Nuclear power plant main pump fault diagnosis method under strong noise background

Abstract

The invention discloses a nuclear power plant main pump fault diagnosis method under a strong noise background. According to the scheme, the method comprises the steps that firstly, acceleration signals of rotor misalignment, shaft end bending, cracks and unbalanced faults of a main pump under strong noise are collected through an experiment bench; then, a white whale optimization algorithm BWO is used for optimizing variational mode decomposition VMD and wavelet transform WT, noise reduction processing is carried out on the signals, and the defect that traditional noise reduction parameters depend on experience is overcome; and finally, performing time domain characteristic parameter extraction by using the denoised signal, constructing a bidirectional long-short-term memory auto-encoder Bi-LSTM-AE model, and realizing fault diagnosis by means of bidirectional time sequence modeling and auto-encoding reconstruction capability of the bidirectional long-short-term memory auto-encoder Bi-LSTM-AE model. According to the scheme, the signal noise reduction effect and the fault feature discrimination capability under strong noise can be improved, the diagnosis precision is enhanced, and safe operation of the main pump of the nuclear power plant is guaranteed.

Description

Technical Field

[0001] The present invention relates to the technical field of nuclear power plant equipment safety, and in particular to a method for diagnosing main pump faults in a nuclear power plant under a strong noise background. Background Art

[0002] As critical equipment for maintaining reactor coolant circulation, the operating status of a nuclear power plant's main pumps directly impacts the safety and stability of the entire nuclear power system. Main pumps are constantly exposed to complex operating conditions characterized by high loads and high noise levels. Even minor faults can rapidly worsen under the harsh operating environment, causing severe equipment damage or even system shutdown, resulting in significant economic losses and safety risks. Therefore, achieving early and accurate diagnosis of potential main pump faults has become a core technical challenge in ensuring the safe and stable operation of nuclear power equipment.

[0003] Among the many fault diagnosis methods, vibration-based diagnostic techniques have become a research hotspot due to their non-invasiveness and rich information. However, in practical applications, environmental noise interference has become a key factor hindering the effective implementation of this technology. Strong noise can mask key device characteristics, significantly reducing the signal-to-noise ratio (SNR), and severely impacting the accuracy of subsequent feature extraction and classification. To improve the reliability of fault diagnosis, noise reduction must be performed on the signal before developing the fault diagnosis algorithm.

[0004] Currently, denoising algorithms for strong noise have been extensively studied in other fields, such as electronic communications and mechanical fault diagnosis, and have achieved some success. However, research in the field of nuclear power plant main pumps is still in its infancy. For example, Xue-ying et al. proposed a denoising method for strong noise signals based on VMD and singular value decomposition for electric gate valves in nuclear power plants, and verified its effectiveness through experiments. Gang et al. proposed a deconvolution and time-frequency pattern decomposition method to remove strong noise from pulse signals. Experimental results show that its denoising ability is stronger than traditional algorithms. Linyu et al. used adaptive wavelet neural network denoising in signal preprocessing for natural gas pipeline defect detection. Verification by signal-to-noise ratio and root mean square error showed that the denoising effect of this model was superior to five traditional denoising algorithms. Tao et al. proposed an end-to-end fault diagnosis framework based on a sliding window sparse Transformer and dynamic threshold denoising, which effectively removed strong noise from bearing signals and demonstrated strong robustness in strong noise environments. To address the challenge of early bearing fault diagnosis in a strong noise environment, Jiaxing et al. proposed a noise reduction model based on an enhanced grey wolf optimization algorithm-particle swarm optimization algorithm and VMD. Their method was validated through testing. While these studies have been successful in their respective application areas, they are difficult to directly apply to fault diagnosis of nuclear power plant main pumps due to the extreme complexity and specificity of the operating environment.

[0005] In terms of fault diagnosis model construction, relatively little research has been conducted on intelligent fault diagnosis algorithms for nuclear power plant main pumps, but extensive exploration has been conducted in other rotating equipment fields. Pengfei et al. combined a long short-term memory neural network (LSTM) with a temporal convolutional network to identify the health status of nuclear power plant rotating equipment from a spatiotemporal correlation perspective. Experimental results showed that this method achieved higher diagnostic accuracy than five traditional algorithms. Xue-Ying et al. used a stacked autoencoder to identify faults in electric gate valves in nuclear power plants, achieving better diagnostic accuracy than an autoencoder (AE). Wenzhe et al. proposed a method based on symmetric point patterns and residual neural networks to accurately identify compound faults in nuclear power plant rotor systems, with diagnostic performance superior to other cutting-edge methods. Xin et al. used a gated recurrent unit (GRU) to improve the diagnostic capability of concurrent faults in electric isolation valves in nuclear power plants, outperforming support vector machines. Addressing the problem that traditional deep learning models require expert experience to optimize hyperparameters, Jianping et al. proposed a hybrid convolutional neural network (CNN) model combined with automatic neural network architecture search for nuclear power plant coolant pump fault identification. Test results showed that its recognition accuracy exceeded that of 1D-CNN and 2D-CNN models. Although these methods have improved the equipment fault diagnosis capabilities to a certain extent, under the realistic conditions of strong noise interference faced by the main pumps of nuclear power plants and increased complexity of timing signals, traditional methods still have obvious limitations in feature expression capabilities and diagnostic accuracy.

[0006] In summary, existing technologies cannot effectively meet the fault diagnosis needs of nuclear power plant main pumps in strong noise environments. Designing a signal processing method that can efficiently extract fault characteristics in strong noise environments and building a fault diagnosis model with strong time series modeling capabilities and high robustness are crucial for improving the effectiveness of early fault diagnosis of nuclear power plant main pumps and ensuring the safe and stable operation of nuclear power systems. Summary of the Invention

[0007] This specification provides a method for diagnosing main pump faults in a nuclear power plant under a strong noise background, so as to overcome at least one technical problem existing in the related art.

[0008] The embodiment of the specification provides a method for diagnosing a main pump fault in a nuclear power plant under a strong noise background, comprising:

[0009] S1. Collect acceleration signals of rotor misalignment fault, rotor shaft end bending fault, rotor crack fault and rotor imbalance fault of the main pump of the nuclear power plant under strong noise background;

[0010] S2. Performing noise reduction processing on the acceleration signal based on a joint noise reduction method of the Beluga Optimization Algorithm (BWO), Variational Mode Decomposition (VMD), and Wavelet Transform (WT), specifically comprising:

[0011] S21. The number of decomposition modes and penalty factor of variational mode decomposition (VMD) and the number of wavelet decomposition layers of wavelet transform (WT) are jointly optimized by the BWO algorithm to obtain the optimal parameter combination.

