11th Iranian and 2nd Regional Tunnelling Conference “Tunnels and the Future” 2-5 November 2015
TBM Performance Prediction in Rock Tunneling Using Various Artificial Intelligence Algorithms * Alireza Salimia , Christian Moormanna, T.N. Singhb, Prasnna Jainc a Institute
of Geotechnical Engineering, University of Stuttgart, Stuttgart, Germany,
[email protected];
[email protected] b Indian Institute of Technology Bombay, Mumbai-400076, India;
[email protected] c National Institute of Rock Mechanics, Kolar Gold Fields-563117, Karnataka, India;
[email protected]
ABSTRACT With widespread increasing applications of mechanized tunneling in almost all ground conditions, prediction of tunnel boring machine (TBM) performance is required for time planning, cost control and choice of excavation method in order to make tunneling economical. Penetration rate is a principal measure of full-face TBM performance and is used to evaluate the feasibility of the machine and predict advance rate of excavation. In this study, a database of actual machine performance from two hard rock tunneling projects in Iran including Zagros lot 1B and 2 with 14.3 km available data has been compiled. To clarify the effective parameters on penetration rate, first principal component analysis (PCA) was performed. Furthermore, well-known Artificial Intelligence (AI) based methods, including artificial neural networks (ANN), adaptive neuro-fuzzy inference system (ANFIS) and support vector regression (SVR) have been employed. As statistical indices, root mean square error (RMSE), correlation coefficient (R2), variance account for (VAF), and mean absolute percentage error (MAPE) were used to evaluate the efficiency of the developed AI models for TBM performance. According to the obtained results, it was observed that AI based methods can effectively be implemented for prediction of TBM performance. Moreover, it was concluded that performance of the SVR model is better than the ANFIS and ANN models. A high conformity was observed between predicted and measured TBM performance for the SVR model.
Keywords: Penetration rate, TBM performance, PCA, ANN, ANFIS, SVR.
1.
INTRODUCTION
Hard rock tunnel boring has become more or less the standard method of tunneling for tunnels of various sizes with lengths over 1.5 to 2 km. estimating the performance of TBM is a vital phase in tunnel design, and for the choice of the most appropriate excavation machine. During the past three decades, numerous TBM performance prediction models for evaluation of TBM have been proposed. In brief, all the TBM performance prediction models can be divided into two distinguished approaches, namely theoretical and empirical ones [1]. Based on rock failure mechanism, theoretical models analyze cutting forces acting on disc cutter to find force equilibrium equations [2-9]. The theoretical models which are primarily developed by using indentation tests or full-scale laboratory cutting tests provide an estimate of cutting forces based on cutter and cutting geometry and spacing and penetration of the cut. It is certainly true that laboratory cutting tests provide the basic understanding of rock fragmentation into the forcepenetration behavior of rocks. The disadvantage of these test is that, it does not completely represent the real rock mass conditions as the TBM disc cutters encounter in the field. On the other hand, an empirical method does have some strength (taking into account rock mass conditions) as well as some shortcomings. The main deficiency of the empirical models is the absence of cutting force, cutter geometry, cutting geometry and ability to match machine thrust and torque/power in various ground conditions. In the last couple of decades, with growing use of TBMs in the world and the necessity to accurately predict performance of machines in different ground conditions, many researchers have worked to develop new prediction models or adjustment factors for the common existing models.
