# Comparison of Optimizers for Fault Isolation and Diagnostics of Control Rod
Drives

Ark Ifeanyi  University of Tennessee
Knoxville, Tennessee 37996, USA
Email<EMAIL_ADDRESS>Jamie Coble  University of Tennessee
Knoxville, Tennessee 37996, USA
Email<EMAIL_ADDRESS>
###### Abstract

This paper explores the optimization of fault detection and diagnostics (FDD)
in the Control Rod Drive System (CRDS) of GE-Hitachi’s BWRX-300 small modular
reactor (SMR), focusing on the electrically powered fine motion control rod
drive (FMCRD) servomotors. Leveraging the coordinated motion of multiple
FMCRDs for control rod adjustments, the study proposes a deep learning
approach, utilizing one-dimensional convolutional neural network (1D
CNN)-based autoencoders for anomaly detection and encoder-decoder structured
1D CNN classifiers for fault classification. Simulink models simulate normal
and fault operations, monitoring electric current and electromagnetic torque.
The training of the fault isolation and fault classification models is
optimized. Various optimizers, including Adaptive Moment Estimation (Adam),
Nesterov Adam (Nadam), Stochastic Gradient Descent (SGD), and Root Mean Square
Propagation (RMSProp), are evaluated, with Nadam demonstrating a relatively
superior performance across the isolation and classification tasks due to its
adaptive gradient and Nesterov components. The research underscores the
importance of considering the number of runs (each run has a different set of
initial model parameters) as a hyperparameter during empirical optimizer
comparisons and contributes insights crucial for enhancing FDD in SMR control
systems and for the application of 1D CNN to FDD.

##  I Introduction

Nuclear reactors play a pivotal role in meeting the world’s energy demands,
and the efficient regulation of thermal power in reactors is critical for
their safe and optimal operation. Control rods (CRs) are fundamental
components in nuclear reactors, modulating the rate of fission by absorbing
neutrons when inserted into the core. This fission rate in turn controls
thermal power output. The control of the CR position (insertion/withdrawal) is
facilitated by control rod drives (CRDs), a mechanism vital for maintaining
the stability and performance of the reactor core.

In the evolution of reactor designs, the paradigm is shifting towards Small
Modular Reactors (SMRs) due to their compact size and operational flexibility.
Notably, GE-Hitachi’s BWRX-300 SMR design represents a significant advancement
by replacing conventional hydraulically-driven CRDs with electrically powered
fine motion control rod drives (FMCRDs). This FMCRD (see Fig. 1) enhances
control precision and responsiveness, contributing to the adaptability and
efficiency of the reactor.

Within the reactor core, CRs operate in banks, collectively influencing the
power profile. Currently, one challenge in monitoring FMCRD conditions is that
the movement of CRs happens intermittently. This means that most of the time,
CRs remain still, only moving when commanded due to changes in reactor
conditions. Additionally, when CRs are stationary, the servomotor, which
controls their movement, doesn’t generate any data that could help assess
their condition. On the other hand, the coordinated motion of multiple FMCRDs
powered by servomotors within these banks creates a unique opportunity for
monitoring and assessing the health of individual components. Leveraging this
coordinated operation, an approach to fault detection and diagnostics (FDD)
that capitalizes on the synchronized behavior of servomotors within a bank was
proposed in a previous work [1]. The paper monitored the torque and position
of the FMCRD to detect and diagnose three fault types: short-circuit fault,
ball-screw jam fault, and ball-screw wear fault.

This current research aims to extend the previous research by achieving two
primary objectives: first, to improve detection and diagnostics results by
monitoring an additional property of the system (current); second, to
systematically investigate the effect of four different first-order optimizers
on these FDD tasks. By addressing these objectives, the study not only
contributes to the understanding of FDD methodologies in the context of FMCRD
servomotors but also provides valuable insights into the optimization of deep
learning models for enhanced performance in fault diagnosis within nuclear
reactor systems.

While SMRs are garnering considerable attention and support for their
promising features, operationalizing these reactors in the United States is
still on the horizon, with the first deployment anticipated in 2029 [2, 3].
The FMCRDs, being integral to the operation of GE-Hitachi’s BWRX-300 SMR,
serve as a backbone for the reactor’s flexibility and efficiency. The ease and
accuracy with which faults in these motors can be detected are critical
factors influencing the economic feasibility of these SMRs. Effective fault
detection not only ensures the safety and reliability of SMRs but also has the
potential to significantly reduce operation and maintenance (O&M) costs,
thereby enhancing the contribution of the global nuclear fleet to energy
security.