[0012] S22, using the decomposition mode number K and penalty factor α obtained after optimization in step S21, decompose the acceleration signal into K intrinsic mode function components IMF through VMD, and the decomposition process satisfies the following formula:

[0013] u k (t) = A k (t)cos(φ k (t))(1)

[0014] In the above formula, the symbol φ k (t) represents the kth IMF modal component u k The instantaneous phase of (t), symbol A k (t) represents the k IMF modal components u k The instantaneous amplitude of (t), A k (t)≥0, symbol t represents the time variable, k=1,2,...,K;

[0015] S23. Perform Hilbert transform on the decomposed IMF components to obtain the analytical signal:

[0016]

[0017] Where δ(t) is the pulse function and j is the imaginary unit;

[0018] S24, introduce the center frequency ω k , modulate the analytical signal:

[0019]

[0020] In the above formula (3), is the correction index;

[0021] S25. Construct variational constraint problem:

[0022]

[0023] In formula (4), {ω k}={ω1,ω2,···,ω K} is the center frequency of each IMF component, {u k}={u1,u2,···,u K} is the IMF component of each signal obtained by VMD decomposition, x is the original decomposed signal, represents the Tikhonov matrix, and the symbol * represents the convolution operation;

[0024] S26. Introduce the Lagrangian multiplication operator λ and the quadratic penalty factor α to construct the augmented Lagrangian function:

[0025]

[0026] S27, iteratively update u by alternating direction multiplication method k 、ω k and λ:

[0027]

[0028] In formulas (6) to (8), n represents the number of iterations, τ is the fidelity coefficient, is the center of gravity of the kth mode power spectrum, They are Fourier transform corresponding to x(t) and λ(t);

[0029] S28. Determine the convergence conditions:

[0030]

[0031] Among them, ε represents the judgment accuracy, ε>0;

[0032] S29, using the wavelet decomposition layer number optimized in step S21, combined with the selected wavelet basis function, first merge the filtered IMF components to reconstruct the signal, then perform multi-scale decomposition on the reconstructed signal, and reconstruct the signal again after thresholding the high-frequency components. The denoising process satisfies the following formula:

[0033] s(n)=f(n)+σe(n)(10)

[0034] In formula (10), s(n) is the noisy signal, f(n) is the effective signal, e(n) is the noise signal, and σ is the noise variance;

[0035] S3. Extracting time domain characteristic parameters of the denoised signal, including mean, root mean square, variance, peak, peak-to-peak value, skewness, kurtosis, impulse factor, margin factor, and shape factor;

[0036] S4: Input the signal after extracting the time domain feature parameters into the bidirectional long short-term memory autoencoder Bi-LSTM-AE, extract the time series features through the bidirectional LSTM network, reconstruct the signal through the encoder-decoder and output the fault classification result, where the forward and backward hidden layer states of the Bi-LSTM are calculated as follows:

[0037]

[0038] Among them, W f represents the forward weight matrix, Wb represents the backward weight matrix, represents the forward hidden layer state, Represents the backward hidden layer state, W o represents the output layer weight matrix, b o represents the output layer bias unit, and f represents the activation function.

[0039] In some optional implementations, the BWO algorithm in step S2 includes a search phase, a development phase, and a whale fall phase. The specific optimization process is as follows:

[0040] The position update formula in the search phase is:

[0041]

[0042] Among them, the symbol T represents the current number of iterations, and the symbol It represents the position of the i-th beluga whale in the j-dimensional space at the T+1-th iteration, and the symbol Indicates that the i-th white whale is at the p-th iteration at the T+1-th iteration. j The position in the dimensional space, symbolized by p j represents a random number selected from the d-dimensional space; The i-th beluga whale is at the p j Position in dimension; is the current position of the rth beluga whale; r1 and r2 are random numbers between the interval (0,1);

[0043] The position update formula in the development phase is:

[0044]

[0045] In formulas (12) to (15), r3 and r4 are random numbers between (0,1). It is the best location for white whales. is the position of the i-th beluga whale, is the position of the random white whale at the current iteration number, J1 is the random jump degree used to represent the flight intensity in the Levy flight strategy, L F is the flight function, μ and v are normally distributed random numbers, is the default constant, Γ is the gamma function, which is used to calculate the scale parameter of the flight function;

[0046] The position update formula during the whale fall phase is:

[0047]

[0048] In formulas (16) to (18), r5, r6, and r7 are random numbers between (0, 1), and x s is the step length of the whale's fall, ub and l b are the upper and lower bounds of the variables, n is the population size of beluga whales, W f is the probability of a white whale falling.

[0049] In some optional implementations, the encoding-decoding process of the autoencoder AE of the Bi-LSTM-AE model is:

[0050]

[0051] Among them, the symbol x represents the input signal, the symbol z represents the hidden layer representation, the symbol x′ represents the reconstructed signal, f and g are the encoding and decoding functions respectively, and W e and W d is the weight matrix;

[0052] The loss function of the autoencoder AE is shown in formula (20):

[0053]

[0054] In formula (20), n is the number of nodes in the input layer or output layer;

[0055] The gate state calculation of the LSTM unit includes:

[0056] f t =σ(W f ·[h t-1 ,x t ]+b f )(twenty one)

[0057] i t =σ(W i ·[h t-1 ,x t ]+b i )(twenty two)

[0058]

[0059] o t =σ(W o ·[h t-1 ,x t ]+b o )(25)

[0060] h t =o t ⊙tanh(c t )(26)

[0061] In formulas (21) to (26), the output of the forget gate is f t ∈[0,1], represents the proportion of information that needs to be retained; σ is the sigmoid activation function; ht-1 is the hidden state at the previous moment; x t is the input unit at the current moment; W f is the weight matrix of the forget gate controller; b f is the bias of the forget gate controller; is a temporary memory unit, W i is the weight matrix of the input gate controller; b i is the bias of the input gate controller; W c is the weight matrix for updating the cell state; b c is the bias for updating the cell state; W o is the weight matrix of the output gate controller; b o is the bias of the output gate controller; h t is the hidden state at the current moment, the symbol ⊙ represents element-by-element multiplication, and tanh represents the hyperbolic tangent function.

