Load Distribution of Evolutionary Algorithm for Complex-Process Optimization Based on Differential Evolutionary Strategy in Hot Rolling Process

1 Key Laboratory of Advanced Control of Iron and Steel Process, School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing 100083, China 2 Institute for Automatic Control and Complex Systems, University of Duisburg-Essen, Duisburg 47057, Germany 3 Beijing Power Electronics and Motor Drives Engineering Research Center, College of Mechanical and Electrical Engineering, North China University of Technology, Beijing 100144, China


Introduction
With the increasing demand for improving the product quality and control accuracy in hot rolling process, the rolling scheduling problem has become an important issue in the steel industry.According to the principle that nominal motor power should be greater than rolling power, the main purpose of hot rolling scheduling problem consists in determining the final thickness for every rolling pass to set other process parameters [1], such as rolling force and bending force.The key point and object of hot rolling shape/gauge control is the shape control of roll gap, in the sense that the load distribution is the basis of strip shape control.Although the classic load distribution is simple and reasonable, it cannot achieve the most optimal setting to shape control [1,2].
The hot rolling process has been optimized with rolling theory or heuristics algorithms [3][4][5].For example, a differential evolution algorithm with space-adaptive idea is applied to several hot strip mills for the optimal design of scheduling.This algorithm expands or shrinks the search space by certain rules and realizes the automatic search for the suitable space and improves the convergence rate and accuracy [3].An intelligent method named variable metric hybrid genetic algorithm was introduced to optimize hot strip mills [4].A genetic algorithm-based optimization was coded and operated for 1370 mm tandem cold rolling schedule.It seems that the performance of the optimal rolling schedule is satisfactory and promising [5].Although the above load distribution is reasonable, it often requires more adjustable parameters during the search for optimal solution, thereby making influence on coverage speed and search efficiency.
Thus, the major objective pursued in this paper is to formulate a better solution on the rolling scheduling optimization.Based on the evolutionary algorithm for complex-process optimization (EACOP) [6][7][8], we improve this algorithm within differential evolutionary strategy and utilize it to optimize load distribution of hot rolling.The DEACOP has the flexible structure which is similar to scatter search and employs some elements of scatter search [9] and path relinking [10].Besides, it makes use of a smaller number of tuning parameters and differential evolutionary strategy among the population members with new strategies.Firstly, according to model's characteristics of load distribution, initial diverse population strategy will be improved with the consideration of latin hypercube uniform sampling.Secondly, differential evolutionary strategy will be presented to replace the original linear combination.Thirdly, a population-update method is introduced to modify the balance between intensification and diversification.Finally, a search intensification strategy called the "go-beyond" to indepth search is established for enhancement of the efficiency of the local optimal solution.This differential evolutionary strategy can generate broader area around the population members and get better intensification and diversification of population members by the go-beyond strategy.
Based on experimental simulation by actual data in hot rolling process, simulation result shows that the application of DEACOP optimizes the gauge reduction for each rolling pass and gives full play to the upstream rolling mill equipment's ability.Meanwhile, DEACOP algorithm regulates crown index of the downstream mills, so it can further improve the efficiency of plate-shaped regulating.

