# Modeling and Optimization for Piercing Efficiency and Energy Consumption Based on Mean Value Substaged KELM-PLS and GA Method

Authors: Dong Xiao; Jichun Wang
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

Piercing manufacture of seamless tubes is the process that pierces solid blank into tube hollow. Piercing efficiency and energy consumption are the important indexes in the production of seamless tubes. Piercing process has the multivariate, nonlinear, cross-coupling characteristics. The complex factors that affect efficiency and consumption make it difficult to establish the mechanism models for optimization. Based on the production process, this paper divides the piercing process into three parts and proposes the piercing efficiency and energy consumption prediction models based on mean value staged KELM-PLS method. On the basis of mean value staged KELM-PLS prediction model, the minimum piercing energy consumption and maximum piercing efficiency are calculated by genetic optimization algorithm. Simulation and experiment prove that the optimization method based on the piercing efficiency and energy consumption prediction model can obtain the optimal process parameters effectively and also provide reliable evidences for practical production.


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## Body

## 1. Introduction

In the production of seamless tube, the rotary reheating is the last working procedure of piercing production process. The tandem rolling production is the next working procedure of piercing production process. They both have higher production efficiency than piercing production process. So, the increase of production piercing efficiency plays a very important role in increasing the efficiency of overall production of seamless tube. Because the factors of affecting the piercing efficiency are rather complicated, it is difficult to build the mechanism model to find out its accurate value. Although the literature [1] analyzes the influence of parameters of piercing production process such as the rotational speed of guide disc and the rotational speed of roll, on the piercing efficiency, it only analyzes the influence of every parameter on the production efficiency statically and singly. Li [2] builds the model of relationship between the piercing efficiency and the wall thickness of shell. Although the literature [3, 4] analyzes the influence of technological parameters such as the percent reduction of roll, the angle of inclination of roll, the rotational speed of roll, and the rotational speed of guide disc, on the piercing energy consumption production, it does not give the comprehensive relational model. The literature [5] analyzes the influence of shape of piercing head on the piercing energy consumption. The literature [6] introduces modeling and optimization for piercing energy consumption, but it did not make a comprehensive consideration for piercing energy consumption and piercing efficiency. The literature [7] introduces a modeling and optimization method for piercing energy consumption and piercing efficiency, but the model is linear which limits the accuracy of model prediction and the results of optimization. By the research for the production practice and the literature, it can be known that the main factors which affect the piercing efficiency include the rotational speed of roll, the percent reduction of roll, the rotational speed of guide disc, the size of tube blank and shell, the temperature of tube blank, and the feed angle of roll. Because the state pays more and more close attention to the iron and steel industry with high energy consumption, every seamless tube manufacturer attaches more importance to the piercing energy consumption. Although to reduce the piercing energy consumption can meet the requirements of the direction of policy of the state and can also reduce the production cost of piercing, it may reduce the production efficiency. By the comprehensive consideration of all kinds of literatures and the actual production situations, this paper draws the following conclusion: the main production process parameters affecting the piercing energy consumption include the rotational speed, the percent reduction and the angle of inclination of roll, the size of tube blank and shell, the rotational speed of guide disc, and the temperature of tube blank.In accordance with the characteristics of actual production of piercing process of tube blank, its working process can be divided into three substages: the primary unstable piercing (the first stage), the stable piercing (the second stage), and the secondary unstable piercing (the third stage), as shown in Figure1. Through the analysis and comparison of the production process and data, it is found that the influences of different process parameter variables in the different piercing stages on the piercing efficiency and energy consumption are different. On this basis, we propose mean value substaged KELM-PLS method for the prediction of piercing efficiency and piercing energy consumption. This method selects the influence variable of process according to the piercing production stage, and the average value of corresponding variables in this stage is used as the input value. It overcomes the following shortcomings of traditional modeling method that the structure is complicated, the number of variables is larger, and the amount of calculation is considerable, and so forth. Finally, by the simulation inspection of the actual production data, the accuracy and rapidity of the introduced method are verified. After the modeling obtains the prediction model of accurate piercing efficiency and energy consumption, the optimum algorithm is utilized to carry out the comprehensive optimization of piercing efficiency and energy consumption on the basis of the model. This paper adopts the GA (genetic algorithm) [8–10]. The algorithm is a simple, fast, accurate, and effective global optimum algorithm. The algorithm can obtain the optimum production process parameters under specified conditions, which can be used to guide the actual production, in order to obtain the maximum economic benefit.Definition of the parts for piercing process.
(a)
The start of the primary unstable piercing stage
(b)
The end of the primary unstable piercing stage
(c)
The end of the stable piercing stage
(d)
The end of the secondary unstable piercing stage


## 2. A Summary of Piercing Efficiency and Energy Consumption

The piercing efficiency of seamless tube is also called the axial sliding friction coefficient, which is the ratio of theoretical pure rolling timetth to the actual pure rolling time tac, as shown in the formula,
(1)y=tthtac.
In the formula, y is the piercing efficiency, %; tth is the theoretical pure rolling time, s; and tac is the actual pure rolling time, s.The main factors to affect the piercing efficiency include the rotational speed of roll, size, the shape and quality of material of tool, the size of tube blank, the size of shell, the feed angle, the temperature of tube blank, and the deformation system. The actual pure rolling timetac is measured directly by the field sensor, and the theoretical pure rolling time tth can be expressed as the following formula:
(2)tth=l+L(πDcnr/60)sinα.
In the formula, nr is the rotational speed of roll, r/min; α is the feed angle, rad; D1 is the average diameter of roll, m; l is the length of deformed zone, m; and L is the length of shell, m. Because the manifestation of deformed zone is completely backward slip, tac is more than tth all the time, and there is y≤1 all the time.The piercing energy consumption can be calculated by using the energy consumption produced in the course of piercing production of a steel tube. It can be found out by accumulating the electric energy consumed in producing this steel tube. The concrete expression is as the following:(3)E=∫t0t1Ppiercingdt=∫t0t1UpiercingIpiercingdt.
In the formula, E is the piercing energy consumption, J; Upiercing is the operating voltage of piercing, V; Ipiercing is the working current of piercing, A; t0 is the starting time of piercing, s; and t1 is the end time of piercing, s.


