The control of slurry pressure aiming to be consistent with the external water and earth pressure during shield tunnelling has great significance for face stability, especially in urban areas or underwater where the surrounding environment is very sensitive to the fluctuation of slurry pressure. In this study, an optimal control method for slurry pressure during shield tunnelling is developed, which is composed of an identifier and a controller. The established identifier based on the random forest (RF) can describe the complex non-linear relationship between slurry pressure and its influencing factors. The proposed controller based on particle swarm optimization (PSO) can optimize the key factor to precisely control the slurry pressure at the normal state of advancement. A data set from Tsinghua Yuan Tunnel in China was used to train the RF model and several performance measures like R2, RMSE, etc., were employed to evaluate. Then, the hybrid RF-PSO control method is adopted to optimize the control of slurry pressure. The good agreement between optimized slurry pressure and expected values demonstrates a high identifying and control precision.

The slurry pressure balanced (SPB) shield tunnelling method has been widely used in underground space development for its strong geological adaptability, small impact on the surrounding environment and high degree of mechanization [

A great number of different factors such as the machine driving state [

To control the balance between the slurry pressure and the external water and earth pressure during shield tunnelling. Firstly, an identifier needs to be established that can adapt to the changes of environmental conditions and predict the slurry pressure during shield tunnelling. Then, based on the proposed identifier model, a controller should be put forward to optimize the key control factor by minimizing the difference between the slurry pressure and the external water and earth pressure. In terms of the identifier model, random forest (RF) is a pattern recognition method based on a “holistic learning” strategy, which has been increasingly favored by researchers in recent years [

In terms of the controller model, particle swarm optimization (PSO) is a powerful optimization algorithm proposed by Kennedy et al. [

The continuation of this paper is organized as follows. In Section 2, we give some preliminaries on the algorithm applied in this paper and then present the methodology of slurry pressure identification and control model. In Section 3, with python language [

During the SPB shield tunnelling, the excavated soil enters the slurry chamber through the opening of the cutter head and was then carried out by slurry. Therefore, different from the EPB shield of which the tunnel face is supported by soil in the soil chamber, the face stability of SPB shield is maintained by the pressurized slurry mixed with soil in the slurry chamber, as shown in

Soil environmental conditions include geometric and geological factors. In terms of geometry, the tunnel diameter and depth have a great influence on slurry pressure. Here, the depth is measured from the tunnel crown to the ground surface. This parameter plays an important role in the control of slurry pressure. As the depth increases (decreases), the slurry pressure has to immediately increase (decrease) otherwise it may cause instability of the excavation face. In terms of geological parameters, many factors such as soil weight and cohesion that has a great influence on the balance of slurry pressure.

The parameters represent the mechanical driving state of shield comprises advance speed (v) and thrust force (F), rotational speed (n), and torque (T). When the formation is harder or heavier, the thrust and torque required are often greater, which means they reflect the difficulty of soil excavation to a certain extent.

The slurry circulation system is the key part of controlling slurry pressure of which the main parameters are air pressure (_{a}_{f}_{d}_{dif}_{f}_{d}_{dif}

Theoretically, according to the supporting principle of excavation face in SPB shield, the desired supporting pressure of the excavation face should be equal to the static earth and water pressure, which can be expressed as [

where

To predict the slurry pressure during shield tunnelling, it is essential to investigate the factors that may affect the slurry pressure and determine which can be used as input parameters of the identification model. According to the analysis in Section 2.1, the specific parameters can be divided into three categories among which the soil environmental factors are cover to diameter ratio(R = Z/D) that comprehensively consider the influence of tunnel geometric conditions. In this study, all the geological parameters are not taken into consideration for the following reasons: 1) The difference of soil weight and coefficient of static lateral earth pressure in this investigated tunnel section is small; 2) Due to the limitation of geological exploration, continuous geological parameters values are unable to be obtained; 3) To a certain extent, shield mechanical drive parameters can reflect the ground conditions. Therefore, through the above analysis, 12 features were taken into consideration in the identifier model. Besides, what we are concerned with is the precise nonlinear relationship between the slurry pressure and its various influencing factors. Thus, the input of this model are the parameters of one step while the output of the model is still the slurry pressure of this step, which is presented as follows:

Random forest (RF) is a pattern recognition method based on a “holistic learning” strategy, which has high nonlinear mapping ability. It is mainly composed of two main components, namely the Decision Tree (DT) algorithm and the bagging algorithm. In DT, the feature space is continuously divided into subspaces to ensure that all samples in the same subspace are as uniform as possible. For regression problems, space division is usually performed by minimizing the following equation:

where _{c} is the predicted value of the terminal leaf node in the tree; _{i} is the output value of sample

Based on the DT algorithm, Breiman [

It is necessary to tune the hyperparameter of the RF model since different hyperparameters result in different performance. The cross-validation method can reduce the overfitting of a model to a certain extent when it was used to evaluate the prediction performance of the model, especially the performance of the trained model on new data [

Usually, the original data set is randomly divided into a training set and a testing set. The training set is used to build a regression model and the test set is used to prove the predictive ability of the model on new data. Generally, the proportions of the training set and the testing set depend on the quality and accuracy of the data as well as the structure of the network itself. If the proportion of the training set is too small, the model will not be able to make predictions; if the proportion of the training set is too large, the model will closely match the results of the training set and will not give good prediction results for new data. According to the optimization analysis, this paper finally uses 70% of the data set for training and the remaining 30% for model testing [

To evaluate the performance of the model during the tuning of the model, a scorer which is a metrics is designated to score training results. All score metrics follow the following principle: a higher return value is better than a lower return value. In this paper, the “explained_variance_score” was applied as shown in

where

In terms of the slurry pressure control methods, the slurry shield can be divided into two basic types, indirect control type (German-style with bubble chamber) and direct control type (Japanese style without bubble chamber). In the early years of SPB shield development, the excavation chamber only contains a slurry chamber which means stability of the excavation face is maintained by controlling the slurry pressure directly (by the flow of feed or discharge slurry pump). However, the slurry pressure would fluctuate greatly with the change of geological conditions which makes the excavation face stability problem uncontrollable. With the development of shield, the excavation chamber of the current SPB shield machine is usually composed of the slurry chamber and bubble chamber which means the control and adjustment of the excavation face are achieved by the slurry circulation control system and air pressure control system together. Generally speaking, the slurry circulation control system is mainly to adjust the change of slurry flow in and out to keep the slurry level near the axis while the air pressure control system is the major execution module to balance the fluctuation of slurry level and control the slurry pressure indirectly. It should be noted that when the slurry level is beyond the limitation, the operators have to stop driving and regulate the feed or discharge slurry pump to readjust the slurry level [

Particle swarm optimization (PSO) is a powerful optimization algorithm based on the swarm behavior of birds or fishes around food, which was applied to search the optimal air pressure in this study. The term “particle” is used here to refer to the individual candidate that defined by velocity (

where

PSO starts with a set of particles randomly generated and initialized. Then, according to pbest and gbest values, all particles update their velocities and positions until the optimal solution is finally reached.

The flowchart of the PSO algorithm is demonstrated in

During the shield tunnelling, the most ideal situation for excavation face stability is that the slurry pressure is equal to the desired slurry pressure. However, the environmental conditions and machine statuses change with the advance of excavation which inevitably results in the fluctuation or even mutation of slurry pressure, especially in complex geological conditions. In order to control slurry pressure, it should be identified first. In other words, the slurry pressure needs to be predicted during shield tunnelling and this is the main purpose of the RF model. Then, based on the proposed identifier model, a controller is established to reduce the fluctuation and make the slurry pressure as close to the expected value as possible. Therefore, the optimization function can be described as follows:

where

To control the slurry pressure during shield tunnelling, a control method based on PSO and RF model mentioned above was developed and its procedure is shown in

The methodology of the RF-PSO model has been explained above. This section demonstrates the application of the proposed model through a practical tunnel project to validate the performance of it.

