An Interpretable Dynamic Feature Search Methodology for Accelerating Computational Process of Control Rod Descent in Nuclear Reactors

Abstract

Within the operational dynamics of a nuclear reactor, the customary approach involves modulating the reactor’s power output by means of control rod manipulation, which effectively alters the neutron density across the core. The descent behavior of the control rod drive lines pertains to the intricate motion exhibited by the control rod components within the reactor during its operational lifespan, characterized by conditions of heightened irradiation, temperature, pressure, and complex fluid dynamics. The precise calculation of the control rod descent process is an integral facet of reactor structural design to ensure the safe and reliable operation of the reactor. However, the current computational fluid dynamics-based simulation methods employed for this purpose necessitate extensive grid computations, imposing significant computational burdens in terms of resources and time. In light of this challenge, we present a novel and interpretative algorithm rooted in dynamic similarity feature search. Through comprehensive validation, this algorithm demonstrates remarkable precision, with the computational results exhibiting an error margin within 10% while simultaneously achieving a substantial enhancement of computational efficiency of nearly three orders of magnitude when compared to conventional computational fluid dynamics techniques and sequence-to-sequence machine learning algorithms. Notably, this algorithm showcases exceptional versatility, holding immense promise for broad applicability across various operational scenarios encountered during the intricate process of nuclear reactor design.

1. Introduction

Global advancements in low-carbon energy are progressing rapidly, and among the diverse array of energy sources, nuclear power exhibits numerous advantages, including efficient electricity generation, minimal carbon emissions, and remarkable versatility. It stands as a pivotal constituent in the realization of a sustainable future. Paramount to the operation of a nuclear reactor is ensuring the utmost safety of both the reactor and its primary loop system, thereby precluding any occurrence of fuel element rupture or consequential radiological contamination in the surrounding environment. Furthermore, under normal operating conditions, the reactor’s power output necessitates meticulous adjustments to cater to load demands.

The thermal energy output of a reactor is contingent upon the neutron density within its core. Consequently, the ultimate objective of power control in any reactor design is to meticulously modulate the reactor’s core power output by adroitly regulating the neutron flux in accordance with the specific operational requirements. In the realm of pressurized water reactors, the principal methodologies employed to modulate neutron density within the core encompass control rod manipulation, the modulation of the boron concentration, the judicious utilization of burnable poisons, and inherent Doppler feedback governed by fuel temperature transients. Control rod manipulation entails the strategic insertion or retraction of solid rod bundles, fabricated from materials exhibiting pronounced neutron absorption capabilities, such as cadmium, hafnium, silver, and indium. This manipulation effectively governs the rate of nuclear fission within the core, thereby enabling precise control over the reactor’s power output. Crucially, control rod movement perturbs localized neutron flux, which alters fuel temperature and activates the Doppler broadening effect. This approach, which capitalizes on control rod motion to dynamically regulate neutron density in the core, boasts notable advantages, including rapid response, exceptional flexibility, and heightened efficiency. Consequently, it finds widespread application in scenarios necessitating swift reactivity adjustments and remains a prevailing technique employed across the vast majority of contemporary nuclear reactor designs [1].

The control rod drive line comprises the control rod drive mechanism, internal components, and fuel assemblies. It stands as the sole equipment unit within the reactor that exhibits relative motion and bears the critical responsibility of reactor initiation, power regulation, and secure shutdown, being aptly referred to as the reactor’s ‘life line’. Through the orchestrated motion of control rod assemblies within the core, the control rod drive mechanism deftly executes a gamut of pivotal functions encompassing reactor start-up, power modulation, power sustainment, routine cessation, and contingency termination. The descent of the control rod drive line denotes the intricate motion of its constituent components within the reactor, transpiring amidst highly irradiated and elevated temperature and pressure conditions, intricately interwoven with multifaceted fluidic dynamics. The descent process entails the collective influence of fluid–solid coupling, manifesting as the interplay between the flow field and the kinematics of the control rod assemblies. Notably, the descent time serves as a pivotal parameter in safety analyses for nuclear power plants, constituting a salient metric in the appraisal of drive line design. However, the descent of control rods entails a complex kinematic journey, governed by the interplay of gravitational forces, buoyancy, fluidic resistance, and the frictional interaction arising from the supple contact between the control rod and the guide tube wall. The descent time of control rods manifests as an intricate nonlinear relationship with key performance parameters, including displacement, velocity, and acceleration. Specifically, the rod position dictates the total displacement required, while acceleration governs the rate of both acceleration and deceleration phases. These parameters interact through dynamic equilibrium, collectively shaping the temporal profile of the descent process.

