A Vibration Trend Prediction Method for Pumped Storage Units Based on ISDO-VMD and Improved Stacked BiLSTM

Abstract

The ISDO-VMD and improved Stacked BiLSTM-based pumped storage unit vibration trend prediction method comprises the following steps: obtaining historical vibration monitoring data of a pumped storage unit, and preprocessing the vibration monitoring data; improving SDO, introducing a Singer chaotic mapping initialization strategy and a Cauchy variation mechanism to obtain an improved sled dog optimization algorithm ISDO; using the improved sled dog optimization algorithm ISDO to jointly optimize the mode number K and the penalty factor alpha in the variational mode decomposition VMD, and obtaining an optimal VMD parameter combination; inputting the obtained optimal VMD parameter combination into a VMD model, and decomposing the preprocessed pumped storage unit vibration sequence; constructing an improved Stacked BiLSTM prediction model; inputting the obtained mode components into the improved Stacked BiLSTM prediction model for training and prediction, and obtaining the prediction results corresponding to the mode components. The prediction results of the mode components are superimposed and reconstructed to obtain the final prediction result of the vibration trend of the pumped storage unit. The method can effectively improve the vibration trend prediction accuracy of the pumped storage unit.

Description

Technical Field

[0001] This invention relates to the field of online prediction technology for vibration signals of pumped storage units, specifically to a method for predicting vibration trends of pumped storage units based on ISDO-VMD and an improved Stacked BiLSTM. Background Technology

[0002] Pumped storage units are crucial equipment for power system peak shaving, frequency regulation, phase regulation, and emergency backup; their operating status directly impacts the safety and stability of the power grid. With the large-scale grid connection of new energy sources, the frequency of pumped storage unit start-ups and shutdowns and the number of operating condition switching operations have increased significantly. During long-term operation, these units are susceptible to vibration anomalies caused by factors such as hydraulic excitation, mechanical imbalance, electromagnetic disturbances, shaft eccentricity, and wear of guide bearings. Vibration signals directly reflect the operating status of key components and are important for condition monitoring, fault diagnosis, and health assessment. Therefore, accurate prediction of pumped storage unit vibration trends is of great significance for early identification of potential faults, reducing the risk of unplanned shutdowns, and ensuring the safe and stable operation of the units.

[0003] However, vibration signals from pumped storage units typically exhibit strong nonlinearity, significant nonstationarity, multi-scale coupling, and noise interference. Especially during frequent start-ups and shutdowns and transitions between multiple operating conditions, the amplitude, frequency, and energy distribution dynamically change, making trend prediction challenging. Traditional mechanistic models, relying on structural parameters and dynamic equations, are limited by the complexity of the unit structure and the variability of operating conditions, making them insufficient for on-site online prediction. In recent years, data-driven methods such as support vector regression, backpropagation neural networks, LSTM, and GRU have been widely used for vibration trend prediction, but they still have shortcomings in addressing complex long-term time-series dependencies, multi-scale feature coupling, and the mining of temporal correlation features. Furthermore, single shallow networks struggle to simultaneously characterize local fluctuations and long-term evolution trends.

[0004] To reduce the impact of non-stationarity and noise, existing techniques often incorporate signal decomposition methods such as empirical mode decomposition, wavelet transform, and variational mode decomposition before prediction. Among these, VMD has good resistance to mode mixing, but its decomposition effect is highly dependent on the number of modes. and penalty factor Improper parameter settings can easily lead to under-decomposition, over-decomposition, or residual noise. Existing empirical settings or traditional intelligent optimization algorithms may still suffer from problems such as uneven initial population distribution, slow convergence speed, and susceptibility to local optima. Furthermore, deep prediction networks are prone to gradient propagation instability, learning rate sensitivity, and generalization performance fluctuations during training, making it difficult to highlight key temporal information. Therefore, there is an urgent need to propose a vibration trend prediction method for pumped storage units that can achieve adaptive optimization of VMD parameters, deep extraction of complex temporal features, and stable convergence during training. Summary of the Invention

[0005] To overcome the problems of strong non-stationarity of vibration signals, empirical dependence of VMD parameters, insufficient extraction of complex time-series features, and poor training stability of deep networks in existing methods for predicting vibration trends of pumped storage units, this invention proposes a vibration trend prediction method for pumped storage units based on ISDO-VMD and an improved Stacked BiLSTM. This method first preprocesses the vibration monitoring data of the pumped storage unit; then, it uses the improved sled dog optimization algorithm ISDO to adaptively optimize the number of modes K and the penalty factor α of variational mode decomposition (VMD); next, it decomposes the vibration signal using the optimal parameters to obtain several modal components with different frequency characteristics; then, it constructs an improved Stacked BiLSTM prediction model that integrates the multi-head attention mechanism (MHA) and the Warmup-CosineDecay dynamic learning rate strategy to predict each modal component separately; finally, it superimposes and reconstructs the prediction results of each modal component to obtain the vibration trend prediction result of the pumped storage unit.

[0006] The technical solution adopted in this invention is as follows: A vibration trend prediction method for pumped storage units based on ISDO-VMD and an improved Stacked BiLSTM includes the following steps: Step 1: Obtain historical vibration monitoring data of the pumped storage unit, preprocess the vibration monitoring data, and divide the preprocessed data into training set, validation set and test set; Step 2: Improve the original Sled Dog Optimization Algorithm (SDO) by introducing the Singer chaotic mapping initialization strategy and the Cauchy mutation mechanism to obtain the improved Sled Dog Optimization Algorithm (ISDO). Step 3: Use the improved sled dog optimization algorithm ISDO to jointly optimize the number of modes K and the penalty factor α in variational mode decomposition (VMD) to obtain the optimal combination of VMD parameters; Step 4: Input the optimal VMD parameter combination obtained in Step 3 into the VMD model to decompose the preprocessed pumped storage unit vibration sequence and obtain several modal components (IMFs). Step 5: Construct an improved Stacked BiLSTM prediction model, which includes a stacked bidirectional long short-term memory network, a multi-head attention mechanism (MHA), and a Warmup-CosineDecay dynamic learning rate strategy. Step 6: Input the modal components obtained in Step 4 into the improved Stacked BiLSTM prediction model for training and prediction, and obtain the prediction results corresponding to each modal component.

[0007] Step 7: Superimpose and reconstruct the prediction results of each modal component to obtain the final prediction result of the vibration trend of the pumped storage unit.

[0008] It also includes step 8: evaluating the prediction results using root mean square error (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE), and coefficient of determination (R²).