Figure 1: Fine Motion Control Rod Drive Systems

Our approach involves the application of deep learning to the FDD tasks.
Investigating the best Keras optimizer to employ in its default state when
applying deep learning to fault detection and diagnostics is a crucial step in
the model development process. Default optimizers in Keras, such as Adam,
RMSprop, or SGD, come with pre-defined hyperparameters that are generally
well-suited for a wide range of tasks. This default state exploration is
particularly valuable in scenarios where computational resources are limited,
as optimizing the hyperparameters of the optimizer can be computationally
expensive. Moreover, the process of tuning optimizer hyperparameters demands a
profound understanding of the specific problem domain, and the optimal
configuration may vary across different fault detection tasks. By initially
assessing the default optimizers, practitioners can gain insights into their
model’s performance without the need for extensive hyperparameter tuning,
allowing for a more efficient and focused optimization process. This pragmatic
approach is especially relevant in fault detection and diagnostics
applications, where the emphasis lies on promptly and accurately identifying
deviations from normal system behavior, and a well-chosen default optimizer
can often provide a strong starting point for model development.

The subsequent section (section II) of this paper delves into a thorough
examination of existing research on fault detection in similar systems and
establishes the foundational knowledge for our investigation. Following this,
section III meticulously details our approach, utilizing Simulink models and
deep learning techniques to simulate and analyze servomotor operations.
Section IV presents findings from the experiments, offering insights into the
effectiveness of various optimizers in fault detection and diagnosis. Finally,
in section V, the paper concludes with a thoughtful synthesis of the outcomes
and their implications in the realm of Small Modular Reactors, emphasizing the
significance of our contributions to the broader discourse on nuclear energy
system reliability and operational efficiency.

##  II Literature Review

FDD encompasses the detection of faults, mapping the detected faults to their
origins (isolation), and identifying the type of fault detected (diagnostics)
[4]. In this research, the focus is on isolation and diagnostics. Although
some researchers have explored FDD within control rod drives (CRDs), which are
primarily used for safety actuation in the case of reactor SCRAM, [5], there’s
a notable gap in understanding FMCRDs, used for normal insertion and
withdrawal to manage power profiles across the reactor core. One prior study
on fault detection in FMCRDs employed displacement monitoring and a
statistical thresholding approach. This involved setting a threshold on the
average difference between demanded and actual rod positions within a bank
[6]. The expected variation in this average difference was meant to
distinguish between healthy and faulty banks [6]. However, this specific
attempt only successfully detected one fault type, namely, short-circuit.
Another recent study applied principal component analysis (PCA) to anomaly
detection in FMCRDs, using T-squared statistics and Q-statistics for this
purpose [7]. This second study extended the observation period to include more
data points in the training data to enable detection and isolation. In
contrast, the approach presented in this paper aims to capture intricate
relationships between variables, potentially enabling the detection of various
fault types with a reduced number of observations in the training data and a
limited number of samples. Additionally, this paper contributes to the FDD
cycle by addressing fault isolation, and diagnostics for this significantly
underrepresented system in the existing literature.

Various anomaly detection techniques have been employed over the years,
encompassing classification-based, clustering-based, nearest neighbor-based,
and statistical approaches [8]. In the context of this paper, fault detection
involves classifying a bank of control rods as either faulty or healthy.
Anomaly detection can be achieved through both traditional machine learning
techniques and more advanced deep learning methods [9]. In the realm of deep
learning, autoencoders stand out as widely utilized architectures for anomaly
detection and fault isolation, particularly in the domain of prognostics and
health management (PHM) [10]. Autoencoder reconstruction has proven effective
in detecting anomalies in industrial motors [11], while variational
autoencoders have been successfully applied to identify anomalies in electric
drives [12]. Given their widespread acceptance and success in similar
applications [13, 14], a variant of the autoencoder was selected as the
architectural choice for fault detection and isolation in this work.