Tsinghua Yuan Tunnel is an urban underground tunnel of the Beijing-Zhangjiakou High-speed Railway in Beijing. Two SPB shield with a diameter of 12.64 m was applied and the external and internal radii of the segmental lining of the tunnel are 6.1 and 5.55 m respectively. The 3#～2# section of the shield tunnel was 1741 m long that launched from shaft 3 and ended in shaft 2. The profile of geology is illustrated in

In this study, a total of 450 sets of data from ring 0 to ring 449 were investigated. The section from the 0th ring to the 225th ring is almost silty clay while the section from the 225th ring to the 449th ring is an interlayer structure of pebble soil, sand, and silty clay. The frequency histogram of each parameter during shield tunnelling is shown in

Pearson correlation coefficient is an effective index to evaluate the correlation which is shown as

where

As can be seen, the input parameters have a strong correlation with slurry pressure especially air pressure, cover to diameter ratio, thrust force and torque, of which the correlation value exceeds 0.5. Furthermore, the correlation between ratio and thrust force, torque respectively were 0.86 and 0.8, which verified that thrust force and torque can reflect the geological conditions to a certain extent. Additionally, if the variance of one feature is much larger than the other, then it may dominate the objective function and result in the model unable to learn from the other features correctly as expected. Therefore, the input data are normalized to the range of [0,1] to speed up the convergence of the RF model. The normalization processing method is as follows:

where

According to section 2.2.2, 70% of the data set (315 groups) was used for training and the remaining 30% (135 groups) was for model testing. The optimal hyperparameters of the slurry pressure identifier model are obtained by five-fold cross-validation. Among all hyperparameters, the number of the estimator (tree) has the most influence on the performance of the RF model, followed by the max depth and max features. Thus, the tuning order of the training model is the number of estimators, max depth and max features. Eventually, the value of these hyperparameters is 1100, 15 and 7, respectively, which were tuned by the five-fold cross-validation method. The final performance score of the random forest model on the training set and test set is 0.963 and 0.946, respectively. To illustrate the advantages of the RF model, the backpropagation (BP) neural network and support vector regression (SVR) were also employed to train the data for comparison. To avoid the influence of determination of the hyperparameters on the comparison results, the hyperparameters of these three models were also tuned by the five-fold cross-validation. The comparison of the measured and predicted values of the slurry pressure on the training set and the test set is shown in

As can be seen, scattered data in both plots are all close to the line of equality (shown as the solid line) in the training set and the testing set, demonstrating the good accuracy of these three models. Furthermore, the width of data of the RF model is narrower than BP and SVR model, which means its deviation from the expected value is smaller and it can also be proved in the following

The network performance evaluation indices of each model on the training set and test set are listed in ^{2}), adjusted coefficient of determination (Adjusted R^{2}) were calculated. A small value of MAE, MSE, RMSE and great values of R^{2} and adjusted R^{2} indicates a good prediction accuracy of the model. The results of the RF model are better than the other two, indicating that the model has high non-linear mapping ability and strong generalization ability. In terms of RMSE, the RF model is 69.7% and 72.0% less than the SVR model and the BP model respectively on the training set while on the test set are 28.6% and 34.5%. In terms of R^{2}, the RF model is 5.4% and 6.6% higher than the SVR model and BP model respectively on the training set while on the test set are 3.0% and 4.1%. The comparative analysis with SVR model and BP model elucidates that the RF model can predict the slurry pressure with reasonable accuracy.

