Online Anomaly Detection for Nuclear Power Plants via Hybrid Concept Drift

Abstract

Timely detection of anomalies in nuclear power plants (NPPs) is essential for operational safety, especially under conditions where process signals deviate gradually or abruptly from nominal patterns. Traditional detection methods often struggle to adapt under transient conditions or in the absence of well-labeled fault data. To address this challenge, we propose KD-ADWIN, an adaptive concept drift-detection framework designed for unsupervised anomaly detection in dynamic industrial environments. The method integrates three core components: a Kalman-based prediction module to extract smoothed signal trends, a multi-channel detection strategy combining statistical and derivative-based drift indicators, and an adaptive thresholding mechanism that tunes detection sensitivity based on local signal variability. Evaluations on a synthetic dataset show that KD-ADWIN accurately detects both abrupt and gradual drifts, outperforming classical baselines. Further validation using full-scope simulation data from a modular high-temperature gas-cooled reactor (MHTGR) demonstrates its effectiveness in identifying concept drifts under realistic actuator and sensor fault conditions.

1. Introduction

As complex and high-reliability systems, nuclear power plants (NPPs) are designed to operate under stringent safety requirements. However, despite rigorous engineering safeguards, uncertainties such as sensor drift, equipment degradation, and human error remain inherent throughout operations [1,2,3]. When such events occur, their effects may propagate through coupled subsystems, potentially compromising plant safety. In many cases, abnormal events manifest as transient processes [4], where system behavior gradually or abruptly deviates from normal operational patterns. During such transients, the statistical characteristics of operating data often shift over time. Timely identification of such changes allows for earlier detection of anomalies and more accurate classification of transient behaviors.

Transient detection in NPPs has been extensively studied through various methodologies, which can be mainly categorized into rule-based, physics-based, and data-driven approaches. Rule-based methods [5,6], such as thresholding and logic inference schemes, offer simplicity and fast response. However, they are often too rigid to accommodate the dynamic and nonlinear nature of transient processes in NPPs, leading to inaccurate or delayed detection. Physics-based methods provide strong interpretability and are effective when physical parameters are well understood [7,8]. However, developing accurate physical models for NPPs is particularly challenging, not only due to their inherent nonlinearities and tightly coupled feedback loops, but also because of interactions across mechanical, thermal, and control subsystems. These challenges are further amplified under transient or degraded operating conditions, where system dynamics can deviate significantly from nominal behavior. As a result, increasing attention has been directed toward data-driven approaches that rely on operational history rather than explicit physical laws. Among these, supervised-learning techniques can offer high detection precision, including hidden Markov models [9], support vector machines [10,11], and artificial neural networks [12,13,14]. However, their deployment in NPPs remains limited by the scarcity of representative fault data, as abnormal events are rare, diverse, and often challenging to label precisely.

Given the difficulty of obtaining well-labeled fault data in NPPs, unsupervised transient detection methods have gained increasing traction. These approaches do not require predefined failure models or large annotated datasets. Instead, they learn normal behavioral patterns from historical or real-time operational data and flag deviations as potential anomalies. Unsupervised methods applied in NPP anomaly detection can be broadly classified into three categories. The first group includes clustering and isolation-based techniques, such as kernel self-organizing maps [15], online dynamic quantum clustering [16], and isolation forest [17]. These methods rely on structural or statistical separation of abnormal samples in feature space without prior labeling. The second group is based on reconstruction models, such as convolutional autoencoders [18] and variational autoencoders combined with isolation mechanisms [19]. These methods compress input signals into latent representations and reconstruct them, detecting anomalies by evaluating reconstruction errors. The third group consists of optimization-based approaches, including quantum evolutionary algorithms [20]. These approaches often assume that the statistical properties of system data remain stationary. In reality, system dynamics may shift during transients, changing the data distribution, which can undermine the effectiveness of detection models and lead to increased false alarms or missed events.

Concept drift detection offers a promising solution to this challenge, which describes the temporal evolution of data distributions in streaming or time-dependent environments [21]. This approach has shown broad applicability in dynamic domains, including IoT [22] and smart grids [23]. Several efforts have also explored the application of concept drift in the nuclear domain. For example, the concept drift window re-training method uses long short-term memory to adapt modular high-temperature gas-cooled reactor (MHTGR) transient models under data distribution changes [24]. Similarly, the bridging distribution adaptive network framework enables adaptive prediction by responding to gradual and abrupt sensor drifts [25]. These studies demonstrate the potential of drift-aware techniques to enhance modeling flexibility and robustness in complex industrial systems. However, existing approaches still face key limitations. Most existing drift detectors are primarily designed to detect either gradual or sudden changes, but not both in a unified framework. In addition, fixed thresholding may further reduce sensitivity under evolving operational regimes.

