## Abstract

Within emerging best-estimate-plus-uncertainty (BEPU) approaches, code output uncertainties can be inferred from the propagation of fundamental or microscopic uncertainties. This paper examines the propagation of fundamental nuclear data uncertainties though the entire analysis framework to predict macroscopic reactor physics phenomena, which can be measured in Canada Deuterium Uranium (CANDU) reactors. In this work, 151 perturbed multigroup cross sections libraries, each based on a set of perturbed microscopic nuclear data, were generated. Subsequently, these data were processed into few-group cross sections and used to generate full-core diffusion models in PARCS. The impact of these nuclear data perturbations leads to changes in core reactivity for a fixed set of fuel compositions of 4.5 mk. The impact of online fueling operations was simulated using a series of fueling rules, which attempted to mimic operator actions during CANDU operations such as studying the assembly powers and selecting fueling sites, which would minimize the deviation in power from some desirable reference condition or increasing or decreasing fueling frequency to manage reactivity. An important feature of this analysis was to perform long-transients (1–3 years) starting with each one of the 151 perturbed full core models. It was found that the operational feedback reduced the standard deviation in core reactivity by 99% from 0.0045 to 2.8 × 10^{−5}. Overall, the conclusions demonstrate that while microscopic nuclear data uncertainties may give rise to large macroscopic variability during simple propagation, when important macrolevel feedback are considered the variability is significantly reduced.

## 1 Introduction

A critical step in the licensing and safe operation of nuclear power plants (NPPs) such as the Canada Deuterium Uranium (CANDU) reactor is predicting the behavior of the plant under both normal operating conditions and possible accident scenarios. Since large-scale experiments, especially those which consider accident conditions, are typically impossible for NPPs most plant safety analysis is performed using computer codes, which synthesize experimentally collected data with physical models to simulate the conditions within the reactor core for a large number of postulated accidents. A key activity in the safety and licensing process is to determine the uncertainty in the models and methods employed as well as the uncertainties in the figure-of-merit used to assess reactor safety under differing scenarios. Such understanding allows the analysis to establish if the results of the simulations meet the acceptance criteria.

The calculation process for a typical safety analysis spans multiple scales of calculation. Microscopic data of nuclide properties, such as cross sections or fission yields, are used to produce a multigroup library, which contains nuclide properties discretized into many energy groups. A lattice-level code solves the neutron transport equation of a small sector (or unit cell) of the reactor and the results can then be used to homogenize the neutronic properties of reactor components such as fuel or reflectors in both space and energy. Finally, a core-level diffusion code can use the homogenized lattice properties to determine macroscopic core properties such as reactivity or power distribution [1]. Each step of the calculation process uses the outputs of the previous step as input, along with other problem-dependent information.

An ongoing area of international research in reactor physics is the quantification and minimization of uncertainties in simulations of NPPs. The Nuclear Energy Agency (NEA) of the Organization for Economic Cooperation and Development (OECD) is administering a benchmark study in Uncertainty Analysis in Modeling for light water reactors (UAM-LWR). Since 2006, participants in the benchmark study have worked to develop and validate techniques to propagate uncertainties in models of light water reactors from the cell physics stage of the calculation up to full-core coupled neutronics/thermal-hydraulics calculations. The benchmark study has defined the three main drivers of uncertainty in NPP simulations as input data uncertainty, modeling uncertainty and numerical uncertainty. Examples of input data uncertainty include uncertainty in nuclear data, material densities, temperatures, thermal hydraulic properties, and cell geometries. Modeling uncertainty considers the simplifications used to model complicated physical processes using computer codes and imperfect knowledge of boundary and initial conditions. Numerical errors result from the nodalization of reactor models and from the numerical methods used to solve the system of equations [2]. Despite the extensive research done on LWR uncertainty quantification throughout the various stages of the calculation, similar studies for CANDU reactors have been limited.