[0062] The beneficial effects of the embodiments of this specification are as follows:

[0063] This technical solution utilizes the Beluga Whale Optimization (BWO) algorithm to jointly optimize the parameters of the variational mode decomposition (VMD) and wavelet transform (WT). BWO simulates the diverse behaviors of a beluga whale, continuously adjusting parameters during the search, development, and whale fall phases. This allows for automated global optimization of parameters. This improvement enables the noise reduction process to precisely adapt to the complex characteristics of nuclear power plant main pump signals, overcoming the limitations of traditional methods.

[0064] During fault diagnosis, the Bi-LSTM network comprehensively captures the temporal dependencies of signals from both the forward and backward dimensions. Compared to traditional unidirectional LSTMs, the Bi-LSTM can more fully exploit the dynamic evolution characteristics of the nuclear power plant's main pump vibration signals over time series. The autoencoder (AE) structure compresses features via the encoder, and then the decoder reconstructs the signal. In this process, the model leverages reconstruction errors to continuously optimize the compactness and discriminability of features. Even in a strong noise environment, the Bi-LSTM-AE model, with its powerful feature processing capabilities, can accurately identify and analyze even faint fault signatures. This significantly enhances the model's ability to distinguish similar fault types, significantly reduces the false positive rate, and comprehensively improves the accuracy and robustness of fault diagnosis. BRIEF DESCRIPTION OF THE DRAWINGS

[0065] In order to more clearly illustrate the technical solutions in the embodiments of this specification or related technologies, the following briefly introduces the drawings required for use in the embodiments or related technical descriptions. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0066] Figure 1 This is a flow chart of a method for diagnosing a main pump fault in a nuclear power plant under a strong noise background provided by an embodiment of this specification;

[0067] Figure 2 Flowchart for WT noise reduction;

[0068] Figure 3 It is the structural diagram of AE;

[0069] Figure 4 This is the network structure diagram of LSTM;

[0070] Figure 5 This is the network structure diagram of Bi-LSTM;

[0071] Figure 6 This is the network structure diagram of Bi-LSTM-AE. DETAILED DESCRIPTION

[0072] The following will be combined with the drawings in the embodiments of this specification to clearly and completely describe the technical solutions in the embodiments of this specification. Obviously, the embodiments described are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0073] It should be noted that the terms "including" and "having" and any variations thereof in the embodiments of this specification and the accompanying drawings are intended to cover non-exclusive inclusions. For example, a process, method, system, product, or apparatus that includes a series of steps or units is not limited to the listed steps or units, but may optionally include steps or units that are not listed, or may optionally include other steps or units that are inherent to the process, method, product, or apparatus.

[0074] Figure 1 This is a flow chart of a method for diagnosing a main pump fault in a nuclear power plant under a strong noise background provided in an embodiment of this specification. The flow chart may include the following steps.

[0075] Step S1: First, data collection is performed.

[0076] The main pump of a nuclear power plant is one of the core equipment in nuclear power plant operation. Its stable operation is crucial to ensuring the safety and normal power generation of the nuclear power plant. However, during long-term operation, the main pump may experience various faults, such as rotor misalignment, rotor shaft end bending, rotor cracks, and rotor imbalance. These faults can cause changes in the vibration characteristics of the main pump. By collecting the main pump's acceleration signal, the vibration characteristics corresponding to these faults can be captured. Furthermore, the operating environment of a nuclear power plant is often accompanied by strong noise, which may come from the operation of other equipment, the flow of fluids, and other sources. Strong noise can interfere with the collected acceleration signals, increasing the difficulty of fault diagnosis. Therefore, to simulate a real operating environment and develop a fault diagnosis method that is effectively resistant to noise interference, the main pump's acceleration signal can be collected under a strong noise background. Therefore, this step can collect the acceleration signals of the main pump of a nuclear power plant under a strong noise background for rotor misalignment, rotor shaft end bending, rotor cracks, and rotor imbalance faults.

[0077] To collect acceleration signals under the strong noise background mentioned in the previous paragraph, a main pump rotor test bench can be constructed. This test bench primarily consists of a motor, coupling, rotor, and accelerometer. The motor and rotor are coupled via the coupling, and the resulting vibration signals are collected via the accelerometer. The motor is the power source of the main pump rotor test bench, providing rotational force for the entire system. The motor converts electrical energy into mechanical energy, driving the rotor. The motor's performance and operating state directly affect the rotor's motion characteristics. For example, the motor's speed stability and torque output can affect the vibration of the main pump. During the experiment, the motor's speed and torque must be precisely controlled according to actual requirements to simulate the main pump's operating conditions under different operating conditions. The coupling connects the motor and rotor, ensuring smooth transmission of the motor's power to the rotor. Furthermore, the coupling can compensate for installation errors and relative displacement between the motor and rotor to a certain extent. Different types of couplings have different characteristics. For example, rigid couplings offer high transmission accuracy but require high installation precision. Flexible couplings provide a certain degree of cushioning and vibration reduction, reducing vibration transmission between the motor and rotor. When selecting a coupling, it's important to consider the specific experimental requirements and the actual conditions of the main pump. The rotor, a core component of the main pump, rotates under the drive of the motor. When rotor faults such as misalignment, shaft end bending, cracks, or imbalance occur, the dynamic characteristics of its rotational motion change, resulting in changes in the characteristics of the vibration signal. For example, rotor misalignment generates additional radial and axial forces during rotation, causing changes in vibration amplitude and frequency. A bent rotor end can shift the rotor's center of gravity, generating unbalanced forces and, in turn, abnormal vibration. By analyzing the vibration signals of the rotor under different fault conditions, the type and severity of the fault can be identified. Accelerometers are key devices for collecting vibration signals from the main pump, converting mechanical vibration into electrical signals. Accelerometers are typically installed in key locations of the main pump, such as the bearing seat and pump body, to accurately measure vibration at various locations. Sensor performance directly impacts the quality of the collected signal. Parameters such as sensor sensitivity, range, and frequency response all need to be appropriately selected based on actual needs. In the presence of strong noise, the sensor's anti-interference capability must also be considered to ensure the collected signal truly reflects the vibration characteristics of the main pump.

