The buried pipelines and metallic structures in subway systems are subjected to electrochemical corrosion under the stray current interference. The corrosion current density determines the degree and the speed of stray current corrosion. A method combining electrochemical experiment with the machine learning algorithm was utilized in this research to study the corrosion current density under the coupling action of stray current and chloride ion. In this study, a quantum particle swarm optimizationneural network (QPSONN) model was built up to predict the corrosion current density in the process of stray current corrosion. The QPSO algorithm was employed to optimize the updating process of weights and biases in the artificial neural network (ANN). The results show that the accuracy of the proposed QPSONN model is better than the model based on backpropagation neural network (BPNN) and particle swarm optimizationneural network (PSONN). The accuracy distribution of the QPSONN model is more stable than that of the BPNN model and the PSONN model. The presented model can be used for the prediction of corrosion current density and provides the possibility to monitor the stray current corrosion in subway system through an intelligent learning algorithm.
Stray current is defined as electric currents flowing along other elements, which are not components of the purposebuilt electric current [
Stray current in the DC mass transit system.
In the face of rapid development of urban rail transit, it is especially important to understand and monitor the process of stray current corrosion that happens in the DC transit system. The existing monitoring of stray current corrosion is mainly conducted through reference electrode. However, problems existed. For example, the polarization potential of reference electrode sometimes cannot represent the corrosion status of buried pipelines accurately due to the influence of IR drop and internal reactants. The faulty reference electrode is hard to replace in the subway system because it is difficult to excavate the electrode from the concrete structure in order to ensure the overall strength of the system. Besides, because the subway system is difficult to make structural changes after completion, accurate corrosion monitoring without excavation is especially important. Therefore, a method for accurately establishing the mapping relationship between the external environment factors and corrosion status is urgently needed. Based on Faraday’s law, corrosion current density
As an effective tool to study the corrosion process, many methods based on the machine learning algorithm have been employed, such as the backpropagation neural network (BPNN) model [
In this paper, stray currents existing in the main structure of the subway systems, buried pipelines, and metal structures around the system are studied using the intelligent algorithmbased method combining with an electrochemical experiment. NaCl is an important composition of the extracted soil solution [
Aiming at stray current corrosion process, the purpose of this paper is to realize the prediction of corrosion current density considering multiple environment factors based on the QPSONN algorithm, and then to provide the possibility of accurately monitoring the corrosion status of buried pipelines without excavation. Thus, the experimental process is firstly conducted to determine the main factors affecting the corrosion current density and then to set up the prediction database. The rest of this paper is organized as follows: The theory of the proposed QPSONN algorithm is described in Section
In 1949, Donald Hebb’s research showed that biological conditioning is generally caused by the nature of neurons, pointing to a learning method of the biological neural unit’s learning mechanism, which is now known as neural plasticity [
The QPSO algorithm based on quantum behavior is generated from the classical PSO algorithm. It is mainly combined with the improvement of quantum theory behavior, which improves the global optimization performance of the algorithm. In quantum mechanics, the state of a particle is described by a wave function
According to the theory of quantum mechanics, the dynamic behavior of particles in quantum space can be described by Schrödinger equation. In the QPSO algorithm, the wave function in the Schrödinger equation is used to express the position of the particles in the quantum particle space. The wave function of a particle with the position of
After the probability density function for a particle appearing at a certain point in the quantum space is calculated, the Monte Carlo method is employed to obtain the position equation of the particle. The position equation is given in the following equation:
The
The mean best position
In equation (
Finally, the evolution formula of the QPSO can be expressed as follows:
The
The flow of the QPSO algorithm is stated in detail as follows:
Randomly initialize the initial position of
Calculate the fitness value of particles according to object function
Update current position of each particle
Compare the last individual optimal values
Compare current global optimal position
Calculate mean best position
For each dimension of the particle, calculate random point
Update new position of each particle
Repeat Step 2 to Step 8, until iteration MaxTimes is met
Specimens fabricated from the sheet of Q235A pipelines were used, and the chemical composition in wt.% is shown in Table
Element composition of Q235A specimen.
C  Si  Mn  P  S  As  Alt  V  Nb  Ti  Cu  Ni  Cr  Mo  B  N  Ceq 

