## Abstract

The use of neutron noise analysis in pressurized water reactors to detect and diagnose degradation represents the practice of pro-active structural health monitoring for reactor vessel internals. Recent enhancements to this remote condition monitoring and diagnostic computational framework quantify the sensitivity of the structural dynamics to different degradation scenarios. This methodology leverages benchmarked computational structural mechanics models and machine learning methods to enhance the interpretability of neutron noise measurement results. The novelty of the methodology lies not in the particular technologies and algorithms but our amalgamation into a holistic computational framework for structural health monitoring. Recent experience revealed the successful deployment of this methodology to pro-actively diagnose different degradation scenarios, thus enabling prognostic asset management for reactor structures.

## 1 Introduction

During a Spring 2019 refueling outage, nuclear power Plant A, a pressurized water reactor (PWR), experienced difficulty remove the reactor vessel upper internals assembly from the reactor vessel. After justifying a lifting loading increase, the upper internals was removed to permit plant refueling. During inspections of likely interference locations, a bolt was found protruding through the core barrel inner diameter at the upper core plate elevation. The bolt was identified as a thermal shield (TS) support block bolt (referred to hereafter as TS bolt). Following the identification of the failed TS bolt, subsequent inspections revealed an additional three failed TS bolts, as well as indications on TS flexures. Two TS flexures were confirmed by visual examination to have crack-like indications at the outer welds connecting the flexures to the TS. Possible/inconclusive indications were also noted at the inner welds at these two flexures, as well as at two neighboring flexures.

Additionally, during core offload for the Spring 2019 outage, fuel assembly grid strap damage was observed at two peripheral fuel assemblies. An inspection of the adjacent baffle plate was performed which revealed degradation of several baffler former bolts (BFBs). The visual examination was then expanded to the entire population of all 832 BFBs in the reactor, which ultimately identified 31 bolts as being visually degraded. A subsequent ultrasonic test examination of all original BFBs identified a total of 196 bolts with ultrasonic test indications and three nontestable bolts, resulting in a total of 230 original BFBs identified as nonfunctional. This level of degradation was unexpected since during two outages prior (i.e., approximately three years), the BFBs were inspected with 192 indications, and 189 were replaced. Therefore, experiencing 232 additional BFB failures over two cycles represented accelerated BFB degradation that does not match industry operating experience for typical irradiation-assisted stress corrosion cracking in BFBs. The vibratory response did not align with baseline data for other PWRs, including the sister unit of Plant A, all of which exhibited a normal forced response to operational flow-induced vibration loads.

That operating experience proved costly to the nuclear industry in terms of costs associated with inspections, replacement hardware, and plant downtime, and serves, in part, to motivate the development and use of structural health monitoring (SHM) methods toward the aim of efficient and credible predictive maintenance strategies. Thus, our emphasis here lies in the novel combination of neutron noise (NN) monitoring with dynamic finite element analyses (FEA), which helped to understand the potential source of degradation in these components and provide diagnostic insight for the four inconclusive thermal shield flexure indications.

Neutron noise monitoring uses ex-core neutron detector signals to measure the dynamics of reactor internals components, which we detail in the Methodology section of this paper.

We additionally note that the two confirmed indications during the Spring 2019 outage were small flaws and not representative of a nonfunctional flexure. Therefore, additional flexure degradation, such as the inconclusive indications being actual indications, was determined to exist at plant A. Plant A re-inspected its thermal shield flexures at the subsequent refueling outage and identified that Plant A had five nonfunctional thermal shield flexures.

In the particular case of plant A, the combined use of NN monitoring and FEA allowed the plant to preemptively prepare replacement contingency hardware and tooling to remediate the damage within the same Fall 2020 outage during which the failures were confirmed. More generally, the methodology used in support of Plant A, and enhanced using machine learning (ML) methods for anomaly perception and diagnostics, provides a novel method to the nuclear industry for SHM. This combined use of dynamic signal processing (i.e., analysis of data acquired from NN monitoring) and diagnostically oriented FEA (i.e., structural analysis posed as an inverse rather than the forward problem) constitutes one of the first published manifestations of a digital twin for a nuclear power plant reactor, as defined in Ref. [1], where the insights inferred through the virtual representation of the system supported the decision-making to efficiently modify the physical system. The novelty of our work herein lies not in the particular algorithms employed (e.g., Gaussian processes or Bayesian networks), but rather in the application of these algorithms into a holistic computational framework for NN-based SHM for nuclear power plants.