This work analyzes the impact of input data uncertainties, specifically nuclear data uncertainties, on full core calculations for a 480 channel CANDU reactor. Since each step of the reactor physics calculation process relies on inputs produced in the previous step, a clear path for uncertainty propagation exists. In this paper, the effect of nuclear data perturbations on core reactivity and core power distribution is examined. It should also be noted that the fuel depletion may also be affected by nuclear data, and hence the bundle-specific depletion will be altered in each perturbed core. A direct propagation of uncertainties, however, must also consider the important feedback and operator actions, which can significantly influence the resultant core simulations. For example, if one were to perturb nuclear cross sections, then propagate them through to few-group libraries and simply run a new diffusion calculation, the resultant core reactivity, peaking factors, etc. would not be representative of a realistic core state because the important optional feedback must be considered. For example, if a given perturbed set of cross sections cause an increase in power in some region of the core then (i) prior to this core state there would have been more depletion in this region of the core thereby limiting the extent to which the power can deviate and (ii) through monitoring the operator would sense more power in this region of the core and would fuel elsewhere, thereby reducing the power imbalance. A further example might examine a given set of perturbed nuclear data, which gives rise to a core state reactivity that is either too high or low for the regulating systems reactivity control, and hence such a state would not occur under normal operation (e.g., in the case of too much excess reactivity the operator would reduce fueling rates such that the reactivity control mechanisms are within normal allowances).

Quantification of uncertainty in simulated NPPs is a key step in the best estimate plus uncertainty (BEPU) or best estimate and uncertainty (BEAU) methodologies for accident analysis. The goal of BEPU analysis is to replace traditional safety analysis, which relied on a set of highly conservative assumptions about plant parameters and initial conditions, with a realistic case, which more closely reflects the actual behavior of the plant. BEAU analysis of CANDU large break loss of coolant accidents (LBLOCAs) has shown that the conservative approach to safety analysis can over significantly overstate the consequences of an accident [3]. Additionally, setting all variables to the most conservative possible values yields a plant configuration which is implausible [1]. Accurate uncertainty propagation studies can be used to further improve the BEPU type studies. Past work [4] has shown that BEPU analysis, which involves sampling of both macroscopic and microscopic properties may not accurately reflect system behavior due to complex covariances and nonlinear relationships between sampled variables. In this study, only microscopic properties are sampled, with all macroscopic properties being derived from the perturbations to the fundamental data, bypassing this problem. Rather observable macroscopic deviations (such as the daily variation in CANDU channel powers) can be measured, and hence these macroscopic measurements can be compared to the variability observed in simulations using propagated microscopic uncertainties.

## 2 Lattice-Level Models

As a result of the UAM-LWR benchmark study, many tools have been developed for uncertainty propagation in reactor physics calculations. One such tool is the SAMPLER module of the SCALE code package produced by Oak Ridge National Laboratory (ORNL). The SAMPLER tool uses a stochastic method for propagating uncertainties, repeating a lattice physics calculation many times with different sets of nuclear data or material/geometry parameters. A set of 1000 premade perturbed nuclear data libraries is provided with the program. In this study, 150 perturbed nuclear data libraries were used, as this will allow a two sided 95/95 Wilks' confidence interval to be generated for the main figure of merit, bundle power [5]. Perturbations to geometry or material properties can be specified by the user and are sampled from user defined distributions [6]. The stochastic method employed by SAMPLER allows for the creation of self-consistent sets of few group constants for each sampled lattice calculation. Additionally, stochastic methods are able to easily track the effect of perturbations on depletion calculations and do not require the user assume that the effect of all perturbations is linear, unlike first-order perturbation methods [7]. In this work, SCALE 6.2.2 has been used.

Use of the SAMPLER module requires that the CANDU lattice cell be modeled in a tool from the SCALE package. Of the available tools, only NEWT meets the requirements of this work (the newer and faster lattice level solver in SCALE, named POLARIS, is currently incompatible with the CANDU lattice). NEWT is capable of performing analysis on circular cluster geometries like those found in CANDU and produced homogenized group constants and performs self-shielding using the CENTRM module. NEWT was then used to model several different lattice configurations including burn-up and homogenize the properties for further use in diffusion calculations.