## 3. Analysis of Modeling Variables and Data Preprocessing

This paper firstly researches and analyzes the factor variables that affect the piercing efficiency and energy consumption. It is found in the research that some production variables affect the final piercing efficiency and energy consumption all the time and some production variables affect a part of the production stage only. The modeling variables of piercing efficiency and energy consumption obtained by the comprehensive comparison are shown in Table1.Table 1
Modeling variable table for piercing efficiency and energy.
Number
Originalvariables
The  first stage variables
The  second stage variables
The  third stage variables
The mean of variables
Efficiency
Energy  consumption
Efficiency
Energy   consumption
Efficiency
Energy  consumption
1
x
1
x
1,1
x
1,1
x
2,1
x
2,1
x
3,1
x
3,1
The  percent reduction of upper roll
2
x
2
x
1,2
x
1,2
x
2,2
x
2,2
x
3,2
x
3,2
The percent reduction of under roll
3
x
3
x
1,3
x
1,3
x
2,3
x
2,3
x
3,3
x
3,3
The angle of inclination of upper roll
4
x
4
x
1,4
x
1,4
x
2,4
x
2,4
x
3,4
x
3,4
The angle of inclination of under roll
5
x
5
x
1,5
x
1,5
x
2,5
x
2,5
x
3,5
x
3,5
The rotational speed of upper roll
6
x
6
x
1,6
x
1,6
x
2,6
x
2,6
x
3,6
x
3,6
The rotational speed of under roll
7
x
7
x
1,7
The position change range of pusher
8
x
8
x
1,8
The actual position of mandrel thrust block
9
x
9
x
1,9
x
2,7
x
3,7
The position of mandrel
10
x
10
x
2,8
x
2,7
x
3,8
x
3,7
The rotational speed of left guide disc
11
x
11
x
2,9
x
2,8
x
3,9
x
3,8
The rotational speed of left guide disc
12
x
12
x
2,10
x
2,9
The temperature of tube blankWhen building the model of efficiency and piercing energy consumption on the basis of mean value substaged KELM-PLS method, the average value of the modeling data is evaluated firstly. The average value of the data variables in three piercing stages is evaluated, respectively, and then the three-dimensional data is changed into the two-dimensional data. For the modeling data of piercing efficiency, the vector[x1,1,x1,2,…,x1,9,x2,1,x2,2,…,x2,10,x3,1,x3,2,…,x3,9] formed by 28 variables can be obtained after it is changed into the two-dimensional data. For the modeling data of piercing energy consumption, the vector [x1,1′,x1,2′,…,x1,6′,x2,1′,x2,2′,…,x2,9′,x3,1′,x3,2′,…,x3,8′] formed by 23 variables can be obtained after it is changed into the two-dimensional data. The data matrix X(I×28) of piercing efficiency and the data matrix X′(I′×23) of piercing energy consumption shown in Figure 2, which considers I production batches, are obtained.Figure 2
Unfolding of process data of three dimensions.


## 4. Building of Prediction Model Based on Mean Value Substaged KELM-PLS

### 4.1. Nonlinear PLS

Since linear PLS (partial least squares) model cannot describe correctly the nonlinear relation between independent variableX and dependent variable Y, nonlinear PLS method is required to solve this issue. Wold et al. extended the PLS method to nonlinear field [11, 12]. There are two feasible methods in nonlinear PLS methods: one is to perform array extension for input matrix, introduce some nonlinear terms of original variable, for example, the square term, and then regress the extended input and output matrix using PLS method. If prior knowledge on the relation of original input variable does not exist, this method cannot guide the selection of combined mode and may lead to oversized dimension of input matrix and the difficulties of processing; the other is to reserve the linear external model of PLS method. Internal model is nonlinear.(1) External relation model is as follows:(4)X=TPT+E=∑a=1AtapaT+E,Y=UQT+F=∑a=1AuaqaT+F,
where A is the number of reserved eigenvectors, ta(n×1) and ua(n×1) are the score vectors of X and Y, respectively, pa(m×1) and qa(p×1) are the load vectors of X and Y, respectively, T(n×A) and U(n×A) are the score matrices of X and Y, respectively, P(m×A) and Q(p×A) are the load matrices of X and Y, respectively, and E and F are the fit residual matrices of X and Y, respectively.(2) Internal relation model is as follows:(5)u^a=f(ta)+ε,
where f(·) is the nonlinear function and ε is the residual.The internal model of PLS method adopts neural network gains extensive application because neural network has the capability of fitting nonlinearity. As traditional feedforward neural network adopts gradient learning algorithm during training, parameters in network need iteration and update. Not only the training time lasts long but also it easily results in the issues of local minimum and excessive training [13].