Concerning the diagnostic assessment of motors, various approaches have been
historically employed, with motor current signature analysis (MCSA) standing
out as the most prevalent [15, 16]. MCSA involves scrutinizing the properties
of the motor’s current signal and aligning them with established behaviors for
different fault types to classify the investigated fault [17]. However, a
notable drawback of this method is its dependency on previously established
behaviors, making fault classification challenging in the absence of such
references. Moreover, similar characteristics among different faults increase
the risk of misdiagnosis. Additionally, the approach relies on monitoring
motor current, rendering classification unfeasible in cases where current
signals are not observed. Addressing these challenges, recent techniques have
turned to the application of neural networks in induction motor fault
diagnostics [18]. In these methods, signals like vibrations may be explored
for diagnostic purposes. Some researchers have also explored motor fault
diagnosis using Fuzzy Logic-based techniques, either as precursors to other
diagnostic methods or as the primary diagnostic algorithm [19]. However,
limited work has been found for diagnostics in FMCRDs, and to the best of our
knowledge, no known published work has discussed their application in within-
bank analysis. It is not uncommon to find anomaly detection in tandem with
diagnosis in various fields including in the health sector where diagnosing
heart disease has been treated as a hierarchical problem [20]. First, an
autoencoder is used to detect anomalies in electrocardiogram (ECG) signals
before transferring the knowledge from the trained autoencoder to a classifier
for diagnosis [20]. In this current study, no knowledge transfer was done and
a 1D-CNN-based classification model was employed to attempt diagnostics within
banks of FMCRDs. 1-dimensional convolutional neural network (1D-CNN)-based
autoencoders were used in this study instead of typical fully connected layer-
based autoencoders to maintain the temporal correlations in the input data. To
add to that, utilization of 1D-CNNs has proven effective for fault diagnosis
in scenarios where the input signals are sequences [21, 22, 23], justifying
their application throughout this paper.

A specially designed deep learning classification model was used for fault
diagnostics. One benefit of deep learning models is that they may reveal
complicated correlations without necessarily needing feature engineering [24].
The constrained coding layer also helps autoencoders extract valuable
information, which is important for the application under investigation
because of the servomotors’ sporadic operations explained in section III-A and
the small number of samples available. One significant benefit of the
suggested technique over the previously discussed statistical methods is its
capacity to concurrently take into account the time dependency of each sensor
variable and the nonlinear correlations between numerous sensor variables.

Moving away from FDD, optimization algorithms determine the speed of
convergence and the predictive performance of models. They do this by updating
the parameters of the model using different update rules. The update rule uses
historical values of parameters and gradients of the loss function, as well as
the hyperparameters of the optimization algorithm, to calculate the model’s
parameters at the current timestep [25, 26]. The four investigated optimizers
and their update rules are shown in Table I. Stochastic gradient descent (SGD)
optimizes parameters using the negative gradient and a learning rate [27].
Root mean square propagation (RMSProp) adapts learning rates per parameter
with moving averages of squared gradients [28] whereas Adam adds momentum,
decaying averages of past gradients, and bias correction [29]. Finally, Nadam
combines Nesterov’s momentum with Adam’s adaptive learning rates for enhanced
stability and convergence [30].

It is worth mentioning that first-order optimizers when optimized, exhibit
inclusion hierarchy relationships where some optimizers can approximate others
as shown in equations (1a), (1b), and (1c). These relationships mean that the
more specific optimizers like SGD, when optimized, should not outperform the
more general optimizers like RMSProp, Adam, and Nadam in their optimized
states [31]. The challenge is that the general optimizers can be more
expensive to optimize because of their increased number of hyperparameters so
achieving this relationship might be impractical for most deep learning
practitioners [31]. As a result, the default Keras versions of all the
optimizers were compared in this research for application in FDD of FMCRDs.

SGD $\displaystyle\subseteq\text{Momentum}\subseteq\text{RMSProp}$ (1a) SGD
$\displaystyle\subseteq\text{Momentum}\subseteq\text{Adam}$ (1b) SGD
$\displaystyle\subseteq\text{Nesterov}\subseteq\text{Nadam}$ (1c)