## 2 Methods

In Secs. 2.1–2.4, we provide an overview of NN monitoring and analysis, then discuss the finite element modeling, and finally, explain the coupling thereof in the context of a monitoring and diagnostic framework. Altogether, the interaction of these methods can be thought of as shown in Fig. 1. In terms of Fig. 1, we accept data acquisition as a “given” to the monitoring and diagnostic framework, although we recognize that data acquisition unto itself constitutes a non-negligible scope of work. We address the data analysis in Sec. 1, the mechanics simulation in Secs. 2 and 3, and the combination thereof in our monitoring and diagnostic framework in Sec. 4. The data analysis informs the mechanics simulation from the standpoint of identifying which vibratory modes pertain to plausible plant degradation scenarios and thus require estimation of parameter sensitives. Those parameter sensitivities subsequently inform a probabilistic diagnostic assessment performed in the context of our asset management platform.

The structural arrangement of a typical PWR, such as discussed herein is shown in Fig. 2 for reference. One aspect of this assembly of specific interest consists of the thermal shield, an annular structure outboard of the core barrel spanning approximately the elevation of the fuel assemblies, and attachment points to the core barrel, where the TS bolts reside near the top and the TS flexures reside near the bottom.

### 2.1 Neutron Noise Analysis.

Neutron noise analysis has been practiced as a means of SHM of reactor vessel internals (RVI) in the commercial nuclear industry for decades, as evidenced in reports that survey its use in traditional PWRs such as Refs. [3] and [4]. Neutron noise analysis has likewise been demonstrated useful for other reactor types such as water-water energetic reactor [5], Canada Deuterium Uranium [6], and even liquid metal reactors [7], but the focus of this paper lies in its application to traditional PWRs. Nonetheless, the dynamic analysis methods discussed herein are readily extensible to other reactor types.

Figure 3 spatially illustrates the ex-core fluence detector locations (shown as black circles about the perimeter) from which the NN signal is acquired. The influence of structural motion on the variance in the dynamic component of the NN signal may be visualized in Fig. 4 wherein detector “D” senses change because the motion of the core support structures changes the moderator (pressurized water) thickness, thus attenuating the observed neutron fluence signal.

The overall signal observable from a fluence detector includes a combination of structural dynamics, acoustics, thermal-hydraulics, and neutronics, and exactly how these signals can be distinguished from one another remains a topic of active research, such as discussed in Refs. [8] and [9]. Nonetheless, as operating plants experience aging related degradation, the acoustic and thermal-hydraulic components remain largely invariant, and the amplitude can be scaled to take account of the neutronics (see $h$ factor discussed in Ref. [10]). Therefore, changes in the NN measurements recorded over a period of time can often serve as a strong indication of possible structural anomalies.

Following the explanation provided in Sec. 2.2.1 of Ref. [10] (i.e., relative movement of core barrel and displaced water volumes within the pressure vessel), consider the simplified image shown in Fig. 4 in which we show a fixed volume within which there is water—core barrel—water, going from left to right, and the core barrel translates to the right by distance $\Delta x$.

where $M1$ and $M3$ represent the water material to the left and right of the core barrel and $M2$ represents the core barrel material, and $xi$, $xj$, $xk$, and $xl$ are the four material positions evolving with respect to $y$. Note that $sM$ represents the sensitivity of material $M$ for a 100% increasing of neutron transport density per [10]. It is thus $\Delta \varphi D(t)\varphi D(t)$ that varies as the core barrel moves back and forth due to flow-induced vibration, and manifests itself as a change in the neutron fluence signal.

Figure 5 shows a typical resultant PSD, denoted “NPSD” since it's derived from a neutron fluence signal.

In accordance with Part 23, Appendix D of Ref. [2], and abiding by the Nyquist criterion (i.e., $fs>2B$ for frequency bandwidth $B$) we employ a sampling frequency $fs$ for the NN signal at a frequency greater than twice the highest frequency of interest to lower internals vibration for a period of at least 60 min.

The amplitude of the NN signal scales with power level, and the vibration levels likewise change with the magnitude of the turbulence-induced forcing functions acting upon the reactor structures [12], so we acquire the NN signals when the power plant operates at hot full power condition.

Upon acquisition of the signal, we process the information most useful to meet condition monitoring objectives, making use of degradation signatures evidenced in the NN signal. That information includes the frequency at which modes amplify (including beam and shell modes), the amplitude of those modes, the frequencies of deterministic signals (e.g., reactor coolant pump pulsations), and identification of whether or not the signal evidences impacting, orbital motion, or wear. One could use numerous methods to extract such features from the NN dynamic dataset, such as those discussed in Refs. [11] and [13].

### 2.2 Finite Element Model Development.

To credibly interpret the source of the anomalous response for plant A, we created a computational model of the core barrel, thermal shield, and baffle-former assembly via the finite element method. Although one can simulate the dynamics associated with lower-order modes of vibration for beams and shells via simplified analytical models, our use of the finite element method intends to capture higher-order modes and otherwise complex structural dynamics that become manifest particularly in the presence of component degradation (i.e., nonaxisymmetric boundary conditions). Our use of this model, in conjunction with the analytical methods presented in the subsequent subsections, to inform assessments based on NN analysis arguably constitutes a novel manifestation of a digital twin for the nuclear power industry, such as defined in Ref. [1]. Concretely, the use of NN monitoring and simulation of the relevant dynamical phenomena indeed combines a *virtual representation* with a *physical system* that is *updated through the exchange of information* (i.e., the three characteristics of a digital twin per [1]), as illustrated in Fig. 6.

