Degradation trend prediction of rail stripping for heavy haul railway based on multi-strategy hybrid improved pelican algorithm

2.2.1. CNN-GRU module

The CNN-GRU module consists of a CNN and a GRU part. The main structure of the basic internal neural network layer of the CNN layer is convolution - pooling - full connection. This kind of structure can reduce the number of weights, extract features of specific objects, and then identify, classify, or predict the corresponding objects according to the features.

The output of the CNN part is used as the input of the GRU part, and the GRU part performs the prediction. GRU is an effective recurrent neural network, which has been improved on the basis of LSTM so that it can better learn the time sequence characteristics of data and suppress gradient learning and gradient explosive. As shown in Figure 2, the BIGRU consists of two layers of GRUs stacked on top of each other, and the final output is jointly determined by the statuses of the two GRUs.

Figure 2. Diagram of BIGRU structure.

The final output state is the combination of the forward propagation output and the back propagation output. The calculation is carried out by:

Where, in the forward propagation process, $$ {x_t} $$, $$ \overrightarrow U\ $$, $$ \overrightarrow W\ $$, and $$ \overrightarrow b\ $$ represent the input at time, the state, the input weight matrix, and the bias item of the hidden layer, respectively. While $$ \overrightarrow U\ $$, $$ \overrightarrow W\ $$, and $$ \overrightarrow b\ $$ are the state, the input weight matrix, and the bias term of the sub-hidden layer, respectively. By using the two-layer GRU structure, the problem of gradient disappearance and gradient explosion in the LSTM network can be effectively solved, and the accuracy of prediction can be improved.

2.2.2. Compression excitation module SE

During the prediction process of stripping development, there is a variety of noise interferences. As the network is trained, these noises will occupy a large amount of memory, and negative impacts could be imposed on the network. By embedding a SE module^[15], the feature weights can be adjusted adaptively to solve the feature loss problem caused by varying channel proportions in the convolution pooling process. This mainly includes Squeeze and Excitation, as shown in Figure 3.

Figure 3. Diagram of SE structure.

The Squeeze part in Figure 3 is expressed as:

The Excitation part is:

Where the parameter $$ T $$ indicates the feature representation, $$ H $$, $$ W $$, and $$ C $$ represent the height, width, and channel of $$ T $$, respectively. $$ \delta $$ and $$ \sigma $$ are the ReLU activation function and the Sigmoid activation function, respectively, and $$ W_1 $$ and $$ W_2 $$ are fully connected layers.

2.3.1. Initialization through the chaos sequence of circle map

At the initialization stage, the chaos sequence of the circle map^[17] is used instead of initialization through a rand function. The initialization definition formula through a circle map is:

Compared to a random distribution, the initial position distribution of the population is more uniform, and the search becomes more convenient. The search range of pelican groups in space is larger. These factors can improve the global search performance of the algorithm to a certain extent.

2.3.2. Dynamic weight factor

At the approaching stage (global search phase), a dynamic weight actor $$ \lambda $$^[18] is introduced, which is defined as:

Where $$ t $$ indicates the current iteration number, and $$ T $$ is the maximum number of iterations. The updated equation of the pelican position becomes:

The dynamic weighting factor helps the pelicans better update their position. At the beginning of the iteration, it is larger, which facilitates a better global search. Towards the end of the iteration, it will be adaptively reduced to improve local search performance while increasing the convergence speed.

2.3.3. Adaptive T distribution mutation

Adaptive $$ T $$ distribution mutation with the iteration number as a degree of freedom^[19] is used as the operator disturbing the position of the solution, and the updated position equation is represented as follows:

The introduction of $$ T $$ distribution mutation not only makes full use of the current position information but also increases the random interference information. This can further improve the ability of the algorithm to avoid local optimal solutions and speed up the convergence of the algorithm.

To sum up, the standard Pelican algorithm is improved by introducing the chaotic sequence of the circle map into the initialization, along with the dynamic weight factor and the adaptive $$ T $$ distribution mutation.