The validation of the control rod drive line’s dropping characteristics currently relies primarily on experimental verification. However, the establishment of experimental setups and associated challenges may result in exorbitant design and testing costs. Therefore, it is imperative to conduct simulation studies to investigate the dropping behavior of the drive line under the influence of springs prior to experimental trials. This approach serves a dual purpose: first, it facilitates structural optimization and mitigates design risks; second, it enables the delineation of specific objectives for experimentation, narrowing down the scope and reducing testing expenses [2]. At present, the simulation of control rod drop behavior in nuclear reactors primarily employs two methods: one-dimensional hydraulic models and dynamic mesh-based computational fluid dynamics (CFDs) models. Donis [3] first introduced an innovative mathematical model specifically designed for analyzing the dropping of individual control rods. This approach simplifies the control rod drive line into a one-dimensional hydraulic model, enabling the establishment of dynamic equations to accurately solve for the control rod drop time. The utilization of one-dimensional hydraulic models has found extensive applications in engineering. To compute the precise dropping time of control rods in pressurized water reactors, Park et al. [4] developed specialized software based on a meticulously crafted one-dimensional hydraulic model. However, when endeavoring to solve the dynamic equations governing the control rod dropping phenomenon using this method, it becomes evident that the hydraulic resistance and mechanical friction coefficient elude theoretical calculations. These parameters necessitate empirical acquisition through corresponding experimental endeavors. Consequently, the method’s capacity for expansion and generalization is notably constrained. Under the pioneering work by Choi et al. [5], CFD methodologies are first employed to estimate the initial control rod drop time in the SMART (System-integrated Modular Advanced ReacTor) system [6]. Since then, CFD methods have become widely prevalent in the analysis of the control rod drop process, encompassing both pressurized water reactors [7,8,9,10,11,12,13,14,15] and Generation IV nuclear reactors [16,17]. The CFD-based simulation method for control rod dropping liberates itself from the reliance on traditional experimental approaches to a great extent. It transcends the limitations imposed by the control rod drive line structure and exhibits remarkable scalability. However, this method demands a substantial number of computational grids, thus engendering time-intensive calculations and imposing high computational requirements. During the nascent stages of new reactor development and design, it becomes imperative to expeditiously validate the drive line structure scheme to provide guidance for thermal–hydraulic design, thereby expediting the formulation of a comprehensive design framework to advance the progress of research and development. In addition, the investigation of error propagation in nuclear systems proved critical for evaluating machine learning method effectiveness [18,19]. The methods mentioned above encounter challenges in accurately simulating the control rod dropping process within a constrained timeframe, prompting the exploration of a rapid computational methodology to complement CFD simulations and expedite the maturation process of novel nuclear reactor designs.

The inverse problem of deducing unknown operational conditions based on partial results from the control rod dropping process, obtained through CFD methods, represents a quintessential generative challenge. While the input space encompasses solely the control rod height, the output space expands into a vast domain, encompassing temporal information such as displacement and velocity throughout the entirety of the control rod dropping process. Traditional machine learning and deep learning techniques excel at discriminative tasks, such as classification and regression, where the input space information surpasses that of the output space. However, their characteristic features are ill suited for our research endeavor. Hence, in response to the exigency for rapid computational analysis of the control rod dropping process in nuclear reactors, this study proposes a generative algorithm founded upon dynamic similarity feature search. Leveraging partial control rod dropping process data, including displacement, velocity, and temporal information acquired through CFD computations, this algorithm enables the expeditious and accurate generation of data pertaining to alternate operational conditions, thereby elucidating the attributes of control rod drive line descent. The computational methodology posited in this study operates independently of the control rod drive line structure or ambient parameters, devoid of any supervised learning training procedures, and demonstrates commendable scalability and generalizability.

