â 11 array). It should be noted that, all the assemblies located within the analyzed cluster are modelled to vibrate in a synchronized manner, with a maximum amplitude of displacement of 0:11cm, for all simulation scenarios. Fig. 13 represents the evolution of the fast (left plot) and thermal (right plot) maximum neutron noise amplitudes with respect to the increase of the size of the vibrating cluster. The curves with blue circles correspond to simulations, based on the lateral vibration of the cluster in the x-direction, whereas, those with the red squares are based on lateral vibration of the cluster in the xydirection. In both plots, the maximum neutron noise level increases, almost, linearly with the increase of the number of vibrating fuel assemblies. In addition, while for the fast group, the direction of vibration does not affect strongly the maximum neutron noise levels (no clear explanation is given), for the thermal group a systematic trend is observed, in which the vibration in two directions results to higher noise levels.
The fast and thermal components of the neutron noise levels are presented, in the following figures, for various perturbation   scenarios. Fig. 14 and Fig. 15 represent the scenarios in which a 3 Â 3 central cluster vibrates in x-and xy-directions, respectively, whereas, Fig. 16 and Fig. 17 present the scenarios in which a central cluster of 11 Â 11 vibrates in the x-and xy-directions, respectively.
As can be seen, the neutron noise behavior, for the four scenarios, is clearly similar to those obtained by modeling simple scenarios (Sections 5.1 and 5.2). In other words, the center of the vibrating cluster exhibits negligible noise, since the relative movement of central assemblies with respect to the entire cluster is zero, while the highest neutron noise levels are observed at both the peripheral locations of the vibrating cluster and its adjacent neighboring bundles. This behavior is to be expected because, on the one hand, the overall modification of the homogenized cross sections will be practically zero for a given central assembly at a time step t due to the synchronized lateral movement of the cen-  tral cluster, as already discussed in Section 3.2 based on Fig. 2 (i.e. there is no modification of the water gap widths for the central located bundles). On the other hand, the synchronized vibration of a central cluster affects only the water gap widths of the peripheral located assemblies of the cluster and their adjacent neighbors. Consequently, the cross sections only of those bundles will be influenced from the cluster vibration, explaining the 2D noise distribution in Fig. 14 to Fig. 17. Vibration of large clusters can explain the large heterogeneities on the neutron noise levels across the core that have been reported by the industry in some real cases (high noise levels in the periphery and lower values at the center core (Bermejo et al., 2017;), as shown in Fig. 14. Amplitude of the fast (left plots) and thermal (right plots) 2D noise distribution at the core-middle axial level induced by the lateral vibration of a 3 Â 3 cluster of fuel assemblies in the x-direction. Fig. 15. Amplitude of the fast (left plots) and thermal (right plots) 2D noise distribution at the core-middle axial level induced by the lateral vibration of a 3 Â 3 cluster of fuel assemblies in the x-y directions. Fig. 16. Amplitude of the fast (left plots) and thermal (right plots) 2D noise distribution at the core-middle axial level induced by the lateral vibration of a 11 Â 11 cluster of fuel assemblies in the x-direction. Fig. 16 and Fig. 17. Finally, the neutron noise has a symmetric spectral phenomenology in both energy groups, as it would be expected.

Conclusions
Neutron noise is a phenomenon that corresponds to the stochastic fluctuation of the neutron population and consequently to the stochastic fluctuation of the reactor power, even at steadystate conditions. The analysis of neutron noise phenomenon contributes in deeper understanding of the fundamental mechanisms that drive the inherent stochastic behavior of a nuclear reactor.
Among other mechanisms, the lateral movement of fuel assemblies is believed to have a significant contribution on the neutron noise phenomenology. Recently, the capabilities of the transient nodal code SIMULATE-3K have been extended, by introducing the so-called fuel assembly vibration model in order to simulate the impact of lateral vibrating fuel assemblies on the neutron noise behavior, which is the main goal of the current research. In this context, a systematic approach has been used in order to study and verify the modelling capabilities, using CASMO-5 and SIMULATE-3K, to simulate the fuel assembly vibrations. To this aim, the preparation of the cross section library on a lattice level is first achieved, using the delta gap model in CASMO-5, which generates the cross sections at various water gap thicknesses, for all the 2D fuel segments of the core. The impact of varying the water gap width on the homogenized two-group macroscopic cross sections is studied and successfully compared against the Monte-Carlo Serpent-2 code, using a reference 2D lattice. The generated cross section library is further utilized by the transient nodal code SIMULATE-3K in order to mimic the fuel assemblies vibration by dynamically modifying the water gap widths between neighboring assemblies. The analysis of various scenarios, i.e. single node vibration; single assembly vibration; and vibration of clusters of fuel assemblies, illustrates the capability of the CMSYS codes to properly model the lateral movement of fuel assemblies.
Further verification analyses and validation studies are planned to be performed in the future. Among others, a comparative study will be performed between SIMULATE-3K code and the already validated code CORE SIM, which is a dedicated neutron noise solver in the frequency domain. In addition, the PSI neutron noise modelling methodology will be applied on real core loading patterns (for a Swiss PWR) in order to estimate the impact of the number of fuel bundles, prone to lateral vibration, on the neutron noise level increase and results will be compared to measured data. Last, the measured neutron noise phenomenology, based on the neutron detector responses, will be attempted to be reproduced and explained using the simulated noise results.