The improved algorithm flow chart is shown in Figure 4 and Table 1, where $$ N $$ and $$ T $$ represent the population of pelicans and the maximum number of iterations, respectively; $$ x_{i, j}^{p_1} $$, $$ x_{i, j}^{p_2} $$, and $$ x_i $$ represent the new status of the $$ {i^{th}} $$ pelican in the $$ {j^{th}} $$ dimension based on phase 1, the new status of the $$ {i^{th}} $$ pelican in the $$ {j^{th}} $$ dimension based on phase 2, and the new position of the pelican after the update, respectively. After all individuals are updated through three stages, namely initialization, approaching, and surface flying, the best candidate solution so far will be obtained, and the algorithm will enter the next iteration. This process will be executed repeatedly based on the position update formula until it is completely over. The obtained optimal parameters will then be outputted to the prediction network for the next step of prediction.

Figure 4. Flow diagram of IPOA.

3.1.1. Preprocessing of experimental data samples

A continuous test spanning several months was conducted on the rail stripping, covering the entire process from the initiation of rail stripping through its development and up to the point of rail replacement. Vibration data from the damaged rail was collected at regular intervals during the same sampling period, amounting to a total of 98 instances. The sampling frequency was set at 4 kHz, with data collection occurring at 2500 sample points each time. The data is depicted in Figure 6, and the corresponding waveforms are sequentially assembled from the vibration sequences that reflect the damage characteristics in a single sampling.

Figure 6. We present Vibration data of rail with stripping in Figure 6A-B. A: Vibration data of rail with stripping 1; B: Vibration data of rail with stripping 2.

In this experiment, two sets of stripping data were utilized, each comprising 98 x 2500 sample points of stripping. Initially, a total of 26 time-frequency domain characteristics were extracted for each collected sample, including the maximum value, kurtosis, peak factor, and others. At this point, the data dimension stood at 98×26. Subsequently, indicators of poor performance were removed based on Spearman's correlation, resulting in 12 remaining features. The data dimension was then reduced to 98×12. Following this, the dimensionality of these 12 features was further reduced through PCA, yielding an eigenvector with a dimension of 98×1. This eigenvector serves as the health indicator capable of characterizing the development trend of rail stripping damage.

The resulting eigenvalues, contribution rates, and accumulative contribution rates corresponding to the remaining 12 principal components after correlation analysis and feature screening are listed in Table 2.

The principal components with characteristic values greater than 0.3 were selected; that is, the top three principal components were used as the final principal components, and the accumulative contribution rate reached 97.895% (far greater than 85%). Their corresponding $$ 12 \times 3 $$ eigenvectors were multiplied by the standardized data to obtain the calculation formula of the top three principal components, as shown in Table 3.

3.1.2. Algorithm validity verification

To validate the accuracy and efficacy of the proposed IPOA, four distinct algorithms, Particle Swarm Optimization (PSO) Algorithm, Whale Optimization Algorithm (WOA), POA, and IPOA, were individually chosen to conduct comparative calculations across eight functions selected from a pool of 23 benchmark functions. These functions encompassed three unimodal benchmark functions, three multimodal benchmark functions, and two mixed benchmark functions, with the definitions outlined in Table 4. Each algorithm underwent 100 iterations for every benchmark function, and the population size for all algorithms was set at 20. Each algorithm was independently executed ten times to obtain average values, and the experimental results were meticulously recorded. Convergence curves illustrating the performance are depicted in Figure 7.

Figure 7. We present Convergence curves of benchmark functions in Figure 7A-H. A: the convergence curves of $$ f_1(x) $$; B: the convergence curves of $$ f_2(x) $$; C: the convergence curves of $$ f_3(x) $$; D: the convergence curves of $$ f_4(x) $$; E: the convergence curves of $$ f_5(x) $$; F: the convergence curves of $$ f_6(x) $$; G: the convergence curves of $$ f_7(x) $$; H: the convergence curves of $$ f_8(x) $$.

In Figure 7, the lines represented by black, green, blue, and red signify the PSO Algorithm, WOA, POA, and IPOA, respectively. It is evident that the red line demonstrates the most rapid descent as the number of iterations progresses. This observation indicates that in equivalent circumstances, IPOA achieves the lowest fitness and the most rapid convergence rate. Hence, the decision was made to utilize the IPOA for optimizing the parameters of the prediction model in subsequent analyses.