### 4.2. KELM Algorithm

In supervised batch learning, the learning algorithms use a finite number of input-output samples for training [14–18]. For N arbitrary distinct samples (xi,ti)∈Rn×Rm, where xi is a n×1 input vector and ti is a m×1 target vector, if an SLFN (single-hidden layer feedforward neural network [19–24]) with N~ hidden nodes can approximate these N samples with zero error, it then implies that there exists βi, ai, and bi, such that
(6)fN~(xj)=∑i=1N~βiG(ai,bi,xj)+εj=tj,
where j=1,…,N, ai, and bi are the learning parameters of hidden nodes (weight vector connecting the input node to the hidden node and threshold of the hidden node) which are randomly selected according to the proof given by Huang et al. and βi is the weight connecting the ith hidden node to the output node. To avoid overfitting the noise in the data, an error term εj is added. G(ai,bi,x) is the output of the ith hidden node with respect to the input x and N~ the number of hidden nodes which can be determined by trial and error or prior expertise. Then, equation can be written compactly as
(7)Hβ=T,
where
(8)H(a1,…,aN~,b1,…,bN~,x1,…,xN)=[G(a1,b1,x1)⋯G(aN~,bN~,x1)⋮⋯⋮G(a1,b1,xN)⋯G(aN~,bN~,xN)]N×N~,β=[β1T⋮βN~T]N~×m,T=[t1T⋮tNT]N×m,
where H is called the hidden layer output matrix of the network; the ith column of H is the ith hidden node’s output vector with respect to inputs x1,x2,…,xN, and the jth row of H is the output vector of the hidden layer with respect to input xj. The hidden node parameters ai and bi need not be tuned during training and may simply be assigned with random values. Equation (7) then becomes a linear system and the output weights β are estimated as
(9)β~=H+T,
where H+ is the Moore-Penrose generalized inverse of the hidden layer output matrix H [25–29].Functiong(·) is usually unknown; we can incorporate kernel functions in g(·) [30]. KELM (kernel extreme learning machine) can solve the problem of random initialization of ELM algorithm, and learning parameters of the model have a good Reuben [31]. It can improve the performance and accuracy of model prediction of piercing efficiency and piercing energy consumption. The kernel matrix K=[K(x;x1),…,K(x;xN)T] (K(·) is the kernel function) is introduced into (9) to estimate the output of the KELM:
(10)o=KT.Herein, the Gaussian kernel function (RBF) is adopted:(11)K(x1;x2)=exp⁡(-λ∥x1-x2∥2),
where λ needs to be specified.


### 4.3. Mean Value Substaged KELM-PLS Modeling Steps

The difference of nonlinear PLS modeling method based on KELM from linear PLS method is that it uses KELM to establish internal nonlinear model and in the meantime achieve the update of internal and external models. This method reserves linear external model, extracts through PLS the attributive information of process, eliminates the colinearity of data, reduces the dimension of input variable, adopts KELM to establish nonlinear internal model between input score vector matrix and output score vector, and raises the nonlinear processing capability of internal model. Thus, KELM-PLS method has the advantages of PLS and KELM, that is, the characteristics of robustness and feature extraction of PLS method and quick nonlinear processing capability of KELM.The modeling and testing steps of nonlinear PLS method based on KELM are as follows.(1) Assign two standardized data matrices,X∈Rn×m and Y∈Rn×p; dynamic nonlinear PLS regression model can be expressed as follows:
(12)X=[x1,x2,…,xp].(2) Deploy the batch data of batch process, use crossvalidation method to determine the number of latent variable, and adopt linear PLS method to calculate the score vector matricesT and U and load vector matrices P and Q for modeling sample X and Y.Consider(13)X=TPT+E=∑a=1AtapaT+E,Y=UQT+F=∑a=1AuaqaT+F.(3) Assign the node number of ELM hidden layer and activation function (e.g., sigmoid function), use ELM to establish nonlinear model between internal modelsT and U, and gain U=fKELM(T), where fKELM(·) is the nonlinear function indicated by KELM. Hidden nodes in a SLFN transform feature space into another feature space. The original ELM defines the number of nodes as a parameter to be defined. We increase the number of hidden nodes until reaching a stop criteria (ex. residual error reduction), meanwhile, the number of hidden nodes is less than N.(4) Use testing data to check model precision. Conduct PLS decomposition on the testing dataX2, gain score vector T2:
(14)X2=T2PT+E.IntroduceT2 into KELM model, gain U2=fKELM(T2), and find out model prediction value through Y^=UQT.(5) The KELM-PLS method is used to build the prediction model of piercing efficiency and energy consumption for the data which is dealing with mean value substaged method according to Figure2 and Table 1, and the model structure is shown in Figure 3.Figure 3
Chart of piercing efficiency and piercing energy model.The regression model between the data matrixX of piercing efficiency and the prediction value y^ of piercing efficiency, which can be obtained by mean value substaged KELM-PLS method, is formula (15), and in the same way, the relational model between the data matrix X′ of piercing energy consumption and the prediction value y^' of piercing energy consumption, which can be obtained by substaged KELM-PLS method, is formula (16):
(15)y^=f(X,θ)=f([x1,1,x1,2,…,x1,9,x2,1,x2,2,…,x2,10,x3,1,x3,2,…,x3,9],θ),(16)y^′=f(X′,θ′)=f([x1,1′,x1,2′,…,x1,6′,x2,1′,x2,2′,…,x2,9′,x3,1′,x3,2′,…,x3,8′],θ′).
Among them, θ and θ′ are the regression coefficients of model of piercing efficiency and energy consumption obtained by utilizing mean value substaged KELM-PLS.


## 4.1. Nonlinear PLS

Since linear PLS (partial least squares) model cannot describe correctly the nonlinear relation between independent variableX and dependent variable Y, nonlinear PLS method is required to solve this issue. Wold et al. extended the PLS method to nonlinear field [11, 12]. There are two feasible methods in nonlinear PLS methods: one is to perform array extension for input matrix, introduce some nonlinear terms of original variable, for example, the square term, and then regress the extended input and output matrix using PLS method. If prior knowledge on the relation of original input variable does not exist, this method cannot guide the selection of combined mode and may lead to oversized dimension of input matrix and the difficulties of processing; the other is to reserve the linear external model of PLS method. Internal model is nonlinear.(1) External relation model is as follows:(4)X=TPT+E=∑a=1AtapaT+E,Y=UQT+F=∑a=1AuaqaT+F,
where A is the number of reserved eigenvectors, ta(n×1) and ua(n×1) are the score vectors of X and Y, respectively, pa(m×1) and qa(p×1) are the load vectors of X and Y, respectively, T(n×A) and U(n×A) are the score matrices of X and Y, respectively, P(m×A) and Q(p×A) are the load matrices of X and Y, respectively, and E and F are the fit residual matrices of X and Y, respectively.(2) Internal relation model is as follows:(5)u^a=f(ta)+ε,
where f(·) is the nonlinear function and ε is the residual.The internal model of PLS method adopts neural network gains extensive application because neural network has the capability of fitting nonlinearity. As traditional feedforward neural network adopts gradient learning algorithm during training, parameters in network need iteration and update. Not only the training time lasts long but also it easily results in the issues of local minimum and excessive training [13].