However, the NEWT module is only capable of performing two-dimensional calculations, meaning it is unable to model the three-dimensional supercell required to characterize CANDU reactivity devices. Given this limitation, the incremental cross sections of the adjuster rods, liquid zone controllers (LZCs), and all guide tubes were determined using the Serpent 2.1.30 Monte Carlo code and the additional uncertainty in the incremental cross sections is not considered. To judge the validity of this simplification, the CANDU reactivity devices were also modeled using the KENO-VI Monte Carlo code from the SCALE package, coupled with the TSUNAMI first-order perturbation tool. Although unable to provide the uncertainty in the incremental cross sections of the devices, the TSUNAMI module of SCALE was used to determine the uncertainty in the lattice-level reactivity worth of the devices. Table 1 shows a summary of the lattice level reactivity worth for each device, calculated using Serpent 2 and KENO-IV. The devices considered are the stainless steel adjuster rods and three possible configurations of the LZCs [8], each in their completely empty (E) and completely full (F) state. Also shown is the lattice-level uncertainty in reactivity worth due to nuclear data uncertainties. The uncertainty in lattice-level reactivity worth was consistently found to be about 1 mk, about the same as in a comparison of two reference (i.e., no reactivity device) cells. The reactivity worth of the adjuster rod and LZCs was found to be in good agreement with values calculated using the DRAGON code presented in the literature [8] and excellent agreement was found between the two Monte Carlo codes. Based on these results, the uncertainty determined with SAMPLER on a standard lattice cell is used in this work to also represent the uncertainty in cells that contain reactivity devices.

Device | Serpent reactivity worth (mk) | Keno reactivity worth (mk) | Uncertainty from nuclear data (mk) |
---|---|---|---|

Adjuster | −61.8 ± 0.2 | −61.1 ± 0.1 | ± 1 |

LZC Type 1 E | −22.9 ± 0.2 | −22.9 ± 0.1 | ± 1 |

LZC Type 1 F | −119.4 ± 0.1 | −119.9 ± 0.1 | ± 1 |

LZC Type 2 E | −9.5 ± 0.2 | −10.4 ± 0.1 | ± 1 |

LZC Type 2 F | −122.4 ± 0.1 | −123.3 ± 0.1 | ± 1 |

LZC Type 3 E | −37.2 ± 0.2 | −38.0 ± 0.1 | ± 1 |

LZC Type 3 F | −116.2 ± 0.1 | −116.1 ± 0.1 | ± 1 |

Device | Serpent reactivity worth (mk) | Keno reactivity worth (mk) | Uncertainty from nuclear data (mk) |
---|---|---|---|

Adjuster | −61.8 ± 0.2 | −61.1 ± 0.1 | ± 1 |

LZC Type 1 E | −22.9 ± 0.2 | −22.9 ± 0.1 | ± 1 |

LZC Type 1 F | −119.4 ± 0.1 | −119.9 ± 0.1 | ± 1 |

LZC Type 2 E | −9.5 ± 0.2 | −10.4 ± 0.1 | ± 1 |

LZC Type 2 F | −122.4 ± 0.1 | −123.3 ± 0.1 | ± 1 |

LZC Type 3 E | −37.2 ± 0.2 | −38.0 ± 0.1 | ± 1 |

LZC Type 3 F | −116.2 ± 0.1 | −116.1 ± 0.1 | ± 1 |

A lattice-level uncertainty propagation study was performed using SAMPLER on the 37 element CANDU fuel bundle over its irradiation cycle. The lattice-level uncertainty in *k*_{∞} is shown in Fig. 1. The irradiation cycle shown in Fig. 1 extends beyond the expected useful life of the fuel (which typically ends at around 7.5 MWd/kg). Uncertainty in *k*_{∞} is highest for fresh fuel and tends to decrease with burnup. The decrease in uncertainty was found to be nonlinear, with a much sharper drop in uncertainty at the beginning of the fuel's irradiation cycle and an almost negligible change in the 6–7.5 MWd/kg range. The results observed using the SAMPLER tool show good agreement with previous uncertainty analysis [9], which utilized the DRAGON code and its own set of uncertainty sampling functions.