TABLE I: Mathematical Update Rules for Keras Optimizers Optimizer | Update Rule
---|---
SGD | $w_{t+1}=w_{t}-\eta\nabla J(w_{t})$
Adam | $m_{t+1}=\beta_{1}m_{t}+(1-\beta_{1})\nabla J(w_{t})$
| $v_{t+1}=\beta_{2}v_{t}+(1-\beta_{2})(\nabla J(w_{t}))^{2}$
| $w_{t+1}=w_{t}-\frac{\eta}{\sqrt{v_{t+1}}+\epsilon}m_{t+1}$
RMSProp | $v_{t+1}=\beta v_{t}+(1-\beta)(\nabla J(w_{t}))^{2}$
| $w_{t+1}=w_{t}-\frac{\eta}{\sqrt{v_{t+1}}+\epsilon}\nabla J(w_{t})$
Nadam | $m_{t+1}=\beta_{1}m_{t}+(1-\beta_{1})\nabla J(w_{t})$
| $v_{t+1}=\beta_{2}v_{t}+(1-\beta_{2})(\nabla J(w_{t}))^{2}$
| $m_{\text{corr}}=\frac{\beta_{1}m_{t+1}+(1-\beta_{1})\nabla
J(w_{t})}{1-\beta_{1}^{t+1}}$
| $w_{t+1}=w_{t}-\frac{\eta}{\sqrt{v_{t+1}}+\epsilon}m_{\text{corr}}$

Table I provides the mathematical update rules for popular first-order
optimizers: Stochastic Gradient Descent (SGD), Adam, RMSProp, and Nadam. Here,
$w_{t}$ represents the model weights, $\eta$ is the learning rate, $\nabla
J(w_{t})$ is the gradient of the loss function, and $\epsilon$ is a small
constant for numerical stability. The parameters $\beta_{1}$ and $\beta_{2}$
control the exponential decay rates for the moving averages in Adam and Nadam.

##  III Methodology

This research employs a Simulink model to simulate the operations of FMCRD
servomotors under normal conditions and various fault modes. Monitoring the
electric current in one of three available phases and the electromagnetic
torque of these servomotors, we utilize deep learning models to isolate and
diagnose faults, encompassing tasks such as anomaly detection and fault
classification. The efficacy of different optimizers, including Adaptive
Moment Estimation (Adam), Nestrov Adam (Nadam), Stochastic Gradient Descent
(SGD), and Root Mean Square Propagation (RMSProp), will be systematically
evaluated. Key metrics, such as minimum loss value and convergence rate, will
guide the selection of the most suitable optimizers for each specific task,
contributing to the robustness and accuracy of the fault detection system.

###  III-A Data

The data used in this study came from an FMCRD simulation model implemented in
MATLAB/Simulink [32]. These data show how control rod banks react to demands
for position changes during the course of a ten-second maneuver. Two
properties of the FMCRD were monitored in this work; the first variable
records the electromagnetic torques (in Newton-meters) of the bank’s
servomotors, and the second one tracks the electric current (in amperes) (see
Fig. 2). Twelve batches of data were generated for each variable, each
displaying a unique motion profile over the observation time (see Fig. 3).
Each batch has four banks: one bank that is in good health and three banks
that are faulty, each of which represents a distinct tested fault type. Each
bank comprises ten rods, with healthy banks containing all ten rods in good
condition, and faulty banks having one faulty rod intentionally placed in
position ’10’. The data breakdown overview deliberately omits some components
for summary purposes. The ’Torque’ branch displays the overall count of rods
and banks in any given batch, although it was specifically demonstrated for
batch ’2’. These sums are evenly distributed, ensuring that each bank, as
exemplified in bank ’1’ of batch ’2’ in the ’Current’ branch, consists of ten
rods. The three investigated faults were short circuit, ball screw wear, and
ball screw jam faults. The ball screw is a mechanical component of the drive
system so the wear and jam faults are mechanical faults whereas the short-
circuit fault is electrical. The wear fault was simulated as a gradual
increase in mechanical load but the jam fault as an instant increment [32].

Figure 2: Data Structure

Additionally, each rod produces five hundred thousand temporally related
observations (excluding the starting point) for each measured property due to
a sampling rate of 50kHz for the ten-second maneuvers. In this work, each
bank, consisting of ten rods with ten-second sequences (500001 data points) of
current or torque, is treated as one sample. The input shape is
$(x,500001,10)$, where $x$ represents the number of banks. A total of forty-
eight generated data banks were created, with twelve for each of the four
categories: healthy (no-fault), short-circuit fault, ramp fault, and step
fault.