The model features include the BFBs, TS flexures, and TS bolt interfaces to accurately represent the numerous connections between the structures in the model. The model mesh uses shell elements to minimize the computational expense associated with performing a dynamic transient analysis. To this end, the model has approximately 125,000 degrees-of-freedom which sufficiently captures the dynamic response (i.e., validated with respect to mode shapes indicative of degradation) but is not so refined to result in unnecessary computation time.

Additionally, the computational model includes hydrodynamic mass to represent the fluid-structure interaction effects in the downcomer. The hydrodynamic mass calculations are based on Fritz's approximation of the effect of liquids on the dynamic motions of immersed solids [14] to calculate mass coefficients and matrices for the downcomer annuli. Specifically, the downcomer includes annuli between the inside of the reactor vessel and the core barrel/thermal shield, and between the core barrel outer diameter and thermal shield inner diameter. Each of these annuli is segmented axially and a mass matrix calculated at each elevation connects to the structure on each side of the annulus.

where the eigenvalues ${\lambda i}i=1n$ represent $n$ structural natural frequencies $\omega 2$, and ${\psi i}i=1n$ are the associated eigenvectors representing $n$ mode shapes.

where $\delta ij$ is the Kronecker delta. In the case of NN analysis, the measurement locations are fixed, as determined by the location of the eight ex-core fluence detectors, and thus the observation matrix $Q$ is quite sparse.

We subsequently parameterized the model to accommodate the following four degradation scenarios, each of which we represent by parameters $p$ in the finite element analysis:

TS bolt and flexure degradation

BFB cracking

Hold-down spring (HDS) relaxation

Core barrel cracking

Note that while the most recent operating experience described in this paper focuses on TS bolt and flexure degradation, BFB degradation has been a material and aging management issue for a number of years [18], as has HDS relaxation, and core barrel cracking is understood to potentially pose a risk to the reliability of reactor structures [19].

From the standpoint of model validation, we adopt the definition of validation from ASME as the “process of determining the degree to which the model is an accurate representation of corresponding physical experiments from the perspective of the intended uses of the model,” wherein we judge the “intended use” of paramount importance to appropriately assess model validity. The intended use of the model aims to support the decision making of plant engineering personnel responsible for maintenance and inspection policies and procedures. As such, we require model validity insofar as the simulated direction of change of observable structural dynamic behavior pertains to reality. For this, we appeal to successful precedent in hot functional testing and companion comprehensive vibration assessment programs, as well as prior NN monitoring campaigns, such as Refs. [4,7], and [10], from which one can map the tracking of dynamic features (i.e., change in core barrel beam mode frequency) with respect to known forms of degradation.

### 2.3 Sensitivity Analysis of Finite Element Model.

With our parameterized finite element model able to accommodate various degradation scenarios, we initiate the following process, which is generally consistent with the methodology described in Ref. [20] in terms of surrogate modeling and sensitivity analysis, making use of ANSYS^{®}optislang and matlab software tools.

First, we define the input parameter space to identify a computationally feasible number of degradation scenarios to simulate. If a given plant design has eight TS flexures and eight TSSBBs, over 800 BFBs, at least three weld seams of different lengths on the core barrel with potential susceptibility to cracking, and an HDS that can be relaxed by varying percentages, the number of plausible degradation scenarios are prohibitively large. That is, if a given bolt or flexure has two possible states, either degraded or undegraded, then there exist at least $2(8+8)$ (TS bolts and flexures) + $2800$ (BFBs) combinations of degradation patterns, for just two of the four degradation scenarios considered in this analysis. Therefore, we discretized the problem into sets of analyses focused on particular degradation phenomena associated with the operating experience and aging management plans for operating nuclear plants [21]. For example, rather than sampling the many plausible degraded patterns for TSSBBs and TS flexures, BFBs, HDS relaxation percentages, and core barrel cracks, the TSSBBs and TS flexures were grouped together into one set of simulations and the BFBs were grouped together into a separate set of simulations. Then, Latin hypercube sampling was employed to sample the parameter space [22]. We found that approximately 1000 simulations provided results for which parameter sensitivities ceased to meaningfully change with additional sampling. To further focus the sampling space and thus represent situations of practical interest to operating plants, we imposed constraints on the input parameter ranges. For example, in simulating TS connection degradation, we limited the total number of TS bolt and flexure failures, as well as the degree of HDS relaxation.