Research works by Rostami and Ozdemir [8], Graham [10], Farmer and Glossop [11], B üchi [12], Hughes [13], Gerhing [14], Bruland [15], Barton [16], Bieniawski van Preinl et al., [17], Gong and Zhao [18], Hassanpour et al., [19-21], Khademi et al., [22], Benato and Oreste [23] are the most common and recent works on this topic as summarized in Table 1. Barton reviewed a wide range of TBM tunnels to establish a database for proposing a new model based on Q rock mass classification system and adding some new parameters to the existing system to be able to use it for TBM applications [16]. This new model, namely QTBM includes many input parameters (such as RQD, joint condition, Stress condition, intact rock strength, quartz content and TBM thrust) as well as some parameters are overlapped in this model [18]. Sapigni et al. [24] studied the empirical relation between RMR and penetration rate. Also Ribacchi and Lembo-Fazio [25] evaluated the relationship between RMR and performance of a double shield machine in the Varzo tunnel. Yagiz [26] performed statistical analyzes on data obtained from Queens’s tunnel in New York and proposed an empirical model to predict TBM penetration rate. He has related four rock mass parameters (UCS, Punch test index or PTI, spacing and orientation of joints) to penetration rate of machine. In a similar research work, Gong and Zhao [18] by performing a nonlinear regression analysis on data obtained from two tunnels excavated in granitic rock masses in Singapore developed an empirical equation to estimate boreability of rock mass. They proposed a relationship between four rock mass parameters (UCS, brittleness, joint count number, and orientation of joints) and boreability index of the rock mass. Hassanpour et al. [20] by performing single and multivariable regression analysis on data obtained from a double shield TBM driven tunnel in predominantly pyroclastic rocks in Iran, developed several empirical equations to estimate TBM field penetration index through the most common rock mass classification including RMR, Q, GSI system. Afterward, Hassanpour et al., [21] on the basis of field penetration index (FPI) as function of RQD and UCS developed a new chart for prediction of rock mass boreability. Khademi et al. [22] in terms of rock mass rating (RMR) presented a multivariable linear regression to estimate performance of TBM in hard rock condition. Bruland [15] updated and improved the NTNU model (introduced by Blindheim [27]) based on field data mainly collected from Norwegian tunnels. NTNU model requires special experiments originated from the drilling. These tests are not commonly available outside Norway. Another well-known method for determination of penetration rate was developed by Rostami et al. [8],[28]. This model leads to the identification of the forces that need to be applied to a disc in order to produce a certain penetration of the rock. This method offers the advantage of being able to consider the geometry of the problem (the diameter of the disc and the distance between the grooves) in detail, whereas the original CSM model does not consider the natural discontinuities of the rock mass, which have an important influence on the net advancement speed on the TBM. In this regards, Yagiz [29] modified the original CSM model adding rock mass properties as input parameters into the model. Ramezanzadeh [30] has also followed up on this work and developed a database of TBM field performance for over 60 km of tunnels. He offered adjustment factors for CSM model to account for joints and discontinuities. Innaurato et. al., [31,32] developed a new model for estimation of penetration rate based on intact rock (presented by uniaxial compression strength) and rock mass condition by considering Rock Structuring Rating (RSR). Innaurato’s model consider the effect of intact and rock mass, but the latter is characterized by infrequently used geomechanical quality index which is rarely available in the geotechnical characterization of a tunnel. Moreover, the penetration rate is estimated without any reference to the force F N acting on each disc [23]. Such a force, as shown by Rostami [9] can have major effect on the penetration rate. Recently, a new empirical formula is presented [23] for estimation of penetration-per-revolution based on UCS, GSI and F N. Due to the complexity of TBM performance prediction, beyond mathematical and empirical solutions, artificial intelligence (AI) methods have been widely utilized by many researchers [33-36]. Iphar [37] employed artificial intelligence network (ANN) and adaptive neuro fuzzy inference systems (ANFIS) for hydraulic impact hammers performance prediction. Also, Tiryaki [38] applied artificial neural network for predicting the cuttability of rocks by drag tools. Moreover, Yagiz et al., [39] utilized ANN method for estimation of tunnel boring machine performance. Besides that, Mahdevari et. al., [40] used a support vector regression analysis (SVR) to predict penetration rate based on data from the Queens Water Tunnel, in New York City. Also, particle swarm optimization (PSO) technique has been utilized by Yagiz and Karahan [41] with the same data from Queens Water Tunnel for prediction of TBM penetration rate. Growth of TBM manufacturing technology and existence of some shortcomings in the prediction models have made it necessary to perform more research on the development of the new models. In this investigation, a database of actual machine performance from two hard rock tunneling projects from Iran including Zagros lot 1B and 2 with 14.3 km available data which were constructed using a double shield (DS) TBM has been compiled. To clarify the effective parameters on penetration rate, first Principal Component Analysis (PCA) was performed; then three different AI methods containing artificial neural networks (ANN), adaptive neuro-fuzzy inference system (ANFIS) and support vector regression (SVR) were developed and the results were compared.