## 4.2. KELM Algorithm

In supervised batch learning, the learning algorithms use a finite number of input-output samples for training [14–18]. For N arbitrary distinct samples (xi,ti)∈Rn×Rm, where xi is a n×1 input vector and ti is a m×1 target vector, if an SLFN (single-hidden layer feedforward neural network [19–24]) with N~ hidden nodes can approximate these N samples with zero error, it then implies that there exists βi, ai, and bi, such that
(6)fN~(xj)=∑i=1N~βiG(ai,bi,xj)+εj=tj,
where j=1,…,N, ai, and bi are the learning parameters of hidden nodes (weight vector connecting the input node to the hidden node and threshold of the hidden node) which are randomly selected according to the proof given by Huang et al. and βi is the weight connecting the ith hidden node to the output node. To avoid overfitting the noise in the data, an error term εj is added. G(ai,bi,x) is the output of the ith hidden node with respect to the input x and N~ the number of hidden nodes which can be determined by trial and error or prior expertise. Then, equation can be written compactly as
(7)Hβ=T,
where
(8)H(a1,…,aN~,b1,…,bN~,x1,…,xN)=[G(a1,b1,x1)⋯G(aN~,bN~,x1)⋮⋯⋮G(a1,b1,xN)⋯G(aN~,bN~,xN)]N×N~,β=[β1T⋮βN~T]N~×m,T=[t1T⋮tNT]N×m,
where H is called the hidden layer output matrix of the network; the ith column of H is the ith hidden node’s output vector with respect to inputs x1,x2,…,xN, and the jth row of H is the output vector of the hidden layer with respect to input xj. The hidden node parameters ai and bi need not be tuned during training and may simply be assigned with random values. Equation (7) then becomes a linear system and the output weights β are estimated as
(9)β~=H+T,
where H+ is the Moore-Penrose generalized inverse of the hidden layer output matrix H [25–29].Functiong(·) is usually unknown; we can incorporate kernel functions in g(·) [30]. KELM (kernel extreme learning machine) can solve the problem of random initialization of ELM algorithm, and learning parameters of the model have a good Reuben [31]. It can improve the performance and accuracy of model prediction of piercing efficiency and piercing energy consumption. The kernel matrix K=[K(x;x1),…,K(x;xN)T] (K(·) is the kernel function) is introduced into (9) to estimate the output of the KELM:
(10)o=KT.Herein, the Gaussian kernel function (RBF) is adopted:(11)K(x1;x2)=exp⁡(-λ∥x1-x2∥2),
where λ needs to be specified.


## 4.3. Mean Value Substaged KELM-PLS Modeling Steps

The difference of nonlinear PLS modeling method based on KELM from linear PLS method is that it uses KELM to establish internal nonlinear model and in the meantime achieve the update of internal and external models. This method reserves linear external model, extracts through PLS the attributive information of process, eliminates the colinearity of data, reduces the dimension of input variable, adopts KELM to establish nonlinear internal model between input score vector matrix and output score vector, and raises the nonlinear processing capability of internal model. Thus, KELM-PLS method has the advantages of PLS and KELM, that is, the characteristics of robustness and feature extraction of PLS method and quick nonlinear processing capability of KELM.The modeling and testing steps of nonlinear PLS method based on KELM are as follows.(1) Assign two standardized data matrices,X∈Rn×m and Y∈Rn×p; dynamic nonlinear PLS regression model can be expressed as follows:
(12)X=[x1,x2,…,xp].(2) Deploy the batch data of batch process, use crossvalidation method to determine the number of latent variable, and adopt linear PLS method to calculate the score vector matricesT and U and load vector matrices P and Q for modeling sample X and Y.Consider(13)X=TPT+E=∑a=1AtapaT+E,Y=UQT+F=∑a=1AuaqaT+F.(3) Assign the node number of ELM hidden layer and activation function (e.g., sigmoid function), use ELM to establish nonlinear model between internal modelsT and U, and gain U=fKELM(T), where fKELM(·) is the nonlinear function indicated by KELM. Hidden nodes in a SLFN transform feature space into another feature space. The original ELM defines the number of nodes as a parameter to be defined. We increase the number of hidden nodes until reaching a stop criteria (ex. residual error reduction), meanwhile, the number of hidden nodes is less than N.(4) Use testing data to check model precision. Conduct PLS decomposition on the testing dataX2, gain score vector T2:
(14)X2=T2PT+E.IntroduceT2 into KELM model, gain U2=fKELM(T2), and find out model prediction value through Y^=UQT.(5) The KELM-PLS method is used to build the prediction model of piercing efficiency and energy consumption for the data which is dealing with mean value substaged method according to Figure2 and Table 1, and the model structure is shown in Figure 3.Figure 3
Chart of piercing efficiency and piercing energy model.The regression model between the data matrixX of piercing efficiency and the prediction value y^ of piercing efficiency, which can be obtained by mean value substaged KELM-PLS method, is formula (15), and in the same way, the relational model between the data matrix X′ of piercing energy consumption and the prediction value y^' of piercing energy consumption, which can be obtained by substaged KELM-PLS method, is formula (16):
(15)y^=f(X,θ)=f([x1,1,x1,2,…,x1,9,x2,1,x2,2,…,x2,10,x3,1,x3,2,…,x3,9],θ),(16)y^′=f(X′,θ′)=f([x1,1′,x1,2′,…,x1,6′,x2,1′,x2,2′,…,x2,9′,x3,1′,x3,2′,…,x3,8′],θ′).
Among them, θ and θ′ are the regression coefficients of model of piercing efficiency and energy consumption obtained by utilizing mean value substaged KELM-PLS.


## 5. Comprehensive Optimization of Piercing Efficiency and Energy Consumption on the Basis of Genetic Algorithm

After building the accurate model, we put the model into the objective function and utilize the genetic algorithm to optimize and find out the optimal production process parameters according to the different conditions of market and production, in order to guarantee the maximum profit of enterprise.