To this end, we propose a hybrid anomaly-detection framework named KD-ADWIN (Kalman-Derivative-Adaptive Windowing). Unlike conventional concept drift approaches that rely on statistical windowing, KD-ADWIN enhances detection sensitivity and robustness through three integrated mechanisms. First, a Kalman filter is applied to extract stable signal trends. Second, KD-ADWIN employs a multi-channel drift-detection strategy. In addition to the classical Adaptive Windowing (ADWIN) algorithm mechanism, which detects gradual distributional changes through adaptive window comparisons, the framework introduces parallel detection channels based on the predicted signal’s first-order and second-order derivatives. Third, an adaptive thresholding mechanism is introduced to tune each detection component’s sensitivity automatically. On the synthetic dataset, KD-ADWIN can accurately capture both abrupt and gradual concept drifts, achieving higher F1-scores than the baseline algorithms ADWIN and Kolmogorov–Smirnov Windowing (KSWIN), with peak memory usage comparable to KSWIN. On the MHTGR simulation data, KD-ADWIN enables accurate detection and early warning of concept drifts in transients caused by operator errors and sensor drift.

The main contributions of this paper are summarized as follows:

(1)

We propose KD-ADWIN, a hybrid concept drift-detection framework that integrates Kalman-based prediction, adaptive windowing, and derivative monitoring. The architecture enables timely and robust detection of distributional shifts in dynamic process environments.

(2)

A unified mechanism is introduced to dynamically adjust detection thresholds and window sizes based on signal volatility. This mechanism improves sensitivity to both gradual and abrupt changes, while reducing false alarms under stable conditions.

(3)

Extensive simulations on synthetic data and full-scope MHTGR simulations demonstrate that KD-ADWIN achieves higher detection accuracy and earlier anomaly identification compared to existing methods, with competitive memory usage suitable for real-time applications.

The remainder of this article is organized as follows. Section 2 formulates the anomaly detection problem as an unsupervised task of detecting distributional changes in streaming process data. Section 3 presents the proposed KD-ADWIN framework, detailing its hybrid architecture that combines Kalman-based forecasting, derivative-based detection, and adaptive thresholding. Section 4 reports the simulation results on both synthetic and simulated nuclear datasets, evaluating the detection accuracy and responsiveness under various drift scenarios. Finally, Section 5 concludes the study and outlines directions for future work.

2. Problem Formulation

In this section, we formalize the anomaly-detection task in NPP monitoring as a distributional change detection problem. We adopt the concept drift to describe evolving system behavior. This leads to an unsupervised problem formulation that seeks to identify time points at which the underlying data distribution undergoes significant changes, without requiring prior knowledge of fault types or labels.

2.1. Data Drift in NPP Operations

NPPs operate under highly dynamic and tightly coupled conditions, where control actions, load demands, environmental variations, and component aging influence system behavior. These factors introduce non-stationary characteristics into the recorded signals, commonly called data drift. For example, gradual shifts in sensor calibration or equipment degradation can cause continuous changes in measured signals. Besides, component malfunctions can cause abrupt changes in system states. In these cases, the key operating variables, such as coolant flow rate or reactor core temperature, may display statistical shifts in mean or variance.

To further illustrate the data drift in NPP operations, Figure 1 shows the evolution of normalized reactor power in a modular high-temperature gas-cooled reactor, based on simulated data under continuous operator disturbance. As shown in Figure 1a, the time series captures both gradual and abrupt changes in reactor output. Gradual drift is reflected in smooth upward or downward trends. In contrast, abrupt transitions appear as sudden jumps or drops. To further analyze the temporal shifts in reactor output, the time-series data were divided into 9 segments. For each segment, the distribution of reactor power values was computed, and a boxplot was constructed to represent the median, interquartile range, and outliers of the distribution. As shown in Figure 1b, the signal distribution is relatively narrow and stable in the early segments (1–4). It shows low variance and a consistent median. In segments 5 through 7, the boxplots reveal a steady elevation in median reactor power along with increasing variability, consistent with a gradual drift. By contrast, the rise in both levels and the spread in segments 8 and 9 shows an abrupt change.

2.2. Modeling with Concept Drift

The NPP example shows that operational data can exhibit both gradual and abrupt drift. To provide a formal description of these temporal changes, we adopt the concept drift framework, which has been widely used to analyze non-stationary data streams. To formalize the notion of drift, let $X\in {\mathbb{R}}^d$ represent the input variable (e.g., sensor or process signals), and $y\in \mathcal{Y}$ denote the corresponding system state or label. Then, concept drift is said to occur between time $t_0$ and $t_1$ if:

$$\exists X:p_{t_0}(X,y)\ne p_{t_1}(X,y),$$

(1)

where $p_t$ denotes the joint distribution of the input variable set X and the target variable y at time t.

Figure 2 provides a schematic illustration of this idea. In Figure 2a, samples from two classes are shown at two time points. At $t_0$, the joint distribution $p_{t_0}(X,y)$ allows for a classification boundary that separates the two groups. At a later time $t_1$, the input–output relationship has changed, resulting in a different distribution $p_{t_1}(X,y)$. Figure 2b illustrates a similar idea in a time-series setting. The top row shows a stream of sequential data, where color changes reflect potential variation in statistical characteristics. The lower part shows how the data distribution shifts over time, including changes in the location and shape of the density functions. These changes may arise gradually or abruptly, and capturing them is necessary for applications involving streaming or time-varying signals.