This work makes four main contributions:

(1) The proposed algorithm addresses the inherent limitations of CFD in simulating control rod descent processes, particularly the prohibitively high computational cost. While maintaining prediction errors ≤ 10%, the computational time is reduced from 9.2 h (CFD) to 61.5 s, a 540-fold improvement. Such a magnitude of efficiency enhancement has not been reported in related studies.

(2) The proposed algorithm enables the real-time generation of rod descent curves under diverse operating conditions using minimal CFD samples, providing engineers with immediate feedback for structural optimization. This capability is uniquely valuable in reactor conceptual design, a scenario not adequately addressed by traditional simulations or standard CFD workflows.

(3) To the best of our knowledge, this is the first study to introduce dynamic similarity feature search into nuclear reactor motion-component simulations, offering a new paradigm for the efficient generative modeling of complex physical processes.

(4) The explicit similarity-matching logic of the proposed algorithm guarantees transparency, offering a key advantage over the opaque “black box” nature of deep learning models, particularly in safety-critical fields such as nuclear engineering.

The structure of this study unfolds as follows: Section 2 delineates the data acquisition protocol adopted for algorithm validation; Section 3 expounds upon the algorithm’s inherent principles, computational workflow, and evaluation metrics; Section 4 conducts a comprehensive analysis of the algorithm’s efficacy; and Section 5 culminates in a summation and outlook encompassing the various facets explored within this exposition.

2. Raw Data Acquisition

The research undertaken in this study centers around the control rod drive line of HPR1000 [20]. To streamline the investigation, three-dimensional numerical simulations utilizing CFD have been employed. All simulations were run with a Vietnam-made Intel Core i7-9700K CPU at 3.60 GHz and 16 GB RAM. These simulations delve into the descent process of individual control rods at different heights. A comprehensive dataset comprising both training and validation samples has been procured. The simulation model illustrating the descent of a single control rod is depicted in Figure 1.

2.1. Control Rod Assembly Dynamics

Assuming that the positive direction of the z-axis is vertically downward, the kinetic equation of the control rod assembly can be written as follows:

$$m_1{\displaystyle \frac{d^{2}z}{d{t}^2}}=m_{1}g-F_{SPRING1}-F_{FLOAT1}-F_{FLUID1}$$

(1)

where m₁g is the gravitational force, $F_{SPRING1}$ is the spring restoring force, $F_{FLOAT1}$ is the buoyant force, and $F_{FLUID1}$ is the fluid resistance.

During the control rod drop, the water temperature changes are negligible. Therefore, the energy equation can be omitted. In this study, the standard k−ε turbulence model is employed. The fluid flow governing equations are summarized below.

Continuity Equation:

$${\displaystyle \frac{\partial \rho }{\partial t}}+\nabla \cdot (\rho U)=0$$

(2)

Momentum Equation:

$${\displaystyle \frac{\partial \rho U}{\partial t}}+\nabla \cdot \left\{\rho U\otimes U\right\}=-\nabla p^{’}+\nabla \left({\mu }_{\mathrm{eff}}\left(\nabla U+(\nabla U{)}^T\right)\right)+S_M$$

(3)

Turbulent Kinetic Energy (k) Equation:

$${\displaystyle \frac{\partial \left(\rho k\right)}{\partial t}}+\nabla \cdot (\rho Uk)=\nabla \left[\left(\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_k}}\right)\nabla k\right]+p_k-\rho \epsilon$$

(4)

Turbulent Dissipation Rate (ε) Equation:

$${\displaystyle \frac{\partial \left(\rho \epsilon \right)}{\partial t}}+\nabla \cdot (\rho U\epsilon )=\nabla \left[\left(\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_k}}\right)\nabla \epsilon \right]+{\displaystyle \frac{\epsilon }{k}}\left(C_{\epsilon }{p}_k-C_{{\epsilon }_t}\rho \epsilon \right)$$

(5)

Viscous Production Term (p_k):

$$p_k={\mu }_t\nabla U\cdot \left(\nabla U+\nabla U^T\right)-{\displaystyle \frac{2}{3}}\nabla \cdot U\left(3{\mu }_t\nabla \cdot U+\rho k\right)$$

(6)

where ρ is the fluid density, t denotes time, U is the time-averaged velocity vector, and μ_eff and μ_t are effective viscosity and turbulent viscosity, respectively. p’ is the modified pressure, S_M is the body force, and the turbulence model constants are assigned as C_ε1 = 1.44, C_ε2 = 1.92, and σ_k = 1.0.