### 5.1. Decision Variable and Optimization Objective of Piercing Efficiency and Energy Consumption

For the upper roll and lower roll and left guide disc and right guide disc, although their real time parameters of actual production are different, the setting values to be controlled are set to the same setting values, and, therefore, they have merged into one production parameter in optimization when carrying out the optimization. As shown in Table2, the decision variables related to the piercing efficiency are x-1,x-2,…,x-17, and the decision variables related to the piercing energy consumption are x-1,x-2,x-3,x-7,x-8,x-9,x-10,x-12,x-13,x-14,x-15,x-17. Their constraint conditions are formulas (18)~(22) and the concrete change range is as follows: the temperature of tube blank: 1200~1300°C, the rotational speed of guide disc: 20~30 r/min, the actual position of mandrel thrust block: 115~170 mm, the position change range of pusher: 0~2550 mm, the position of mandrel: 200~300 mm, the rotational speed of roll: 105~195 r/min, the angle of inclination of roll: 10~14° mm, and the percent reduction of roll: 120~170 mm. In order to obtain the maximum profit of production, it is necessary to carry out the comprehensive optimization of piercing efficiency and energy consumption and take the following formula as the optimization model to seek the optimum decision variable value of x-1,x-2,…,x-17:
(17)max⁡y=k1f(x,θ)+k2f(x′,θ′),
where f(x,θ) and f(x′,θ′) are piercing efficiency and energy consumption, respectively, which are calculated by mean value substaged KELM-PLS method; k1 and k2 are the weighted values of adjustment of optimization objective, the adjustment range of k1 is 0~13, and the adjustment range of k2 is 0~11. When the rate of capacity utilization is lower, it is necessary to maximize k2, and in this way the energy consumption in production can be reduced and at the same time the production cost can be reduced, thus maximizing the production benefit. Similarly, when the rate of capacity utilization is higher and the supply and marketing of product are flourishing, it is necessary to maximize k1, in order to guarantee that the production line of seamless tube operates in the state of maximum efficiency, and in this way, the maximum benefit of production can also be obtained. In other conditions, it is necessary to carry out the adjustment according to the experience and concrete conditions.Table 2
Decision making variable table of optimization.
Number
Variables
Decision variables of piercing efficiency
Decision variables of piercing energy consumption
The mean of variables
1
x
-
1
√
√
The rotational speed of roll at first stage
2
x
-
2
√
√
The percent reduction of roll at first stage
3
x
-
3
√
√
The angle of inclination of roll at first stage
4
x
-
4
√
The position change range of pusher
5
x
-
5
√
The actual position of mandrel thrust block
6
x
-
6
√
The position of mandrel at first stage
7
x
-
7
√
√
The temperature of tube blank at second stage
8
x
-
8
√
√
The rotational speed of roll at second stage
9
x
-
9
√
√
The percent reduction of roll at second stage
10
x
-
10
√
√
The angle of inclination of roll at second stage
11
x
-
11
√
The position of mandrel at second stage
12
x
-
12
√
√
The rotational speed of guide disc at second stage
13
x
-
13
√
√
The rotational speed of roll at third stage
14
x
-
14
√
√
The percent reduction of roll at third stage
15
x
-
15
√
√
The angle of inclination of roll at third stage
16
x
-
16
√
The position of mandrel at third stage
17
x
-
17
√
√
The rotational speed of guide disc at third stageThe constraint condition of production optimization is as follows:(1) the change range of production process parameters of piercing equipment(18)s.t.1105≤x-1,x-8,x-13≤195,0≤x-4≤2550,115≤x-5≤170,1200≤x-7≤1300,200≤x-6,x-11,x-16≤300,120≤x-2,x-9,x-14≤170,10≤x-3,x-10,x-15≤14,20≤x-12,x-17≤30;(2) the bite condition of seamless tube piercing production(19)s.t.22Ti>Qi+2Pi.
Among them, i is the number of substage of piercings, i=1,2,3; Ti is the axial dragged-into frictional force Ti=pifbiLHicos⁡θi of one roll to affect the tube blank, N; Pi is the positive pressure axial component Pi=pibiLHisinθi of one roll to affect the tube blank, N; and Qi is the resistance of piercing head, N.Consider(20)Qi=piπrH2×(2πni60DGdGJsinθi+1+(dKJ/dGJ)21+dKJ/dGJ)(2πni60)-1.In the formula,pi is the mean specific roll pressure, Kg/mm2; f is the friction coefficient; θi is the angle of inclination of roll, rad; ni is the rotational speed of roll, rad/min; rH is the radius of nose of piercing head, mm; LHi is the length of deformed zone, mm; bi is the percent reduction of roll, mm; DG is the diameter of roll waist, mm; dGJ is the distance of roll at the place of inlet, mm; and dKJ is the distance of roll at the place of bore throat, mm.In the constrained conditions shown in formula (17), the rotational speed of rolls n1,n2,n3 corresponds to the decision variables x-1,x-8,x-13, respectively; the percent reduction of roll b1,b2,b3 corresponds to the decision variables x-2,x-9,x-14, respectively; the angle of inclination of rolls θ1,θ2,θ3 corresponds to the decision variables x-3,x-10,x-15, respectively; and the position of mandrels LH1,LH2,LH3 corresponds to the decision variables x-6,x-11,x-16, respectively.(3) The quality of shell produced by the cross piercing should meet the requirements of site(21)s.t.3MP=fquality(x)<MN.
In the formula, MN  = the requirements for quality, %; MP  = the quality of shell, %; and fquality()  = the quality of shell calculated by the quality prediction model [32].Finally, by formula (17)~formula (22), the model of comprehensive optimization of piercing efficiency and energy consumption can be expressed as follows:
(22)max⁡y=k1xθ+k2x′θ′s.t.105≤x-1,x-8,x-13≤195,0≤x-4≤2550,115≤x-5≤170,1200≤x-7≤1300,200≤x-6,x-11,x-16≤300,120≤x-2,x-9,x-14≤170,10≤x-3,x-10,x-15≤14,20≤x-12,x-17≤30,2Ti>Qi+2Pi,MP=fquality(x)<MN.