[0009] Step 1 includes the following steps: Step S1.1: Collect vibration monitoring data of the pumped storage unit. The vibration monitoring data includes the upper guide bearing swing signal, shaft vibration signal, guide bearing vibration signal or other vibration monitoring signals that can reflect the operating status of the unit. Step S1.2: Remove outliers and delete invalid periodic data from the collected raw vibration monitoring data; Step S1.3: Use interpolation to complete the missing data points. The calculation formula is as follows: ; In the formula, The sampling time corresponding to the missing data. The vibration data after interpolation and completion. , The effective sampling times adjacent to the missing point. , These are the valid raw vibration monitoring values ​​at the corresponding time.

[0010] Step S1.4: Normalize the vibration data using the Min-Max normalization method. The normalization formula is as follows: ; In the formula, This represents the original vibration data. This represents the normalized vibration data. and These represent the minimum and maximum values ​​in the vibration data, respectively. Step S1.5: Divide the preprocessed vibration data into a training set, a validation set, and a test set. Preferably, the ratio of the training set, validation set, and test set is 8:1:1.

[0011] Step 2, the improved sled dog optimization algorithm ISDO, includes the following steps: Step S2.1: Set the population size N, the maximum number of iterations T, the dimension of the optimization variable Dim, and the upper and lower bounds of the parameters to be optimized; where the parameters to be optimized include the number of VMD modes K and the penalty factor α; Step S2.2: Initialize the sled dog population using Singer chaotic mapping to improve the uniformity and diversity of the initial population distribution in the search space. The initialization formula is as follows: ; In the formula, Dog represents the position of an individual sled dog, Lb and Ub represent the lower and upper bounds of the optimization variable, respectively, and chaos represents the chaotic sequence value generated by the Singer chaotic map; Step S2.3: Use the envelope entropy after VMD decomposition as the fitness function to calculate the fitness value for each sled dog individual. The formula for calculating the envelope entropy is: ; ; In the formula, E k For the first k Envelope entropy of the first modal component; K The number of VMD decomposition modes; This represents the normalized probability distribution of the envelope amplitude of the k-th mode; Fit This represents the fitness value for an individual sled dog. A smaller envelope entropy indicates a more concentrated energy distribution of the modal components after signal decomposition, resulting in a better decomposition effect. Step S2.4: Update the individual sled dog positions using the sled dog selection mechanism, travel model, obstacle avoidance model, disorientation model, and training model, including: The initial population for SDO is generated through random distribution to ensure that the initial solution covers the entire search space. Its expression is: ; In the formula: Dog for N × Dim dimensional population matrix; N Population size; Dim To optimize dimensions; Lb and Ub These are the lower and upper bounds of the optimization variable, respectively; rand Generate uniform random numbers within the interval [0,1].

[0012] The algorithm determines the number of dogs participating in the core task (pulling the sled) using a formula. N 1. Focus on the region with the best solution to improve search efficiency: ; In the formula: k 1 is the adjustment coefficient. k These are parameters related to the task.

[0013] Depending on the sled dog's position in the team, the speed update formula is divided into three categories, with position updates uniformly following the "speed accumulation" logic: when i When =1,2 (dogs at the front of the line): ; when i ∈ (3, N 1-2 (with the dog in the middle of the group): ; when i = N 1-1, N 1 (dogs at the rear of the group): ; In the formula: ω Inertial weights; c 1, c 2 represents the learning factor; r 1. r 2 and r 3 is a random number within the interval [0, 1]. Dog Gi , Dog Z Dogs in reference positions; Dog i-2 , Dog i+2 For adjacent dogs; Dog t , Dog r A randomly selected reference individual; l It is a binary control variable (0 or 1).

[0014] ; When the search encounters a local optimum, the position is updated through random perturbation to achieve obstacle avoidance. The specific expression is as follows: ; In the formula: r 4 and r 5 is a random number between [0, 1]; it jumps out of the local optimum region through bidirectional perturbation.

[0015] When the population reaches a search standstill, it is guided to reset its position using global information, the specific expression of which is as follows: ; In the formula: r 6 and r 7 is a random number between [0, 1]. C The guiding coefficient; For a sign random variable, Dog z The globally optimal individual. Dog r It is a random individual.

[0016] The expression for training and optimizing dogs with poor performance is: ; In the formula: r 8~ r 11 A random number within the range [0,1]. F This is a learning ability factor used to adaptively adjust training intensity; its expression is: ; In the formula: For the current individual fitness, and These are the minimum and maximum fitness values ​​of the population, respectively. X 1. X 2. X 3 represents the training direction vector; D 1. D 2. D 3 represents the performance gap; its definition is as follows: ; In the formula: Dog Better To be an outstanding individual; Dog Worse Individuals awaiting training.

[0017] Step S2.5: After updating the individual sled dog positions, a Cauchy mutation mechanism is introduced to enhance the algorithm's ability to escape local optima. The improved position update method is as follows: ; In the formula, This represents the position of the i-th sled dog. This represents the current globally optimal individual position. This represents the speed of the i-th sled dog. This indicates element-wise multiplication. Let r represent a random number that follows a uniform distribution, p represent the mutation probability threshold, preferably p is 0.5, and r i Represents the Cauchy perturbation vector; The travel model focuses on a large-scale global search; the obstacle avoidance model performs a fine-grained search of the optimal neighborhood; the disorientation model introduces random perturbations to reduce the risk of the algorithm getting stuck in local optima; and the training model gradually shrinks the search range as it iterates. These multiple update models work together to balance the ISDO algorithm's global exploration and local exploitation capabilities, efficiently searching for the optimal combination of VMD parameters.

[0018] Step S2.6: Determine whether the maximum number of iterations has been reached. If it has, output the position of the sled dog individual with the smallest fitness value as the optimal parameter combination for VMD. If it has not been reached, return to step S2.3 to continue iterating.

[0019] Step 3, which uses the Improved Sled Dog Optimization Algorithm (ISDO) to jointly optimize the VMD parameters, includes the following steps: Step S3.1: Set the search range of the number of VMD modes K and the penalty factor α. Preferably, the search range of K is [3,15] and the search range of α is [100,5000]. Step S3.2: Represent the position of each individual sled dog as a two-dimensional vector: ; In the formula, Indicates the first The number of modalities corresponding to each individual Indicates the first The penalty factor corresponding to each individual; Step S3.3: Since the number of modes K is an integer, the K value obtained during the optimization process is rounded down. The expression is: ; In the formula: K raw is the original floating-point modality number output for the individual sled dog position; Round(·) is the rounding function; K is the number of valid modalities available for VMD after rounding.