For a moving control volume V(t), the conservation equation for scalar ϕ can be expressed in integral form as follows:

$${\displaystyle \frac{d}{dt}}{\int }_V\rho \varphi dV+{\int }_{\partial V}\rho \varphi (\overrightarrow{u}-{\overrightarrow{u}}_g)\cdot d\overrightarrow{A}={\int }_{\partial V}\mathsf{\Gamma }\nabla \varphi \cdot d\overrightarrow{A}+{\int }_V{S}_{\varphi }dV$$

(7)

where V(t) represents the deforming control volume with time-dependent geometry, $dV\left(t\right)$ denotes its moving boundary, u_g is the grid velocity (mesh motion velocity), ρ is the fluid density, Γ is the diffusion coefficient, and S_ϕ denotes the source term of scalar ϕ per unit volume.

Using the first-order backward difference, the time derivative term in above equation becomes the following:

$${\displaystyle \frac{d}{dt}}{\int }_V\rho \varphi dV={\displaystyle \frac{(\rho \varphi V{)}^{n+1}-(\rho \varphi V{)}^n}{\mathsf{\Delta }t}}$$

(8)

where superscripts n and n + 1 denote the current and next time steps, respectively.

To satisfy the conservation laws on the discretized grid, the time derivative of the control volume (dV/dt) can be calculated using the following equation:

$${\displaystyle \frac{dV}{dt}}={\int }_{\partial V}{\overrightarrow{u}}_g\cdot d\overrightarrow{A}={\sum }_{j=1}^{n_f}{\overrightarrow{u}}_{g,j}\cdot {\overrightarrow{A}}_j$$

(9)

where n_f is the number of cell face, and A_j is the area vector of face j. The face velocity term ${\overrightarrow{u}}_{g,j}\cdot {\overrightarrow{A}}_j$ is calculated as

$${\overrightarrow{u}}_{g,j}\cdot {\overrightarrow{A}}_j={\displaystyle \frac{\delta V_j}{\mathsf{\Delta }t}}$$

(10)

where $\delta V_j$ is the volume swept by face j during Δt.

2.2. Simulation Results

The maximum drop height of the control rods is 3.618 m. Simulations are conducted on control rod dropping models at varying heights H₁/H₂/H₃...H_n within its maximum range. The mass of the entire control rod assembly is evenly distributed among each individual rod. The main structural parameters of the CFD model can be found in

Table 1. Main parameters of CFD model.

Number	Model Structure	Parameters
1	Guide tube inner diameter (d1), mm	11.2
2	Buffer section inner diameter (d2), mm	10.04
3	Control rod diameter (d3), mm	9.70
4	Buffer section length (h), mm	765
5	Flow hole diameter (d4), mm	2.3
6	Drainage hole diameter (d5), mm	1
7	Drop height (H), mm	H1/H2/H3…Hn [0~3618]

.

Under the condition of a cold reactor, the temperature is 20 °C and the pressure is 0.1 MPa. At the same time, the density of water is $\rho =998\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$, and the kinematic viscosity is $\mu =1.003\times {10}^{-3} \mathrm{k}\mathrm{g}/(\mathrm{m}·\mathrm{s})$. The boundary condition at the fluid domain exit is set as a pressure outlet with a gauge pressure of 0 Pa. The initial velocity of the about-to-fall control rod assembly is set at ${\mathrm{v}}_0=0 \mathrm{m}/\mathrm{s}$. Utilizing dynamic mesh technology, the dropping process is simulated using 2020 ANSYS fluent fluid dynamics software [21]. A pressure contour plot of the fluid flow during the control rod drop is shown in Figure 2.