Table 1. Review of some TBM performance models Prediction value
Model
Rock mass factors
Machine factors
Penetration rate (m/h)
Graham [10]
Uniaxial compressive strength
Cutter force
Penetration rate (m/h)
Farmer and Glossop [11]
Tensile strength
Cutter force
Penetration rate (m/h) Advance rate (m/h) Some TBM parameters
Büchi [12]
Penetration rate (m/h)
Hughes [13]
Penetration rate (m/h) Rostami Advance rate (m/h) Ozdemir [8] Some TBM parameters Penetration (mm/rev)
rate
Compressive and tensile strength Cutter spacing, cutter tip Correction factors for rock anisotropy, joint width, cutter radius, cutter spacing, mica content force, TBM diameter, RPM Uniaxial compressive strength and
Gerhing [14]
Cutter force, Fn, diameter
cutter
Uniaxial compressive strength, Tensile strength
Cutter spacing, cutter tip width, cutter radius, cutter force, TBM diamater, RPM Uniaxial compressive strength, correction factors Cutter force, for joints, specific fracture energy, etc. Fn
Penetration rate (m/h) Advance rate (m/h)
Bruland [15]
Uniaxial compressive strength, drilling rate index Cutter force, RPM, cutter spacing, (DRI), number of joint sets, joint frequency and joint cutter size and shape, installed orientation, porosity cutterhead power
Penetration rate (m/h) Advance rate (m/h)
Barton [16]
RQD0, Jn, Jr, Ja, Jw, SRF, rock mass strength, cutter life index (CLI), quartz content, induced biaxial stress at the face, porosity
Cutter force
Penetration rate (m/h) Bieniawski von Uniaxial compressive strength, abrasivity, rock TBM diameter, Total cutter Advance rate (m/h) Preinl et al., [17] mass jointing at the face, stand-up time, water flows head thrust, RPM and torque Specific energy (kJ/m3) Borability (kN/mm/rev)
index
BI
Gong and Zhao [18]
Field Penetartion Index Hassanpour FPI (kN/mm/rev) al. [19]
Compressive strength, volumetric joint count, brittleness index, angle between main discontinuities and tunnel axis et Uniaxial compressive strength and RQD
Field Penetartion Index Khademi et al. FPI (kN/mm/rev) [22]
Uniaxial compressive strength, RQD, Joint condition, angle between main discontinuities and tunnel axis
Penetartion (mm/rev)
Uniaxial compression strength, GSI
rate
Benato Oreste [23]
and
Cutter force
Cutter force, RPM
Cutter force, RPM
Fn
2. GEOLOGY DESCRIPTION OF PROJECTS AREA (ZAGROS LONG TUNNEL; LOTS 1B & 2) Zagros long tunnel with total length of 49 km, located in Kermanshah Province in the west of Iran, is one of the largest tunneling projects in Iran. Its construction purpose is to transfer water of Sirvan River to the west and southwest plains of Iran in order to extension of irrigated agriculture and modern water-based industries. The project comprises three water conveyance tunnels including lot 1A (14 km) as the northeast section, 1B (9 km) as the middle section, and lot 2 (26 km) as the southwest section. Zagros long tunnel is situated within the Zagros fold-thrust belt with considerable geological complexity. The Zagros tunnel route includes several geological formations with wide range rock mass qualities. During the tunneling operation, changes in rock quality were frequent, with rock masses ranging from poor to very good. The lithology of the route of lot 1B consists of limestones, dolomitic limestone, bituminous shale and marl layers that these rock units belong to Surmeh (Jurassic) and Gurpi (late Cretaceous) formations [42] (Fig.1). Also, according to 1:100,000 Geological Map of Kermanshah (Fig. 2), geological formations in the route of lot 2 are Jurassic units (Ilam Formation), Cretaceous limestone units, Gurpi Formation, Garu Formation, Khami Group, and Pabdeh Formation (Fig.2). These formations mainly consist of dark gray shale, shaly limestone and limestone rocks. The study area is located in the Zagros Fold-Thrust Belt, where Arabian plate compressional tectonic forces have created several folds and faults in the study area [43]. Structurally, the geological units around the tunnel route are moderately folded and severely faulted. As shown in Fig.1 and 2, the tunnel has passed through some synclines and anticlines with multiple faults. Lack of uniformity in weathering and erosion of Zagros Mountain due to changing physical and mechanical properties of geological units, has led to changes in the depth of overburden along the tunnel route. The maximum depth of tunnel is 1000 m with the average depth equal 400 m [44]. Large part of the tunnel route is located beneath the water table whereby groundwater level varies from 30 to 340 m above the tunnel crown, but considering the significant thickness of overburden and closing joints in depth, water leaking into the tunnel limited only to the karstic cavities and crushed tectonic zones. There are geological formations such as Pabdeh and Gurpi that are main area containing oil (gas) in some parts of the tunnel
[45]. It has been observed that the liquid has leak through the holes and voids between the primary lining segments of the tunnel into inside the tunnel during tunnel excavation. Also in some conditions, gathering more than 100 ppm H 2S gas has been recorded and led to stop working and decreasing speed of drilling operations [46].