[0078] To collect data, the experimental equipment, including the motor, coupling, rotor, and accelerometer, must first be installed and debugged according to design requirements, ensuring that all components are securely connected and operating properly. The motor is then started, and the rotor is brought to a predetermined speed, simulating the actual operating conditions of the main pump. During rotor operation, the accelerometer collects the main pump's vibration signals in real time and converts them into electrical output signals. These signals are recorded and stored by the data acquisition system for subsequent analysis and processing.

[0079] To improve the accuracy and reliability of fault diagnosis, it is necessary to collect a large number of acceleration signals under different fault conditions. Therefore, during the experiment, it is necessary to simulate various fault conditions such as rotor misalignment, shaft end bending, cracks, and imbalance, and collect sufficient sample data for each fault condition. At the same time, it is also necessary to consider the impact of different operating conditions such as speed and load on the vibration signal and collect data under multiple operating conditions.

[0080] Step S2: Denoising the acceleration signal of the nuclear power plant main pump based on the Beluga Whale Optimization (BWO)-Variational Mode Decomposition (VMD)-Wavelet Transform (WT) algorithm.

[0081] The VMD algorithm, due to its excellent adaptive time-frequency decomposition capabilities, is widely used for noise reduction analysis of non-stationary signals. However, practical applications of the VMD algorithm face difficulties in parameter selection, especially for key parameters such as the decomposition mode number K and the penalty factor α. Traditionally, these parameters have been set using trial and error or fixed empirical values. This lacks versatility and makes it difficult to achieve optimal decomposition results, limiting its application in engineering applications under complex conditions.

[0082] Furthermore, while VMD can effectively separate signals of different frequency bands, problems such as modal aliasing or residual noise leakage may still exist in situations where high-frequency noise is significant and modal boundaries are unclear, resulting in limited noise reduction effectiveness of a single VMD. To further improve noise reduction performance, the technical solution of this application introduces WT on the basis of VMD, leveraging its multi-scale decomposition and high-frequency compression characteristics to perform secondary noise reduction on the filtered reconstructed signal, thereby effectively removing residual noise and improving signal smoothness and structural integrity.

[0083] To achieve automatic parameter optimization and optimal combination search, the technical solution of this application constructs a multi-parameter joint optimization model based on BWO. K, α, and the number of wavelet decomposition layers are used as variables to be optimized, and the optimal parameter combination is obtained under the global search framework.

[0084] The following describes the principle of VMD. VMD is based on classical Wiener filtering, Hilbert transform, and frequency mixing. It integrates variational decomposition in the signal processing process and decomposes the signal in a non-recursive mode. During the decomposition process, the center frequency and effective bandwidth of each modal component are determined, and the signal is decomposed into a series of intrinsic mode function components (IMF). In the variational mode decomposition process, the original signal is decomposed into K IMF modal components. The specific representation of IMF is shown in formula (1):

[0085] u k (t) = A k (t)cos(φ k (t))(1)

[0086] In formula (1), the symbol φ k (t) represents u k The instantaneous phase of (t), symbol A k (t) represents u k The instantaneous amplitude of (t), A k (t)≥0.

[0087] VMD transforms signal decomposition into solving a variational problem within a variational framework. Therefore, the essence of VMD is the construction and solution of variational problems. When VMD decomposes a signal, the detailed steps of establishing and solving the variational model are as follows:

[0088] (1) In order to obtain the analytical signal and unilateral spectrum of each IMF, each IMF obtained by decomposition is first subjected to Hilbert transform.

[0089]

[0090] In formula (2), u k (t) is the kth IMF modal component, δ(t) is the impulse function, and * is the convolution symbol.

[0091] (2) Modulate the spectrum to the corresponding baseband and introduce exponential terms for adjustment. Assume that the center frequency of each IMF is ω k .

[0092]

[0093] In formula (3), is the correction index.

[0094] (3) Use Gaussian smoothness to calculate the signal bandwidth of K modes, and then calculate the L of the modulated signal gradient 2 The norm is solved to obtain the variational constraint problem shown in formula (4).

[0095]

[0096] In formula (4), {ω k}={ω1,ω2,···,ω K} is the center frequency of each IMF component, {u k}={u1,u2,···,u K} is the IMF component of each signal obtained by VMD decomposition, x is the original decomposed signal, represents the Tikhonov matrix.

[0097] (4) The Lagrange multiplication operator and quadratic penalty factor are introduced to solve the problem of constrained variational models.

[0098]

[0099] In formula (5), α is the quadratic penalty factor and λ is the Lagrange multiplication operator.

[0100] (5) Using the multiplication operator alternating direction method to k},{ω k},λ are iterated alternately to solve the saddle point problem of the Lagrangian expression. The calculation process is shown in Equations (6) to (8).

[0101]

[0102]

[0103] In formulas (6) to (8), n represents the number of iterations, τ is the fidelity coefficient, is the center of gravity of the kth mode power spectrum, They are Fourier transform corresponding to x(t) and λ(t).

[0104] The specific iterative solution process of VMD is:

[0105] 1) Initialization And let n = 0;

[0106] Represents the kth IMF component u k The initial estimate after Hilbert transform, which represents the initial shape of the IMF component in the frequency domain; It is the initial center frequency estimate of the kth IMF component. The center frequency determines the main distribution position of the IMF component in the frequency domain. is the initial Lagrange multiplier estimate, which is used to handle the variational constraint problem constructed in VMD and balance different objective terms.

[0107] 2) Update the number of iterations: execute n=n+1;

[0108] In this step, the number of iterations n increases by 1 for each round of iterative calculation. This step is a counting identifier of the iterative process, which is used to distinguish different rounds of iterative calculations. It is also the basis for judging whether the iterations have reached the specified number or met the convergence conditions.

[0109] 3) Component and parameter update: For k = 1:K, when ω ≥ 0, calculate and update and λ;

[0110] Fixed current center frequency and the Lagrange multiplier λ n In the case of , for each k (i.e., each IMF component), a specific calculation method (based on the solution principle of the VMD variational problem) is used to update the estimated value of the IMF component in the frequency domain This step is to optimize the specific form of each IMF component under the current parameter settings to make it more in line with the requirements of signal decomposition.

[0111] After getting the update After that, fix and λ n , calculate and update the center frequency of the kth IMF component It is usually based on the principle of energy center of gravity, so that the energy of each IMF component is more concentrated near the new center frequency, thereby more accurately separating signals with different frequency components.