[0020] The optimization algorithm searches for continuous domain values, while the number of modes in the VMD decomposition must be a positive integer. A constraint interval of K∈[3,15] is set to avoid insufficient signal decomposition due to too few modes, or over-decomposition and spurious components due to too many modes. Penalty factor. α Since the values ​​are continuous, no rounding operation is required.

[0021] Step S3.4: Set each set of parameters Input the VMD model, decompose the vibration signal, and calculate the corresponding envelope entropy fitness value. Fitness Fit i As a quantitative indicator for evaluating the performance of VMD parameters in the ISDO algorithm, the optimization objective is to minimize... Fit i The algorithm iterates through all sled dog individuals in the population, compares the fitness values ​​corresponding to each set of parameters, and continuously updates the globally optimal parameter combination until the iteration termination condition is met. Step S3.5: Iteratively update the parameter combination with the minimum fitness value using the improved sled dog optimization algorithm ISDO, and use it as the optimal parameters for VMD: ; In the formula, Represents the number of optimal modes This represents the optimal number of penalty factors.

[0022] Step 4 includes the following steps: Step S4.1: Decompose the vibration signal into K modal components with finite bandwidths, and minimize the sum of the estimated bandwidths of each modal component, constructing the following variational constraint model: ; In the formula, For the set of all modal components, This is the set of central angular frequencies corresponding to each modal component; The number of VMD decomposition modes; Represents the Dirac delta function; The imaginary unit; For time-series sampling time; The k-th modal component is the time-domain signal. This is a complex exponential demodulation term used to shift the modal spectrum to the baseband. For time t The first-order partial differential operator; The L2 norm is used to measure modal bandwidth; * indicates convolution operation. The constraints are: ; In the formula, Indicates the first One modal component, Indicates the first The center frequency of each modal component Represents the original vibration signal; Step S4.2: Introducing the quadratic penalty factor α and the Lagrange multiplier λ, the constrained variational problem is transformed into an unconstrained variational problem, yielding the augmented Lagrange function: ; In the formula, Represents the augmented Lagrangian function constructed from the VMD model; Represents time-domain Lagrange multipliers; This refers to the inner product operation of functions; Step S4.3: The alternating direction multiplier method is used to iteratively update each modal component, center frequency, and Lagrange multiplier. This method decomposes the VMD variational optimization problem into multiple simple subproblems that are solved alternately, continuously optimizing the modal components and their corresponding center frequencies. Iteration stops when the convergence condition is met, outputting K independent, non-overlapping modal components (IMFs) in the frequency domain. Each modal component corresponds to a different characteristic frequency range, effectively separating the low-frequency trend term, fault characteristic components, and high-frequency noise in the original vibration signal, reducing the non-stationary characteristics of the time-series signal, and providing high-quality input data for subsequent prediction models.

[0023] Step S4.4: Output the decomposition results of the vibration signal: .

[0024] In the formula, This represents the k-th modal component obtained from VMD decomposition, corresponding to the time-domain vibration component within a specific center frequency range.

[0025] In step 5, the improved Stacked BiLSTM prediction model includes: 5.1: Construct a stacked bidirectional long short-term memory (BLSTM) network, wherein the stacked BLSTM network comprises at least two layers of BiLSTM network; for the input time series: ; In the formula, This represents a one-dimensional time-series sample representing the network input. These represent the vibration sampling values ​​at each moment within the time series; The BiLSTM network extracts temporal features of historical and future directions using forward LSTM and backward LSTM respectively, and its hidden state is updated as follows: ; In the formula, This represents the hidden state output by the forward LSTM at time t; This represents the hidden state of the inverse LSTM output at time t; This represents the operation function of the forward LSTM unit; This represents the operation function of the inverse LSTM unit; This represents the network input time-series sample value at time t; This represents the hidden state of the forward LSTM at time t-1; This represents the reverse LSTM hidden state at time t+1.

[0026] Concatenate the forward hidden state and the backward hidden state to obtain: ; In the formula, This represents the bidirectional fused hidden state after BiLSTM splicing at time t; This represents the vertical concatenation operation of vectors, which combines the forward hidden state and the backward hidden state into a complete feature vector.

[0027] 5.2: Stack multiple BiLSTM networks, with the output of the previous BiLSTM layer serving as the input of the next BiLSTM layer, to enhance the model's ability to extract complex nonlinear temporal features and multi-scale dynamic features. 5.3: Introducing Multi-Head Attention (MHA) at the output of Stacked BiLSTM; Let the output of Stacked BiLSTM be H, then the query matrix Q, key matrix K, and value matrix V are obtained through linear mapping: ; In the formula, , , The weight matrix is ​​trainable. 5.4: Calculate attention weights using scaled dot product attention: ; In the formula, This indicates the scaled dot product attention output feature; This represents the Softmax activation function, which normalizes the similarity matrix into attention weights; Indicates the dimension of the key vector; 5.5: The temporal feature correlations in different subspaces are computed in parallel using multiple attention heads, and the outputs of the multiple attention heads are concatenated. ; In the formula, This indicates that the multi-head attention mechanism ultimately outputs a fused weighted feature. This represents the vector concatenation operation; This represents the output of the i-th attention head. i =1, 2, ..., m ; m represents the number of attention heads, Indicates the output mapping matrix; 5.6: The multi-head attention mechanism allocates weights based on temporal relevance to select key and effective features. The fully connected layer completes the feature dimension mapping, transforming high-dimensional features into one-dimensional temporal prediction results and outputting the prediction sequence of a single modality component, providing a foundation for subsequent superposition and reconstruction of multimodal prediction results.

[0028] In step 5, the Warmup-CosineDecay dynamic learning rate strategy includes: Let the total number of training steps be The current training steps are n, and the warm-up phase steps are n. The initial learning rate is The maximum learning rate is The minimum learning rate is ; During the preheating stage, i.e. At that time, the learning rate increases gradually in a linear manner: ; In the formula, This represents the real-time learning rate corresponding to the nth iteration step; After the preheating phase ends, i.e. At that time, the learning rate decays according to a cosine function: ; By employing the dynamic learning rate strategy described above, the model can avoid gradient oscillations in the early stages of training and achieve fine-tuned parameter adjustments in the later stages of training, thereby improving the model's convergence stability and prediction accuracy.

[0029] Step 6 includes the following steps: Step S6.1: Implement time-series data supervised sample transformation based on a sliding window, using historical continuous vibration information as input and future modal values ​​as prediction targets. Modal-specific sample construction can match the frequency division characteristics of VMD. The network learns features at a single frequency scale, reducing multi-scale feature coupling interference and improving the prediction accuracy of each component.