## 3 Full Core Models

The 480 channel CANDU core was modeled with 12 bundles per channel in the diffusion code PARCS v32m21co.^{2} A hypothetical starting point for the reactor core was generated using a pseudo-random burnup distribution. The CANDU core was modeled with adjuster rods, adjuster rod guide tubes, liquid zone controllers, and the guide tubes for the mechanical control absorbers and shut off rods. The poison injection nozzles for shutdown system 2 (SDS2) were not included in the model. An iterative matlab code was developed to simulate the reactor regulating system's (RRSs) control of the fourteen liquid zone controllers by reading in PARCS power levels and comparing them against a reference power for the core zone. A target channel power reference map consistent with operating stations was adopted for both LZC modeling and for subsequent fueling operations. The diffusion model is shown in Fig. 2 and the position of the liquid zone controllers in the core is shown in Fig. 3. Fuel cells are shown in light gray and reflector cells are shown in black (surrounding the outside of the core). The three liquid zone type indexes in the figure indicate the three distinct liquid zone configurations used within the model [8] and separate SERPENT simulations were used for each configuration.

## 4 Results

### 4.1 Initial-Stage Core Results.

The core was initially modeled in PARCS using one reference set (best estimate values) of few group cross sections and 150 perturbed cross section sets and no operational feedback (fueling rate and/or location) were considered. For each cross section set, the RRS response was simulated using liquid zone controllers only (the adjuster rods were assumed to be always inserted and mechanical control absorbers were always withdrawn as per normal operating conditions in a CANDU). While the LZC operation can compensate for some of the bulk and spatial reactivity imbalances observed in the perturbed cores, their limited worth cannot compensate for some of the large cross section perturbation cases. A histogram of the core *k*_{eff} values obtained using the perturbed cross sections is shown in Fig. 4 at this stage of the analysis. The mean value of *k*_{eff} was found to be 1.0012, with a standard deviation of 0.0045. The relative standard deviation in channel powers is shown in Fig. 5. The average relative standard deviation in channel power was found to be 3.89%. Large peaks in the channel power relative standard deviation were absorbed in the cells surrounding the liquid zone controllers in the bottom half of the core.

Significant variation was also found in the maximum channel and bundle powers calculated using each dataset. The maximum channel power for each nuclear dataset is shown in the scatter plot in Fig. 6. A total of 45 cases exceed the maximum allowable channel power of 7.3 MW. The average maximum channel power was found to be 7.18 MW, with a standard deviation of 0.17 MW. The maximum bundle power for each nuclear dataset is shown in Fig. 7. A total of 42 cases exceed the maximum allowable bundle power, with 25 of these cases being different from the cases with channel powers above the maximum allowable limit (i.e., a total of 70 cases exceed either the maximum channel power limit, maximum bundle power limit, or both). The average maximum bundle power was found to be 0.897 MW, with a standard deviation of 0.033 MW. Given that these variations are atypical of any operating CANDU, these intermediate stage results emphasize the importance of the fueling operations in managing not just excess reactivity but also core power shape.

### 4.2 Canada Deuterium Uranium Operational Feedback—Intermediate Stage Results.

Canada Deuterium Uranium reactor fueling operations provide unique feedback control wherein the operators' actions to balance core bulk and spatial reactivity as well as channel powers, avoids large spatial power tilts or excess core reactivity (e.g., avoids abnormally high/low LZC levels, avoids LZC to LZC imbalances, and also acts to limit fueling in regions where channel powers are higher than the target). As such although each of the 150 perturbed cores has a random sample of potential nuclear data, the effect of such changes on core evolution including operator actions must be considered prior to assessing the impact of these perturbations of quantities like channel power and zone levels. Specifically, online refueling in operating CANDU reactors is performed by an operator after assessing data on core power distribution, liquid zone levels, and fuel burnup [10]. Figures 4 and 5 show that the perturbations to nuclear data sometimes result in significant perturbations to core reactivity and core power distribution, which would not align with the fueling engineers practice. Knowing this, it is reasonable to assume that perturbations to nuclear data will affect the manner in which the fueling engineer refuels the reactor. The flexible fueling pattern for CANDU makes it possible that decisions by the fueling engineer may suppress such distortions to the core than might otherwise have occurred due to nuclear data uncertainties.