[0112] Update λ: After the update is completed and After that, these two quantities are fixed and the Lagrange multiplier λ is updated. The update rule is based on the solution logic of the variational problem. The purpose is to gradually adjust the value of λ to satisfy the variational constraints and balance the signal reconstruction error and the bandwidth of each mode.

[0113] 4) Convergence judgment: Determine whether to stop iteration. If the convergence condition shown in formula (9) is met, then stop iteration; otherwise, return to step 2).

[0114]

[0115] In formula (9), ε is the judgment accuracy, ε>0.

[0116] The convergence condition determines whether the iteration can be stopped by comparing the sum of the relative changes in the frequency domain estimated values of each IMF component in two adjacent iterations. Measures the difference between the estimated values of the k-th IMF component at the n+1th iteration and the nth iteration (the square of the L2 norm), is the amplitude measure of the estimated value of the k-th IMF component at the n-th iteration (the square of the L2 norm). The two are divided to obtain the relative change. The relative changes of all k IMF components are summed. If this sum is less than the preset judgment accuracy ε, it means that after this iteration, the changes of each IMF component are small enough, that is, the iteration has converged and reached a relatively stable state, and the iteration can be stopped. If this condition is not met, it means that the iteration has not converged, and it is necessary to return to step 2 to continue the next round of iterative calculation to further optimize each IMF component and related parameters.

[0117] The principle of WT is introduced below. WT noise reduction is mainly achieved by decomposing the signal into high-frequency components and low-frequency components, and then performing wavelet reconstruction. Its core principle is based on the multi-resolution analysis capability of wavelet transform on the signal in the time-frequency domain, which can effectively separate the noise components and effective components in the signal. The process is shown in Equation (10).

[0118] s(n)=f(n)+σe(n)(10)

[0119] In equation (10), s(n) represents the signal reconstructed by IMF decomposition and energy fraction calculation. In the nuclear power plant main pump fault diagnosis scenario, this signal is the intermediate signal after pre-processing the collected noisy vibration signal and is used for further wavelet transform noise reduction. f(n) represents the effective signal, which is the signal portion that can reflect the operating status and fault characteristics of the nuclear power plant main pump. For example, the specific vibration signal characteristics generated by the main pump when faults such as rotor misalignment, shaft end bending, cracks, or imbalance occur are included in the effective signal. e(n) represents the noise signal, which is the useless signal component that interferes with the extraction and analysis of the effective signal. Environmental noise in the nuclear power plant and other interference noise generated by equipment operation all fall into this category. σ represents the noise variance, which is used to measure the strength or fluctuation of the noise signal. The larger the noise variance, the greater the noise energy and the greater the interference with the effective signal.

[0120] WT noise reduction process is as follows Figure 2As shown, WT first decomposes the noisy signal s(n) into high-frequency and low-frequency components at different scales by selecting appropriate wavelet basis functions and the number of denoising layers. The selection of wavelet basis functions requires comprehensive consideration of factors such as signal characteristics (including smoothness, symmetry, and frequency characteristics; for example, smooth signals should use wavelets with a large number of vanishing moments, symmetric signals should use symmetric wavelets, and signals with specific frequencies should use wavelets with good frequency resolution), the denoising objective (selecting wavelets with strong separation capabilities when noise intensity is high, or minimizing the impact on signal characteristics when preserving them), and computational complexity (selecting wavelet basis functions with limited computing resources or high real-time requirements). The appropriate wavelet basis functions are selected after comprehensively weighing these factors. The number of denoising layers determines the scale at which the signal is decomposed. More decomposition layers yield more detailed information at different scales, but this also increases the computational complexity. Generally speaking, the appropriate number of layers can be determined based on the signal characteristics and noise intensity. For example, if the noise intensity is large, you may need to select more decomposition layers to separate the noise and signal more finely; while for signals with relatively weak noise, fewer layers may be sufficient to achieve better noise reduction effects.

[0121] After the previous steps, the signal is decomposed into high-frequency and low-frequency components. The high-frequency component typically contains signal details and noise, as noise generally has high-frequency characteristics. The low-frequency component primarily reflects the overall trend and key features of the signal, essentially, the outline of the valid signal. This multi-scale decomposition is achieved based on the multi-resolution analysis properties of the wavelet transform. By varying the scaling factor, the signal can be observed at different time-frequency resolutions. After obtaining the high-frequency and low-frequency components, the high-frequency component is thresholded. Since noise is primarily concentrated in the high-frequency portion, by setting an appropriate threshold, the high-frequency component below the threshold can be treated as noise and suppressed or removed, while the component above the threshold (considered to be high-frequency details of the valid signal) is retained. Common thresholding functions include soft and hard thresholding. Soft thresholding shrinks coefficients below the threshold to a small non-zero value, while hard thresholding sets coefficients below the threshold to zero. Wavelet reconstruction is then performed on the thresholded high-frequency component and the unprocessed low-frequency component, recombining these processed components to obtain the de-noised signal. This process can minimize noise interference while preserving effective signal features, thereby improving signal quality and providing a more reliable data foundation for subsequent fault diagnosis. In nuclear power plant main pump fault diagnosis, noise in the main pump vibration signal can seriously interfere with the extraction and identification of fault features. WT noise reduction can effectively suppress noise and highlight characteristic information related to the main pump fault. This enables vibration signal-based fault diagnosis methods (such as the fault diagnosis model based on bidirectional long-short-term memory autoencoders used later) to more accurately analyze signals, improving the accuracy and reliability of fault diagnosis, thereby ensuring the safe and stable operation of the nuclear power plant main pump, promptly identifying potential faults and taking appropriate measures to avoid major accidents.

[0122] The following describes the principles of BWO. The Beluga Whale Optimization (BWO) algorithm is inspired by the hunting behavior of beluga whales in nature and is designed to solve a variety of complex optimization problems. By simulating the swimming, feeding, and whale fall behaviors of beluga whales, the algorithm constructs three distinct processes: exploration, exploitation, and whale fall.

[0123] (1) Search phase

[0124] The purpose of the search phase is to simulate the behavior of a beluga whale exploring the ocean in search of prey. This phase involves the beluga whale randomly searching different locations to find potential prey (i.e., the best solution in the solution space). The beluga whale's position update is shown in Equation (11).