3.1.3. Network parameter setting

In this experiment, the historical health indicator data of rail stripping was used to predict the health indicator of future momentum. The dimensionality of input data is 12, and 1 for the output data. Through multiple experiment runs, the optimal settings of the main parameters for this model were obtained and are shown in Table 5.

3.1.4. Performance test

In order to better assess the accuracy of the prediction performance of the proposed model, three indicators of MAE, RMSE, and $$ {R^2} $$ were used to evaluate it, and their specific expressions are:

The three traditional prediction network models were selected for comparative experiments at the same conditions: (1) BP, (2) LSTM, and (3) GRU. The results are demonstrated in Figure 8, Table 6, and Table 7.

Figure 8. We present Prediction results of stripping development in Figure 8A-B. A: results of stripping 1; B: results of stripping 2.

Out of the whole dataset, the first 75% of data points were used as the training set, and the last 25% as the test set. The prediction accuracy of the model was determined by comparing the stripping health indicator value of the test set with the actual health indicator. It can be seen in Figure 8, Table 6, and Table 7 that the prediction accuracy of the proposed method is significantly better than that of the other five traditional models. For the prediction of stripping 1, MAE and RMSE of the proposed model decreased by 32.5% and 56.8%, respectively, and $$ {R^2} $$ increased by 34.3%, compared with those of GRU, the model with the best performance indicators out of the five traditional models. For the prediction of stripping 2, MAE and RMSE of the proposed model decreased by 50% and 50.7%, respectively, and $$ {R^2} $$ increased by 35.3%, compared with those of GRU, the model with the best performance indicators out of the three traditional models.

3.1.5. Ablation experiment

In order to make the research results more objective, compare with the latest related research work, and evaluate the role and effect of several key modules in the proposed model, the key modules were removed or replaced in turn for ablation research, while other parameters were kept unchanged. The following four different prediction models were designed for comparative analysis.

(1) POA-CNN-SE-BIGRU. While keeping the other structures of the network unchanged, the multi-strategy hybrid IPOA was replaced with the standard POA. This model was mainly used to verify the role of the improved part of the algorithm.

(2) CNN-SE-BIGRU. While keeping the other structures of the network unchanged, the IPOA was removed. This model was used to evaluate the effect of the optimization algorithm.

(3) CNN-BIGRU. While keeping the other structures of the network unchanged, the SE module was removed. This model was used mainly to evaluate the role of the SE attention module.

(4) BIGRU. The regular BIGRU.

The comparisons of results are shown in Figure 9, Table 8, and Table 9.

Figure 9. We present Convergence results of IPOA and POA in Figure 13A-B. A: convergence results of IPOA and POA of bearing 1; B: convergence results of IPOA and POA of bearing 2.

Clearly, the convergence curves of stripping 1 and stripping 2 are shown in Figure 9: the black line represents the IPOA, and the red line represents the POA. This shows that IPOA has a low fitness value and fast convergence speed.

It can be seen in Table 8 and Table 9 that the predictive ability of the proposed model is better than that of other comparison models. The improvement of prediction results is attributed to the IPOA and the introduction of SE attention mechanisms. Taking the prediction results of stripping 1 as an example, it is shown in the ablation experiment that from BIGRU to CNN-BIGRU, MAE and RMSE decreased by 22% and 15.7%, respectively, and $$ {R^2} $$ increased by 9.5%; When SE was introduced, as in CNN-SE-BIGRU, compared with CNN-BIGRU, MAE decreased by 3%, RMSE decreased by 6.7%, and $$ {R^2} $$ increased by 2.5%. This means that the introduction of the SE attention mechanism has enhanced the ability of the network to capture effective features and reduce the interference of useless information to a certain degree. After POA was added, as in POA-CNN-SE-BIGRU, compared with CNN-SE-BIGRU, MAE and RMSE decreased by 3% and 17.1%, respectively, and $$ {R^2} $$ increased by 7.2%. This means POA is effective in optimizing model parameters. Furthermore, compared with those of POA-CNN-SE-BIGRU, MAE and RMSE of the proposed model decreased by 15.6% and 29.3%, respectively, and $$ {R^2} $$ increased by 5.6%. This is achieved from the multi-strategy hybrid improvement of IPOA.