To test the impact of the fueling engineer's feedback on the reactor core, a simulated/automated fueling simulator was implemented using matlab. The matlab fueling simulator provides similar feedback as the REFUEL module of RFSP, documented by Gray [11] and used to simulate fueling in several CANDU analyses. The algorithm assigns all 480 channels a weight of 1.0 and then proceeds through a list of 17 rules, adjusting the channel weights up or down based on whether the channel passes or fails the rule. At the completion of the process, the highest weighted channel is chosen for refueling (only one channel is selected at a time). If multiple channels have the same weight a tiebreaking criteria is used to select the optimal channel. This process is repeated until the desired number of channels is selected (i.e., typically 2 channels for a given operational day). Within the full-core PARCS model, bundle depletion values are adjusted such as to mimic the fresh fuel loaded into the core at one of the channel ends and the shift of the fuel along the channel. If the results are considered successful the simulation records moves to the next day (i.e., the limits on channel power, LZC level, etc. are maintained). If the results are not successful, the selection of channels is repeated and the previously selected channel has its weight reduced.^{3}

For the reference cross section set and each of the 150 perturbed cross sections sets, the initial reactor state was simulated using the pseudo-random burnup pattern in order to establish an initial condition. The fueling algorithm was then applied in a two-stage process (as outlined below) with two parameters used to determine the acceptability of the core:

being within 0.5 mk of criticality;

all zone powers within 0.7% of target zone powers (determined from Darlington operational data).

The first stage of the process was used to correct for obvious distortions that could not be compensated by the LZC system. During this first stage of this process, the LZC levels were fixed at 50% and the core was fueled according to the rules discussed above until the core was within 0.5 mk of criticality. The number of channels to fuel was determined based on the reactivity error. If the core exceeded the reactivity target, the core was simulated without refueling until the excess reactivity was less than 0.5 mk (typically for 2 to 3 days). For cases with negative reactivity, the algorithm determined a combination of channels, which reduced the reactivity error to within 0.5 mk. During this stage of the analysis, up to eight channels were allowed to be refueled by the algorithm in a single day to eliminate large reactivity errors. Such a large number of fueling operations do not occur in a CANDU; however, to bring the initial core closer to a viable and acceptable core within a limited number of cycles, a larger number of fueling was allowed in this stage.

The second step of the algorithm started with the core properties determined from the previous stage and had a goal to produce a viable/representative core which included RRS and fueling operations. In this stage, RRS control of the LZCs was simulated. The number of channels to be fueled ranged between 0 and 4, based on the average level of the LZCs on the previous simulation day. The fueling algorithm proceeded to the next simulation day when a combination of channels, which reduced the maximum zone error was determined. Once this second step of the algorithm was completed, the matlab code checked the excess reactivity of the core and if too large, the entire two-stage fueling algorithm was repeated until a core, which satisfied both criteria, was found.

Although both steps of this initial fueling algorithm used the same general method to select channels for fueling, different weights were placed on the various criteria at each step. For the first step of the fueling process, the importance of channel burnup was increased relative to the other fueling criteria, while the weight of local LZC level was completely ignored (recall that the LZCs were fixed at 50%). The tiebreaker for this stage was ratio of current exit burnup to target exit burnup, with the highest value being preferred. During the second step, the weight of the zone power to target power ratio and local liquid zone level were increased. The tiebreaker for this step was the ratio of channel power to target channel power, with the lowest value being preferred.

Feedback from fueling operations had a significant impact on the distribution of core *k*_{eff}. Figure 8 shows a histogram of core *k*_{eff} after the fueling operations were completed. After fueling, all 151 trials fall within 5.0 × 10^{−4} of 1.0000. The mean value of *k*_{eff} was found to be 0.99999, with a standard deviation of 2.8 × 10^{−5}. The standard deviation is core *k*_{eff} was reduced 99% by fueling operations as compared to the case where only perturbed cross sections were considered.

The fueling feedback also reduced the relative standard deviation in the 480 channel powers of the CANDU core. Figure 9 shows the updated relative standard deviation in channel power. The average relative standard deviation was reduced to 2.51%. The large peaks in channel power standard deviation, which were previously observed in the vicinity of the liquid zone controllers, have been mostly eliminated. The maximum relative standard deviation in channel power has been reduced from 9.18% to 7.13%.