[0125]

[0126] In formula (11), T is the number of iterations of the current algorithm; is the new position of the i-th beluga whale in the j-th dimension; p j is a random number selected from the d-dimensional space; The i-th white whale is at p j Position in dimension; is the current position of the rth beluga whale (r is a randomly selected beluga whale); r1 and r2 are random numbers between (0,1) used to enhance the algorithm's optimization ability during the search phase; sin(2πr2) and cos(2πr2) indicate that the mirror beluga whale's fin is facing the water surface.

[0127] (2) Development stage

[0128] At this stage, the algorithm updates the position of the white whale based on the Levy flight strategy. The updated position of the white whale As shown in formulas (12) to (15).

[0129]

[0130] In formulas (12) to (15), r3 and r4 are random numbers between (0, 1), which introduce randomness into the position update. It is the best position of the White Whale in the current iteration, guiding the search area to move closer to the high-quality area. is the current position of the i-th beluga whale, is the position of the randomly selected white whale at the current iteration number, and the difference between the two Provides a direction reference for position update. J1 is the random jump degree used to represent the flight intensity in the Levy flight strategy, which is used to reflect the flight intensity. F is the flight function, and as the number of iterations T increases in formula (13), Increase, This decreases, allowing J1 to adjust dynamically. This means that in the early stages of the iteration, the algorithm allows for larger jumps to expand the search range. As the iteration progresses, the jumps decrease, and the algorithm focuses more on local fine search, balancing global exploration and local development. In formula (14), μ and v are random numbers that follow a normal distribution, introducing randomness. is a default constant with a value of 1.5. σ is calculated from Equation (15) using the gamma function Γ and is F Providing parameter support ensures the mathematical rigor of the Levy flight strategy, so that the algorithm can not only use the long-step jump characteristics of Levy flight to quickly approach the optimal solution during the development phase, but also achieve fine search through parameter adjustment to avoid falling into local optimality and improve the overall optimization performance of the algorithm.

[0131] (3) Whale Fall Stage

[0132] The position update of the white whale is shown in equations (16) to (18).

[0133]

[0134] In formula (16), r5, r6 and r7 are random numbers between (0, 1) to introduce randomness into the position update; is the current position of the i-th beluga whale; is the randomly selected white whale position, and the difference between the two is used to adjust the direction of position update; x s is the step size of the whale's fall, which determines the amplitude of the position update. In formula (17), the symbol u b and l b are the upper and lower bounds of the variable, respectively, limiting the range of the step size; n is the population size of beluga whales, W f is the probability of the white whale falling, T is the current number of iterations, T max Is the maximum number of iterations. Exponential function part Let step length x s With W f , n, T change dynamically to control the decay speed of the step length, thereby adjusting the precision of the search. Formula (18) The probability of the white whale falling W f It is related to the number of iterations T. When the iteration starts (T=0), the whale fall probability W f The maximum value is 0.1. As the number of iterations T increases, W f Gradually decreases, when T=T max When W f This trend indicates that in the early stages of iterative optimization, the algorithm allows for a higher probability of whale falls, increasing the randomness and scope of search. As iterations progress, the probability of whale falls decreases, and the algorithm prioritizes local search, mimicking the characteristic that "the closer a beluga whale is to its food, the less dangerous it is," balancing global exploration with local development and improving the algorithm's optimization efficiency.

[0135] In general, the whale fall phase uses these three formulas, combining random numbers with the falling probability W that changes with iterations. f , dynamically adjust the position of the beluga whale and simulate the whale falling behavior, so that the algorithm can maintain a certain degree of randomness in the search process to avoid falling into the local optimum, and adjust the search strategy according to the iterative process, enhance the ability to mine the optimal solution, and thus improve the performance of parameter optimization (such as VMD and WT parameters) in the fault diagnosis of the main pump of the nuclear power plant under strong noise background.

[0136] After signal denoising is completed using the Beluga optimization algorithm (BWO) to optimize variational mode decomposition (VMD) and wavelet transform (WT), time-domain feature parameters can be extracted. Specifically, these parameters may include mean, root mean square (RMS), variance, peak, peak-to-peak value, skewness, kurtosis, impulse factor, margin factor, and shape factor. These parameters characterize the signal's time-domain characteristics from different perspectives. The mean reflects the signal's average level; the RMS reflects the signal's energy; the variance measures the signal's volatility; the peak and peak-to-peak values highlight the signal's extreme values; the skewness and kurtosis describe the signal's probability distribution; and the impulse factor, margin factor, and shape factor reflect the signal's impact and waveform characteristics from different dimensions. By extracting these time-domain feature parameters, fault information in the signal can be effectively captured, providing comprehensive and valuable feature input for subsequent fault diagnosis. Subsequently, in step 3, a bidirectional long-short-term memory autoencoder (Bi-LSTM-AE) model is constructed, leveraging its bidirectional time series modeling and autoencoder reconstruction capabilities for fault diagnosis.

[0137] Step 3: Fault diagnosis of the main pump of a nuclear power plant based on the Bidirectional Long Short-Term Memory Autoencoder (Bi-LSTM-AE) algorithm.

[0138] First, the principle of autoencoder AE is introduced. Autoencoder (AE) is an unsupervised learning neural network with the following structure: Figure 3 As shown in Figure 2, an autoencoder consists of an input layer, a hidden layer, and an output layer. Connecting the input layer to the hidden layer and applying an activation function to the layers is called encoding. Using the features extracted by the hidden layer to restore the data, the result is the output layer, a process called decoding. All layers are connected using fully connected operations.

[0139] AE first encodes the input data x to obtain the compressed feature z, and then decodes z to obtain the reconstructed data x′, as shown in Equation (19).

[0140] x′=g(z)=g(f(x))(19)

[0141] In formula (19), f and g represent activation functions.

[0142] Formula (19) can be expressed in another form as follows:

[0143]

[0144] Among them, the symbol x represents the input signal, the symbol z represents the hidden layer representation, the symbol x′ represents the reconstructed signal, f and g are the encoding and decoding functions respectively, and We and W d is the weight matrix;

[0145] The loss function of AE is shown in formula (20).

[0146]

[0147] In formula (20), n is the number of nodes in the input layer or output layer.

[0148] The following describes the principles of the Long Short-Term Memory (LSTM) neural network. This is a special type of recurrent neural network (RNN) designed specifically for processing temporal dependencies in long sequences of data. By introducing forget gates, input gates, and output gates, LSTM effectively alleviates the vanishing and exploding gradient problems that plague traditional RNNs during long sequence training.