2.3. Drift-Based Problem Formulation

Operational data in NPPs are non-stationary and can exhibit distributional shifts due to transients, control actions, degradation, or faults [26]. We therefore formulate anomaly detection as identifying significant distributional changes in the observed data stream. This aligns with the concept drift, where changes in the joint distribution $p(X,y)$ can be partially captured by monitoring the marginal distribution $p\left(x\right)$. Accordingly, we model anomaly detection as the task of detecting time points where $p\left(x_t\right)$ changes significantly.

Let ${\left\{x_t\right\}}_{t=1}^T$, with $x_t\in {\mathbb{R}}^d$, denote a multivariate time series of d-dimensional sensor or system variables recorded over time. The goal is to identify one or more drift points $t_d\in \{1,\dots ,T\}$, such that:

$$p\left(x_{t_d^{-}}\right)\ne p\left(x_{t_d^{+}}\right),$$

(2)

where $x_{t_d^{-}}$ and $x_{t_d^{+}}$ represent samples from time intervals immediately before and after $t_d$, respectively. A detected drift point $t_d$ is then interpreted as a potential indicator of abnormal system behavior.

In practice, these distributional changes may occur gradually or abruptly, affecting statistical properties such as the mean, variance, correlation structure, or higher-order dependencies. The formulation accommodates both types of changes without assuming a specific failure model or labeled events. This formulation does not assume prior knowledge of specific fault patterns or labeled abnormal data. Instead, it provides a general framework for unsupervised detection based on distributional monitoring. In the NPP, such a formulation is beneficial under transient conditions or during early degradation stages, where system labels may be unavailable but distributional signals still reflect deviations from nominal behavior.

3. Methodology

ADWIN is a classical concept drift-detection approach that maintains a variable-length sliding window and detects drift based on changes in the mean between subwindows [27]. The window size expands under stable conditions and contracts when fluctuations are observed, allowing it to adapt to varying drift rates. ADWIN effectively detects gradual shifts, as it accumulates sufficient evidence before declaring a change. However, detection may be delayed for abrupt changes. KSWIN offers an alternative by applying the Kolmogorov–Smirnov test to detect changes in the empirical distribution within a sliding window [28]. Its nonparametric nature allows for sensitivity to subtle and continuous changes in data structure. However, similar to ADWIN, it suffers from delayed detection when the drift is abrupt, as enough post-change samples must be collected before a shift is confirmed.

These limitations motivate the development of more flexible and responsive mechanisms that can capture both gradual and sudden changes in NPP data. To this end, we propose a hybrid framework named KD-ADWIN (Kalman-Derivative-Adaptive Windowing). As illustrated in Figure 3, the method integrates Kalman-based prediction, hybrid-drift detection, and an adaptive thresholding mechanism. These modules are designed to jointly enhance sensitivity and robustness under both gradual and abrupt drift scenarios. Each module is introduced with its motivation, functional role, and implementation as follows.

3.1. Kalman-Based Prediction

To improve the responsiveness of drift detection, we introduce a Kalman-based prediction module that generates short-horizon forecasts of monitored signals. The predicted values themselves are incorporated into the drift-detection process, allowing the system to respond to potential changes before they fully manifest in the raw data.

The Kalman filter is formulated in the standard discrete-time state-space representation:

$$\begin{array}{cc}\hfill {\mathbf{z}}_t& =\mathbf{A}{\mathbf{z}}_{t-1}+{\mathbf{w}}_t,\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill {\mathbf{x}}_t& =\mathbf{H}{\mathbf{z}}_t+{\mathbf{v}}_t,\hfill \end{array}$$

(4)

where ${\mathbf{z}}_t$ is the hidden state at time t, ${\mathbf{x}}_t$ is the observed variable (e.g., a sensor signal), $\mathbf{A}$ is the state transition matrix, and $\mathbf{H}$ is the observation matrix. The process noise ${\mathbf{w}}_t$ and observation noise ${\mathbf{v}}_t$ are assumed to follow zero-mean Gaussian distributions with covariances $\mathbf{Q}$ and $\mathbf{R}$, respectively.

At each time step, the Kalman filter computes the predicted observation ${\widehat{\mathbf{x}}}_t=\mathbf{H}{\widehat{\mathbf{z}}}_t^{-}$, where ${\widehat{\mathbf{z}}}_t^{-}$ is the prior estimate of the state based on the model. These predicted values ${\widehat{\mathbf{x}}}_t$ are then passed to the downstream drift-detection modules. This predictive mechanism enables the framework to monitor changes in the expected trajectory of the signal, providing a forward-looking capability that improves sensitivity to emerging drifts. In scenarios where real measurements have not yet changed significantly, the forecasted values may offer early warnings of potential anomalies.

3.2. Hybrid Drift Detection

To detect changes in the data distribution, we construct a hybrid drift-detection mechanism that combines both statistical monitoring and derivative-based information. This module operates on the outputs of the Kalman-based predictor, using both the original signals and short-horizon forecasts.