[0030] Step S6.2: Input the training samples corresponding to each modal component into the improved Stacked BiLSTM prediction model for training; Step S6.3: The training set is used for model parameter gradient updates, while the validation set does not participate in parameter backpropagation and is only used to evaluate the model's generalization performance. By saving the model parameters with the smallest validation error, the network weights with the best generalization ability are selected to avoid overfitting the model on the training set. This model is a multimodal prediction model, and each set of VMD modal components corresponds to an independently trained and independently saved set of optimal model parameters.

[0031] Step S6.4: Input the test set into the trained model to obtain the prediction results for each modal component: ; In the formula, This represents the prediction time series results for the first-order VMD modal components; This represents the prediction time series results of the second-order VMD modal components. This represents the prediction time series result of the Kth order VMD mode component.

[0032] In step 7, the final vibration trend prediction result is obtained by superimposing the prediction results of each mode: ; In the formula, This represents the final predicted result of the original vibration signal obtained from superposition and reconstruction.

[0033] In step 8, the evaluation indicators for the prediction results include: ; In the formula, n Indicates the number of predicted samples. Indicates the first i The actual value of each sample Indicates the first i The predicted value for each sample, This represents the average of the actual values.

[0034] This invention discloses a vibration trend prediction method for pumped storage units based on ISDO-VMD and an improved Stacked BiLSTM, with the following advantages: 1) This invention uses the improved sled dog optimization algorithm ISDO to jointly optimize the number of modes K and the penalty factor α in VMD, avoiding the problem that traditional VMD parameters rely on manual experience setting, and improving the adaptability and decomposition quality of VMD decomposition.

[0035] 2) This invention introduces the Singer chaotic mapping initialization strategy into the original sled dog optimization algorithm, which makes the initial population more evenly distributed in the search space, which is beneficial to improving the algorithm's global search capability and convergence speed.

[0036] 3) This invention introduces the Cauchy mutation mechanism in the process of updating the individual position of sled dogs, which enhances the algorithm's ability to escape local optima and improves the accuracy of VMD parameter optimization.

[0037] 4) This invention decomposes complex non-stationary vibration signals into multiple modal components with relatively concentrated frequency features through VMD, which reduces the complexity and noise interference of the original vibration sequence and is beneficial for subsequent prediction models to extract effective features.

[0038] 5) This invention employs a Stacked BiLSTM structure to simultaneously extract forward and backward temporal information of vibration sequences, and enhances the model's ability to express long-term dependency features, local fluctuation features, and multi-scale dynamic features through multi-layer stacking.

[0039] 6) This invention introduces a multi-head attention mechanism (MHA) at the output of the Stacked BiLSTM, enabling the model to adaptively focus on key time steps that contribute significantly to the prediction results from multiple feature subspaces, thereby improving the feature representation capability of complex vibration signals.

[0040] 7) This invention adopts the Warmup-CosineDecay dynamic learning rate strategy, which can steadily increase the learning rate in the early stage of training and smoothly decrease the learning rate in the later stage of training, thereby improving the gradient oscillation and convergence instability problems in the training process of deep neural networks and improving the generalization ability of the model.

[0041] 8) This invention combines ISDO-VMD, Stacked BiLSTM, multi-head attention mechanism and dynamic learning rate strategy to form a complete vibration trend prediction technology solution for pumped storage units, which can effectively improve the accuracy of vibration trend prediction and provide technical support for pumped storage unit status monitoring, fault early warning and intelligent operation and maintenance. Attached Figure Description

[0042] The present invention will be further described below with reference to the accompanying drawings and examples; Figure 1 This invention is based on ISDO A schematic diagram of the overall process of VMD and the improved Stacked BiLSTM method for predicting vibration trends in pumped storage units.

[0043] Figure 2 This is a graph showing the convergence curve of the fitness function of the ISDO algorithm.

[0044] Figure 3 This is a schematic diagram illustrating the comparison between predicted and actual values ​​in an embodiment of the present invention.

[0045] Figure 4 This is a diagram showing the comparison of evaluation results across multiple indicators.

[0046] Figure 5 A graph showing the comparison of prediction errors for different models. Detailed Implementation

[0047] The present invention will now be described in further detail with reference to the accompanying drawings: This invention proposes a vibration trend prediction method for pumped storage units based on ISDO-VMD and an improved Stacked BiLSTM. First, the ISDO algorithm is used to adaptively and jointly optimize the number of modes K and the penalty factor α in VMD to determine the optimal parameter combination. Then, the optimal parameters are substituted into VMD to decompose the vibration signal, obtaining several modal components with different frequency distributions. Next, a Stacked BiLSTM network is constructed, and MHA is introduced at its output, using a Warmup-CosineDecay dynamic learning rate strategy for training. Finally, each modal component is input into the prediction model, and the prediction results of each component are superimposed to obtain the final vibration trend prediction value. The prediction process is as follows: Figure 1 As shown, the specific steps include the following: Step a: Experimental analysis was conducted using monitoring data on the X-axis swing of the upper guide bearing of Unit 3 at a pumped storage power station in a certain city of a certain province. The selected time period was from December 1st of one year to March 27th of the following year, with data recording intervals of 15 minutes, resulting in a total of 1460 valid data points. The data was preprocessed and divided into three parts: the first 1168 data points were assigned to the training set, the middle 146 data points to the validation set, and the last 146 data points to the test set.

[0048] Step a1: Outlier removal and invalid period data deletion are performed on the collected raw vibration monitoring data. Missing data points are filled using interpolation. The vibration data are then normalized using the Min-Max normalization method, with the normalization formula as follows: ; In the formula, This represents the original vibration data. This represents the normalized vibration data. and These represent the minimum and maximum values ​​in the vibration data, respectively.

[0049] Step a2: Use the VMD decomposition method to decompose the acquired vibration signal into a series of relatively stable subsequences.

[0050] Step a2.1: The specific operation of VMD decomposition is to construct a variational problem and decompose the vibration signal into... A finite-bandwidth intrinsic mode function (IMF) is derived, and the decomposed intrinsic mode function (IMF) is obtained. The variational constraint model constructed to minimize the sum of the estimated bandwidths of the IMFs is shown below: ; The constraints are: ; In the formula, This represents the k-th modal component obtained from the decomposition; This represents the center frequency of the k-th mode obtained from the decomposition; It is the Dirac function; This is the original vibration signal; This represents the convolution operation.

[0051] Step a2.2: To obtain the optimal solution of the above variational constraint model, a quadratic penalty function is introduced. and Lagrange multipliers This transforms the constrained problem into an unconstrained variational problem, and the resulting Lagrangian function is: ; In the formula, This represents the Lagrange multiplier.