Although bringing the maximum channel and bundle powers below the maximum allowable limits at this intermediate stage was not an explicit goal of the fueling algorithm, it was found that the fueling operations brought all cases within the allowed limits and reduced the variation in both maximum channel powers and maximum bundle powers. The maximum channel and bundle powers for each nuclear dataset are shown after fueling operations in Figs. 10 and 11. The average maximum channel power was reduced to 6.97 ± 0.06 MW, while the average maximum bundle power was reduced to 0.864 ± 0.016 MW. No case was found to have either a bundle power or a channel power, which exceeded the allowable limit.

### 4.3 Detailed Core Follow Results.

After the generation of 151 viable cores above, additional transients covering 1 to 2 years of operations were performed wherein normal fueling and LZC movement was simulated. The results from this final stage of analysis were then analyzed for reactivity and power variability for ∼730 days (2 yr) of core follow simulation. Fueling decisions during the core follow simulation were made using the same algorithm described previously. The algorithm was allowed to fuel between zero and five channels per day, with a target of two channels per day for fueling. During the core follow simulations, the following operational limits were enforced:

core states, which deviated by more than 0.5 mk from criticality, were not allowed as these would involve liquid zone levels beyond those normally encountered in operations, and

core states, which exceeded the maximum channel or bundle power limits, were not allowed.

All 151 data sets were simulated for the full 730-day period using these operational limits and the fueling rules discussed previously.

The average, maximum, and minimum core *k*_{eff} value over the 151 nuclear data sets for each of the 730 simulated days is shown in Fig. 12. This figure shows the average *k*_{eff} on each day of the core follow along with the maximum and minimum *k* value obtained from the 150 perturbed set of nuclear data. During the entire 2 yr of simulations, the maximum absolute deviation from criticality was 0.152 mk. The maximum channel power for each simulated day is shown in Fig. 13 and demonstrates that the largest maximum channel power observed was 7.27 MW. On average over the 730 days, the maximum channel power was 7.10 MW. Finally, the maximum bundle power is shown in Fig. 14 with the maximum bundle power over the 730 simulated days of 0.93 MW, equal to but not exceeding the maximum acceptable limit. The bundle power limit is reached but not exceeded on 32 simulated days over the entire dataset (i.e., in the 110,230 unique reactor configurations simulated^{4}). The average maximum bundle power over the 2 yr of core follow was 0.919 MW. Although the liquid zone controller level was allowed to vary between 15% and 85% full, during the core follow simulation the maximum and minimum observed LZC level for any nuclear dataset was 61% and 31%, respectively, with an average fill level of approximately 40% (similar to the variability and averages of a typical 900 MW CANDU plant). The relatively small changes in LZC level over the entire dataset and their value being significantly below the design limits indicates that the fueling simulator was managing reactivity very well during these simulations.

## 5 Conclusions

This work has examined the impact of perturbations to the so-called microscopic uncertainty in nuclear data on full core models of a 480 channel CANDU reactor under normal operating conditions. The nuclear data perturbations were propagated from the microscopic level to the few group lattice-level cross sections used in the full core model, leading to perturbations in macroscopic core properties. Additional uncertainties in the incremental cross sections of CANDU reactivity devices were not considered due to limits in the lattice physics propagation tools; however, based upon the similarity observed with other work, the lattice-level uncertainties derived for a regular lattice cell were applied to irregular cells that contained reactivity control devices. It was found that nuclear data perturbations when considered in isolation create a standard deviation in *k*_{eff} of 4.5 mk and an average relative standard deviation in channel power of 3.89%. The uncertainty in the macroscopic quantity of channel power was observed to peak in the vicinity of liquid zone controllers at the bottom of the core. Most importantly, for these cases where nuclear data are propagated through core calculation in isolation from fueling or LZC feedback, the maximum operating limit on channel power is exceeded in a large number of cases. The remainder of the work explored the effect of the reactor regulating system's liquid zone controllers and the effect of online fueling on the observed variability.

To quantify the impact of feedback from the fueling engineer on the variation in macroscopic properties such as core reactivity and power distribution a simulated fueling program was developed, which could either attempt to simulate fueling as closely as possible, or accelerate the fueling operations to “fuel out” any large unrealistic inhomogeneity, which resulted from direct perturbation of the nuclear data. The fueling simulator attempts to balance core reactivity, zone controller level, and power tilts while ensuring that operating limits are maintained. The differences in the accelerated or normal modes of the simulator are that the accelerated mode does not require strict adherence to the operating limits and also performs some initial fueling operations with the zone levels frozen to aid in solution stability at the outset of the process.