Let $x_t$ denote the observed value at time t, and ${\widehat{x}}_t$ be the corresponding prediction. The effective input to the detection mechanism is the combined stream ${\tilde{x}}_t$, which includes both actual observations and predicted values, ${\tilde{x}}_t:=\{x_t,{\widehat{x}}_t\}$. A sliding window ${\mathcal{W}}_t=\{{\tilde{x}}_{t-w+1},\dots ,{\tilde{x}}_t\}$ is maintained. At each time step t, the current window ${\mathcal{W}}_t=\{{\tilde{x}}_{t-w+1},\dots ,{\tilde{x}}_t\}$ of size w is partitioned into two adjacent subwindows:

$${\mathcal{W}}_0=\{{\tilde{x}}_{t-w+1},\dots ,{\tilde{x}}_{t-k}\},{\mathcal{W}}_1=\{{\tilde{x}}_{t-k+1},\dots ,{\tilde{x}}_t\},$$

(5)

where k is the splitting point satisfying $1<k<w$. Let ${\mu }_0$ and ${\mu }_1$ denote the sample means and ${\sigma }_0^2$ and ${\sigma }_1^2$ denote the sample variances, of the left and right subwindows, respectively.

To detect a drift in the mean, we compute the absolute difference between ${\mu }_0$ and ${\mu }_1$, and check if it exceeds a time-dependent threshold ${\delta }_{\mu }\left(t\right)$, derived using Hoeffding’s inequality [29]:

$${\delta }_{\mu }\left(t\right)=\sqrt{{\displaystyle \frac{1}{2n}}ln\left({\displaystyle \frac{4N}{\delta }}\right)},n={\displaystyle \frac{w_L{w}_R}{w_L+w_R}},$$

(6)

where N is the total number of samples in the window, and $w_L$ and $w_R$ are the sample sizes in the left and right subwindows, respectively. Similarly, to detect drift in the variance, we compute the absolute difference between ${\sigma }_L^2$ and ${\sigma }_R^2$ and check if it exceeds a time-dependent threshold ${\delta }_{{\sigma }^2}\left(t\right)$, derived using Bernstein’s inequality [30]:

$${\delta }_{{\sigma }^2}\left(t\right)=\sqrt{{\displaystyle \frac{2{\widehat{\sigma }}^{2}ln(4N/\delta )}{m}}}+{\displaystyle \frac{2ln(4N/\delta )}{3m}},$$

(7)

where ${\widehat{\sigma }}^2$ is the maximum variance in the two subwindows, and m is the harmonic mean of the sample sizes from the two subwindows.

To improve the detection responsiveness, particularly for abrupt changes, we introduce derivative-based information. The first-order and second-order differences are computed from the predicted values:

$$\mathsf{\Delta }{\widehat{x}}_t={\widehat{x}}_t-{\widehat{x}}_{t-1},{\mathsf{\Delta }}^2{\widehat{x}}_t=\mathsf{\Delta }{\widehat{x}}_t-\mathsf{\Delta }{\widehat{x}}_{t-1}.$$

(8)

To suppress noise and improve robustness, these derivatives are further smoothed using a simple moving average over the current window of size w:

$${\overline{\mathsf{\Delta }\widehat{x}}}_t={\displaystyle \frac{1}{w}}\sum _{i=0}^{w-1}\mathsf{\Delta }{\widehat{x}}_{t-i},{\overline{{\mathsf{\Delta }}^2\widehat{x}}}_t={\displaystyle \frac{1}{w}}\sum _{i=0}^{w-1}{\mathsf{\Delta }}^2{\widehat{x}}_{t-i}.$$

(9)

A drift alert is triggered if either the smoothed values exceed dynamically adjusted thresholds ${\tau }_t^{\left(1\right)}$ and ${\tau }_t^{\left(2\right)}$. The final detection decision at time t is obtained by fusing the results of the window-based and derivative-based detectors:

$$D_t=\mathbb{I}\left(|{\mu }_0-{\mu }_1|>{\delta }_{\mu }\left(t\right)\vee |{\sigma }_0^2-{\sigma }_1^2|>{\delta }_{{\sigma }^2}\left(t\right)\right)\vee \mathbb{I}\left(\left|{\overline{\mathsf{\Delta }\widehat{x}}}_t\right|>{\tau }_t^{\left(1\right)}\vee \left|{\overline{{\mathsf{\Delta }}^2\widehat{x}}}_t\right|>{\tau }_t^{\left(2\right)}\right),$$

(10)

where $\mathbb{I}(·)$ denotes the indicator function, which outputs 1 if the condition inside is true, and 0 otherwise.

3.3. Adaptive Thresholding Mechanism

To maintain responsiveness and robustness under dynamic signal conditions, KD-ADWIN uses a sliding window ${\mathcal{W}}_t$ with adaptive size control. The window grows over time until it reaches a maximum size $w_{max}$. When this limit is reached, the oldest sample is removed as new data arrives. If a drift is detected, the window is reset, as illustrated in Figure 4. In this case, all data before the drift point is discarded, and the new window is initialized with the most recent samples. The window size is then set to a minimum value $w_{min}\left(t\right)$ to support faster detection of subsequent changes.