For each task, the training and test sets have a distinct composition, which
is explained in detail in the corresponding sections. For clarity, the
defective rod was always located at the same variable index (’10’) of the bank
in all test scenarios, while it could theoretically be at any index and still
cause the detection and isolation procedures to operate as intended.

Figure 3: Healthy position profile of a single batch

For brevity, this section specifically discusses the behavior of batch ’1’,
although similar patterns were observed in the healthy banks of all other
batches (see Fig. 3). The rods in that bank began to progressively rise one
second after batch ’1’ was being monitored. This was the result of a request
for a position modification. The rods overshot at the required point before
stabilizing at the predetermined location, which was around 1.735 cm. Then, at
the $5^{th}$ second, there was another ascent request, which resulted in
another slow upward movement and a subsequent equilibrium settlement. The
drive mechanism’s operating periods are defined by these intervals during
which the rods travel to reach the intended position. The fact that the
servomotor only runs for a portion of the observation period means that there
are few data points from the rod position signals and other monitored response
signals that can be used to identify faults, making the FDD tasks more
difficult. The step profile shows that the position demand is immediate, but
the actual rod movements take time to unfold, therefore there is a temporal
lag between the requested and actual locations.

###  III-B Fault Isolation

In addressing the intricate challenge of fault isolation within banks of
control rod drive servomotors, our research undertook a methodical
exploration, leveraging advanced deep learning techniques. The initial pivotal
step involved the selection of a first-order Keras optimizer, a critical
influence on the subsequent training of our 1D-CNN autoencoder models (Fig.
4). These models, expected to be adept at capturing nuanced temporal
dependencies within the electric current and electromagnetic torque signals of
fine motion control rod drives (FMCRDs), formed the core of our fault
isolation methodology.

Figure 4: 1D-CNN autoencoder

The training phase commenced with datasets exclusively derived from healthy
control rod drive banks (nine samples), enabling the models to learn and
encode the subtle features inherent to normal operational conditions. This
foundational training set the stage for subsequent fault isolation tasks,
where deviations from the learned normalcy would serve as indicators of
potential faults. To bolster the models’ robustness, a separate set (three
samples) of control rod drive banks in a healthy state was utilized for
validation. This step, crucial for assessing generalization capabilities,
aimed to ensure the methodology’s effectiveness across diverse datasets
representative of normal operational conditions.

Validation laid the groundwork for the establishment of a reconstruction error
threshold, a pivotal parameter guiding the identification of anomalies during
subsequent testing. Rigorous testing involved exposing the trained and
validated models to datasets from control rod drive banks exhibiting various
faults. The primary focus was on pinpointing rods with significantly higher
reconstruction errors, indicative of potential deviations from the learned
healthy operational patterns.

Recognizing the diversity of operational parameters and the potential impact
of optimizers, our methodology embraced an iterative approach. The fault
isolation task was systematically repeated for each first-order Keras
optimizer, allowing for a comprehensive assessment of their influence on the
performance of the 1D-CNN Autoencoder models. Furthermore, this process was
replicated for both electric current and electromagnetic torque properties,
providing a nuanced understanding of fault isolation across distinct
operational parameters.

###  III-C Fault Diagnostics

In our endeavor to advance fault diagnostics for control rod drive
servomotors, our methodology seamlessly transitions into a comprehensive
approach, integrating a structured encoder-decoder 1D-CNN classifier (Fig. 5).
This section unfolds a strategic sequence of steps designed to proficiently
diagnose faults within the operational parameters of control rod drives.

The process initiates with the selection of a first-order Keras optimizer,
echoing the foundational importance observed in the fault isolation task. This
optimizer, with default hyperparameters, significantly shapes the training of
our encoder-decoder structured 1D-CNN classifier, acting as a core step for
our fault diagnostics methodology.

Figure 5: Encoder-decoder structured 1D-CNN classifier

The training phase assumes a central role, utilizing a substantial dataset
comprising 70% labeled faulty data. This extensive dataset exposes the
classifier to a diverse array of fault scenarios, fostering its ability to
discern patterns indicative of specific FMCRD faults. The encoder-decoder
structure facilitates the extraction and representation of critical features
essential for precise fault diagnosis.

Following the intensive training phase, the methodology progresses to critical
validation and testing stages. The validation set, constituting 15% of the
samples, serves as a pivotal checkpoint for fine-tuning the model parameters
and ensuring its robustness in recognizing fault patterns. Subsequently, the
model undergoes a comprehensive test phase, classifying an unseen set of 15%
samples to evaluate its generalization capabilities and real-world
applicability.