Model | MAE | MSE | RMSE | R2 | Adjusted R2 | |
---|---|---|---|---|---|---|

Formula | ||||||

Train | RF | 0.0172 | 0.0005 | 0.023 | 0.9945 | 0.9948 |

SVR | 0.0610 | 0.0057 | 0.076 | 0.9424 | 0.9438 | |

BP | 0.0640 | 0.0068 | 0.082 | 0.9316 | 0.9334 | |

Test | RF | 0.0437 | 0.0031 | 0.055 | 0.9681 | 0.9700 |

SVR | 0.0581 | 0.0060 | 0.077 | 0.9384 | 0.9421 | |

BP | 0.0650 | 0.0070 | 0.084 | 0.9274 | 0.9317 |

where

Based on the proposed RF model, a control model that keeps the slurry pressure in line with the desired one as much as possible during shield tunnelling was established. The tuning parameters of PSO including the maximum number of iterations (IterMax), the population size (PopSize), the lower and upper boundaries of a variable (VarMin and VarMax) have been selected as follows: IterMax = 1000, PopSize = 100, VarMin = 0, and VarMax = 1.5. The two parameters of the personal learning coefficient (c1) and the global learning coefficient (c2) are set to be 1.5 and 1.5, respectively, which were determined by several trial and error runs. The optimization results are illustrated in

The main job of the controller model is to minimize the difference between slurry pressure and desired support pressure of excavation face by searching for the optimal air pressure. The suggested air pressures solved by PSO are shown in

As mentioned before, the impact of features on model predictions can be obtained based on the importance analysis of RF. Generally, in the random forest regression model, the statistic R^{2} is used to characterize the relative importance. The importance score of the feature is normalized (the sum of all importance scores is 1) and the result is shown in

In practical applications, the proposed model can provide initial estimations of slurry pressure. With the prediction model of slurry pressure, control factors can be optimized and the suggested value will be given. Briefly, the improved model in this research is expected to provide insightful suggestions to support operators in the control of face stability during slurry shield tunneling. However, in this research, only air pressure is considered which means the given suggestion is limited during slurry shield tunneling and slurry parameters should also be taken into consideration in the future study. Besides, more data should be trained to improve the accuracy of the proposed model.

In this study, with python language, a dynamic identifier is constructed based on RF to perform the complex relationship between slurry pressure and its affecting factors. Then, combined with the trained RF model, a PSO-based controller was designed to optimize the air pressure to control slurry pressure during shield tunnelling. The proposed model is applied to a case study of Tsinghua Yuan Tunnel in Beijing. To illustrate the advantages of the RF model, the SVR model and BP model were also employed for comparison. Finally, the hybrid RF-PSO control method was applied in the optimization of slurry pressure. Major conclusions are obtained as follows:

1. Based on the RF algorithm, a reasonable relationship between slurry pressure and three main aspects including soil environmental factors, shield mechanical driving state and slurry circulation system was obtained. The performance of this identifier model on the training set and test set were 0.963 and 0.946, respectively, indicating the high nonlinear mapping ability and strong generalization ability of this model.

2. Through comparative analysis with the BP model and SVR model, the RF model has smaller RMSE and greater R^{2}, which indicates that the RF model demonstrates better prediction accuracy of slurry pressure than the SVR model and BP model.

3. According to the importance analysis, the soil environmental parameters, shield mechanical driving parameters and slurry circulation parameters account for the total importance of 0.342, 0.143 and 0.515, respectively. Among them, air pressure has the greatest influence on slurry pressure, which accounts for almost half of its importance and is followed by the cover to diameter ratio, which accounts for 0.342.

4. The optimized slurry pressure shows good agreement with the target support pressure, which means the proposed model has great performance in the control of slurry pressure during shield tunnelling. Compared with the field data in Tsinghua Yuan Tunnel project, the fluctuation of optimized slurry pressure was significantly reduced.

5. The proposed method is recommended as a useful tool to provide suggestions for slurry pressure control during shield tunnelling. Further work is needed to enlarge the database and promote the applicability of the hybrid RF-PSO-based control model for slurry pressure.