To determine the appropriate value of $w_{min}\left(t\right)$, we introduce a relative indicator of signal dynamics based on the smoothed second-order derivative, ${\overline{A}}_2\left(t\right):=\left|{\overline{{\mathsf{\Delta }}^2\widehat{x}}}_t\right|$. Instead of setting fixed thresholds, we compute the logarithmic change in this quantity to capture relative variation:

$${\mathsf{\Delta }}_{log}{A}_2\left(t\right):=log({\overline{A}}_2\left(t\right)+1)-log({\overline{A}}_2(t-1)+1),$$

(11)

where ${\overline{A}}_2\left(t\right)$ and ${\overline{A}}_2(t-1)$ denote the smoothed second-order derivatives computed from the current and previous windows, respectively. The update rule for $w_{min}\left(t\right)$ is then defined as:

$$w_{min}\left(t\right)=\left\{\begin{array}{cc}max(w_{min}(t-1)-\lambda ,w_{\mathrm{lower}}),\hfill & \mathrm{if}{\mathsf{\Delta }}_{log}{A}_2\left(t\right)>\delta ,\hfill \\ min(w_{min}(t-1)+\lambda ,w_{\mathrm{upper}}),\hfill & \mathrm{if}{\mathsf{\Delta }}_{log}{A}_2\left(t\right)<-\delta ,\hfill \\ w_{min}(t-1),\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

(12)

where $\lambda$ is a fixed step size, and $w_{\mathrm{lower}},w_{\mathrm{upper}}$ are the allowed bounds. The threshold $\delta$ controls the sensitivity to relative changes. When the signal changes rapidly, ${\overline{A}}_2\left(t\right)$ increases, leading to a positive ${\mathsf{\Delta }}_{log}{A}_2\left(t\right)$ and a smaller window size to improve reaction speed. When the signal becomes smoother, a negative log change enlarges the window to suppress false positives. If the change is minor, the window size remains stable.

In addition, we adjust the dynamic detection thresholds for the first- and second-order derivatives. For each derivative order $n\in \{1,2\}$, the threshold is computed as:

$${\tau }_t^{\left(n\right)}={\mu }_{{\mathsf{\Delta }}^n}+k_n\left(t\right)·{\sigma }_{{\mathsf{\Delta }}^n},$$

(13)

where ${\mu }_{{\mathsf{\Delta }}^n}$ and ${\sigma }_{{\mathsf{\Delta }}^n}$ are the mean and standard deviation of the n-th derivative within the current window, and $k_n\left(t\right)$ is a scaling factor that determines detection sensitivity. We update $k_n\left(t\right)$ using the same relative change indicator ${\mathsf{\Delta }}_{log}{\overline{A}}_2\left(t\right)$:

$$k_n\left(t\right)=\left\{\begin{array}{cc}{k}_n(t-1)+{\delta }_k,\hfill & \mathrm{if}{\mathsf{\Delta }}_{log}{\overline{A}}_2\left(t\right)>\delta ,\hfill \\ k_n(t-1)-{\delta }_k,\hfill & \mathrm{if}{\mathsf{\Delta }}_{log}{\overline{A}}_2\left(t\right)<-\delta ,\hfill \\ k_n(t-1),\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

(14)

where ${\delta }_k$ is a small step size for gradual tuning. When signal dynamics increase, the threshold factor $k_n\left(t\right)$ is raised to suppress transient alerts. When the signal becomes smoother, the threshold is lowered to improve sensitivity and reduce missed detections. This joint adaptation of window and thresholds provides a unified mechanism for tuning detection behavior under diverse operating conditions.

3.4. Anomaly-Detection Workflow

Figure 3 illustrates the overall system architecture and data flow of KD-ADWIN. The framework operates in an online fashion and sequentially processes time-series data to detect distributional changes associated with system anomalies. At each time step, raw sensor data enters the Kalman-based prediction module, which applies a state-space model to estimate the short-horizon signal trajectory. The predicted values ${\widehat{x}}_t$ are used alongside the observed signal $x_t$ to form a combined stream, which is passed into the hybrid drift-detection module. Within this module, the combined stream is analyzed by two parallel mechanisms. The first applies a statistical drift-detection scheme based on a two-part sliding window and adaptive statistical tests on the mean and variance. The second computes smoothed first- and second-order derivatives from the predicted data to enhance sensitivity to abrupt changes. The outputs from both paths are forwarded to the joint drift decision module, where a change is declared if any of the criteria are met.

Once a drift is detected, the adaptive thresholding mechanism updates key parameters in real time. This includes adjusting the minimum window size $w_{min}\left(t\right)$ and the derivative thresholds ${\tau }_t^{\left(1\right)}$ and ${\tau }_t^{\left(2\right)}$ based on recent changes in signal behavior. These updates are fed back into the detection loop, enabling the system to maintain appropriate sensitivity under different drift patterns. The final output is a binary anomaly flag at each time step. This closed-loop process allows the KD-ADWIN framework to online operate continuously and adaptively, adjusting its detection strategy in response to evolving system dynamics.