Fig. 1 Longitudinal geological profile of Zagros long tunnel (lot 1B) [47]
Fig. 2 Longitudinal geological profile of Zagros long tunnel (lot 2) [22]
3. DATA PROCESSING 3. 1. TBM Performance Database To obtain the required data for analysis of TBM performance at Zgaros lot 1B& 2, results of studies performed during the pre-construction phase and construction phase have been compiled into a database. During the construction phase and through back-mapping of the tunnel, predicted geological and geomechanical properties of rock mass along
the tunnels were examined by detailed investigation of tunnel face. In this stage, information such as rock type, rock mass fracturing, joint condition, characteristics of fault zones, weathering/alteration characteristics, ground water condition and rock stability information were recorded on mapping sheets. In addition to sampling from surface outcrops and boreholes in the pre-construction phase, during back-mapping of the tunnel many samples were taken from the muck and tunnel face to perform tests such as point load index and petrographic analysis. In the construction phase, machine performance data and operating parameters such as applied thrust, RPM, torque and etc. were also recorded continuously in special sheets and analyzed separately. Zagros tunnels (lot 1B &2) were constructed by double-shield machine (manufactured by Herrenknecht) and lined with precast segmental lining. Hence, there were some limitations on the accessibility of geological feature in the tunnels. To such that, an attempt was made in this investigation to select parts of the tunnels were sufficient and reliable geological data were available. Data were collected from the following general locations within the tunnels: • Locations where exploration borings extended to the tunnel level • Tunnels sections where the rock face was investigated during geological back-mapping • Places where extrapolation of surface geological parameters to tunnels level were possible with high degree of reliability. Descriptive statistical distribution of variables in the data base and input parameters for generated models is summarized in Table 2.The main specifications of Double-shield TBM are listed in Table 3. The data includes the first 5.3 km of Zagros lot 2 plus 9 km of lot 1B with more than 75 sections of bored tunnel were selected based on the above criteria for more analyses. Also, the most important performance parameters including average rate of penetration (ROP), penetration per revolution (P), and Field Penetration Index or FPI, [48] have been calculated using formula (1)-(3) as listed below:
ROP Lb t b P
(1)
ROP *1000
(2)
RPM * 60 FPI
Fn P
(3)
Where ROP is rate of penetration (m/h), Lb is boring length (m), tb is boring time (h), P is cutter penetration in each cutterhead revolution (mm/rev), RPM is cutterhead rotational speed (rev/min), FPI is Field Penetration Index (kN/cutter/mm/rev), Fn is cutter load or normal force (kN). Table.2 Descriptive statistics of generated database for this study [19,47] Variable
N
Min
Max
Mean
Std. deviation
Variance
UCS (MPa) BTS Js (m)
75 75 75
15 1 0.1
150 13.7 0.5
49.14 5.26 0.24
37.76 2.99 0.1
1425.928 8.977 0.11
RQD (%)
75
15
95
61.08
18.21
331.912
Alpha (0 )
75
1
75
34.14
23.69
561.478
RMR (Basic) Q GSI
75 75 75
21 0 20
75 8 67
49.06 3.2 44.09
10.73 2.2 10.87
115.225 4.855 118.315
Table 3. Main specifications of TBM Parameter
Value
Machine diameter
6.73 m
Cutters diameter Number of disc cutters
432 mm 42
Disc nominal spacing Max. operating cutterhead thrust
90 mm 28,134 kN at 350 bar
Cutterhead power Cutterhead speed
2100 kW 0-11
Cutterhead torque (nominal) Thrust cylinder stroke Conveyer capacity Total TBM weight
4450 kN.m at 9 rpm 1700 mm 690 t/h 573 t
3. 2. Principle Component Analysis In order to establish the predictive models among the parameters obtained in this study, principal component analysis (PCA) was performed in the first stage of the analysis. PCA is a classical method that provides a sequence of the best linear approximations to a given high-dimensional observation, and it has received much more attentions in the literature. PCA is used frequently in different types of analysis (from neuroscience to computer graphics) because it is a simple, nonparametric method of extracting relevant information from confusing data sets. With minimal additional effort, PCA provides a roadmap on how to reduce a complex data set to a lower dimension. For instance, Fig.3 represents a two-variable data set which has been measured in the X-Y coordinate system. The principal direction in which the data varies is shown by the U axis and the second most important direction is the V axis orthogonal to it. If one transforms each (X, Y) coordinate into its corresponding (U, V) value, the data is de-correlated, meaning that the co-variance between the U and V variables is zero. For a given set of data, principal component analysis finds the axis system defined by the principal directions of variance (i.e., the U-V axis system in Fig.4). The directions U and V are called the principal components. In this new reference frame, note that variance is greater along axis U than it is on axis V. PCA computes new variables which are obtained as linear combinations of the original variables. These variables are found by calculating the covariance (or correlation) matrix of the data patterns [49, 50]. In this paper, PCA was performed on a set of output and factors (input parameters), and the ratio of variance of first component to total variance (variance ratio) were calculated. Accordingly, this ratio can be determined by the similarity among the output and a set of input factors. In the present study, in order to quantify the performance of TBM, the Field Penetration Index (FPI) was computed from the raw data. The FPI has been utilized for analysis of TBM performance by many researchers [51-53], [19-22]. Several analyses with two, three, and four as well as five features were performed to obtain the effective parameters on the TBM performance (Fig.5). As can be seen from Fig.5, the factor containing two inputs (UCS, Js) were shown to be more effective and FPI has been considered as a function of these inputs; hence, these parameters were selected as input parameters for the predictive models.