### 5.2. Comprehensive Optimization of Piercing Efficiency and Energy Consumption

The comprehensive optimization of piercing efficiency and energy consumption is an optimization issue with constraint condition, and this paper selects the genetic algorithm as the algorithm to find the solution of model of comprehensive optimization of piercing efficiency and energy consumption. Flow chart of piercing efficiency and energy optimization is shown in Figure4.Figure 4
Flow chart of optimization.(1) The definition of the fitness function of piercing efficiency and energy consumption optimization is as follows:(23)max⁡y=k1f(x,θ)+k2f(x′,θ′).
In the formula, θ is the coefficient of model of piercing efficiency; θ′ is the coefficient of model of piercing energy consumption; x is the decision variable of optimization of piercing efficiency; and x′ is the decision variable of optimization of piercing energy consumption. f() is the model output of mean value substaged KELM-PLS method.(2) Because the comprehensive optimization of piercing efficiency and energy consumption is an issue of continuous parameters optimization, float encoding is adopted as the encoding mode. It avoids the length limitation of binary encoding which reduces the performance and solution accuracy. Float encoding does not require coding and decoding operation which improves the computing speed and accuracy to solve. At the same time, the integral arithmetic crossover algorithm is adopted [33, 34].(3) Punishment technology is commonly used for constrained optimization problems in genetic algorithm. Penalty function methods transform a constrained optimization problem into a sequence of unconstrained optimization problems. The constraints are appended to the objective function via a penalty parameter and a penalty function. In general, a feasible penalty function should admit a positive penalty for infeasible points and no penalty for feasible points. The fitness function is designed for(24)F=y-m1×max⁡{0,g1(x)}-m2×max⁡{0,g2(x)}-m3×max⁡{0,g3(x)}-m4h(x),
where g1 is the inequality constraint of formula (18), g2 is the inequality constraint of formula (19), g3 is the inequality constraint of formula (21), and h is the equality constraint of formula (20).(4) GA algorithm stopping convergence condition is as follows. When the algorithm is run continuously for 20 generations, it stops if the fitness value changes to less than 10−6. And it stops when the algorithm is run continuously for 1000 generations.


## 5.1. Decision Variable and Optimization Objective of Piercing Efficiency and Energy Consumption

For the upper roll and lower roll and left guide disc and right guide disc, although their real time parameters of actual production are different, the setting values to be controlled are set to the same setting values, and, therefore, they have merged into one production parameter in optimization when carrying out the optimization. As shown in Table2, the decision variables related to the piercing efficiency are x-1,x-2,…,x-17, and the decision variables related to the piercing energy consumption are x-1,x-2,x-3,x-7,x-8,x-9,x-10,x-12,x-13,x-14,x-15,x-17. Their constraint conditions are formulas (18)~(22) and the concrete change range is as follows: the temperature of tube blank: 1200~1300°C, the rotational speed of guide disc: 20~30 r/min, the actual position of mandrel thrust block: 115~170 mm, the position change range of pusher: 0~2550 mm, the position of mandrel: 200~300 mm, the rotational speed of roll: 105~195 r/min, the angle of inclination of roll: 10~14° mm, and the percent reduction of roll: 120~170 mm. In order to obtain the maximum profit of production, it is necessary to carry out the comprehensive optimization of piercing efficiency and energy consumption and take the following formula as the optimization model to seek the optimum decision variable value of x-1,x-2,…,x-17:
(17)max⁡y=k1f(x,θ)+k2f(x′,θ′),
where f(x,θ) and f(x′,θ′) are piercing efficiency and energy consumption, respectively, which are calculated by mean value substaged KELM-PLS method; k1 and k2 are the weighted values of adjustment of optimization objective, the adjustment range of k1 is 0~13, and the adjustment range of k2 is 0~11. When the rate of capacity utilization is lower, it is necessary to maximize k2, and in this way the energy consumption in production can be reduced and at the same time the production cost can be reduced, thus maximizing the production benefit. Similarly, when the rate of capacity utilization is higher and the supply and marketing of product are flourishing, it is necessary to maximize k1, in order to guarantee that the production line of seamless tube operates in the state of maximum efficiency, and in this way, the maximum benefit of production can also be obtained. In other conditions, it is necessary to carry out the adjustment according to the experience and concrete conditions.Table 2
Decision making variable table of optimization.
Number
Variables
Decision variables of piercing efficiency
Decision variables of piercing energy consumption
The mean of variables
1
x
-
1
√
√
The rotational speed of roll at first stage
2
x
-
2
√
√
The percent reduction of roll at first stage
3
x
-
3
√
√
The angle of inclination of roll at first stage
4
x
-
4
√
The position change range of pusher
5
x
-
5
√
The actual position of mandrel thrust block
6
x
-
6
√
The position of mandrel at first stage
7
x
-
7
√
√
The temperature of tube blank at second stage
8
x
-
8
√
√
The rotational speed of roll at second stage
9
x
-
9
√
√
The percent reduction of roll at second stage
10
x
-
10
√
√
The angle of inclination of roll at second stage
11
x
-
11
√
The position of mandrel at second stage
12
x
-
12
√
√
The rotational speed of guide disc at second stage
13
x
-
13
√
√
The rotational speed of roll at third stage
14
x
-
14
√
√
The percent reduction of roll at third stage
15
x
-
15
√
√
The angle of inclination of roll at third stage
16
x
-
16
√
The position of mandrel at third stage
17
x
-
17
√
√
The rotational speed of guide disc at third stageThe constraint condition of production optimization is as follows:(1) the change range of production process parameters of piercing equipment(18)s.t.1105≤x-1,x-8,x-13≤195,0≤x-4≤2550,115≤x-5≤170,1200≤x-7≤1300,200≤x-6,x-11,x-16≤300,120≤x-2,x-9,x-14≤170,10≤x-3,x-10,x-15≤14,20≤x-12,x-17≤30;(2) the bite condition of seamless tube piercing production(19)s.t.22Ti>Qi+2Pi.
Among them, i is the number of substage of piercings, i=1,2,3; Ti is the axial dragged-into frictional force Ti=pifbiLHicos⁡θi of one roll to affect the tube blank, N; Pi is the positive pressure axial component Pi=pibiLHisinθi of one roll to affect the tube blank, N; and Qi is the resistance of piercing head, N.Consider(20)Qi=piπrH2×(2πni60DGdGJsinθi+1+(dKJ/dGJ)21+dKJ/dGJ)(2πni60)-1.In the formula,pi is the mean specific roll pressure, Kg/mm2; f is the friction coefficient; θi is the angle of inclination of roll, rad; ni is the rotational speed of roll, rad/min; rH is the radius of nose of piercing head, mm; LHi is the length of deformed zone, mm; bi is the percent reduction of roll, mm; DG is the diameter of roll waist, mm; dGJ is the distance of roll at the place of inlet, mm; and dKJ is the distance of roll at the place of bore throat, mm.In the constrained conditions shown in formula (17), the rotational speed of rolls n1,n2,n3 corresponds to the decision variables x-1,x-8,x-13, respectively; the percent reduction of roll b1,b2,b3 corresponds to the decision variables x-2,x-9,x-14, respectively; the angle of inclination of rolls θ1,θ2,θ3 corresponds to the decision variables x-3,x-10,x-15, respectively; and the position of mandrels LH1,LH2,LH3 corresponds to the decision variables x-6,x-11,x-16, respectively.(3) The quality of shell produced by the cross piercing should meet the requirements of site(21)s.t.3MP=fquality(x)<MN.
In the formula, MN  = the requirements for quality, %; MP  = the quality of shell, %; and fquality()  = the quality of shell calculated by the quality prediction model [32].Finally, by formula (17)~formula (22), the model of comprehensive optimization of piercing efficiency and energy consumption can be expressed as follows:
(22)max⁡y=k1xθ+k2x′θ′s.t.105≤x-1,x-8,x-13≤195,0≤x-4≤2550,115≤x-5≤170,1200≤x-7≤1300,200≤x-6,x-11,x-16≤300,120≤x-2,x-9,x-14≤170,10≤x-3,x-10,x-15≤14,20≤x-12,x-17≤30,2Ti>Qi+2Pi,MP=fquality(x)<MN.