4. Simulations and Analysis

To evaluate the performance of the KD-ADWIN framework, simulations were conducted on both a synthetic dataset and the MHTGR simulator dataset. On the synthetic dataset, KD-ADWIN can accurately capture both abrupt and gradual concept drifts, achieving higher F1-scores than the baseline algorithms ADWIN and KSWIN, with memory usage comparable to KSWIN. On the MHTGR simulation data, KD-ADWIN enables accurate detection and early warning of concept drifts in transients caused by operator errors and sensor drift.

4.1. Simulations on Synthetic Dataset

To evaluate the performance of the proposed KD-ADWIN framework in non-stationary environments, we first conduct simulations on a synthetic dataset designed to simulate concept drift. Compared to real-world benchmarks, synthetic data allows precise control over the timing, type, and severity of drift events, thus providing a clear and repeatable basis for performance comparison [31].

The dataset is constructed as a time series with ${10}^4$ time steps. It comprises 11 segments with distinct statistical characteristics, including both abrupt and gradual drifts. Each segment is generated from a Gaussian distribution with independently sampled mean and variance. For abrupt drifts, the mean is randomly selected from $[-2,2]$ and the variance from $[0.05,0.5]$. For gradual transitions, the mean and variance change linearly from the values of the preceding segment to new values sampled from $[-3,3]$ and $[0.5,2.0]$, respectively. The resulting stream includes 6 abrupt drift points and 4 gradual drifts. These statistical variations simulate both rapid and progressive operational changes commonly observed in sensor-based industrial monitoring.

To assess detection performance, we adopt a tolerance-based matching strategy [32,33] between detected and ground-truth drift points. A detected drift is counted as a true positive (TP) if it falls within a fixed tolerance window around a true drift point. Otherwise, it is considered a false positive (FP). If no detection occurs within the tolerance window of a ground-truth point, it is labeled as a false negative (FN). This matching scheme accounts for the inherent uncertainty in detecting gradual drifts and reduces the penalization of small detection delays. We use four quantitative metrics to evaluate each method: precision, recall, F1-score, and peak memory usage. Precision is defined as the fraction of correct detections among all reported drifts. Recall reflects the fraction of actual drift events that are successfully detected. The F1-score provides a harmonic mean between precision and recall, summarizing the overall accuracy. Peak memory usage is measured as the maximum resident memory footprint throughout the execution, indicating the algorithm’s resource efficiency under streaming conditions.

The baseline methods include ADWIN [27] and KSWIN [28], both of which are model-free detectors commonly used in streaming data scenarios. ADWIN detects distributional changes by maintaining an adaptive sliding window and comparing means across subwindows. In contrast, KSWIN uses the Kolmogorov–Smirnov test to identify statistical shifts in empirical distributions. For consistency, standard implementations from the River library are adopted [34]. Figure 5 shows the detection results of these three methods on a synthetic dataset. The blue curve represents the input signal, which consists of 11 segments generated from different distributions. Vertical green lines mark the true drift points. For evaluation purposes, two types of tolerance windows are defined around each true drift point: gray bands indicate abrupt drifts ($\pm 50$ time steps), while yellow bands mark gradual drifts over the same interval. Red dashed lines denote the drift points detected by each method. As shown, the proposed KD-ADWIN successfully detects all eight ground truth drift events, including both abrupt and gradual types, without missing any. ADWIN detects several gradual drifts but fails to respond to some abrupt changes, likely due to the time it needs to accumulate sufficient evidence in its window-based structure. KSWIN captures most abrupt events but generates more false positives, suggesting that its sensitivity to short-term distributional fluctuations may affect precision. Compared with the baselines, KD-ADWIN provides more comprehensive and reliable coverage across different drift types.

Memory usage and computational performance were measured on the computational platform (Python CPython 3.11.1; macOS-15.5-arm64; 8 cores; 8 GB RAM). Figure 6a illustrates the memory usage over time for the three detection algorithms. ADWIN (blue) maintains the lowest memory footprint throughout the entire test window, with peak usage at $222.72$ MB. KSWIN (orange) and KD-ADWIN (green) both show elevated memory consumption, with peak usage reaching $375.28$ MB and $387.33$ MB, respectively. The slight increase in KD-ADWIN compared to KSWIN is attributed to the added cost of derivative-based monitoring and adaptive thresholding. Despite this, KD-ADWIN remains within the same magnitude as KSWIN and well below practical limits for online deployment. Figure 6b provides a quantitative comparison of detection performance under a fixed tolerance window. KD-ADWIN achieves the highest F1-score of $1.0$, reflecting a perfect balance between precision and recall. Specifically, its precision and recall both reach $1.0$, indicating that all detected events are true positives with no missed or spurious detections. KSWIN yields moderate performance, with an F1-score of $0.6364$, precision of $0.5$, and recall of $0.875$, suggesting strong sensitivity but reduced specificity. ADWIN performs worst among the three, with an F1-score of $0.4$, precision of $0.33$, and recall of $0.5$, consistent with its underperformance in identifying abrupt drifts. Computational efficiency analysis (Figure 6c) reveals per-update processing times of $0.002$ ms for ADWIN, $0.118$ ms for KSWIN, and $0.033$ ms for KD-ADWIN. Despite integrating multiple techniques, KD-ADWIN maintains sub-millisecond processing times, positioning between the two baselines, and confirming its suitability for real-time applications. Overall, KD-ADWIN demonstrates superior detection accuracy while maintaining manageable memory overhead and acceptable computational cost, validating its effectiveness and practical feasibility for real-time drift-detection in streaming applications.