The entire fault diagnostics methodology is systematically repeated for each
selected first-order Keras optimizer and operational property. This iterative
approach facilitates a comprehensive comparative analysis, offering insights
into the varied performances of classifiers and highlighting optimal
configurations for distinct fault diagnosis scenarios.

##  IV Results

In this section, the explained methods are implemented and different
visualizations are employed for the different tasks. The speed of convergence
for the different optimizers was inferred from single runs of multiple epochs
but predictive abilities were compared from multiple runs where each run has
different initial parameters. The final validation loss for each run was
compared for the isolation tasks whereas the minimum (best) validation loss
for each run of multiple epochs was compared for diagnostics. The decision of
the metric to compare for each task was made from assessing preliminary runs
whose results were not reported in this work.

###  IV-A Fault Isolation

This section employs bar charts to illustrate the average contribution of each
control rod across multiple banks in order to identify and pinpoint faults.
Specifically, if a control rod exhibits significantly higher average
contributions than the other rods within the same bank, it implies that the
fault originates from the particular rod and its associated FMCRD in that
position. To elaborate further, in banks experiencing faults, certain rods are
anticipated to display higher average contributions than their counterparts in
corresponding positions in healthy banks. Notably, all faults in the studied
banks are intentionally placed in position ’10’. Consequently, the rods
located in position 10 are expected to be the primary contributors to
reconstruction errors in the faulty banks.

a)

b)

Figure 6: Validation loss profile from current monitoring. a) Single Run; b)
Thirty runs.

Fig. 6 shows the performances of the different optimizers when current is the
monitored variable for the isolation task. When the single run of training
with 150 epochs (Fig. 6a) is analyzed to measure the speed of convergence, the
validation loss profiles show that Nadam was the quickest to reach its minimum
possible loss with Adam as a close second. RMSProp in third has yet to
converge completely but equally achieves low loss values. Over thirty runs
(Fig. 6b) and ignoring outliers, Nadam has the lowest range of losses across
runs with rmsprop and Adam coming close with equally low loss values.
Considering how quickly Nadam converges before RMSProp and its low median
final loss values across runs, Nadam is selected as the best performer in this
case, with Adam as a close second.

During testing, all the general models (RMSProp, Nadam, and Adam) were able to
detect and correctly isolate the faults in the 10th rod of the three different
faulty banks. However, the mechanical faults were harder to isolate as
displayed in Fig. 7 for the RMSProp optimizer.

a)

b)

c)

d)

Figure 7: Fault isolation with current monitoring. a) - healthy; b) - short
circuit fault; c) - step fault; d) - ramp fault.

In the case of isolation from torque monitoring, Fig. 8a, the single run of
100 epochs, shows rmsprop as the quickest to converge whereas the thirty runs
plot (Fig. 8b) also suggests that rmsprop guarantees low loss values but Nadam
gave the best median and loss range. Here, Nadam is selected as the overall
best performer with test results similar to the previous case (see Fig. 9).

a)

b)

Figure 8: Validation loss profile from torque monitoring. a) Single Run; b)
Thirty runs.

a)

b)

c)

d)

Figure 9: Fault isolation with torque monitoring. a) - healthy; b) - short
circuit fault; c) - step fault; d) - ramp fault.

Overall, the adaptive and more general optimizers can adequately detect and
isolate the FMCRD faults for the three investigated fault types with either
monitored property but rmsprop and Nadam outperformed Adam, especially with
torque monitoring.

###  IV-B Fault Diagnostics

Since fault diagnostics is a classification task, confusion matrices are used
to show the test results in this section. Optimization algorithms are compared
with loss values and convergence speed as in the preceding section.