[0052] Step a2.3: Use the alternating direction multiplier method to solve the unconstrained variational problem above. Solve the optimal solution of the constrained variational problem by alternately updating each modal component, center frequency and Lagrange multiplier.

[0053] Step a2.4: Given a precision for determining whether the iteration has stopped. If the iteration stopping condition is met, the iteration stops and the result is output; otherwise, the iteration continues. Ultimately, k modal function components (IMFs) with different center frequencies can be obtained.

[0054] Step a3: After VMD decomposition, each modal component is obtained. Input-output pairs are constructed for each modal component using a sliding window method to form supervised learning samples.

[0055] Step b: Construct an improved Stacked BiLSTM prediction model. The unit structure of Stacked BiLSTM is as follows: Figure 1 As shown, the specific construction process is as follows: Step b1: Construct a stacked bidirectional long short-term memory network, Stacked BiLSTM.

[0056] For the input time series BiLSTM extracts temporal features of historical and future directions using forward LSTM and backward LSTM respectively, and its hidden state is updated as follows: ; Concatenate the forward hidden state and the backward hidden state to obtain: ; Step b2: Stack the two BiLSTM layers, using the output of the previous BiLSTM as the input of the next BiLSTM. Each hidden layer has 128 units, and the Dropout ratio is 0.3.

[0057] Step b3: Introduce a multi-head attention mechanism (MHA) at the output of the Stacked BiLSTM. Let the output of the Stacked BiLSTM be... Then, the query matrix Q, key matrix K, and value matrix V are obtained through linear mapping: ; In the formula, , , This is a trainable weight matrix.

[0058] Step b4: Calculate attention weights using scaled dot product attention: ; In the formula, This represents the dimension of the key vector.

[0059] Step b5: Calculate the temporal feature correlations in different subspaces in parallel using 8 attention heads, and then concatenate the outputs of multiple attention heads: ; In the formula, This represents the output of the i-th attention head, where m represents the number of attention heads, which is 8 in this embodiment. This indicates the output mapping matrix.

[0060] Step b6: Input the weighted features output by the multi-head attention mechanism into the fully connected layer to obtain the predicted values ​​of the corresponding modal components.

[0061] Step b7: Train the model using the Warmup-CosineDecay dynamic learning rate strategy. Specific parameter settings are as follows: Initial learning rate Maximum learning rate Minimum learning rate Total number of training rounds Number of rounds in the preheating stage The batch training size is 64, the optimizer type is Adam, and the loss function is MSE.

[0062] During the preheating stage ( The learning rate increases linearly: ; After the preheating phase ends ( The learning rate decays according to a cosine function: ; Step c: Improve the original sled dog optimization algorithm by introducing the Singer chaotic mapping initialization strategy and the Cauchy mutation mechanism to obtain the improved sled dog optimization algorithm ISDO. The specific process is as follows: Step c1: Set the population size of the Improved Sled Dog Optimization Algorithm (ISDO) to 50, the maximum number of iterations to 20, and the dimension of the optimization variables to 2. Number of VMD modalities. The search scope is set to Punishment factor The search scope is set to .

[0063] Step c2: Initialize the sled dog population using Singer chaotic mapping. The Singer chaotic mapping expression is as follows: ; In the formula, chaos is the value generated by the Singer chaotic sequence.

[0064] Step c3: Use the envelope entropy after VMD decomposition as the fitness function. Envelope entropy reflects the complexity of the signal envelope. When the envelope entropy is small, the signal energy distribution is more concentrated and the modal characteristics are more obvious, indicating a better decomposition effect. Calculate the fitness value for each individual sled dog.

[0065] Step c4: Update the individual sled dog positions using a sled dog selection mechanism, a travel model, an obstacle avoidance model, a disorientation model, and a training model, and introduce a Cauchy mutation mechanism after the position update. The improved position update method is as follows: ; In the formula, Indicates the first The location of each individual sled dog This represents the current globally optimal individual position. This indicates element-wise multiplication. denoted as a random number that follows a uniform distribution, and p represents the mutation probability threshold (0.5 in this embodiment). This represents the Cauchy perturbation vector. The position parameters Set to 0, scale parameter Take 1.

[0066] Step c5: Determine if the maximum number of iterations has been reached. If it has, output the position of the sled dog individual with the smallest fitness value as the optimal parameter combination for VMD. If it has not been reached, return to step c3 to continue iterating.

[0067] The convergence curve of the fitness function of the ISDO algorithm is as follows: Figure 2As shown. By Figure 2 It can be seen that as the number of iterations increases, the envelope entropy gradually decreases and tends to stabilize. When the number of iterations reaches 13, the envelope entropy drops to its minimum value. At this point, the fitness function converges to the optimum, and the parameter optimization is complete. The final optimal parameters for VMD are: number of modes. Punishment factor .

[0068] Step d: Using ISDO and validation set data, determine the number of modes in VMD. and penalty factor The two parameters are optimized to obtain the optimal parameter combination for VMD. The specific operation steps are as follows: Step d1: Initialize the ISDO optimization algorithm by setting the population size of sled dogs to 50, the maximum number of iterations to 20, and the optimization range of the parameters. Simultaneously, the number of VMD modes K and the penalty factor are... These two parameters serve as two dimensions of the sled dog's position in ISDO.

[0069] Step d2: Randomly initialize the positions of individuals in the population, use the envelope entropy as the fitness value of each sled dog individual, and calculate the fitness value of each individual at present.

[0070] Step d3: Update the position of each sled dog and use the updated position of each sled dog as a new parameter combination for VMD. The vibration signal was decomposed using VMD, and the fitness value of each sled dog individual was calculated under the new parameters.

[0071] Step d3.1: Sled dog position update. The number of dogs participating in pulling the sled is determined by the sled dog selection mechanism, and the position is updated through the travel model, obstacle avoidance model, disorientation model and training model.

[0072] Step d3.2: Introduce the Cauchy mutation mechanism after updating the individual sled dog positions to enhance the algorithm's ability to escape local optima.

[0073] Step d4: Select the sled dog individual with the smallest fitness value as the optimal individual in the current iteration.

[0074] Step d5: Determine if the maximum number of iterations has been reached. If so, output the position of the best individual in this iteration and use it as the optimal parameter combination for VMD. If the condition is not met, proceed to step d3 and increment the iteration count by 1.

[0075] Step e: Input the optimized hyperparameters into the VMD model to decompose the preprocessed vibration signal, obtaining 11 modal components with different frequency distributions. Input each modal component into the improved Stacked BiLSTM-MHA model to obtain the prediction results of each modal component in the test set.