where $\psi pqr$ is the modal coefficient for reference $p$ (indicated as $c$ or $d$ in the $MAC$ definition), degree-of-freedom $q$, and mode $r$, $N0$ is the number of outputs, and $*$ indicates the complex conjugate. The closer the MAC is to 1.0 the greater the consistency between two mode shapes while a MAC closer to zero implies inconsistent mode shapes. The use of the MAC thus made it possible to identify the frequency at which a given mode shape occurred $f(\psi )$. Provided identification of those frequencies, we can then compute the change in mode frequencies subject to imposing degraded conditions upon the finite element model.

where $m(p)$ is a scalar mean function that captures the global trend in the data and $k(p,p\u2032)$ is a scalar correlation function to account for the deviation of the data from the global trend. We based the choice of a Kriging surrogate modeling method on the successful precedent of this methodology in conjunction with sensitivity analysis for simulation of reactor stochastic dynamics (e.g., Ref. [20]), although we recognize alternate viable methods exist, such as polynomial chaos expansions.

The specific way we trained and verified the surrogate model (using ANSYSoptislang) employs a “coefficient of prognosis,” as described in Ref. [25]. The TS bolt and flexure connections have two possible states, either degraded or undegraded, which we represented using binary input parameters. To successfully quantify sensitivities, we defined the binary input parameters as ordinal discrete (versus nominal discrete) to maintain the relationships required to train the surrogate model.

The majority of degradation cases in the DOE include a subset of frequencies for which mode shapes were not identified given the MAC threshold because the shapes were significantly distorted or ceased to exist. For example, for a given design point, the MAC identified six of ten possible degraded mode shapes as similar to the undegraded mode shape, and thus the frequency change is quantified. However, because four of the expected output values (i.e., frequencies at which some mode shape occurs) are missing due to a lack of similarity to any undegraded mode shape, the design point would be considered incomplete and unusable. Therefore, to make full use of the generated DOE data, we incorporated the “incomplete design point” functionality of the ANSYSoptislang tools. That is, instead of a design point being marked as altogether failed (thus omitting some informative frequency changes), the found mode shape frequencies for that design point were used to train the surrogate models while the unfound frequencies were disregarded. To accomplish this, we defined a second layer of output parameters to identify the unfound frequencies (i.e., as not-a-number) and then subsequently deselected those parameters from the surrogate model training inputs.

### 2.4 Remote Condition Monitoring Enhancements.

As demonstrated for similar applications in other industries, such as wind energy [27] or train tracks [28], one can accomplish SHM by the accumulation of historical data, the signal processing thereof, and the use of ML methods for pattern recognition analysis and fault detection, even amidst significant data sparsity. The work we present herein provides a means to develop and assess fault signatures for RVI structures, analogous to what has been done for nuclear plant generators in Ref. [29], and building upon prior work characterizing event signatures on NN data [30].

The specific method used for NN informed RVI SHM is based on Ref. [31]. Over the course of $mt$ time periods $t$ during which samples of NN data are acquired, we extract $nf$ features $\lambda $ from processing the PSDs per Eq. (2), such as a frequency, vibration amplitude, or a binary variable (e.g., contact or not), denoted $\lambda 1$ through $\lambda nf$. Thus, if we let $j\u0302$ range from $1$ to $nf$ and $i\u0302$ range from $1$ to $mt$, we track $\lambda j\u0302(ti\u0302)$ through the monitoring framework. For example, $\lambda 1$ could be the core barrel beam mode frequency which may vary between approximately 6 and 10 Hz, as time varies from $t1$ to $tmt$. In terms of $\lambda 1$, $\lambda 2$, $\u2026$, $\lambda nf$, we do not show specific values in order to protect intellectual property. In terms of the differences between $t1$, $t2$, $\u2026$, $tmt$, one period of data acquisition may be separated from a subsequent period of data acquisition by a few hours (i.e., multiple samples taken on a given day), a few months (i.e., multiple samples taken through a fuel cycle), or multiple years (i.e., samples only taken coincident with some meaningful period of plant life). Upon aggregation of the relevant data in that format, training of a regression-based ML model (e.g., support vector machine or artificial neural network per [32]) serves to recognize the high-dimensional data patterns associated with a normal vibration signature. Once the machine-learned that normal vibration signature, we established statistical fault detectors to identify potentially anomalous behavior. We based the fault detectors on permitting the difference between observed and predicted to fall within a particular number of standard deviations of the distribution established by quantifying uncertainty in the observed quantities.