## 5.2. Comprehensive Optimization of Piercing Efficiency and Energy Consumption

The comprehensive optimization of piercing efficiency and energy consumption is an optimization issue with constraint condition, and this paper selects the genetic algorithm as the algorithm to find the solution of model of comprehensive optimization of piercing efficiency and energy consumption. Flow chart of piercing efficiency and energy optimization is shown in Figure4.Figure 4
Flow chart of optimization.(1) The definition of the fitness function of piercing efficiency and energy consumption optimization is as follows:(23)max⁡y=k1f(x,θ)+k2f(x′,θ′).
In the formula, θ is the coefficient of model of piercing efficiency; θ′ is the coefficient of model of piercing energy consumption; x is the decision variable of optimization of piercing efficiency; and x′ is the decision variable of optimization of piercing energy consumption. f() is the model output of mean value substaged KELM-PLS method.(2) Because the comprehensive optimization of piercing efficiency and energy consumption is an issue of continuous parameters optimization, float encoding is adopted as the encoding mode. It avoids the length limitation of binary encoding which reduces the performance and solution accuracy. Float encoding does not require coding and decoding operation which improves the computing speed and accuracy to solve. At the same time, the integral arithmetic crossover algorithm is adopted [33, 34].(3) Punishment technology is commonly used for constrained optimization problems in genetic algorithm. Penalty function methods transform a constrained optimization problem into a sequence of unconstrained optimization problems. The constraints are appended to the objective function via a penalty parameter and a penalty function. In general, a feasible penalty function should admit a positive penalty for infeasible points and no penalty for feasible points. The fitness function is designed for(24)F=y-m1×max⁡{0,g1(x)}-m2×max⁡{0,g2(x)}-m3×max⁡{0,g3(x)}-m4h(x),
where g1 is the inequality constraint of formula (18), g2 is the inequality constraint of formula (19), g3 is the inequality constraint of formula (21), and h is the equality constraint of formula (20).(4) GA algorithm stopping convergence condition is as follows. When the algorithm is run continuously for 20 generations, it stops if the fitness value changes to less than 10−6. And it stops when the algorithm is run continuously for 1000 generations.