Using simulation, key performance curves of time, displacement, speed, and acceleration were obtained during the control rod dropping process. A total number of 290 cases are calculated (i.e., initial height H₀ = [0 m, 0.01 m,…2.90 m]), where each consists of approximately 2000 to 6000 data points. Figure 3 illustrates a segment of the simulation data, sampled at intervals of 0.2 m. In Figure 3, the time history curves were plotted using different colors, where a light color represents simulation results with a small initial drop height, and a dark color was used for simulation results under large initial drop height conditions. The total volume of the data is roughly between 100,000 and 300,000. As displacement, velocity, and other parameters are tightly time-dependent, this study focuses on typical time series data. The data obtained from the simulations in this section provide a solid foundation for the research in the following section.

3. Algorithm

3.1. Dynamic Similar Feature Searching Algorithm

The data in this study are time series in nature. Moreover, distinct data sequences often contain segments with similar features. Consequently, this study introduces a dynamic search algorithm based on similarity features. This algorithm combines a data sequence’s own historical observations with the nearest neighbor sequence segments in the dataset. It dynamically generates subsequent data sequences using a sliding window approach.

Assume there exist $k$ data sequences in an n-dimensional space. Denote the $i\mathrm{th}$ data sequence as $F_i$. Define the set of data sequences a $D$, where $D=\{F_1,F_2,\dots ,F_k\}$. Each data sequence comprises multiple sequence segments. The sequence segment is denoted by $E$. The $i\mathrm{th}$ point in $E$ is indicated as $E_i$. $E_{ij}$ represents the data at the $j\mathrm{th}$ dimension of point $E_i$, for two sequence segments $E_1=\left(\right(E_1{)}_1,(E_1{)}_2,\dots (E_1{)}_t)$ and $E_2=\left(\right(E_2{)}_1,(E_2{)}_2,\dots (E_2{)}_t)$ with the length of $t$. The dissimilarity between E₁ and E₂ is computed as the sum of the metric evaluations at corresponding points, according to metric $d$ (defined in Appendix A).

$$d\left(E_1,E_2\right)=\sqrt{{\sum }_{i=0}^n{\sum }_{j=0}^t{p}_i({{\left(E_1\right)}_{ij}-{\left(E_2\right)}_{ij})}^2}$$

(11)

A smaller $d\left(E_1,E_2\right)$ indicates less dissimilarity between sequence segments $E_1$ and $E_2$.

For the target sequence $F_m$, due to the continuity in the data sequence, the $v\mathrm{th}$ point in $F$ is related to the preceding sequence segment of length $t$. Furthermore, owing to the similarity between sequence segments, the target sequence $F$ can be synthesized by incorporating similar sequence segments from the known dataset $D$.

In the target sequence $F_m$, the $v\mathrm{th}$ point is associated with the preceding $t$ points. Thus, the $v\mathrm{th}$ point can be represented as follows:

$${(F}_m{)}_v=f\left(\right(F_m{)}_{v-1},(F_m{)}_{v-2},(F_m{)}_{v-3},\dots ,\left(F_m{)}_{v-t}\right),t\le v$$

(12)

where $f$ is the generating function of the sequence:

$$f={\sum }_{i=1}^m{q}_i(E_m{)}_v,{\sum }_{i=1}^m{q}_i=1,E_m\in G$$

(13)

where $G$ is the set of sequence segments $E_m$ identified through traversal in the known sequence set $D$, where $E_m$ consists of the $m$ sequence segments with sequentially minimal dissimilarity values based on metric $d$. $(E_m{)}_v$ represents the last point in the sequence segment $E_m$. $q_i$ is a scale factor chosen based on the situation.

3.2. Algorithm Design

In this study, a data generation algorithm (i.e., Algorithm 1) is established based on the differences in data categories and the internal temporal correlation of the data.