4.2. Simulations on MHTGR Dataset

To further assess the applicability of the proposed framework under operating conditions, we conducted simulations on a full-scope dynamic simulator of HTR-PM600. The HTR-PM plant entered commercial operation on 6 December 2023. As the world’s first built and operational MHTGR plant, HTR-PM represents a significant achievement in advanced nuclear technology with inherent safety features and enhanced safety margins during transient conditions [35]. As a subsequent commercial version of HTR-PM, the HTR-PM600 plant aims to use the proven HTR-PM reactor and steam generator modules, with six modules connected in parallel to form a 600 MWe commercial-scale MHTGR plant [36]. The simulator models the coupled behavior of reactor physics, thermal-hydraulics, and instrumentation systems, and supports fault injection within a closed-loop control environment. Previous studies on HTR models have demonstrated the complexity of whole-core simulation and the challenges in maintaining numerical stability [37].

Two representative scenarios were examined. The first involves misacceleration of the main helium circulator, characterizing an actuator fault induced by operational error. The second introduces an exponential drift in sensor readings, simulating progressive sensor degradation. These scenarios reflect typical but challenging anomalies in NPPs, where underlying distributional changes may manifest either gradually or abruptly, posing difficulties for timely and accurate detection.

4.2.1. Scenario A: Main Helium Circulator Misacceleration

The detection performance of KD-ADWIN is first evaluated under the main helium circulator misacceleration scenario, where 9 manual steps incrementally increase the flow rate during a single operation cycle. This scenario simulates an anomaly caused by operator misoperation. From a stable $100\%$ RFP (Relative Full Power) state, the speed of the main helium circulator in one module is manually increased in small increments. This action leads to a progressively increasing coolant flow rate, disrupting the system’s thermal balance. Consequently, the reactor power may unexpectedly rise due to positive reactivity feedback, while temperature differences across the steam generator decrease. If unaddressed, this condition can trigger protective alarms and ultimately lead to a reactor trip. The key events are summarized in

Table 1. Test log for main helium circulator misacceleration scenario.

Time Step (s)	Operations and HTR-PM600 Status
0–305	Initial state: Six-module operation at 100% RFP, coordinated control active.
306–3479	Test start: Nine manual speed adjustments (1% each) performed at intervals of 5 to 8 min.
3480	First light board alarm: High normalized mass flow ratio in primary/secondary loops ($\ge 1.2$).
3635	Test conclusion: Reactor trip triggered by high mass flow ratio alarm ($\ge 1.3$).

.

Figure 7 illustrates the drift-detection results for the primary loop flow rate under the circulator misacceleration scenario. A total of 20 drift points are identified along the normalized trajectory. These points align closely with both the operator-induced setpoint changes and the gradual reactor response. However, no single detection module captures all events. Several drift points are exclusively flagged by the ADWIN-based channel, while others arise solely from the first- or second-order derivative signals. This complementarity highlights the benefit of the hybrid detection strategy, where the integration of multiple mechanisms improves overall coverage and reduces missed events. The lower panels show the absolute values of the first- and second-order derivatives computed from the smoothed signal, alongside their respective dynamic thresholds. These thresholds vary over time in response to signal volatility. Notably, following closely spaced or frequent drifts, such as those around $t=400$ s, and $t=1100$ s, the thresholds increase temporarily. This adaptation reduces spurious triggering during high-activity intervals. In contrast, during more quiescent phases, the thresholds decline, enabling the detection of more subtle deviations. The observed dynamics confirm that the adaptive thresholding mechanism effectively regulates the sensitivity of each channel based on context, mitigating the risk of over-detection while maintaining responsiveness.

To investigate the internal behavior of the adaptive detection mechanism, Figure 8a illustrates the evolution of the window size over time. Each red dashed line marks a detected drift. As expected, the window size drops sharply after each drift, indicating that stale data are discarded and the algorithm prioritizes new distributional patterns. Unlike fixed reset-window schemes, however, the window’s minimum size after each reset is not constant. It is dynamically modulated by the recent drift profile. Specifically, when consecutive drifts are abrupt and closely spaced, the minimum window size is temporarily increased to improve robustness and avoid false positives. In contrast, under more gradual transitions, the minimum window size is reduced to enhance sensitivity. This behavior is observable in the figure, where some post-drift windows reset to higher sizes than others. For example, around t = 1500 s and t = 2800 s, the reset level is noticeably elevated compared to earlier points. This suggests that the system has internally classified the surrounding changes as high-volatility, prompting it to increase memory depth.