Upon perception of a potential anomalous condition, the next logical question arises has to do with the cause for that unexpected behavior, thus the task of diagnostics. One may think of this as analogous to a medical setting in which some observable symptoms exist and the objective is to identify the sickness causing those symptoms. A Bayesian belief network (BBN), such as described by Ref. [33], provides a convenient computational structure for diagnostic model formulation. Within a BBN, the FEA-based sensitivity analysis results inform a conditional probability table (CPT) for NN informed RVI SHM, such as shown in Table 1. In Table 1, the first column lists four degradation mechanisms modeled via FEA. We infer the existence of one of those conditions based on the probabilities of the magnitude of certain features falling in a low, normal, or high state. As an example, arbitrary probability values shown in **gray** text within Table 1 illustrate our use of conditional probabilities used to diagnose HDS relaxation, based on features $\lambda 1$ and $\lambda nf$ (e.g., the frequency and amplitude of some mode shape). The $\lambda $ parameters employed for diagnostic modeling in Table 1 coincide with those used for monitoring (and correspondingly anomaly detection). In this example, if the HDS is not relaxed (i.e., in a “normal” state), then there are 98% and 94% chances that features $\lambda 1$ and $\lambda nf$ likewise fall within the normal range, respectively. Correspondingly, if the HDS is relaxed (i.e., in an “abnormal” state), then there are 85% and 70% chances that features $\lambda 1$ and $\lambda nf$ fall in the low range, respectively. We express the BBN structure as shown in Eq. (16), where the probability $P$ of each degradation scenario (i.e., $dHDS$ indicating hold down spring degradation) is computed based on multiplying the probabilities of the presence of that degradation scenario conditioned upon the value of extracted features $\lambda 1,\lambda 2,\u2009\u2026\lambda nf$. We estimate the specific conditional probabilities from the parameter sensitivities revealed through exercising the finite element models. When $Sij$ (the computed sensitivity) falls close to zero, the conditional probability that feature $\lambda j\u0302$ shows a high or low value likewise is nearly zero. Conversely, if $Sij$ approaches 1.0, the conditional probability that feature $\lambda j\u0302$ shows a high or low value likewise approaches 1.0, and we examine the local parameter sensitivity (i.e., one factor at a time analysis) to determine whether the expected trend tends toward high or low.

Degradation | $\lambda 1$ | $\u2026$ | $\lambda nf$ | ||||
---|---|---|---|---|---|---|---|

HDS relaxation | Low | Normal | High | $\u2026$ | Low | Normal | High |

Normal | 0.01 | 0.98 | 0.01 | 0.03 | 0.94 | 0.03 | |

Abnormal | 0.85 | 0.10 | 0.05 | 0.70 | 0.20 | 0.10 | |

BFB degradation | Low | Normal | High | $\u2026$ | Low | Normal | High |

Normal | |||||||

Abnormal | |||||||

TS flexure / Bolt | Low | Normal | High | $\u2026$ | Low | Normal | High |

Normal | |||||||

Abnormal | |||||||

Core barrel (CB) cracking | Low | Normal | High | $\u2026$ | Low | Normal | High |

Normal | |||||||

Abnormal |

Degradation | $\lambda 1$ | $\u2026$ | $\lambda nf$ | ||||
---|---|---|---|---|---|---|---|

HDS relaxation | Low | Normal | High | $\u2026$ | Low | Normal | High |

Normal | 0.01 | 0.98 | 0.01 | 0.03 | 0.94 | 0.03 | |

Abnormal | 0.85 | 0.10 | 0.05 | 0.70 | 0.20 | 0.10 | |

BFB degradation | Low | Normal | High | $\u2026$ | Low | Normal | High |

Normal | |||||||

Abnormal | |||||||

TS flexure / Bolt | Low | Normal | High | $\u2026$ | Low | Normal | High |

Normal | |||||||

Abnormal | |||||||

Core barrel (CB) cracking | Low | Normal | High | $\u2026$ | Low | Normal | High |

Normal | |||||||

Abnormal |

Arbitrary conditional probabilities are only shown here in italics for one degradation mechanism for illustration purposes.

Despite the apparent simplicity of Table 1 in assigning low, normal, and high conditional probabilities for each feature, by studying the variation in features $1$ through $nf$ relative to the candidate degradation mechanisms, we observe distinct patterns in terms of which features depart from normal for a given form of degradation. Therefore, identifying whether each feature falls into a low, normal, or high state in conjunction with the multitude of features extracted from the NN signal enables us to provide reasonable diagnostic conclusions.

We then compare the monitoring results at a given point in time $ti\u0302$ with the logic within the CPT to estimate the probability that one of the candidate degradation scenarios exists. If the probability $P$ for the existence of a degradation mechanism (e.g., $dHDS$ or $dCB$ for hold down spring relaxation or core barrel cracking, respectively) exceeds a user-defined threshold, we conclude that the degradation form is present.

## 3 Results and Discussion

We benchmarked the dynamic finite element model developed to interpret the NN data for Plant A to confirm the appropriate modal and forced response before implementing degradation scenarios. A sample of the benchmarking results is shown in Table 2 and demonstrates the modal is accurately capturing the response of lower internals, and thus suitably validated for diagnostic SHM purposes.