Algorithm 1: Pseudocode of implementation for sequence data generation

Input: Initial t points $x_1,x_2,x_3,\dots ,x_t$ of sequence F

Output: Next node $x_{t+1}$, to be added to sequence F

1: While the length m of sequence F does not meet the stopping criterion, do

2: Segment $E\leftarrow$ Extract a segment $(x_{m-t+1},x_{m-t+1},\dots ,x_m)$ of length t from the end of sequence F

3: Compute the distance from sequence segment F to sequence segment E under metric d

4: Set $G\leftarrow$ Identify m closest neighboring sequence segments to segment E under metric d

5: The next sequence point $x_{t+1}\leftarrow f\left(x_j\right),x_j\in G$, and append it to sequence F

6: End while

3.3. Evaluation Metrics for Computational Results

In this study, a comprehensive analysis of the model calculations is conducted, leveraging four key metrics: Mean Absolute Percentage Error (MAPE), Root Mean Square Error (RMSE), Mutual Information (MI), and computation time.

(1) Mean Absolute Percentage Error (MAPE) is the average of the absolute values of the relative percentage errors, and it serves as an indicator of the model’s predictive accuracy. The computation formula is as follows:

$$MAPE\left({\tilde{y}}_i,y_i\right)={\displaystyle \frac{1}{n}}{\sum }_n^i\left|{\displaystyle \frac{y_i-{\tilde{y}}_i}{y_i}}\right|$$

(14)

In this context, $y_i$ represents the true value, while ${\tilde{y}}_i$ is the predicted value generated by the algorithm.

(2) Root Mean Square Error (RMSE) is employed to assess the deviation between observed and true values and is commonly used as a standard for evaluating model predictions. The calculation formula is as follows:

$$RMSE\left({\tilde{y}}_i,y_i\right)=\sqrt{{\displaystyle \frac{1}{m}}{\sum }_{i=1}^m{\left({\tilde{y}}_i,y_i\right)}^2}$$

(15)

(3) Mutual Information (MI) is used to evaluate the existence and strength of the relationship between two variables $X$ and $Y$. The defining formula for MI is as follows:

$$I\left(X,Y\right)={\iint }_{XY}P(X,Y)log{\displaystyle \frac{P(X,Y)}{P\left(X\right)P\left(Y\right)}}$$

(16)

MI can be reformulated as follows:

$$I\left(X;Y\right)=H\left(Y\right)-H\left(Y\right|X)$$

(17)

In this context, $H\left(Y\right)$ represents the entropy of $Y$, defined as $H\left(Y\right)=-{\int }_{Y}P\left(Y\right)\mathrm{l}\mathrm{o}\mathrm{g}P\left(Y\right)$, which quantifies the uncertainty of $Y$. The higher the dispersion of $Y$’s distribution, the greater the value of $H\left(Y\right)$. $H\left(Y\right|X)$ indicates the uncertainty of $Y$ given $X$. $I(X,Y)$ quantifies the reduction in Y’s uncertainty due to the introduction of $X$. Therefore, the greater the value of $I(X,Y)$, the stronger the relationship between $X$ and $Y$. In this study, the Ml between the predicted and true values is calculated to assess the performance of the generative model.

4. Analysis of Computational Results

4.1. Assessment of Local Scenarios

Four different drop heights (0.55 m, 2.2 m, 2.8 m, 2.9 m) are set as the starting point for the target sequence, where displacement–time and velocity–time curves are generated. At this initial position of 0.55 m and 2.2 m, as shown in Figure 4a,b, the synthetic displacement data (red curve) nearly overlap entirely with the original data (green curve); the synthetic velocity data (blue curve) exhibit minor fluctuations in the initial stages but stabilize in the later phases. Moreover, the synthetic data demonstrate excellent alignment with the original dataset (green curve), indicating that the algorithm effectively simulates the sequence changes in the original data at this starting position. Meanwhile, at drop heights of 2.8 m and 2.9 m, which are close to the limit height of 3.618 m, the synthetic displacement data in Figure 4c,d show greater bias compared with the original data. Even more, the gap between original and simulated velocity tends to be less neglectable as the initial position of the rod becomes lower.