The robust performance of KD-ADWIN also depends on the quality of its input signals. In addition to the observed data stream, the Kalman-based prediction module supplies a forecast of the next signal value, which is jointly fed into the detection pipeline. This predictive input enriches the detection window with forward-looking context, allowing both the ADWIN-based and derivative-based branches to respond to incipient changes before fully manifesting in the measurement stream. Figure 8b compares the normalized primary loop flow rate with the corresponding Kalman-predicted values during the misacceleration scenario. The predicted curve closely tracks the true signal throughout the test, including each of the nine step-like perturbations. The absence of noticeable lag or overshoot confirms that the predictor remains responsive without excessive smoothing, even under frequent state changes. This anticipatory design is similar to the derivative term in classical feedback control. In KD-ADWIN, the Kalman-enhanced input sequence supports both timely drift recognition and stable operation under varying system dynamics, where the predictive component contributes to the sensitivity of the framework.

4.2.2. Scenario B: Exponential Sensor Drift

This scenario simulates a gradual anomaly caused by sensor degradation. An exponential drift of the form $y\left(t\right)=y_0·e^{\alpha t}$ is injected into the measured helium flow rate of one reactor module. Since the control system is unaware of the fault, it attempts to compensate by reducing the actual circulator speed, resulting in a continuous decline in physical flow. As shown in

Table 2. Test log for exponential sensor drift scenario.

Time Step (s)	Operations and HTR-PM600 Status
0–119	Initial state: Stable operation at 100% RFP.
120	Test start: Exponential drift begins on the helium flow measurement.
560	First alarm: High rate of change in main helium pressure drop.
1050	Reactor trip: Hot helium temperature exceeds safety threshold (790 °C).

, the fault begins at $t=120\mathrm{s}$, eventually triggering an alarm at $t=560\mathrm{s}$ and causing a thermal protection trip at $t=1050\mathrm{s}$ when the helium temperature exceeds 790 °C.

Figure 9 illustrates the detection performance of KD-ADWIN under the exponential sensor drift scenario. In contrast to the abrupt changes in Scenario A, the primary loop flow rate in this case exhibits a smooth and continuous downward trend. As expected, the first-order and second-order derivative channels remain largely inactive during the early and middle stages of the drift, as shown in the bottom two panels. No sharp variations in gradient are present, and the thresholds are not exceeded for most of the timeline. However, at the initiation of the exponential drift ($t=120$ s), the second-order derivative briefly crosses its adaptive threshold. This early signal, although short-lived, marks the first detectable deviation from steady-state behavior. Neither the first-order derivative nor the statistical channel registers a response at that moment. This observation demonstrates that the second-order derivative is sensitive to minor curvature changes and can provide a leading indicator under certain conditions.

Subsequently, the statistical detection component (ADWIN) begins to detect a series of low-amplitude but consistent distributional shifts. These are reflected as gradual memory adjustments and multiple confirmed drift points well before the system-level alarm ($t=560$ s) and reactor protection trip ($t=1050$ s). In total, 14 drift events are identified, indicating that KD-ADWIN is capable of tracing the unfolding fault trajectory in real time. This result highlights the complementary behavior of the hybrid detection design. While derivative-based channels are tuned for abrupt responses, the statistical pathway maintains sensitivity to long-term gradual changes. Together, they enable unified drift coverage under complex and evolving system conditions.

To further evaluate input fidelity during the drift process, Figure 10 compares the normalized original signal and the Kalman-predicted flow rate. The two curves remain closely aligned despite the growing discrepancy between measurement and ground truth. This confirms that the prediction module preserves temporal coherence and enables consistent signal interpretation throughout the anomaly progression. Notably, the earliest concept drift is flagged well before the fixed-threshold alarm, confirming the framework’s capability for early warning even under slowly manifesting anomalies.

5. Conclusions

This paper presents KD-ADWIN, an adaptive concept drift-detection framework for real-time anomaly detection in dynamic process environments. The method integrates Kalman-based prediction, statistical monitoring, and derivative-based sensitivity enhancement within a unified sliding window structure. An adaptive thresholding mechanism is introduced to adjust detection sensitivity and window parameters in response to observed signal volatility. Controlled simulations on a synthetic dataset show that KD-ADWIN achieves higher detection accuracy than two widely used baselines, offering a better balance between precision and recall while maintaining competitive memory usage. Additional validation using full-scope simulations of the MHTGR nuclear power plant further supports its practical applicability. In both abrupt actuator fault and gradual sensor drift scenarios, KD-ADWIN provides early and consistent drift identification, confirming its effectiveness for high-stakes monitoring tasks. Future work will focus on improving the method’s performance in noisy environments by integrating noise-reduction techniques and adaptive filtering to enhance its robustness in real-world conditions. This will help the system better adapt to the dynamic, real-time monitoring requirements of NPPs.