The Gauge and Shape Model
2.1.System Description.In order to determine rolling force of each stand, as well as the other settings, the key point of load distribution is that the exit thickness of each stand should be distributed reasonably.Thus, in this section the optimal load distribution with the consideration of overall performance on shape and gauge is proposed.The optimal load distribution of finishing mill group can be divided into three stages [11][12][13].
The first phase requires that the 1st stand's reduction should be left some room, as the steel billet's thickness may fluctuate when steel billet goes into rolling mill.
In the second phase, the 2nd and 3rd stands should make full use of equipment power, therefore making the amount of the reduction as large as possible.
In the third phase, the rolling force in the last stage should gradually decrease from the 4th to the last stand, so that the accuracy and performance of the shape and gauge can be synthesized properly.Meanwhile, the relative crown of the last four stands should be equal.
With the consideration of those steps, the objective function is derived as follows, which constructs with the desire of above three stages with DEACOP optimizes load distribution: where   is the rolling force of the th stand, CR  and ℎ  separately represent exit crown and thickness of the th stand,   denotes weighted coefficient, and Δ  is the compensation coefficient, which maintain the equality of relative crown from the 4th to the last stand.From the view of engineering, some related variables of rolling process can be restricted, such as ℎ +1 < ℎ  , 0 ≤   ≤  max . 1 and  2 denote the proportional coefficient about rolling force and both coefficients are changed according to technological condition, where  1 is set as 0.9 and  2 is set as 1 in this paper.Apart from the above constraints, the Shohet discriminant [1] about sheet deformation is also necessary.Since the rolling process of hot strip mill is different from the cold rolling, to some extent, Shohet discriminant may relax the requirement of relative crown: where , ℎ denote the entry and exit strip thickness,   / and  ℎ /ℎ separately stand for relative crown of entry and exit,  is the strip width, and  = 2 or 1.86.
The main purpose of load distribution system is seeking a set of data ℎ  , which not only meets the Shohet equation, but also can fulfill those technological conditions.Meanwhile, in order to get the minimum value of objective function, the rolling force model, thickness distribution model, and the temperature model have to be established.

Rolling Force Model.
According to [1], the classic rolling force equation can be expressed as follows: where the subscript  denotes the rolling pass number, B is the strip width, and    denotes the horizontal projection length of contact arc between roll and workpiece.
, where R is roller radius,   is roller radius after deformation [12], Δℎ is the reduction for every rolling pass, ] is Poisson's ratio, and  is Young's modulus.Relative deformation degree and average thickness are denoted as  and ℎ  in (4).Deformation resistance introduces (5), where  = ( + 273)/1000 and t is rolling temperature. = ln(ℎ −1 /ℎ  ) and  = (V  /   ) separately stand for deformation degree and rate about workpiece.

Thickness Distribution Model. Consider the following:
where ℎ   is the experiential thickness value,  1 ,  2 denote the site statistics coefficient,   is cumulative energy distribution coefficient,  0 is initial thickness when workpieces go into the first finishing mill, and ℎ  is exit thickness when workpieces go through the last stand.

Temperature Model.
Temperature is an important factor in hot rolling, which can directly impact on the rolling force value of each pass.Equation ( 8) expresses temperature drop model from roughing exit to finishing entrance, while the next equation denotes slab temperature drop caused by going through finishing mill: 0 means entry temperature when slab goes into the first finishing mill. is blackness,  is Boltzmann constant,  is density,  is specific heat capacity, and  is the time when strip is transferred from the exit of roughing mill to the entrance of the finishing mill. RC is steel temperature after strip going through roughing mill.
The exit temperature of each finishing mill is denoted as   .  is water spray temperature between mills.  is cooling coefficient, the interstand distance (  ) is indicated, and ℎ  V  is the product that multiplies exit thickness by rolling speed.

Evolutionary Algorithm for Complex-Process Optimization
In this section, the evolutionary algorithm for complexprocess optimization based on differential evolutionary strategy (DEACOP) is proposed to solve load distribution problem of the hot rolling scheduling.The DEACOP is innovative strategy embedded in various submethods within the flexible.This algorithm improves path relinking to generate a new combination method which considers a broader area around the population members.Meanwhile DEACOP improves the balance between intensification and diversification with a population-update method.The above strategies can escape from suboptimal solutions and advance the search efficiency.The algorithm consists of five parts: (1) building the initial population, (2) determining similarity solution, (3) differential evolutionary strategy, (4) population update, and (5) deep search feasible solutions.Its principle is to deeply explore new population members near individuals with minimum fitness.The optimization process will be repeatedly executed unless the stop conditions were met.