3.2.1. Preprocessing of experimental data samples

The data was collected on the accelerated aging platform PRONOSTIA, as shown in Figure 10. The goal of PRONOTIA is to provide real experimental data to describe the degradation of ball bearings throughout their entire service life (until complete failure). The PRONOSITIA platform provides almost all types of defects (balls, rings, and cages) for every degraded bearing.

Figure 10. PRONOSTIA Bearing Data Collection Platform.

Two accelerometers were installed on the bearing box, and the accelerometer was used to collect raw vibration signal data in the horizontal and vertical directions of the bearing. The data is shown in Figure 11. Record 2560 data points every ten seconds, or 0.1 seconds of data, with a sampling rate of 25.6 kHz. Two datasets were selected here: bearing 1 and bearing 2.

Figure 11. We present Vibration data of bearings in Figure 11A-B. A: vibration data of bearing 1; B: vibration data of bearing 2.

3.2.2. Performance test

The following three traditional prediction network models were selected for comparative experiments at the same conditions: (1) BP, (2) LSTM, and (3) GRU. The prediction results are demonstrated in Figure 12, Table 10, and Table 11.

Figure 12. We present Prediction results of bearing damage development in Figure 12A-B. A: prediction results of bearing 1; B: prediction results of bearing 2.

Out of the whole dataset, the first 75% of data points were used as the training set, and the last 25% as the test set. The prediction accuracy of the model was determined by comparing the bearing health indicator value of the test set with the actual health indicator. It can be seen in Figure 12, Table 10, and Table 11 that the prediction accuracy of the proposed method is significantly better than that of the other five traditional models. For the prediction of bearing 1, MAE and RMSE of the proposed model decreased by 46.3% and 48.1%, respectively, and $$ {R^2} $$ increased by 66.0%, compared with those of GRU, the model with the best performance indicators out of the five traditional models. For the prediction of bearing 2, MAE and RMSE of the proposed model decreased by 32.2% and 34.5%, respectively, and $$ {R^2} $$ increased by 61.2%, compared with those of GRU, the model with the best performance indicators out of the three traditional models.

3.2.3. Ablation experiment

In the same way, in order to make the research results more objective, compare with the latest related research work, and evaluate the role and effect of several key modules in the proposed model, the key modules in the proposed model were removed or replaced in turn for ablation research, while other parameters were kept unchanged. The following four different prediction models were designed for comparative analysis: (1) POA-CNN-SE-BIGRU, (2) CNN-SE-BIGRU, (3) CNN-BIGRU, and (4) BIGRU. The results are shown in Figure 13, Table 12, and Table 13.

Figure 13. We present Convergence results of IPOA and POA in Figure 9A-B. A: convergence results of IPOA and POA of stripping 1; B: convergence results of IPOA and POA of stripping 2.

Clearly, the convergence curves of bearing 1 and bearing 2 are shown in Figure 13; the black line represents the IPOA, and the red line represents the POA. This shows that IPOA has a low fitness value and fast convergence speed.

It can be seen in Table 12 and Table 13 that the predictive ability of the proposed model is better than that of other comparison models. The improvement of prediction results is attributed to the IPOA and the introduction of the SE attention mechanism. Taking the prediction results of bearing 1 as an example, it is shown in the ablation experiment that from BIGRU to CNN-BIGRU, MAE and RMSE decreased by 8.1% and 6.4%, respectively, and $$ {R^2} $$ increased by 6.3%; When SE was introduced, as in CNN-SE-BIGRU, compared with CNN-BIGRU, MAE decreased by 8.8%, RMSE decreased by 12.5%, and $$ {R^2} $$ increased by 11.8%. This means that the introduction of the SE attention mechanism can enhance the ability of the network to capture effective features and reduce the interference of useless information to a certain degree. After POA was added, as in POA-CNN-SE-BIGRU, compared with CNN-SE-BIGRU, MAE and RMSE decreased by 11.3% and 13.0%, respectively, and $$ {R^2} $$ increased by 7.9%. This means POA is effective in optimizing model parameters. Compared with those of POA-CNN-SE-BIGRU, of the proposed model, MAE and RMSE decreased by 20% and 10.7%, respectively, and $$ {R^2} $$ increased by 8.2%. This is achieved from the multi-strategy hybrid improvement of IPOA.