[0076] Step f: Sum the prediction results of each modal component to obtain the final vibration trend prediction result of the pumped storage unit, and use the root mean square error (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE), and coefficient of determination as the basis for the prediction. These four evaluation metrics are used to perform error analysis on the model's prediction results. The specific formulas for the evaluation metrics are as follows: ; ; ; ; In the formula, n represents the number of predicted samples; Indicates the first The actual value of each sample; Indicates the first Predicted values ​​for each sample; This represents the average of the actual values ​​of n predicted samples.

[0077] To fully demonstrate the effectiveness of the proposed model, ablation comparison experiments were conducted against seven other models: LSTM, VMD-LSTM, SDO-VMD-LSTM, ISDO-VMD-LSTM, ISDO-VMD-BiLSTM, ISDO-VMD-Stacked BiLSTM, and ISDO-VMD-Stacked BiLSTM-MHA. Simultaneously, a horizontal comparison experiment was performed against three mainstream baseline models: BP, Transformer, and GRU. All models were tested using the same dataset. The experimental results are as follows: Figure 3 As shown.

[0078] from Figure 3 As can be seen, the vibration trend prediction curve of the hybrid model proposed in this invention is largely consistent with the actual vibration curve, demonstrating high prediction accuracy and proving the superiority and effectiveness of this invention.

[0079] Table 1 shows a comparison of the performance of different prediction models in the ablation experiment.

[0080]

[0081] As shown in Table 1, each fusion module played a positive role in improving model performance. The LSTM model, as the baseline model, had a relatively large prediction error, with an RMSE of 0.7161. After introducing VMD, the RMSE of the VMD-LSTM model decreased to 0.5962, and R... 2 The R² value increased to 0.9835, indicating that VMD can effectively reduce the modeling complexity of vibration signals. After optimizing the VMD parameters using SDO and ISDO, the model performance was further improved. The RMSE of ISDO-VMD-LSTM decreased to 0.5142, demonstrating that ISDO has stronger global search capabilities and parameter optimization accuracy. Introducing BiLSTM reduced the RMSE from 0.5142 to 0.4445; by constructing a Stacked BiLSTM structure, the RMSE further decreased to 0.3937; and by introducing MHA, the RMSE further decreased to 0.3783. Finally, the model proposed in this invention, combined with the Warmup-CosineDecay dynamic learning rate scheduling strategy, achieved RMSE, MAE, and MAPE of only 0.2716, 0.2172, and 0.1186, respectively, with an R² of 0.9966. Compared with the suboptimal model, RMSE, MAE and MAPE are reduced by 28.21%, 27.43% and 27.99% respectively, which indicates that the improved model of this invention has higher prediction accuracy and better stability than other traditional models.

[0082] As can be clearly seen from Table 1, the error analysis evaluation index of the model proposed in this invention is significantly better than that of other comparative models. This indicates that the ISDO optimization algorithm proposed in this invention has a stronger optimization capability than the original SDO optimization algorithm, and can find a better parameter combination for VMD. At the same time, the proposed improved Stacked BiLSTM-MHA model has a stronger temporal feature extraction capability than the single LSTM model, which further improves the vibration trend prediction performance. The performance index of the comparative models is shown in Table 2.

[0083]

[0084] As shown in Table 2, the model proposed in this invention performs well in terms of RMSE, MAE, MAPE, and R... 2 The proposed model outperforms mainstream prediction models such as BP, Transformer, and GRU in all performance metrics. Specifically, the RMSE, MAE, and MAPE of the proposed model are reduced by 61.9%, 61.6%, and 61.7% respectively compared to the second-best performing GRU model, while the R² is improved to 0.9966. These results validate the comprehensive effectiveness of the proposed method, demonstrating its significant advantage in prediction accuracy.

[0085] Radar chart comparing multiple performance indicators of different models, as shown below Figure 4 As shown. From Figure 4 It can be clearly seen that the model proposed in this invention achieves minimum values ​​in key indicators such as RMSE, MAE, and MAPE, and in R... 2 Its performance across indicators is the best, fully demonstrating its overall advantages in prediction accuracy and stability.

[0086] Comparison of prediction errors of different models as follows Figure 5 As shown. From Figure 5 The error distribution results show that the prediction error of the proposed model is mainly concentrated near zero, with a relatively concentrated distribution range and good symmetry, indicating that its prediction deviation is small at most sample points. In contrast, the error distribution of the comparative model is more dispersed, and it is more prone to larger deviations in sections with large vibration amplitudes.

[0087] In summary, the vibration trend prediction method for pumped storage units based on ISDO-VMD and improved Stacked BiLSTM proposed in this invention achieves adaptive optimization of VMD parameters through ISDO, enhances the ability to extract complex time-series features through improved Stacked BiLSTM-MHA, and improves training stability through the Warmup-CosineDecay dynamic learning rate strategy. This method can effectively improve the accuracy of vibration trend prediction for pumped storage units and provide reliable technical support for condition monitoring, fault early warning, and intelligent operation and maintenance of pumped storage units.

Claims

1. A method for predicting the vibration trend of pumped storage units based on ISDO-VMD and improved Stacked BiLSTM, characterized in that... Includes the following steps: Step 1: Obtain historical vibration monitoring data of the pumped storage unit, preprocess the vibration monitoring data, and divide the preprocessed data into training set, validation set and test set; Step 2: Improve the Sled Dog Optimization Algorithm (SDO) by introducing the Singer chaotic mapping initialization strategy and the Cauchy mutation mechanism to obtain the improved Sled Dog Optimization Algorithm (ISDO). Step 3: Use the improved sled dog optimization algorithm ISDO to jointly optimize the number of modes K and the penalty factor α in variational mode decomposition (VMD) to obtain the optimal combination of VMD parameters; Step 4: Input the optimal VMD parameter combination obtained in Step 3 into the VMD model to decompose the preprocessed pumped storage unit vibration sequence and obtain several modal components (IMFs). Step 5: Construct an improved Stacked BiLSTM prediction model, which includes a stacked bidirectional long short-term memory network, a multi-head attention mechanism (MHA), and a Warmup-CosineDecay dynamic learning rate strategy. Step 6: Input the modal components obtained in Step 4 into the improved Stacked BiLSTM prediction model for training and prediction, and obtain the prediction results corresponding to each modal component; Step 7: Superimpose and reconstruct the prediction results of each modal component to obtain the final prediction result of the vibration trend of the pumped storage unit.