Fig. 10 shows the diagnostics results for current monitoring and as seen, the
speed of convergence is similar for all optimizers and the prediction
performances across twenty runs also appear close. However, Adam is selected
because of its higher confidence, ignoring the outlier values. The confusion
matrix for the test data (Fig. 11) shows 100% identification of all faults
with SGD but all optimizers produced the same test result.

a)

b)

Figure 10: Validation loss profile from current monitoring. a) Single Run; b)
Twenty runs. Figure 11: Diagnostics test results with current monitoring

For torque (Fig. 12), the single run shows signs of overfitting with all
optimizers but the lowest validation loss was achieved with Nadam around the
15th epoch. Over multiple runs, however, a similar low median of minimum loss
across epochs was achieved with SGD, rmsprop, and Nadam but SGD had the lowest
median leading to its selection as the best. Testing with the best models, all
optimizers achieved 100% test accuracies (see Fig. 13 for rmsprop) but from
the multiple runs, it can be deduced that Adam would find it most challenging
to replicate that result.

a)

b)

Figure 12: Validation loss profile from torque monitoring. a) Single Run; b)
Twenty runs. Figure 13: Diagnostics test results with torque monitoring

Overall, both torque and current were useful properties for the investigated
tasks. This aligns with what is seen in the literature since the most used
diagnostics method for nuclear reactors (MCSA) relies on monitoring and
analyzing electric current signals [17]. In this research, however, predefined
behaviors of the current signal under the investigated faults were not
required. Moreover, in an electric motor, the relationship between electric
current and electromagnetic torque is described by the motor’s torque equation
(equation 2) [33], offering a possible explanation for the effectiveness of
torque.

$T_{\text{em}}=K_{\text{torque}}\cdot\Phi\cdot I_{a}$ (2)

Here, $T_{\text{em}}$ represents the electromagnetic torque,
$K_{\text{torque}}$ is the torque constant, which characterizes the motor’s
torque-producing ability. $\Phi$ denotes the magnetic flux, a quantity
dependent on the motor’s design and the field current. $I_{a}$ is the armature
current, signifying the electric current supplied to the motor’s armature
winding. As the armature current ($I_{a}$) increases, the electromagnetic
torque ($T_{\text{em}}$) produced by the motor also increases approximately
linearly, provided that the magnetic flux ($\Phi$) remains constant.

TABLE II: Summary of Performances Task |  Monitored Property |  Best Optimizer |  Worst Optimizer
---|---|---|---
Isolation |  Current |  Nadam |  SGD
Isolation |  Torque |  Nadam |  SGD
Diagnostics |  Current |  Adam |  None
Diagnostics |  Torque |  SGD |  Adam

Table II shows the summary of the results so far and Nadam was mostly
preferred for the different highlighted reasons. It should be noted that in
this work, the convergence rate was a secondary metric to validation loss
distribution because it depended on a single run. Hence, the convergence rate
was only considered when there was no clear difference in validation loss
performance (Fig. 6b). In addition, SGD performed poorly for the isolation
tasks but showed comparable performances for diagnostics.

Across both isolation and diagnostics tasks, Nadam produced the best and most
generalizable predictors. This is important because Adam is usually the
default optimizer for most practitioners [34]. A possible reason for the
overall performance of Nadam is that its adaptive gradient component which
allows it to adjust step size for each parameter and enables it to better
handle various data distributions and complexities. Additionally, its Nesterov
accelerated gradient component, allows it to anticipate future gradient
updates, leading to smoother convergence and helping escape local minima more
efficiently.

SGD performed better for classification tasks than reconstruction tasks with
autoencoders possibly due to the nature of the optimization process and the
challenges associated with each task. In classification tasks, the goal is to
learn a decision boundary between different classes, and SGD’s simplicity and
efficiency in updating model parameters based on the gradients of the loss
function make it effective for this purpose. On the other hand, in
reconstruction tasks with autoencoders, the objective is to accurately
reconstruct input data, which involves capturing intricate relationships and
dependencies among variables. The fixed learning rate of SGD, without adaptive
adjustments, can hinder its ability to navigate the complex and high-
dimensional loss landscape associated with autoencoder-based reconstruction,
potentially leading to suboptimal convergence and performance [27]. Adaptive
optimizers are often more suitable for autoencoder-based tasks as they
dynamically adjust learning rates based on the historical behavior of each
parameter, facilitating smoother convergence in intricate reconstruction
scenarios [11].