The accelerated algorithm was used to fuel a CANDU core model using the 151 different homogenized cross section sets from the same pseudo-random starting and through the use of the accelerated algorithm was able to generate a pseudo equilibrium core for all 151 cases with relatively low computational cost. These fueling operations were found to reduce the standard deviation in core *k*_{eff} by 99% to 0.03 mk. The relative standard deviation in channel power was found to decrease by 35% to 2.51%. The peaks in channel power standard deviation, which were observed near some LZCs in the core before fueling, were not present after the fueling operations. It was found that the standard deviation in zone powers was reduced by between 80% and 99% after the fueling operations as compared to core variability observed when perturbing the nuclear data in isolation. The standard deviation in LZC % fill was also found to decrease after fueling, suggesting a lesser reliance on the RRS to maintain the core within its acceptable conditions. Finally, fueling operations caused the 70 cases which initially exceeded the maximum allowable channel or bundle power to fall within acceptable limits.

Finally, an extensive and detailed set of simulations were performed using the fueling algorithm without acceleration for a 730 day period for all set of nuclear data. It was found that despite the perturbations to nuclear data leading to high variability in the core when viewed in isolation, the combination of fueling feedback and RRS feedback maintained the core *k*_{eff} within 0.152 mk of criticality for all trials over the entire 730 simulated days. The maximum channel and bundle powers were found to stay within the set operational limits for the full 2 yr simulation for all nuclear data sets.

The results presented in this study show that the unique feedback mechanisms of online fueling and liquid zone control response significantly reduce the magnitude of perturbations to macroscopic core properties resulting from the uncertainty in microscopic nuclear data during long-term operations. The work demonstrates the strong interrelationships between the microscale phenomena and uncertainties (e.g., in nuclear data) to macroscopic phenomena (channel power) and how operational and other feedback help to reduce variability.

## Acknowledgment

The authors would like to thank the University Network of Excellence in Nuclear Engineering (UNENE) and the Natural Sciences and Engineering Research Council of Canada (NSERC) for their support of this work.

## Nomenclature

### Abbreviations

- BEAU =
best estimate and uncertainty

- BEPU =
best estimate plus uncertainty

- CANDU =
CANada Deuterium Uranium

- CENTRM =
Continuous Energy Transport Module, a tool within the SCALE code package used to evaluate the problem specific neutron flux spectrum for self-shielding calculations

- LWR =
light water reactor

- LZC =
liquid zone controller

- NEA =
Nuclear Energy Agency

- NEWT =
New ESC-based Weighting Transport code, a two-dimensional discrete ordinates lattice level physics solver included in the SCALE code package

- NPP =
nuclear power plant

- OECD =
Organization for Economic Cooperation and Development

- ORNL =
Oak Ridge National Laboratory

- PARCS =
Purdue Advanced Reactor Core Simulator

- RFSP =
Reactor Fuelling Simulation Program

- RRS =
reactor regulating system

- SAMPLER =
a stochastic sampling super sequence within the SCALE which controls the repeated execution of a SCALE code sequence with randomly sampled perturbations to the input

- SCALE =
a reactor physics analysis code package produced by Oak Ridge National Laboratory

- SDS2 =
shutdown system 2

- UAM-LWR =
uncertainty analysis in modeling for light water reactors

## Footnotes

Note that as of this writing the newest version of PARCS (3.3.1) has a bug in the treatment of Xe and Sm in the finite difference solver and was therefore note used in this work. Future versions of PARCS will correct the bug. Additionally, PARCS treats poison levels in freshly fuelled channels as having reached equilibrium immediately—a minor, but notable approximation.

Similar simulation checks for fuelling are performed by the engineers prior to the physical fuelling machine operations. If the results of the simulation checks show acceptable core properties, then the actual fuelling takes place. If the results are not as anticipated the fuelling engineer would select a different channel and repeat the checks.

Based upon 151 unique sets of nuclear data (i.e., 151 viable initial cores) simulated on each of the 730 days of core follow simulations.