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, crosscoupling 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 KELMPLS method. On the basis of mean value staged KELMPLS 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.
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 [
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 Figure
Definition of the parts for piercing process.
The start of the primary unstable piercing stage
The end of the primary unstable piercing stage
The end of the stable piercing stage
The end of the secondary unstable piercing stage
The piercing efficiency of seamless tube is also called the axial sliding friction coefficient, which is the ratio of theoretical pure rolling time
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 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:
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 Table
Modeling variable table for piercing efficiency and energy.
Number 
Original 
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 







The percent reduction of upper roll 
2 







The percent reduction of under roll 
3 







The angle of inclination of upper roll 
4 







The angle of inclination of under roll 
5 







The rotational speed of upper roll 
6 







The rotational speed of under roll 
7 


The position change range of pusher  
8 


The actual position of mandrel thrust block  
9 




The position of mandrel  
10 





The rotational speed of left guide disc  
11 





The rotational speed of left guide disc  
12 



The temperature of tube blank 
When building the model of efficiency and piercing energy consumption on the basis of mean value substaged KELMPLS 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 threedimensional data is changed into the twodimensional data. For the modeling data of piercing efficiency, the vector
Unfolding of process data of three dimensions.
Since linear PLS (partial least squares) model cannot describe correctly the nonlinear relation between independent variable
(1) External relation model is as follows:
(2) Internal relation model is as follows:
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 [
In supervised batch learning, the learning algorithms use a finite number of inputoutput samples for training [
Function
Herein, the Gaussian kernel function (RBF) is adopted:
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, KELMPLS 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,
(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 matrices
Consider
(3) Assign the node number of ELM hidden layer and activation function (e.g., sigmoid function), use ELM to establish nonlinear model between internal models
(4) Use testing data to check model precision. Conduct PLS decomposition on the testing data
Introduce
(5) The KELMPLS 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 Figure
Chart of piercing efficiency and piercing energy model.
The regression model between the data matrix
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.
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 Table
Decision making variable table of optimization.
Number  Variables  Decision variables of piercing efficiency  Decision variables of piercing energy consumption  The mean of variables 

1 



The rotational speed of roll at first stage 
2 



The percent reduction of roll at first stage 
3 



The angle of inclination of roll at first stage 
4 


The position change range of pusher  
5 


The actual position of mandrel thrust block  
6 


The position of mandrel at first stage  
7 



The temperature of tube blank at second stage 
8 



The rotational speed of roll at second stage 
9 



The percent reduction of roll at second stage 
10 



The angle of inclination of roll at second stage 
11 


The position of mandrel at second stage  
12 



The rotational speed of guide disc at second stage 
13 



The rotational speed of roll at third stage 
14 



The percent reduction of roll at third stage 
15 



The angle of inclination of roll at third stage 
16 


The position of mandrel at third stage  
17 



The rotational speed of guide disc at third stage 
The constraint condition of production optimization is as follows:
(1) the change range of production process parameters of piercing equipment
(2) the bite condition of seamless tube piercing production
Consider
In the formula,
In the constrained conditions shown in formula (
(3) The quality of shell produced by the cross piercing should meet the requirements of site
Finally, by formula (
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 Figure
Flow chart of optimization.
(1) The definition of the fitness function of piercing efficiency and energy consumption optimization is as follows:
(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 [
(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
(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.
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:
First, the production data of 90 shells are handled to obtain the input data
Test result of piercing efficiency model.
Test result of piercing energy consumption model.
As shown in Figures
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 KELMPLS  92.14  0.39  93.14  0.32 
MICR  91.12  0.36  90.53  0.30 
MPLS  90.21  0.52  89.56  0.48 
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 

The rotational speed of roll at first stage  136.72  120.91  r/min 
2 

The percent reduction of roll at first stage  144.75  140.13  mm 
3 

The angle of inclination of roll at first stage  13.71  12.33  ° 
4 

The position change range of pusher  2549.63  2547.32  mm 
5 

The actual position of mandrel thrust block  115.64  112.42  mm 
6 

The position of mandrel at first stage  279.32  279.43  mm 
7 

The temperature of tube blank at second stage  1286.23  1251.54  °C 
8 

The rotational speed of roll at second stage  171.82  154.43  ° 
9 

The percent reduction of roll at second stage  144.86  142.75  mm 
10 

The angle of inclination of roll at second stage  13.47  12.49  ° 
11 

The position of mandrel at second stage  280.56  277.78  mm 
12 

The rotational speed of guide disc at second stage  26.74  26.15  r/min 
13 

The rotational speed of roll at third stage  152.51  137.69  r/min 
14 

The percent reduction of roll at third stage  146.43  142.74  mm 
15 

The angle of inclination of roll at third stage  13.73  12.34  ° 
16 

The position of mandrel at third stage  280.64  278.87  mm 
17 

The rotational speed of guide disc at third stage  28.65  26.78  r/min 
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 KELMPLS model  Unit 

Piercing efficiency  84.56  88.93  90.24  % 
Piercing energy consumption  11.352  8.341  8.268  KWh 
Optimization result with maximum piercing efficiency.
Optimization result with minimum piercing energy consumption.
For the process of producing the shell of seamless tube by piercing, this paper adopts the mean value substaged KELMPLS 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.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research is supported by National Natural Science Foundation of China (Grant nos. 61203214), National Natural Science Foundation of China (Grant nos. 61203103, 61374146, 61374147) and Provincial Science and Technology Department of Education Projects (L2013101).