## 6. Simulation and Experiment

In order to verify the accuracy of the method, this paper selects the data of production of 90 piercing tubes of Diescher Mannesmann piercer of seamless tube factory of Baosteel company in January 2014. The first 65 shells are used to build the prediction model of piercing efficiency and energy consumption and the last 25 shells are used for the inspection of accuracy of model. The test process conditions are as follows: the diameter of roll:Dg=1000 mm, the cone angle of inlet: β=2.75°, the diameter of guide disc: Dp=1800 mm, the toe angle: Φ=0°, the distance of guide disc: Lp=173 mm, the rotational speed of guide disc: 26 revolutions per minute, the initial temperature of guide disc: 100°C, the rotational speed of roll: 185 revolutions per minute, the initial temperature of roll: 150°C, the percent reduction of roll: 150 mm, the change range of angle of inclination of roll: 13°, the temperature of piercing head: 100°C, the temperature of tube blank: 1250°C, the specification of tube blank: Φ178 mm, the specification of shell: Φ179 mm × 30.25 mm, and the quality of material: C22 (20# steel).First, the production data of 90 shells are handled to obtain the input dataX(65,28) and X′(65,23) for modeling of piercing efficiency and energy consumption as well as the input data Xnew(25,28) and Xnew′(25,23) for inspection of piercing efficiency and energy consumption according to Figure 2 and Table 1. After using the mean value substaged KELM-PLS method to build the prediction models of piercing efficiency and energy consumption, respectively, the input data Xnew(25,28) and Xnew′(25,23) for inspection of piercing efficiency and energy consumption are put into formula (5), respectively, and their precision of model is compared with the precision of the model of actual values y(25,1) and y^′(25,1). And at the same time, the prediction model of efficiency and piercing energy consumption of traditional MPLS method is built. The following 12 process variables that affect the piercing efficiency are selected: the intermesh of upper roll and lower roll, the values of angles of inclination of upper and lower rolls, the rotational speed of upper roll and lower roll, the position of pusher, the actual position of mandrel thrust block, the position of mandrel, the rotational speed of left and right discs, and the temperature of tube blank. Similarly, the production data of 25 shells are utilized to carry out the inspection of model. Figures 5 and 6 show the results of comparison among the inspections of the models of piercing efficiency and the piercing energy consumption of three kinds of methods which are mean value substaged KELM-PLS, MICR [7], and MPLS method.Figure 5
Test result of piercing efficiency model.Figure 6
Test result of piercing energy consumption model.As shown in Figures5 and 6, the prediction accuracies of model of piercing efficiency on the basis of mean value substaged KELM-PLS, MICR, and MPLS model are 92.14%, 91.12, and 90.21%, respectively, and the input variables of their models are 28 and 1101 variables, respectively. The calculation times used by the models are 0.39 seconds, 0.36 seconds, and 0.52 seconds. The prediction accuracies of model of piercing energy consumption on the basis of mean value substaged KELM-PLS, MICR, and MPLS model are 93.14%, 90.53%, and 89.56%, respectively, and the input variables of their models are 23 and 801 variables, respectively. The calculation times of models are 0.32 seconds, 0.30 seconds, and 0.48 seconds, respectively. Table 3 indicates the comparison among the three kinds of methods of piercing efficiency and energy consumption. It can be found from this condition that the model of mean value substaged KELM-PLS method increases the prediction accuracy of model to a certain extent, and at the same time, the reduction of input quantity of model also greatly reduces the calculation time used by the model, which makes the model more easy to use online. Figure 7 shows that maximum piercing efficiency is taken as the strategy of seeking optimization when the influence of the piercing energy consumption is not considered. The maximum piercing efficiency obtained by the optimization is 90.24%. Figure 8 shows that the minimum energy piercing consumption is taken as the strategy of seeking optimization when the piercing efficiency is not considered. The minimum piercing energy consumption obtained by the optimization is 8.268 Kwh. And at the same time, the production process parameter values in every stage that minimize the piercing energy consumption are found out. Table 4 indicates the parameter values of production process in every stage and Table 5 indicates the comparison between the indexes of piercing efficiency and energy consumption before and after the optimization.Table 3
Comparison of the production index after optimization of piercing efficiency and energy consumption.
Method of modeling
Piercing efficiency
Piercing energy consumption
Accuracy of model
Time of modeling
Accuracy of model
Time of modeling
Mean value substaged KELM-PLS
92.14
0.39
93.14
0.32
MICR
91.12
0.36
90.53
0.30
MPLS
90.21
0.52
89.56
0.48Table 4
Optimization result of piercing efficiency and energy consumption.
Number
Decision variables
The mean of variables
Optimization results in the highest efficiency piercing
Optimization results in the lowest energy consumption
Unit
1
x
-
1
The rotational speed of roll at first stage
136.72
120.91
r/min
2
x
-
2
The percent reduction of roll at first stage
144.75
140.13
mm
3
x
-
3
The angle of inclination of roll at first stage
13.71
12.33
°
4
x
-
4
The position change range of pusher
2549.63
2547.32
mm
5
x
-
5
The actual position of mandrel thrust block
115.64
112.42
mm
6
x
-
6
The position of mandrel at first stage
279.32
279.43
mm
7
x
-
7
The temperature of tube blank at second stage
1286.23
1251.54
°C
8
x
-
8
The rotational speed of roll at second stage
171.82
154.43
°
9
x
-
9
The percent reduction of roll at second stage
144.86
142.75
mm
10
x
-
10
The angle of inclination of roll at second stage
13.47
12.49
°
11
x
-
11
The position of mandrel at second stage
280.56
277.78
mm
12
x
-
12
The rotational speed of guide disc at second stage
26.74
26.15
r/min
13
x
-
13
The rotational speed of roll at third stage
152.51
137.69
r/min
14
x
-
14
The percent reduction of roll at third stage
146.43
142.74
mm
15
x
-
15
The angle of inclination of roll at third stage
13.73
12.34
°
16
x
-
16
The position of mandrel at third stage
280.64
278.87
mm
17
x
-
17
The rotational speed of guide disc at third stage
28.65
26.78
r/minTable 5
Comparison of the production index after optimization of piercing efficiency and energy consumption.
Production index of piercing
Before optimization
After optimization with MICR model
After optimization with KELM-PLS model
Unit
Piercing efficiency
84.56
88.93
90.24
%
Piercing energy consumption
11.352
8.341
8.268
KWhFigure 7
Optimization result with maximum piercing efficiency.Figure 8
Optimization result with minimum piercing energy consumption.


## 7. Conclusions

For the process of producing the shell of seamless tube by piercing, this paper adopts the mean value substaged KELM-PLS modeling method, builds the prediction model of piercing efficiency and energy consumption, and provides the online prediction of piercing efficiency and energy consumption of tube blank. On this basis, the genetic algorithm is adopted for the comprehensive optimization of piercing efficiency and energy consumption and the optimum production process parameters are obtained according to the different market demands and constraint conditions. The test indicates that when the minimum energy consumption is required, the optimized energy consumption of production is reduced obviously, and at the same time, the cost of production is reduced, and when the high production efficiency is required, the optimized production increases the piercing efficiency to a certain extent, which can guarantee that the production task is finished smoothly. Meanwhile, the method can also be spread to the application to the index predication and optimization of other multistage batch processes.



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*Source: 132654-2014-06-17.xml*