[0149] LSTM is mainly composed of three gates, which are the forget gate f t , input gate i t and output gate o t These gate units continuously update the cell state c at time t using recursive equations t , its structure is as follows Figure 4 shown.

[0150] Figure 4 In the equation (21) to (26), the calculation methods of various parameters are shown in (21) to (26).

[0151] f t =σ(W f ·[h t-1 ,x t ]+b f )(twenty one)

[0152] i t =σ(W i ·[h t-1 ,x t ]+b i )(twenty two)

[0153]

[0154] o t =σ(W o ·[h t-1 ,x t ]+b o )(25)

[0155] h t =ot ⊙tanh(c t )(26)

[0156] In formulas (21) to (26), the output of the forget gate is f t ∈[0,1], represents the proportion of information that needs to be retained; σ is the sigmoid activation function; h t-1 is the hidden state at the previous moment; x t is the input unit at the current moment; W f is the weight matrix of the forget gate controller; b f is the bias of the forget gate controller; is a temporary memory unit; W i is the weight matrix of the input gate controller; b i is the bias of the input gate controller; W c is the weight matrix for updating the cell state; b c is the bias for updating the cell state; W o is the weight matrix of the output gate controller; b o is the bias of the output gate controller; h t Hide status for the current moment.

[0157] The following is an introduction to the principle of Bi-LSTM. Although LSTM can effectively model long-term dependencies in time series, its unidirectional structure limits the use of information after the current position in the sequence, making it difficult to fully capture the evolution of the signal in the entire time dimension. Bidirectional Long Short-Term Memory (Bi-LSTM) introduces two LSTM networks in opposite directions in the structure to model the sequence from the forward and backward directions respectively, thereby capturing more complete time series features. The principle of Bi-LSTM is as follows: Figure 5 shown.

[0158] Figure 5 In LSTM f Represents forward, LSTM b Reverse stands for reverse. In Bi-LSTM, the forward LSTM calculates the hidden layer information along the forward order of the time data, and the backward LSTM calculates the hidden layer information along the reverse order of the time data. Then, the hidden layer information of the two LSTMs is fused to obtain the output information of the Bi-LSTM. The Bi-LSTM transfer calculation process is shown in Equations (27), (28), and (29).

[0159]

[0160] In formulas (27), (28), and (29), W f and W bRepresent the weight matrices for forward propagation and backward propagation respectively; and Represents the hidden layer memory unit information of the model's forward propagation and backward propagation outputs respectively; o t Represents the output information of the Bi-LSTM model; W o is the weight matrix of the model output layer; b o is the bias unit of the model output layer.

[0161] The following describes the construction of the Bi-LSTM-AE model. Combining the structural advantages of Bi-LSTM and AE, Bi-LSTM-AE demonstrates superior performance in fault diagnosis tasks compared to using either Bi-LSTM or AE models alone. This performance is primarily reflected in the following aspects:

[0162] 1) More comprehensive time series feature extraction capabilities. Bi-LSTM-AE uses a bidirectional LSTM structure and simultaneously utilizes both forward and backward information of the time series to more comprehensively capture the dynamic characteristics of fault signals and improve classification accuracy.

[0163] 2) It possesses both feature learning and data reconstruction capabilities. Bi-LSTM-AE introduces the AE structure into the time series model, extracting deep features while preserving key structural information of the input through the reconstruction process. This combined feature and reconstruction approach helps the model learn more discriminative features, effectively improving its ability to distinguish similar fault types.

[0164] 3) Improved adaptability to complex fault patterns. In multi-category, nonlinear, or fuzzy boundary fault classification tasks, Bi-LSTM-AE can better identify subtle differences between different categories and reduce the misclassification rate through the synergy of sequence modeling and compression reconstruction.

[0165] 4) Enhanced robustness and fault tolerance. Compared to the limitations of AE in processing time series or the sensitivity of Bi-LSTM to outliers, Bi-LSTM-AE can more stably handle noisy or partially missing data, which helps improve the reliability of the model in practical applications.

[0166] The Bi-LSTM-AE network structure is as follows Figure 6 shown.

[0167] Figure 6 In the example, the encoder consists of a Bi-LSTM network and the input is the time series data X=[x1,x2,...,x T The encoder maps the time series data to the hidden layer state h T and provides input to the decoder.

[0168] The decoder consists of an LSTM network and a fully connected layer. The decoder stage starts from the initial input y init Start by receiving the hidden layer state vector h from the encoder T , and use it as the initial state h of the decoder T (i.e. c'0=h T ), then the LSTM calculation step calculates the output value y1 and uses it as the input value for the next moment, and then repeats the above calculation process. Finally, the fault classification function is achieved by connecting the fully connected layer and the softmax activation function.

[0169] Traditional noise reduction algorithms often struggle to achieve ideal noise reduction results when processing vibration signals from nuclear power plant main pumps due to their reliance on empirically defined parameters. This technical solution leverages the Beluga Whale Optimization (BWO) algorithm to jointly optimize the parameters of the variational mode decomposition (VMD) and wavelet transform (WT). BWO simulates the diverse behaviors of a beluga whale, continuously adjusting parameters during the search, development, and whale fall phases. This automated global optimization of parameters enables the noise reduction process to precisely adapt to the complex characteristics of nuclear power plant main pump signals, overcoming the limitations of traditional methods.

[0170] During fault diagnosis, the Bi-LSTM network comprehensively captures the temporal dependencies of signals from both the forward and backward dimensions. Compared to traditional unidirectional LSTMs, the Bi-LSTM can more fully exploit the dynamic evolution characteristics of the nuclear power plant's main pump vibration signals over time series. The autoencoder (AE) structure compresses features via the encoder, and then the decoder reconstructs the signal. In this process, the model leverages reconstruction errors to continuously optimize the compactness and discriminability of features. Even in a strong noise environment, the Bi-LSTM-AE model, with its powerful feature processing capabilities, can accurately identify and analyze even faint fault signatures. This significantly enhances the model's ability to distinguish similar fault types, significantly reduces the false positive rate, and comprehensively improves the accuracy and robustness of fault diagnosis.

[0171] Those skilled in the art will appreciate that the accompanying drawings are merely schematic diagrams of an embodiment, and the modules or processes in the accompanying drawings are not necessarily required to implement the present invention.