Fig.3 Principal Components for data representation
Fig. 4 Principal components for dimension reduction
Fig. 5 Principal components analysis for some features in this study
4. ARTIFICIAL INTELLIGENCE METHODS Due to various geotechnical conditions encountered along the tunnel alignment, prediction of the performance of a TBM is a non-linear and complex problem [54],[39]. Recently artificial intelligence (AI) based models are successfully employed by some researchers to solve this difficult non-linear problem in geotechnical projects. In this regard, in order to predict PR, three different AI methods named artificial neural networks (ANN), adaptive neurofuzzy inference system (ANFIS) and support vector regression (SVR) are developed. 4. 1. Artificial neural network (ANN) Neural networks function is similar to the biological structure of human brains. They are layered structure networks with highly interconnected processing elements (neurons) that exist in the network layers. ANN has the ability of transforming a set of inputs to a set of desired outputs to learn and set itself to the environment. To do so, the connections or weights between the elements are modified by extracting a generalized correlation available in the inputs–outputs. During the learning phase, the experimental examples are used as signals for input and output layers. After learning, in the recall phase, prediction can be made for new inputs [55-58]. In this method, the output signals from one layer, which are adjusted by weighting factors, are transmitted to subsequent layer. In this way, the net input to each element is the sum of the weighted output of the elements in the former layer. An activation function, such as, sigmoidal logistic function is used to calculate the output of the elements. The number of hidden layers and attributed elements depends on the complexity of the problem to be solved. Till date, many learning algorithms have been developed for neural networks. It is well established fact that the back-propagation (BP) method is the most efficient technique for learning in multi-layer neural networks [59]. This type of the network consists of at least three layers: input layer, hidden layer and output layer [60, 61]. In BP algorithm, the learning phase includes a forward pass and a reverse pass. In the forward pass, a set of input-output pairs is introduced to the model and then output related to the initiated patterns is calculated by the model at the end of this pass. In the reverse pass, the calculated output is compared with that of target pattern. If the obtained difference (error) is lower than a predefined threshold, the learning phase is finished. Otherwise, the error is back propagated through the network, which results in connection weights adjustment [62]. Feed-forward back-propagation neural network (FBPNN) is normally used for solving input–output mapping problems where closer mapping is required. Using this technique the network is able to precisely predict target pattern for a given input pattern. As mentioned earlier, 75 datasets were prepared for this study. The datasets must be grouped into training and testing sets. To do so, available data sets were divided into two subsets randomly, i.e., 80 % data sets for training and 20 % data sets for testing (the same as ANFIS and SVR). As such, maximum efforts should be made to consider all the pertinent parameters or inputs [63]. To recognize the optimum network, different topologies were tried and compared by calculating root mean square of error (RMSE) (Eq. 4), for each of the models.