2. The vibration trend prediction method for pumped storage units based on ISDO-VMD and improved Stacked BiLSTM as described in claim 1, characterized in that: Step 1 includes the following steps: Step S1.1: Collect vibration monitoring data of the pumped storage unit, including the upper guide bearing swing signal, shaft vibration signal, and guide bearing vibration signal; Step S1.2: Remove outliers and delete invalid periodic data from the collected raw vibration monitoring data; Step S1.3: Use interpolation to complete the missing data points. The calculation formula is as follows: ； In the formula, The sampling time corresponding to the missing data. The vibration data after interpolation and completion. , The valid sampling times adjacent to the missing point are: , These are the valid raw vibration monitoring values ​​at the corresponding time. Step S1.4: Normalize the vibration data using the Min-Max normalization method. The normalization formula is as follows: ； In the formula, This represents the original vibration data. This represents the normalized vibration data. and These represent the minimum and maximum values ​​in the vibration data, respectively. Step S1.5: Divide the preprocessed vibration data into training set, validation set and test set.

3. The vibration trend prediction method for pumped storage units based on ISDO-VMD and improved Stacked BiLSTM as described in claim 2, characterized in that: Step 2, the improved sled dog optimization algorithm ISDO, includes the following steps: Step S2.1: Set the population size N, the maximum number of iterations T, the dimension of the optimization variable Dim, and the upper and lower bounds of the parameters to be optimized; where the parameters to be optimized include the number of VMD modes K and the penalty factor α; Step S2.2: Initialize the sled dog population using Singer chaotic mapping to improve the uniformity and diversity of the initial population distribution in the search space. The initialization formula is as follows: ； In the formula, Dog represents the position of an individual sled dog, Lb and Ub represent the lower and upper bounds of the optimization variable, respectively, and chaos represents the chaotic sequence value generated by the Singer chaotic map; Step S2.3: Use the envelope entropy after VMD decomposition as the fitness function to calculate the fitness value for each sled dog individual. The formula for calculating the envelope entropy is: ； ； In the formula, E k For the first k Envelope entropy of the first modal component; K The number of VMD decomposition modes; This represents the normalized probability distribution of the envelope amplitude of the k-th mode; Fit This represents the fitness value corresponding to an individual sled dog; the smaller the envelope entropy, the more concentrated the energy distribution of the modal components after signal decomposition, and the better the decomposition effect. Step S2.4: Update the individual sled dog positions using the sled dog selection mechanism, travel model, obstacle avoidance model, disorientation model, and training model, including: The initial population for SDO is generated through random distribution to ensure that the initial solution covers the entire search space. Its expression is: ； In the formula: Dog for N × Dim dimensional population matrix; N Population size; Dim To optimize dimensions; Lb and Ub These are the lower and upper bounds of the optimization variable, respectively; rand Generate uniformly random numbers within the interval [0,1]. The algorithm determines the number of dogs participating in the core task using a formula. N 1. Focus on the region with the best solution to improve search efficiency: ； In the formula: k 1 is the adjustment coefficient. k For task-related parameters; Depending on the sled dog's position in the team, the speed update formula is divided into three categories, with position updates uniformly following the "speed accumulation" logic: when i When =1,2: ； when i ∈ (3, N 1-2) time: ； when i = N 1-1, N 1 o'clock: ； In the formula: ω Inertial weight; c 1, c 2 represents the learning factor; r 1. r 2 and r 3 is a random number within the interval [0, 1]. Dog Gi , Dog Z Dogs in reference positions; Dog i-2 , Dog i+2 For adjacent dogs; Dog t , Dog r A randomly selected reference individual; l Binary control variables; ； When the search encounters a local optimum, the position is updated through random perturbation to achieve obstacle avoidance. The specific expression is as follows: ； In the formula: r 4 and r 5 is a random number between [0, 1]; it escapes the local optimum region through bidirectional perturbation; When the population reaches a search standstill, it is guided to reset its position using global information, the specific expression of which is as follows: ； In the formula: r 6 and r 7 is a random number between [0, 1]. C The guiding coefficient; For a sign random variable, Dog z It is the globally optimal individual. Dog r For random individuals; The expression for training and optimizing dogs with poor performance is: ； In the formula: r 8~ r 11 A random number within the range [0,1]. F This is a learning ability factor used to adaptively adjust training intensity; its expression is: ； In the formula: For the current individual fitness, and These are the minimum and maximum fitness values ​​of the population, respectively. X 1. X 2. X 3 represents the training direction vector; D 1. D 2. D 3 represents the performance gap; its definition is as follows: ； In the formula: Dog Better To be an outstanding individual; Dog Worse Individuals to be trained; Step S2.5: After updating the individual sled dog positions, introduce the Cauchy mutation mechanism. The improved position update method is as follows: ； In the formula, This represents the position of the i-th sled dog. This represents the current globally optimal individual position. This represents the speed of the i-th sled dog. This indicates element-wise multiplication. Let p represent a random number that follows a uniform distribution, and r represent the mutation probability threshold. i Represents the Cauchy perturbation vector; Step S2.6: Determine whether the maximum number of iterations has been reached. If it has, output the position of the sled dog individual with the smallest fitness value as the optimal parameter combination for VMD. If it has not been reached, return to step S2.3 to continue iterating.

4. The vibration trend prediction method for pumped storage units based on ISDO-VMD and improved Stacked BiLSTM as described in claim 3, characterized in that: Step 3, which uses the Improved Sled Dog Optimization Algorithm (ISDO) to jointly optimize the VMD parameters, includes the following steps: Step S3.1: Set the search range for the number of VMD modes K and the penalty factor α; Step S3.2: Represent the position of each individual sled dog as a two-dimensional vector: ； In the formula, Indicates the first The number of modalities corresponding to each individual Indicates the first The penalty factor corresponding to each individual; Step S3.3: Since the number of modes K is an integer, the K value obtained during the optimization process is rounded down. The expression is: ； In the formula: K raw The original floating-point modal number output for the individual sled dog position; Round(·) is the rounding function; K is the number of valid modalities available for VMD after rounding; Step S3.4: Set each set of parameters Input the VMD model, decompose the vibration signal, and calculate the corresponding envelope entropy fitness value; fitness Fit i As a quantitative indicator for evaluating the performance of VMD parameters in the ISDO algorithm, the optimization objective is to minimize... Fit i The algorithm iterates through all sled dog individuals in the population, compares the fitness values ​​corresponding to each set of parameters, and continuously updates the globally optimal parameter combination until the iteration termination condition is met. Step S3.5: Iteratively update the parameter combination with the minimum fitness value using the improved sled dog optimization algorithm ISDO, and use it as the optimal parameters for VMD: ； In the formula, Represents the number of optimal modes This represents the optimal number of penalty factors.