Mode | Modal benchmarking ratio^{a} | Forced response benchmarking ratio^{b} |
---|---|---|

n = 2 shell mode | 1.01 | 0.97 |

1.03 | ||

n = 3 shell mode | 1.05 | 0.91 |

1.09 | ||

n = 4 shell mode | 0.93 | 1.14 |

1.06 |

Mode | Modal benchmarking ratio^{a} | Forced response benchmarking ratio^{b} |
---|---|---|

n = 2 shell mode | 1.01 | 0.97 |

1.03 | ||

n = 3 shell mode | 1.05 | 0.91 |

1.09 | ||

n = 4 shell mode | 0.93 | 1.14 |

1.06 |

Modal benchmarking ratio is the ratio of the calculated to the nominal expected frequency based on test data. Two ratios are provided due to the presence of orthogonal modes.

Forced response benchmarking ratio is the ratio of the calculated narrowband response around the mode of interest to the expected response magnitude over that frequency range from test data.

With the finite element model validated for its context of use, degradation scenarios could be introduced with the confidence that the model accurately simulates the effect of the degradation. Through a DOE, we determined that failure of at least 2–4 thermal shield flexures was required to match the modal and response characteristics of the anomalous response recorded for plant A. When TS flexures were inspected at the following refueling outage, it was discovered that five failed flexures existed at plant A, confirming the results of the finite element model. A representative figure of the changes in response spectra is shown in Fig. 7 which shows shifts in both response frequencies and amplitude.

Generalizing the work done for plant A to support proactive monitoring of RVI structures to numerous candidate degradation mechanisms, Fig. 8 depicts the variation in the frequency of one lower internals mode frequency plotted versus the number of in-tact thermal shield flexures $NFLEX$. Each marker on this plot represents one modal analysis solution. The significant variation in mode frequency for a given $NFLEX$ has to do with two things. First, other parameters besides $NFLEX$ are perturbed by the computational DOE. For example, the number of TS bolts, the specific bolts and flexures, and the HDS relaxation contribute to how much variation in frequency can occur for this mode. Second, there exist numerous unique patterns of TS flexures that constitute a given $NFLEX$. For example, if $NFLEX=2$, the two in-tact flexures could be adjacent to one another, diametrically opposed across the reactor, or anywhere in between. Furthermore, Fig. 8 depicts just one mode of many modes tracked through the DOE. The solid line depicts the mean trend in mode shape frequency with respect to in-tact TS flexures (suggesting weak local sensitivity) whereas each marker represents a given simulation result (i.e., eigenvalue from modal analysis). Thus, while the specific mode frequency plotted in Fig. 8 exhibits significant variance for a given $NFLEX$, and that variance does not appear particularly sensitive to changes in $NFLEX$, the change of each mode with respect to each perturbed parameter (e.g., each individual flexure) uniquely differed for each scenario.

In cases of significant TS connection degradation (at least half of the flexures or bolts failed), we observed an increase in the frequency of the core barrel beam mode in some limited scenarios, counter to intuition that would expect a ubiquitous decrease in frequency upon any loss of stiffness. However, with such significant TS connection degradation, additional modes resembling a core barrel beam mode appear at more frequencies. This finding of instances of an increased core barrel beam mode frequency, with a beam mode present at the lower mode frequencies, reflects the outcome of the work done to determine the 2–4 flexure failure requirement to match the anomalous plant A modal response where, with three failed flexures, the amplitude of the core barrel beam mode decreased and a core barrel beam mode appeared at a higher frequency. We confirmed the existence of mode shapes resembling the undegraded core barrel beam mode at multiple frequencies due to significant TS connection degradation upon inspection of the MAC matrices where multiple modes met or exceeded the MAC value threshold (i.e., the MAC value over which a mode was identified as a match).

As per Eq. (15), the sensitivity of all modes was quantified with respect to degradation parameters, and a typical result of that is shown in Fig. 9. From Fig. 9 one may see that eight different TS flexures, denoted flex_{{}}, and eight different TS bolts, denoted tsb_{{}}, where the number in brackets indicates the circumferential location, as well as the summation of in-tact TS flexures and TS bolts, denoted *n*_{flex} and *n*_{tsb}, respectively, contribute differently to one mode frequency. The left-most (positive valued) “first-order” bars represent the $Si$ values for mode frequency $j$ per Eq. (15), whereas the center and right-most (positive and negative valued) bars, denoted $\rho Rank$ and $\rho Linear$, denote the traditional rank (Spearman) and linear (Pearson) correlation coefficients, respectively. While the sensitivity index and correlation coefficients all provide general measures of relative importance, the particular meaning of each metric differs. Specifically, the sensitivity indices provide a measure of the contribution to the overall parameter variance from each variable, whereas the Spearman and Pearson correlation coefficients provide a measure of monotonicity or linearity, respectively, between the variables and output parameter. In some cases, the taller bars, whether positive or negative, show some consistency between sensitivity metrics (e.g., tall bars for all three metrics suggest the importance to this mode from TS bolts oriented at 247 deg and 292 deg). However, in other cases, we observe that despite a high sensitivity index, the correlation coefficients are nearly zero for the TS flexure at 236 deg. As such, one must exercise caution in choosing appropriate sensitivity metrics with which to inform diagnostic models. In our present context, we chose to primarily employ the sensitivity indices in lieu of relying solely on correlation coefficients for populating the diagnostic model logic, because the variance-based metric most directly pertains to monitoring for changes in observable dynamics of the reactor structures.