Regarding computational efficiency, let us delve into the instance of a generated target sequence commencing from an initial height of 0.55 m. Employing the proposed dynamic similarity feature search algorithm for simulation, the entire calculation process for the descent of the rod was accomplished within a mere 61.5 s, while the three-dimensional dynamic simulation using CFD demanded a significant duration of 9.2 h. This remarkable discrepancy reveals a staggering 540-fold improvement in computational efficiency, spanning approximately three orders of magnitude. The substantial enhancement of computational efficiency serves as a poignant validation of the algorithm’s original intent, which is to significantly augment the efficiency of the overall design process for nuclear reactors.

4.2. Overall Algorithmic Assessment

To better evaluate the efficacy of the proposed algorithm, a comprehensive assessment of the algorithm’s performance across all drop heights is conducted in this study. Evaluation metrics (RMSE, MAPE, MI) are statistically analyzed for data generated with a starting position of 0 and an interval of 0.12 m, constituting a total of 30 groups.

The distribution of RMSE for control rod drop velocity and displacement is shown in Figure 5. Meanwhile, the MAPE distribution is shown in Figure 6. For control rod drop starting positions ranging from 0 to 2.853 m (prior to the buffering phase), the RMSE and MAPE for velocity are concentrated within the range of 3% to 8% and 1% to 2%, respectively. The RMSE and MAPE for displacement are mainly targeted between the range of 1% to 4% and 0% to 1%. Moreover, these values tend to decrease as the initial drop height increases, indicating that the generated data deviate minimally from the actual data, and that the computational results of the intelligent model in this study are highly accurate. However, for control rod drop starting positions beyond 2.853 m (after the buffering phase), the RMSE values for both velocity and displacement show a sharp increase, suggesting that the intelligent model’s data fitting capabilities weaken in this segment.

The distribution of mutual information (MI) for control rod drop displacement and velocity is illustrated in Figure 7. In the range of control rod drop starting positions from 0 to 2.853 m (prior to the buffering phase), MI values for displacement are concentrated between 7.2 and 7.7, and for velocity they are concentrated between 6.1 and 7.2. This indicates a high degree of correlation between the generated and original data. As the starting position increases, the MI values gradually decrease. Beyond the 2.853 m starting position (after the buffering phase), the MI for both displacement and velocity drops sharply, signifying a rapid decrease in correlation between the generated and original data.

According to evaluation metrics such as RMSE, MAPE, and MI, the generated data for control rod drop starting positions ranging from 0 to 2.853 m (before the buffering phase) accurately fit the original data, demonstrating the algorithm’s practical applicability. However, for starting positions beyond 2.853 m (after the buffering phase), all metrics show a declining trend. This decline is attributed to the fact that the generated data are predicated on a preset initial velocity for a given number of steps. Due to the presence of a buffering point, it is challenging to obtain the nearest actual dataset within this range, leading to reduced data fitting efficacy.

In summary, the algorithm developed in this study demonstrates strong applicability in the range of control rod drop starting positions from 0 to 2.853 m, that is, before the control rod enters the buffering phase. The data generated by the intelligent model can effectively simulate the original data.

4.3. Comparison with Different Methods

To further evaluate the performance of the proposed dynamic feature search method, we compared it with an LSTM-based sequence-to-sequence deep learning method. The LSTM structure is an extension of recurrent neural networks (RNNs), which solves the exploding gradients and vanishing gradients problems. Mathematically, the LSTM operations are governed by the following:

$$\begin{array}{c}{f}_t=\sigma (W_f\cdot [h_{t-1},x_t]+b_f)\\ i_t=\sigma (W_i\cdot [h_{t-1},x_t]+b_i)\\ \begin{array}{c}{o}_t=\sigma (W_o\cdot [h_{t-1},x_t]+b_o)\\ {\tilde{c}}_t=\mathit{tanh}(W_c\cdot [h_{t-1},x_t]+b_c)\\ \begin{array}{c}{c}_t=f_t\odot c_{t-1}+i_t\odot {\tilde{c}}_t\\ h_t=o_t\odot \mathit{tanh}(c_t)\end{array}\end{array}\end{array}$$

(18)

where f_t, i_t, and o_t denote the forget, input, and output gates, respectively. c_t represents the memory cell state, and σ is the sigmoid activation function. W and b correspond to trainable weight matrices and biases. $\odot$ indicates element-wise multiplication.