RMSE
1 n
n
(A
imeas Aipred )
2
(4)
i 1
As it can be seen from the table 4, the network, which had one hidden layer with the neural network architecture of 2-4-1, was considered as the optimum model for TBM performance prediction. In the developed ANN, UCS (MPa) and Js (cm) have been considered as inputs and FPI (kN/cutter/mm/rev) selected as output. The network is shown in Fig.6. A graphic comparison of measured and predicted TBM performance is depicted in Fig.7. Table 4.Results of a comparison between some of the models No
Transfer function
Model
RMSE
1 2 3
LOGSIG-LOGSIG-PURESLIN (L-L-P) LOGSIG-LOGSIG-PURESLIN (L-L-P) TANSIG-LOGSIG-TANSIG-PURESLIN (T-L-T-P)
2-4-1 2-6-1 2-6-10-1
2.53 2.67 2.81
4 5
TANSIG-LOGSIG-PURESLIN (T-L-P) LOGSIG-LOGSIG-LOGSIG-PURESLIN (L-L-L-P)
2-10-1 2-13-28-1
2.96 3.01
6 7
TANSIG-TANSIG-PURESLIN (T-T-P) LOGSIG-LOGSIG-PURESLIN (L-L-P)
2-13-5-1 2-30-1
2.78 2.57
8
LOGSIG-LOGSIG-LOGSIG-PURESLIN (L-L-L-P)
2-9-7-1
3.76
Fig.6 the optimum architecture of ANN used in this study
Fig.7 Correlation coefficient for ANN model
4. 2. Adaptive neuro-fuzzy inference system (ANFIS). Artificial inference systems such as neural networks and fuzzy logic have been used widely in recent years. Each system and its associated method have its own advantages and disadvantages. Artificial neural network has an
advantage of recognizing pattern and adapting the method to cope with the changing environment. Fuzzy logic has an advantage of incorporating human knowledge and expertise to deal with uncertainty and imprecision. Therefore, recent efforts have been made to take advantage of both approaches. As a result of these studies an integration of these systems, ANFIS has become a popular tool in the rock and soil engineering as well as engineering geology in recent years [64-70]. A neuro-fuzzy system is, in fact, a neural network that is functionally equivalent the fuzzy inference model. It can be trained to develop IF-THEN fuzzy rules and determine membership functions for input and output variables of the system. The ANFIS is a fuzzy Sugeno model put in the framework of adaptive systems to facilitate learning and adaptation [71]. Such framework makes the ANFIS modeling more systematic and less reliant on expert knowledge. Subsequently, we briefly explain an ANFIS system by using a model with two inputs as an example (Fig.8).To construct the ANFIS model, five layers were used, as demonstrated in Fig.8. Each layer has some nodes described by a node function. The circles in the network represent nodes with no variable parameters, while the squares indicate nodes with adaptive parameters determined by network during training. The nodes in the first layer represent the fuzzy sets in the fuzzy rules. It has parameters that control the shape and the location of the center of each fuzzy set which are called premise parameters. In the second layer, every node computes the product of its inputs. In layer 3, normalization of the firing strength of the rules occurs by calculating the ratio of the ith rule’s firing strength to the sum of all rules’ firing strengths. Nodes in the fourth layer are adaptive, where each node function represents the first-order model with consequent parameters. Layer 5 is called the output layer where each node is fixed. It computes the overall output as the summation of all the inputs from the previous layer. Optimizing the values of the adaptive parameters is the most important step for the performance of the adaptive system. Specially, the supposed parameters in layer 1 and the consequent parameters in layer 4 need to be determined. Jang [71] proposed a hybrid learning algorithm to determine the parameters of an ANFIS model. A hybrid learning algorithm uses the gradient descent and least square techniques to optimize the network parameters. The least squares estimation can be used to determine consequent parameters assuming that the layer 1 parameters are fixed. Then, the layer 4 parameters can be fixed, and a back propagation approach is used to fit the premise parameters in layer 1. By iterating between the layer 1 parameters and the layer 4 parameter optimization, the optimal values for all free parameters are computed [72, 73]. In this study, the available data sets were divided into two subsets randomly, i.e., 80 % data sets for training and 20 % data sets for testing (the same as ANN and Support Vector Regression). Subtractive clustering has an autogeneration capability to determine the number and initial location of the cluster centers in a set of data. This method partitions the data into groups called clusters by specifying a cluster radius and generates a Sugeno-type fuzzy inference system (FIS) with the minimum number of rules according to the fuzzy qualities associated with each of the clusters. Hybrid learning algorithm, a combination of least squares and back propagation gradient, was applied to identify the membership function parameters of a single output, Sugeno-type fuzzy inference systems (FIS).Several models with two input parameters and one output parameter were constructed and trained. To evaluate models with different structures (FIS division) and then to determine the best model, RMSE was calculated for these models. The proposed ANFIS model for predicting performance of TBM has three membership functions for each input parameter and three rules. Other parameter types and their values used for the constructed ANFIS model can be seen in Table 5. Fig.9 shows the relationship between measured and predicted values obtained from the ANFIS model in the testing stage.