For the particular mode frequency shown in Fig. 9, the positive correlation coefficients generally indicate that the number flex_{{}} or tsb_{{}} parameter tends to be intact (i.e., equal to 1 rather than 0), and that as more flexures or bolts are intact (i.e., n_{flex} and n_{tsb} increasing), the frequency tends to be higher. Conversely, as these parameters decrease indicating degradation, the frequency likewise decreases.

The “incomplete design point” method from optislang [25] provides the opportunity to infer further diagnostic information from the design points with unidentified mode shape frequencies. The computational TS connection DOE results were compiled to examine possible correlations that exist between model parameters and the ability, or lack thereof, to find a mode shape. We performed a correlation analysis to assess the extent to which the TS bolt and TS flexure model parameters correlate to a binary response variable which indicates if a mode is found [1] or not (0), such as illustrated in Fig. 10 for a typical mode shape. A strongly positive correlation coefficient (*ρ*) indicates the model parameter is statistically correlated to that mode being found and, therefore, suggests relatively insignificant mode distortion imparted by degradation at that specific bolt or flexure (since the mode can be found). Conversely, a strongly negative *ρ* indicates the model parameter is statistically correlated to that mode not being found and, therefore, suggests significant mode distortion (since it could not be found). Indicated parameters, when degraded, exhibit a particularly strong correlation to mode shape distortion and are thus of keen interest for diagnostics. Indicated parameters, when intact, exhibit a particularly strong correlation to mode shape retention.

By aggregation of historically acquired NN data and the analysis thereof, an illustration of the observed frequency for one mode of interest is plotted versus time with the blue markers in Fig. 11. The red markers shown in Fig. 11 represent the magnitude of that frequency predicted by the ML-based pattern recognition analysis. Our use of ML generates a deterministic result, and thus no error bars are plotted about the predicted frequencies. That said, upon accumulation of both acquired and computational samples, our ML approach evaluates the expected range of uncertainty and thus alerts the analyst of possible anomalous behavior if new observations fall outside of that expected range (i.e., if the red predicted markers fall far enough away from the observed blue markers). For one typical vibration mode, Fig. 12 depicts the probability distribution between observed and predicted samples. In the case of Fig. 12, the difference between observed and predicted distributions is small, but in other cases, the predicted distribution may significantly differ from the observed in which case our ML framework provides an alert of possible anomalous behavior.

The specific data acquisition times are not shown in Fig. 11, but the span of time covers approximately two years. At some points in time, the observed and predicted frequencies fall close to one another, whereas at other times the observed and predicted frequencies diverge. The times during which the observed and predicted frequencies differ suggest the possible presence of some abnormal state. At those times, our predictive maintenance ML framework triggers the diagnostic model (i.e., BBN using the CPT shown in Table 1) to make an assessment as to the cause that best explains the observed behavior.

The degree of core barrel cracking degradation analyzed in our current work resulted in mode frequency changes generally so small that direct detection via monitoring of ex-core NN could prove difficult. However, the trends seen in the sensitivity analysis results informed how one could disposition core barrel cracking indications for diagnostic use. The sensitivity of the core barrel cantilever beam and shell mode frequencies is cyclical with respect to the orientation of a crack around the lower girth (circumferential) weld, as shown in Figs. 13 and 14. The solid lines in Figs. 13 and 14 represent the mean of the $GP$ surrogate trend, whereas the markers indicate the finite element simulation realizations used to train the surrogate. This relates to the bending direction of the orthogonal beam modes and the node locations of the shell modes. The multifaceted directions of bending create a situation where it proves difficult to detect flaws around the core barrel periphery because identification depends on the location of the flaw(s) and the orientation of the mode shape, which cannot be predicted. Therefore, using only individual mode responses through NN monitoring for core barrel crack detection could yield unreliable diagnostic conclusions. The high dependence of the core barrel beam frequency or shell modes on the location and magnitude of a core barrel crack can render it sometimes detectable, but often not detectable.

While flaw identification considering individual modes could provide unreliable diagnostic conclusions, evaluation of the mode shapes (and distorted variants thereof) together, could greatly improve the ability to detect and monitor core barrel weld cracks. The circumferential node spacing of individual shell modes would improve the “coverage” around the core barrel perimeter and thus improve the ability to detect a core barrel crack. Given the high number of plausible crack locations, a more computationally robust means of tracking significantly distorted mode shape variation (and companion uncertainty quantification) may prove necessary and thus could constitute a form of future research.