In the LSTM-based sequence-to-sequence algorithm, a dual-component architecture comprising an encoder LSTM and a decoder LSTM is employed. The encoder LSTM processes sequential input $c_{input}=\left[c_1,c_2,\dots ,c_k\right]$ and propagates its final hidden state h_k to the decoder as the initial context. Then, the decoder LSTM subsequently generates predictions $c_{output}=\left[c_{k+1},c_{k+2},\dots ,c_N\right]$ through autoregressive computation. In this study, the LSTM-based sequence-to-sequence architecture takes the initial control rod height (i.e., H₀) as the input feature and outputs the predicted displacement (i.e., H₁, H₂,…H_n) and velocity (i.e., v₁, v₂,…v_n) time series.

To compare the proposed dynamic feature search method and LSTM-based method more comprehensively, five different data ratios, ranging from 10% to 30%, are used. More specifically, when using the proposed dynamic feature search method, data ranging from 10% to 30% were used to construct the search space, while the remaining 70% to 90% of unseen data were used for validation. On the other hand, the implementation of the LSTM method consists of two steps. In the first step, 10% to 30% of the data were used for model training. In the second step, the remaining 70% to 90% of unseen data were used for model testing.

Figure 8 presents a comparison between the proposed dynamic feature search method and the LSTM method under varying data ratios. In Figure 8, four different error metrics (i.e., MAPE of displacement, MAPE of velocity, RMSE of displacement, and RMSE of velocity) were used for validation. The results show that the proposed dynamic feature search method and LSTM method have similar prediction accuracies. It should be noted that the accuracy of the proposed dynamic feature search method increases with the increase in used data. This implies that when more data are available, the performance of the proposed method can be further improved. However, this trend is not observed when using the LSTM-based method. For example, Figure 8b illustrates that increasing the data ratio for training negatively impacts prediction accuracy, as the MAPE of velocity rises from approximately 3% to around 10%. Another important feature worth highlighting is the total time required by these two methods. As can be seen from Section 3, no training step is needed when using the proposed method. On the other hand, the time required for LSTM training ranges from 0.5 to 1 h, depending on the data ratio used. After training (only for LSTM), the prediction time for the proposed method and the LSTM method is close, ranging from 17 s to 80 s. The above results show that the proposed dynamic feature search method is much more efficient than the LSTM-based method.

5. Conclusions

(1) This study endeavors to address the protracted computational time associated with the descent of control rods in nuclear reactors. To this end, we propose a dynamic similarity feature search algorithm capable of swiftly generating comprehensive computational results for control rod descent under varying operating conditions. Notably, these results exhibit minute disparities when compared to those obtained through CFD simulations, thereby lending robust support to the quest for heightened efficiency in reactor structural design.

(2) Within an equivalent computational framework, the dynamic similarity feature search algorithm surpasses traditional CFD simulations by significantly augmenting computational efficiency, thus achieving an astounding three orders of magnitude enhancement. This remarkable advancement serves as incontrovertible evidence of the algorithm’s inherent capacity to bolster design efficiency. Moreover, unlike conventional supervised machine learning models, this algorithm obviates the need for a training process, thereby mitigating concerns pertaining to suboptimal generalization across diverse datasets.

(3) The algorithm proposed in this study capitalizes on a subset of generated simulated data to extract a wealth of information from a finite vector space, thereby catering to specific real-world applications. It exemplifies the quintessential characteristics of a physics-informed data-driven approach, offering superior interpretability in contrast to conventional machine learning methodologies. Compared with complex sequence-to-sequence machine learning algorithms, the proposed method shows similar accuracy and significantly lower complexity. Additionally, the algorithm’s streamlined workflow underscores its inherent versatility and remarkable potential for broad applicability.

In the context of nuclear reactor design, researchers frequently encounter analogous challenges, such as insufficient dataset richness for training crucial equipment fault diagnostic models or inadequate experimental data to validate program physics during start-up. In light of such typical generative conundrums, the remarkable adaptability of the proposed algorithm renders it exceptionally well suited for transformative modifications, thereby unleashing its untapped potential across diverse domains.