Fig.8 Architecture of ANFIS
Table.5 the ANFIS information used in this study ANFIS parameter type
Value
MF type Number of MFs Number of fuzzy rules
Gaussian 3 3
output function
Linear
Number of nodes Number of linear parameters Number of nonlinear parameter
23 9 12
Total number of parameters Training RMSE
21 2.43
Fig.9 Correlation coefficient for the ANFIS model
4. 3. Support vector regression (SVR) A novel kind of machine learning (ML), support vector machine (SVM) was developed for solving both classification and regression problems, which maximize predictive accuracy and avoids over-fitting simultaneously. Over the period of time many techniques and methodologies were developed for ML tasks. Amongst them SVM is relatively new method which is based on structural risk minimization (SRM) [74]. The term SVM refers to both classification and regression methods, and the terms support vector classification (SVC) and support vector regression (SVR) is used for specification. Obviously only SVR is capable to solve extrapolative problems by building a predictive model. SVR estimates a continuous-valued function that encodes the fundamental interrelation between a given input and its corresponding output in the training data. This function then can be used to predict outputs for given inputs that were not included in the training set. This is similar to a neural network. However, a neural network’s solution is based on empirical risk minimization. In contrast, SVR introduces structural risk minimization into the regression and thereby achieves a global optimization, while a neural network achieves only a local minimum [75]. For example a generic model can be written as: y f (X ) WT X b (5) Where W is the weight vector corresponding to X, and b is the bias. The generalization performance of such linear function f(X) is fairly limited and unable to reflect the true regression procedure. In order to overcome such weakness, a standard mathematical solution is the introduction of kernel function φ(X), which is a non-linear mapping function from the input space to a higher dimensional feature space. By using φ(X), we can reach infinite dimensions for a more expressive f. Four of the commonly used kernel functions are listed in Table 6.
Table.6 Admissible kernel functions Name
Definition
Parameter T
Linear
K ( Xi , X j ) = ( Xi ) X j
----
Polynomial
K ( X i , X j ) = [( X i )T X j +1]d
d
Radial Basis Function (RBF)
K ( Xi , X j ) = e
-γ xi - x j
2
K ( X i , X j ) = tanh[( X i )T X j + r ]
Sigmoid*
γ r
(*: For some r values, the kernel function is invalid)
With the help of φ(X), linear regression function Eq. (5) is extended to non-linear function Eq. (6): y f ( X ) W T( X ) b
(6)
Where W is the weight vector corresponding to φ(X). The goal is to estimate the coefficients (W and b) following two rules at the same time. First, in order to achieve the best performance, f(Xi) should be as close as possible to the truth yi for all training samples. Second, in order to prevent over-fitting, f(X) should be as flat as possible. These are equivalent to the following programming problem, namely primal problem of SVR: min
1 T 1 W W C 2 l
l
(
i
i )
i 1
W T ( X i ) b) yi i , s.t. yi (W T ( X i ) b) i i , i 0, i 1,...., l.
(7)
In the above formulation, slack variables of i and i are included to cope with otherwise infeasible constraint of the optimization problem and constant C>0 determines the tradeoff between the parameter norm (used to measure the “flatness”: smaller norm means smoother function) and deviations from target greater than ε (Fig.10). This problem is usually solved introducing using Lagrange multipliers, leading to the minimization of LP
n 1 W i i yi W T X i b 2 i 1
n
n
i 1
i 1
ii i i yi W T X i b n
n
i 1
i 1
(8)
i i C i i
Considering W, b, i and i * and its maximization with respect to the Lagrange multipliers, i , i *, i and i *. In order to solve this problems one needs to compute the Karush-Kunh-Tucker conditions [76], that states some conditions over the variables in Eq. (8), and n LP W i i yi X i 0 W i 1
(9)
n LP i i 0 b i 1
(10)
n LP i i 0 b i 1
(11)
LP C 0 i i i
(12)
i , i , i , i 0
(13)
i W T X i b yi 0 i i yi W T X i b 0 i
ii 0 and
(14)
(15)
ii 0
(16)
The usual procedure to solve the SVR introducing Eqs. (9-12) into Eq. (7), leads to the maximization of
n Ld i i i 1 n n Τ X X i j j i j i 1 j 1 i
(17)
Subject to Eq. (10); 0 i and i * C. This procedure can be solved using QP and Iterative Re-Weighted Least Squares (IRWLS) procedures. Support vector regression was trained by using the input variables selected by the PCA model and the FPI as the output of the model. The available data sets were divided into two subsets randomly, i.e., 80% data sets for training and 20 % data sets for testing (The same as ANFIS and ANN). The details of the topology selected for the SVR model are listed in Table 7. In order to obtain the parameters of the topology that are listed in Table 7, several configurations were tested with different kernel types (radial basis function, polynomial and hyperbolic tangent) and parameter values. These tests were performed in the same way as the methodology proposed by Sánchez Lasheras et. al. [77]. The problem was solved by using the popular suite of machine learning software written in Java called Weka and developed at the University of Waikato [78]. The correlation coefficient between measured and predicted FPI by SVR in testing stage is shown in Fig.11. According to Fig.11, correlation coefficient between measured and predicted FPI is 0.92. This R 2 showed a good correlation between these two sorts of FPI.