5. The vibration trend prediction method for pumped storage units based on ISDO-VMD and improved Stacked BiLSTM as described in claim 4, characterized in that: Step 4 includes the following steps: Step S4.1: Decompose the vibration signal into K modal components with finite bandwidths, and minimize the sum of the estimated bandwidths of each modal component, constructing the following variational constraint model: ； In the formula, For the set of all modal components, This is the set of central angular frequencies corresponding to each modal component; The number of VMD decomposition modes; Represents the Dirac delta function; The imaginary unit; For time-series sampling time; The k-th modal component is the time-domain signal. This is a complex exponential demodulation term used to shift the modal spectrum to the baseband. For time t The first-order partial differential operator; The L2 norm is used to measure modal bandwidth; * indicates convolution operation. The constraints are: ； In the formula, Indicates the first One modal component, Indicates the first The center frequency of each modal component Represents the original vibration signal; Step S4.2: Introducing the quadratic penalty factor α and the Lagrange multiplier λ, the constrained variational problem is transformed into an unconstrained variational problem, yielding the augmented Lagrange function: ； In the formula, Represents the augmented Lagrangian function constructed from the VMD model; Represents time-domain Lagrange multipliers; This refers to the inner product operation of functions; Step S4.3: The alternating direction multiplier method is used to iteratively update each modal component, center frequency, and Lagrange multiplier. The alternating direction multiplier method decomposes the VMD variational optimization problem into multiple simple subproblems that are solved alternately, continuously optimizing the modal components and their corresponding center frequencies. When the convergence condition is met, the iteration stops, and K independent modal components (IMFs) with non-overlapping frequency domains are output. Step S4.4: Output the decomposition results of the vibration signal: ； In the formula, This represents the k-th modal component obtained from VMD decomposition, corresponding to the time-domain vibration component within a specific center frequency range.

6. The vibration trend prediction method for pumped storage units based on ISDO-VMD and improved Stacked BiLSTM as described in claim 5, characterized in that: In step 5, the improved Stacked BiLSTM prediction model includes: 5.1: Construct a stacked bidirectional long short-term memory (BLSTM) network, wherein the stacked BLSTM network comprises at least two layers of BiLSTM network; for the input time series: ； In the formula, This represents a one-dimensional time-series sample representing the network input. These represent the vibration sampling values ​​at each moment within the time series; The BiLSTM network extracts temporal features of historical and future directions using forward LSTM and backward LSTM respectively, and its hidden state is updated as follows: ； In the formula, This represents the hidden state output by the forward LSTM at time t; This represents the hidden state of the inverse LSTM output at time t; This represents the operation function of the forward LSTM unit; This represents the operation function of the inverse LSTM unit; This represents the network input time-series sample value at time t; This represents the hidden state of the forward LSTM at time t-1; This represents the reverse LSTM hidden state at time t+1; Concatenate the forward hidden state and the backward hidden state to obtain: ； In the formula, This represents the bidirectional fused hidden state after BiLSTM splicing at time t; This represents the vertical concatenation operation of vectors, which combines the forward hidden state and the backward hidden state into a complete feature vector. 5.2: Stack multiple BiLSTM networks, with the output of the previous BiLSTM layer serving as the input of the next BiLSTM layer; 5.3: Introducing Multi-Head Attention (MHA) at the output of Stacked BiLSTM; Let the output of Stacked BiLSTM be H, then the query matrix Q, key matrix K, and value matrix V are obtained through linear mapping: ； In the formula, , , The weight matrix is ​​trainable. 5.4: Calculate attention weights using scaled dot product attention: ； In the formula, This indicates the scaled dot product attention output feature; This represents the Softmax activation function, which normalizes the similarity matrix into attention weights; Indicates the dimension of the key vector; 5.5: The temporal feature correlations in different subspaces are computed in parallel using multiple attention heads, and the outputs of the multiple attention heads are concatenated. ； In the formula, This indicates that the multi-head attention mechanism ultimately outputs a fused weighted feature. This represents the vector concatenation operation; This represents the output of the i-th attention head. i =1, 2, ..., m ; m represents the number of attention heads, Indicates the output mapping matrix; 5.6: The multi-head attention mechanism allocates weights based on temporal relevance and selects key effective features; the fully connected layer completes the feature dimension mapping, converts high-dimensional features into one-dimensional temporal prediction results, and outputs the prediction sequence of a single modality component.

7. The vibration trend prediction method for pumped storage units based on ISDO-VMD and improved Stacked BiLSTM as described in claim 6, characterized in that: In step 5, the Warmup-CosineDecay dynamic learning rate strategy includes: Let the total number of training steps be The current training steps are n, and the warm-up phase steps are n. The initial learning rate is The maximum learning rate is The minimum learning rate is ; During the preheating stage, i.e. At that time, the learning rate increases gradually in a linear manner: ； In the formula, This represents the real-time learning rate corresponding to the nth iteration step; After the preheating phase ends, i.e. At that time, the learning rate decays according to a cosine function: 。 8. The vibration trend prediction method for pumped storage units based on ISDO-VMD and improved Stacked BiLSTM as described in claim 7, characterized in that: Step 6 includes the following steps: Step S6.1: Implement time series data supervised sample conversion based on sliding window, using historical continuous vibration information as input and future modal values ​​as prediction targets; Step S6.2: Input the training samples corresponding to each modal component into the improved Stacked BiLSTM prediction model for training; Step S6.3: The training set is used for model parameter gradient updates, and the validation set is only used to evaluate the model's generalization performance; by saving the model parameters with the smallest validation error, the network weights with the best generalization ability are selected; this model is a multimodal prediction model, and each set of VMD modal components corresponds to an independently trained and independently saved set of optimal model parameters; Step S6.4: Input the test set into the trained model to obtain the prediction results for each modal component: ； In the formula, This represents the prediction time series results for the first-order VMD modal components; This represents the prediction time series results of the second-order VMD modal components. This represents the prediction time series result of the Kth order VMD mode component.

9. The vibration trend prediction method for pumped storage units based on ISDO-VMD and improved Stacked BiLSTM as described in claim 8, characterized in that: In step 7, the final vibration trend prediction result is obtained by superimposing the prediction results of each mode: ； In the formula, This represents the final predicted result of the original vibration signal obtained from superposition and reconstruction.

10. The vibration trend prediction method for pumped storage units based on ISDO-VMD and improved Stacked BiLSTM according to claim 9, characterized in that: It also includes step 8: evaluating the prediction results using root mean square error (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE), and coefficient of determination (R²). Evaluation indicators for prediction results include: ； In the formula, n Indicates the number of predicted samples. Indicates the first i The actual value of each sample Indicates the first i The predicted value for each sample, This represents the average of the actual values.

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