[0172] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or make equivalent replacements for some of the technical features therein. However, these modifications or replacements do not deviate the essence of the corresponding technical solutions from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for diagnosing main pump faults in a nuclear power plant under strong noise background, characterized in that: The following steps are involved: S1. Collect acceleration signals of rotor misalignment fault, rotor shaft end bending fault, rotor crack fault and rotor imbalance fault of the main pump of the nuclear power plant under strong noise background; S2. Performing noise reduction processing on the acceleration signal based on a joint noise reduction method of the Beluga Optimization Algorithm (BWO), Variational Mode Decomposition (VMD), and Wavelet Transform (WT), specifically comprising: S21. The number of decomposition modes and penalty factor of variational mode decomposition (VMD) and the number of wavelet decomposition layers of wavelet transform (WT) are jointly optimized by the BWO algorithm to obtain the optimal parameter combination. S22, using the decomposition mode number K and penalty factor α obtained after optimization in step S21, decompose the acceleration signal into K intrinsic mode function components IMF through VMD, and the decomposition process satisfies the following formula: u k (t)=A k (t)cos(φ k (t)) (1) In the above formula, the symbol φ k (t) represents the kth IMF modal component u k The instantaneous phase of (t), symbol A k (t) represents the k IMF modal components u k The instantaneous amplitude of (t), A k (t)≥0, symbol t represents the time variable; S23. Perform Hilbert transform on the decomposed IMF components to obtain the analytical signal: Where δ(t) is the pulse function and j is the imaginary unit; S24, introduce the center frequency ω k , modulate the analytical signal: In the above formula (3), is the correction index; S25. Construct variational constraint problem: In formula (4), {ω k }={ω1,ω2,···,ω K } is the center frequency of each IMF component, {u k }={u1,u2,···,u K } is the IMF component of each signal obtained by VMD decomposition, x is the original decomposed signal, represents the Tikhonov matrix, and the symbol * represents the convolution operation; S26. Introduce the Lagrangian multiplication operator λ and the quadratic penalty factor α to construct the augmented Lagrangian function: S27, iteratively update u by alternating direction multiplication method k 、ω k and λ: In formulas (6) to (8), n represents the number of iterations, τ is the fidelity coefficient, is the center of gravity of the kth mode power spectrum, They are Fourier transform corresponding to x(t) and λ(t); S28. Determine the convergence conditions: Among them, ε represents the judgment accuracy, ε>0; S29, using the wavelet decomposition layer number optimized in step S21, combined with the selected wavelet basis function, first merge the filtered IMF components to reconstruct the signal, then perform multi-scale decomposition on the reconstructed signal, and reconstruct the signal again after thresholding the high-frequency components. The denoising process satisfies the following formula: s(n)=f(n)+σe(n) (10) In formula (10), s(n) is the noisy signal, f(n) is the effective signal, e(n) is the noise signal, and σ is the noise variance; S3. Extracting time domain characteristic parameters of the denoised signal, including mean, root mean square, variance, peak, peak-to-peak value, skewness, kurtosis, impulse factor, margin factor, and shape factor; S4: Input the signal after extracting the time domain feature parameters into the bidirectional long short-term memory autoencoder Bi-LSTM-AE, extract the time series features through the bidirectional LSTM network, reconstruct the signal through the encoder-decoder and output the fault classification result, where the forward and backward hidden layer states of the Bi-LSTM are calculated as follows: Among them, W f represents the forward weight matrix, W b represents the backward weight matrix, represents the forward hidden layer state, Represents the backward hidden layer state, W o represents the output layer weight matrix, b o represents the output layer bias unit, and f represents the activation function.

2. The method for diagnosing main pump faults in a nuclear power plant under a strong noise background according to claim 1, characterized in that: The BWO algorithm in step S2 includes a search phase, a development phase, and a whale fall phase. The specific optimization process is as follows: The position update formula in the search phase is: Among them, the symbol T represents the current number of iterations, and the symbol It represents the position of the i-th beluga whale in the j-dimensional space at the T+1-th iteration, and the symbol Indicates that the i-th white whale is at the p-th iteration at the T+1-th iteration. j The position in the dimensional space, symbolized by p j represents a random number selected from the d-dimensional space; The i-th beluga whale is at the p j Position in dimension; is the current position of the rth beluga whale; r1 and r2 are random numbers between the interval (0,1); The position update formula in the development phase is: In formulas (12) to (15), r3 and r4 are random numbers between (0,1). It is the best location for white whales. is the position of the i-th beluga whale, is the position of the random white whale at the current iteration number, J1 is the random jump degree used to represent the flight intensity in the Levy flight strategy, L F is the flight function, μ and v are normally distributed random numbers, is the default constant, Γ is the gamma function, which is used to calculate the scale parameter of the flight function; The position update formula during the whale fall phase is: In formulas (16) to (18), r5, r6, and r7 are random numbers between (0, 1), and x s is the step length of the whale's fall, u b and l b are the upper and lower bounds of the variables, n is the population size of beluga whales, W f is the probability of a white whale falling.

3. The method for diagnosing main pump faults in a nuclear power plant under a strong noise background according to claim 1, characterized in that: The encoding-decoding process of the autoencoder AE of the Bi-LSTM-AE model is: Among them, the symbol x represents the input signal, the symbol z represents the hidden layer representation, the symbol x′ represents the reconstructed signal, f and g are the encoding and decoding functions respectively, and W e and W d is the weight matrix; The loss function of the autoencoder AE is shown in formula (20): In formula (20), n is the number of nodes in the input layer or output layer; The gate state calculation of the LSTM unit includes: f t =σ(W f ·[h t-1 ,x t ]+b f ) (21) i t =σ(W i ·[h t-1 ,x t ]+b i ) (22) the t =σ(W o ·[h t-1 ,x t ]+b o ) (25) h t =o t ⊙tanh(c t ) (26) In formulas (21) to (26), the output of the forget gate is f t ∈[0,1], represents the proportion of information that needs to be retained; σ is the sigmoid activation function; h t-1 is the hidden state at the previous moment; x t is the input unit at the current moment; W f is the weight matrix of the forget gate controller; b f is the bias of the forget gate controller; is a temporary memory unit, W i is the weight matrix of the input gate controller; b i is the bias of the input gate controller; W c is the weight matrix for updating the cell state; b c is the bias for updating the cell state; W o is the weight matrix of the output gate controller; b o is the bias of the output gate controller; h t is the hidden state at the current moment, the symbol ⊙ represents element-by-element multiplication, and tanh represents the hyperbolic tangent function.

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