10^{–2}  10^{–2}  10^{–2}  10^{–3}  10^{–3}  10^{–3}  10^{–3}  10^{–3}  10^{–3}  10^{–3}  10^{–2}  10^{–2}  10^{–2}  10^{–4}  10^{–4}  10^{–4}  10^{–2} 
13  19  49  28  19  10  1  2  2  1  1  1  3  1  1  37  22 
In this paper, the accelerated simulation corrosion was conducted by the experimental system shown in Figure
Experimental system of electrochemical corrosion under the coupling excitation of stray current and Cl^{−} ion.
The counter electrode used in this experiment is a 4 cm^{2} Ti electrode, and the reference electrode is a saturated calomel electrode (SCE). The potential values shown here and in the next are of the relative SCE. There is a maintained gap of 15 mm between the counterelectrode and counterelectrode during each measurement. A CHI604e electrochemical workstation was employed as the electrochemical measurement instrument. During electrochemical testing, the test loop is independent of the applied DC loop. Each time before the testing of Tafel polarization curves, the opencircuit potential (OCP) needs to be measured to ensure scanning range. The potential scanning direction in the Tafel method is from negative to positive. The potential range of each scan is based on the testing result of OCP. The scanning rate utilized in the Tafel method was set to 2 mV/s. All experiments were performed in an indoor environment at 20°C. The solution remained undisturbed during polarization curve measurement.
In the case of excitation amplitude of 0.025 A/cm^{2}, the polarization curve under stray current corrosion in the initial oxidationreduction potential of 230 mV and 0.2 mol/L NaCl is shown in Figure
Experimental results of stray current corrosion on Q235A steel test samples.
Electrochemical parameters during stray current corrosion on Q235A steel test samples.
The experimental database is obtained from the electrochemical corrosion experiment stated above. The stray current density
Correlation coefficient between input variables and
The whole prediction algorithm can be divided into preparation phase, optimization phase, training phase, and prediction phase. The backpropagation (BP) learning is to generate a final prediction model. The QPSO algorithm is employed to search for the suitable connection weights and biases for the initial structure of the neural network. The framework of the QPSONN method for predicting corrosion current density is shown in Figure
Prepare the training and validation data sets and set the training ratio, validation ratio, and testing ratio in data normalization process
Initialize the input weights, output weights, and hidden biases of NN structure, establish the initial NN model, and initialize the vector containing optimization object
Update particles according to the QPSO evolution equations (
Update the parameters for each particle according to the global optimal solution
When the weights and the hidden biases are updated, the NN model will be correspondingly updated with new parameters
Calculate the fitness according to the prediction results with the training data sets. (the fitness calculation method will be explained detailedly in equation (
Judge whether meeting end of criterion, if yes, obtain the optimal input weights, output weights, and the hidden biases; if not, jump to step 3
Establish the optimal NN model based on the optimization results returning from QPSO (optimized weights and biases)
Predict corrosion current density based on the built QPSONN model
Evaluate prediction results of corrosion current density
Structure of the ANN and encoding details in QPSO.
The ratio of training set, validation set, and testing set were preliminarily set as 70%, 15%, and 15%, respectively. The error minimization tolerance was set to 1%. The ANN training target was set to MSE of 0.01. The establishment, optimization, training, and testing of the QPSONN were carried out in the environment of MATLAB R2015b. The parameters of the QPSONN are given in Table
Parameters of the QPSOANN algorithm.



Population size  80 
Max iterations  300 
Contractionexpansion coefficient  0.5 


Number of input neurons  4 
Number of output neurons  1 
Transfer function in hidden layer 

Transfer function in output layer 

Learning rate  0.01 
Training times  300 
The input weights, hidden biases, output weights, and output biases are the optimization target of the QPSO algorithm. The details of the QPSONN algorithm are shown in Figure
Encoding for particle in QPSO and pretreatment for data set in the NN: the structure of the ANN model is shown in Figure
Data normalization: before the training and optimization process in the QPSONN model, the data set needs to be normalized to induct statistical distribution of unified samples and eliminate the dimensional impact between input indicators. In this model, linear minmax normalization was employed. Taking the input parameter stray current density
where
Optimization process in QPSO: in general, two termination conditions can be chosen: the max generation
Fitness function in the QPSONN model: when a search process is finished and a particle is updated with the determined weights and biases, the fitness of this particle
where
Structure of the ANN and encoding details in QPSO.
In this section, the effectiveness of the QPSONN is validated through experiments on the experimental data. The work in this paper is mainly to improve the performance of the traditional NN for the
Fitness versus number of hidden nodes in the ANN.
The experiments were performed on a computer running 64 bits Windows 10 system. In order to perform quantitative comparison, the following indexes are used to measure the performance of each prediction model, such as the precision, mean absolute error (MAE), mean square error (MSE), root mean square error (RMSE), proportional error (PERR), and coefficient of determination:
Comparative study needs to be conducted first to determine the proposed model for corrosion current density is effective. While comparing the QPSONN with the BPNN and PSONN, 100 runs are repeated to get the average value of prediction results. Based on the prediction results in Table
Comparison on prediction among the QPSONN, PSONN, and BPNN.
Ratio of training set (%)  Prediction precision and ANOVA test  