###  IV-C Effect of Runs

A single run might not capture the variability inherent in complex tasks,
potentially leading to misleading conclusions. Conversely, an excessive number
of runs may incur unnecessary computational costs. Thus, finding the optimal
balance in the number of runs is imperative to strike a balance between
computational efficiency and result reliability, allowing for more informed
conclusions in empirical model comparisons. Consequently, the effect of the
number of runs for performance evaluation was investigated with two workloads:
isolation with current; and diagnostics with torque. For the first workload,
with the single run in Fig. 6a, Nadam and Adam achieved similar loss values,
but with ten runs as shown in Fig. 14a, Adam became the favorite in terms of
median final validation loss. With forty runs (Fig. 14b), Nadam provided a
better median loss value with Adam and rmsprop not too far behind.

a)

b)

Figure 14: Validation loss profile from current monitoring for isolation with
the different number of runs. a) Ten Runs; b) Forty runs.

a)

b)

c)

Figure 15: Best validation loss of the diagnostics classifier from torque
monitoring with different numbers of runs. a) - Five runs; b) - Ten runs; c) -
Forty runs.

For the second workload, with one run, Nadam achieved the lowest minimum loss
(Fig. 12a). With five runs, the best median was seen with Adam in Fig. 15a.
Ten runs favored rmsprop (Fig. 15b), whereas forty runs suggested SGD as the
best (Fig. 15c).

From the analyses in this section, two deductions can be made. First, the best
optimizer choice for any workload depends on the number of runs. Second, the
optimal number of runs in this work was thirty for the isolation task and
twenty for the diagnostics task. Comparing Fig. 6b and Fig. 14b respectively
representing thirty runs and forty runs of the same workload as well as Fig.
12b and Fig. 15c respectively representing twenty runs and forty runs of
similar workloads, it is seen that the median values of the different
optimizers did not significantly change and the best and worst optimizers
remained the same. This justifies the number of runs used for comparisons in
the preceding sections (IV-A and IV-B).

While the proposed classification algorithm exhibits promising results within
the simulated environment, its feasibility in real-world experimental data
warrants thorough consideration. Although some measurement noise was added to
the simulation to make it more realistic, transitioning from simulated to
real-world data introduces complexities such as uncertainties, and unforeseen
operating conditions. Additionally, the precise replication of fault scenarios
encountered in real-world scenarios within simulations may be challenging.
Hence, the algorithm’s robustness and adaptability to real-world dynamics need
validation through comprehensive experimental testing. Real-world experimental
data collection from the BWRX-300 SMR, incorporating diverse operating
conditions and fault scenarios when they become operational, would provide
invaluable insights into the algorithm’s performance. Moreover, the
algorithm’s sensitivity to sensor placement and the availability of sensor
data in practical SMR setups must be assessed. Addressing these challenges and
validating the algorithm’s efficacy in real-world scenarios is crucial for
ensuring its successful deployment in enhancing fault detection and
diagnostics in SMR control systems.

##  V Conclusion

Firstly, it was shown from the test results that fault isolation and
diagnostics of the BWRX-300 FMCRD can be consistently achieved with the
proposed methods. Secondly, from the brief comparison of the monitored
properties for each task, the electric current signal appears to be a silver
bullet useful for all the investigated tasks. Therefore, a sensor that
measures current or proxy properties should be strategically placed in
relevant areas in the BWRX-300 SMR to swiftly and accurately detect, isolate,
and diagnose its FMCRD faults to potentially ultimately drive down its O&M
costs. Thirdly, based on the selection criteria, dataset, and methods applied,
Nadam was the best optimizer for most of the investigated tasks. Lastly, it
was also shown that empirical comparison of optimizers should treat the number
of runs for model assessment as hyperparameters to be optimized.

As an extension to this work, tuning the hyperparameters of each optimization
algorithm before comparing the best models should be considered. Furthermore,
setting a target validation error and measuring the speed of convergence to
that target point across multiple runs for different optimizers could be
examined.

## Acknowledgment

This material is based upon work supported by the Department of Energy under
Award Number DE-NE0009278.

This report was prepared as an account of work sponsored by an agency of the
United States Government. Neither the United States Government nor any agency
thereof, nor any of their employees, makes any warranty, express or implied,
or assumes any legal liability or responsibility for the accuracy,
completeness, or usefulness of any information, apparatus, product, or process
disclosed, or represents that its use would not infringe privately owned
rights. Reference herein to any specific commercial product, process, or
service by trade name, trademark, manufacturer, or otherwise does not
necessarily constitute or imply its endorsement, recommendation, or favoring
by the United States Government or any agency thereof. The views and opinions
of authors expressed herein do not necessarily state or reflect those of the
United States Government or any agency thereof.

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