Appealing to a 2 × 2 confusion matrix of actual condition versus predicted condition, one could frame the diagnostic problem in terms of sensitivity (true positive rate) and specificity (true negative rate), where NN monitoring offers a potentially viable means of detecting a crack with low sensitivity. If the core barrel cracks are truly not healthy, and NN cannot detect the presence of a crack (indicating a healthy structure), then that would produce a false negative. However, when considering multiple modes together one may reasonably expect reliable instances of a true positive test result (i.e., the unhealthy structure and the test says so).

While our work here shows one of the first implementations of a coupled digital twin (i.e., proactive and combined data + mechanics-based analysis) to support the prognostic health management of reactor structures, the general use of NN analysis for monitoring reactor structures has enjoyed successful use in PWRs for multiple decades. As such, we evaluated the reliability of this monitoring and diagnostic framework by surveying multiple past plant data sets to ask the question “if such a modeling framework were in place at that time, would we have concluded what was then found?” We directed that question specifically at a European three-loop Westinghouse-designed plant that experienced both BFB degradation and relaxed HDS and had periodically sampled NN data throughout approximately, the last 20 years [34]. Indeed, the application of our monitoring and diagnostic framework to data acquired from that plant revealed a level of engineering insight commensurate with the reports published at the time of NN analysis, thus lending credence to the reliability of this diagnostic framework.

## 4 Conclusion

The recent experience at plant A dealing with TS bolt and TS flexure degradation further substantiates the viability of NN analysis as a means to proactively monitor and assess the condition of RVI structures. Availability of information related to existing component degradation provides utilities with a unique opportunity to prepare for contingency repair/replacement activities before a refueling outage where one could discover degradation by traditional means of inspection. Our work herein enables industry stakeholders to learn how otherwise disparate engineering disciplines of computational structural mechanics, data-driven modeling, probabilistic analysis, and dynamic signal analysis may combine to yield results of practical use to operating power plant decision makers.

The novelty of our present work lies in the combination of NN monitoring and the mechanics-informed analysis thereof. That combination constitutes one of the first published manifestations of a digital twin, in accordance with Ref. [1], as well as [35] for dynamics problems, to support the prognostic health management of reactor structures, and enables the general desire of the nuclear industry to move more toward a digitally enhanced paradigm for asset management. Moreover, NN monitoring viewed from a value-of-information perspective can quantify commercial risk and thus further motivate industry movement toward proactive asset monitoring [36].

As possible future work, we recognize that the state of data available from operating nuclear plants for SHM is often less than ideal, from the standpoint of sparsity and general disparity across different plants, periods of time, applications, and questions of interest. Thus, enriching the data available via physics-informed ML methods, or employing advanced probabilistic modeling methods could be expected to yield improvements in the ability of industry practitioners to realize the greatest value from otherwise sparse data sets with SHM related objectives. For example, recognizing the practical constraint of the long time scale for data availability, one could more exhaustively validate numerical models with existing data sets, including plant operational data and developmental subscale testing, and thus impute the results gained from exercising validated computational models into the ML framework used herein. Instances of particularly severe degradation may fall outside the range by which this framework could provide reliable diagnostic conclusions, but we view this SHM framework as one portion of an overall aging management plan whereby plants can proactively mitigate such significant conditions.

Particularly as additional data is accumulated from operating PWRs, future work is expected to involve robust uncertainty quantification, involving methods of verification and validation (V&V), and predictive capability assessments per Refs. [37] and [38], to further substantiate the credibility of NN analysis to support remote condition monitoring and thus asset management of RVI structures.

## Acknowledgment

We acknowledge our managers Matthew Kelley and Kaitlyn Musser, our project managers Nathan Lang and M.E. Countouris, our co-workers with subject matter expertise in neutron noise analysis, including Gregory Meyer, Richard Basel, Jeremy Koether, and David DiBasilio, and our co-workers contributing to the FEA including Margarita Wiebe, Brett Bailie, and Richard Vollmer.

## Nomenclature

- $d$ =
degradation mechanism

- $f$ =
frequency, where $\omega =2\pi f$

- $G\u0302(f)$ =
power spectral density function

- $GP$ =
gaussian process

- $h$ =
modal frequency response function

- $K$ =
stiffness matrix

- $M$ =
mass matrix

- $MAC$ =
model assurance criterion

- $p$ =
parameter used to simulate degradation

- $S$ =
sensitivity index

- $t$, $\Delta t$ =
time and time increment

- $Z$ =
Fourier components

- $\lambda $ =
feature extracted from dynamic data

- $\rho $ =
density

- $\chi $ =
recorded signal

- $\varphi $ =
neutron flux

- $\psi $ =
mode shape (eigenvector)

## References

*Operation and Maintenance of Nuclear Power Plants*