BPNN ( 
PSONN ( 
QPSONN ( 



 
40  80.12  82.87  89.51  6.64  8.65 × 10^{−7}  9.39  4.13 × 10^{−13} 
50  80.23  84.03  89.65  5.62  1.12 × 10^{−7}  9.42  1.39 × 10^{−9} 
60  81.26  84.26  90.11  5.85  1.03 × 10^{−7}  8.85  1.68 × 10^{−11} 
70  82.75  85.10  90.23  5.13  5.99 × 10^{−8}  7.48  1.22 × 10^{−11} 
80  83.33  85.80  91.34  5.54  3.49 × 10^{−5}  8.01  1.26 × 10^{−11} 
90  83.41  86.02  91.95  5.93  1.67 × 10^{−6}  8.54  6.29 × 10^{−18} 
Evaluation indexes on prediction results.
Error  BPNN  PSONN  QPSONN 

MAE  0.0837  0.0720  0.0454 
MSE  0.0110  0.0097  0.0039 
RMSE  0.1047  0.0983  0.0624 
PERR  0.5478  0.4830  0.1949 
Although the precision of the QPSONN model is proved to be better than the PSONN and the BPNN model through the general statistics analysis like mean value, the comparison conducted above is rough. Furthermore, a deeper analysis is indispensable to be carried out. In this analysis, the accuracy distribution of each algorithm is conducted after running 100 times. The corresponding results of the BPNN, PSONN, and QPSONN with six different proportions of training set are illustrated in Figure
Precision distributions of three algorithms in different training set ratios: (a) 40%; (b) 50%; (c) 60%; (d) 70%; (e) 80%; (f) 90%.
Next, the effect on prediction precision due to different proportions of training set is analysed in this section. The effect of ratio on the precision results is shown in Figure
The impact on precision caused by ratio of training set. (a) Population size = 60; (b) population size = 80.
When the ratio of training set is 80% and parameters are used as given in Table
Prediction results of the QPSONN model, PSONN model, and BPNN model.
The regression lines between tested and predicted values of three algorithms are displayed in Figure
Regression lines between measured values and predicted values of three algorithms. (a) BPNN model; (b) PSONN model; (c) QPSONN model.
Based on the comparison analysis from multiple aspects in this section, a higher precision is achieved by the QPSONN model regardless of the ratio of the training set. In terms of computational efficiency, the QPSONN model costs a relatively longer time than the BPNN due to the optimization process of the QPSO algorithm. Since the corrosion is a longterm process, realtime performance is not a major factor in corrosion current density prediction, which means the offline analysis can be performed based on the collected data. Therefore, the model based on the QPSONN algorithm is feasible and more resultful for this prediction problem.
The parameter of training times represents the times of iterations for optimizing the configuration of a neural network during the training process [
Prediction precision versus training times of NN. (a) Ratio of training set = 60%; (b) ratio of training set = 80%.
Based on the results in Figure
In terms of distribution of precision, the mean precision is looser when the training times of the NN is small. With the increase in training times, the distribution of mean prediction is more concentrated, which indicates that the algorithm is more stable.
Generally speaking, the population size of whole particle is a significant parameter that has an impact on the convergence speed, accuracy, and stability of QPSO. In order to evaluate its impact on the QPSONN, the population size varies from 20 to 80, and then the results corresponding to each parameter are compared for difference.
As can be seen from Figure
Prediction precision versus population size of QPSO. (a) Ratio of training set = 40%, 60%, and 80%; (b) ratio of training set = 30%, 50%, and 70%.
The max generation of QPSO affects the optimization process to a large extent, which determines the effectiveness and efficiency of the search process. Therefore, it is essential to study the impact of the maximum generation
Prediction precision versus max generation of QPSO. (a) Ratio of training set = 60%; (b) ratio of training set = 80%.
It can be seen from Figure
In addition, the mean precision is relatively loosedistributed in the low
In this paper, a QPSONN model was proposed to predict the corrosion current density
The proposed model indicates that the QPSONN model exhibits a theoretical value in the prediction of corrosion current density
The proposed model is designed for the prediction of corrosion current density for Q235A steel. However, buried pipelines contain a variety of metals, stray current corrosion characteristics of which are different. Future work will concentrate on predicting the corrosion current density under the coupling action of stray current and multiple ions regardless of the type of buried pipeline metal. To establish the prediction model dealing with different pipeline metals, new database has to be created through extra electrochemical experiments.
The experimental data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The authors acknowledge the project funded by the National Natural Science Foundation of China (NSFC) (51607178); special project supported by the China Postdoctoral Science Foundation (2018T110570); and project funded by the China Postdoctoral Science Foundation (2019M652005), the Natural Science Research Project of Jiangsu Higher Education Institutions (18KJB460003), and Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD) for the financial support to this research.