Fig.10 Prespecified accuracy and slack variable
in SVR
Table.7 Parameters of the SVR model Parameter
Value
Type
ε-SVR
Kernel
Radial Basis Function (RBF)
Degree Γ Tolerance of stopping criterion ε
2 1 0.0001 0.1
Fig.11 Correlation coefficient for SVR model
5. COMPARISON OF THE AI MODELS In the present study, the developed AI models which are constructed to predict the TBM performance are compared. Here, the performances of these models were evaluated according to statistical criteria such as correlation coefficient (R2), the root mean square error (RMSE) (Eq.4), mean absolute percentage error (MAPE) and variance account for (VAF). Root mean square error (RMSE), a measure of the goodness-of-fit, best describes an average measure of the error in predicting the dependent variable. However, it does not provide any information on phase differences. Mean absolute percentage error (MAPE), which is a measure of accuracy in a fitted series value in statistics, was also used for comparison of the prediction performances of the models. MAPE usually expresses accuracy as a percentage: MAPE
1 n Aimeas Aipred 100 n i 1 Aimeas
(18)
Variance Account for (VAF), performance index is used to investigate to what degree the model can explain the variance in data. VAF (1
var( Aimeas Aipred ) var( Aimeas )
) 100
(19)
Where var denotes the variance, Aimeas is the ith measured element, Aipred is the ith predicted element. The results of applying these models are summarized in Table 8. Table.8 Performance indices for AI models
6.
Model
R2
RMSE
MAPE
VAF
SVR
0.92
1.36
8.12
91.97
ANFIS
0.88
2.37
10.15
87.84
ANN
0.86
2.53
18.64
85.67
DISCUSSION AND CONCLUSION
As can be seen from the Table 1, the most frequent input parameters used in the previous studies are: the uniaxial compressive strength of intact rock (used by 70% of the models), distance and the orientation of discontinuities (used by 50% of the models), the assumed thrust per cutter (used by 40% of the models) and the cutter diameter (used by 30% of the models). In this investigation, the results of PCA have good agreement with previous investigation for prediction of TBM performance. In general, the penetration rate depends on the toughness of the rock material and on the characteristics of the joints in rock mass as well as TBM operating parameters. In this regards, the rock strength has major impact on rock behavior under compression, as noted by many others in the past. When the rolling cutter indents the rock, the stress applied must be higher than the rock strength. So, the rock strength is directly relevant to the performance of TBM. Therefore, UCS has often been used as representative parameters of rock toughness which is influenced by many characteristics of rocks such as constitutive minerals and their spatial positions, weathering or alteration rate as well as porosity and density. It is worth to be mentioned that boreability of rock decreases with
increase of UCS. In addition, the joint conditions certainly affect the rock breakage process. It is easy to be recognized that discontinuities can facilitate rock breakage because cracks induced by TBM cutters easily develop and propagate along with the existing discontinuities, so it has major impact on the TBM performance. Based on the Bruland [15], with the decrease of joint spacing, the TBM penetration increases. It is good to be noted that, although the influence of the joint orientation on TBM PR was widely observed in the tunneling projects, quantifying the impact of the joint orientation on TBM performance has not been very successful and formulas offered in many studies have limited application. Theoretically, orientation of discontinuities (bedding and joint planes) can play significant role in the TBM boring process. Angle α which is defined as the smallest angle between the tunnel axis and the discontinuity surface can be a good parameter to evaluate influence of joint/bedding orientation on TBM performance. On the other hand, experiences gained from similar studies revealed that finding a reasonable relationship between discontinuity orientation alone and TBM parameters is not easy; since the influence of joint orientation is not monotonic and peaks around 45-60 [30]. In blocky and layered rock masses with two or three joint sets, the effect of individual joint orientation can be neglected. It seems that the orientation of discontinuities can affect the boreability and TBM performance most significantly in rock masses with one main discontinuity set (such as thin bedded, foliated and schistose rock masses). Also, this matter has been confirmed by Hassanpour et al. [20] which found a weak correlation between Angle α and FPI. In addition to rock material properties and rock mass characteristics, TBM parameters including thrust and power are main parameters used for TBM performance estimation. The machine specifications and in particular operational parameters including thrust and power represent the amount of forces and torque delivered to rock via cutterhead and disc cutters to initiate fracture propagation in rock. In this investigation, three different AI methods containing ANN, ANFIS and SVR have been developed based on the database of TBM performance in Zagros tunnel projects. As a result of the comparison of VAF, RMSE, MAPE and coefficient of correlations (R2) for predicting TBM performance (Table.9), shows that the prediction performance of SVR model is better than the ANFIS and ANN models. However, the developed artificial intelligence models in present study have great potential for predicting the TBM performance with a great degree of accuracy, robustness and minimum error.
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