Building the Initial Population.
In this subsection a latin hypercube uniform sampling (LHS) is first used to generate the initial population.To illustrate how LHS works, during the following description we will explain the building process of LHS.N is variable dimension that is set as 2; sampling size  is 10.The distribution procedure of LHS is as follows.
Through LHS procedure, an initial set Pop of Psize diverse vectors is generated, whose size is set as 10 × Nvar (Nvar is defined as a number of variables which need to be optimized).Meanwhile high-quality solution set Pop1 is composed in terms of better fitness.Its number is 1.Diversity set Pop2 (its size is b2) includes individual selected randomly from the remaining m−b1 vectors in Pop.According to the above completion strategy, the population size is  = 1+2.McKay et al. [14] pointed out that the total average received LHS than a simple random sampling mean has smaller total variance.

Check for Similarity Solution.
The purpose of similarity determining is to help escape from (possible) local optimal area.Algorithm 1 checks for duplicity with Euclid distance in the population before performing the next subsection (combination method).If the Euclidean distance between the two solutions is less than set value dist, then one of two solutions will be replaced with random solution within the search space.Otherwise, reserve two solutions and continue to the next judge.

Differential Evolutionary Strategy of Population.
In the traditional EACOP, the reference set was usually based on linear combination method, which has advantage in some aspects.Unfortunately, there is difficult to solutions of complex issue.Thus, an improved differential variation method was introduced as follows: where  +1  is individual set after variation,   1 is the current best individual populations, and  is scaling factor.(  1 −   ) item was used in this strategy in order to increase the algorithm coverage speed.Meanwhile (  2 −  3 ) was introduced as disturbance, which can make difference for each variation individual, therefore maintaining the population diversity.
Besides, the crossover strategy is used for better evolutionary effects.After the crossover operation on    and  +1  , thereby generating the new individual  +1  .The crossover strategy equation can be expressed as follows: where CR can be defined as crossover probability factor between 0 and 1 and rand is uniform random number in the same interval.

Population Update.
As described in the combination method, we incorporate each member of the reference set with the rest  − 1 members, resulting in  − 1 new solution.
Best quality solutions among new solutions were chosen and compared with their parent, if their value is better than the parent, and then the latter is replaced in the population.The principle of population regeneration is that new solutions are generated along with the path formed by the parent superrectangular.

Deep Search Feasible Solutions.
By the previous steps, the new population members are surrounded by hyperrectangle in accordance with update strategy.For enhancing the search intensification to exploit better feasible solutions, the evolutionary algorithm has implemented go-beyond strategy which consists in exploiting promising directions.Go-beyond strategy (Figure 2) means that new solution is created in the light of direction defined by the child and its parent (Algorithm 3).Deep search step is shown as follows: firstly, create a new solution  child in hyperrectangle which is generated by a pair of solutions (  ,   ) and estimate if fitness value ( child ) outperforms parent fitness value (  or   ) (  ,   is selected after deciding which of them is combined with other  − 1 population individuals).If ( child ) > (x  or x  ), new solution  Nchild is created according to go-beyond strategy over again.Now once more the program determines if ( child ) is greater than ( child ); if the result holds, determine update solution  child with go-beyond strategy again.In the end, last subsection  is introduced to make  child replace one of   or   .

Optimization Process of DEACOP.
According to description about above five parts subsection, all of subsections will be integrated to build an evolutionary algorithm.Optimization steps of DEACOP are shown as follows.
Step 1. Set initial parameters that include variable dimension vars,  size diverse vectors of the initial set (normally  size = 10 × vars), the number of high-quality solution b1, and random set size b2.Initial population whose size  is the sum of b1 and b2.To escape from suboptimal solution, the number of consecutive iterations  stick is defined as a vector. stick = [ 1 ,  2 , . . .,   ] = [0, 0, . . ., 0].  change is denoted as logo whether get suboptimal solution.
Step 2. This step uses a latin hypercube uniform sampling to generate initial set of diverse solutions.But the set should meet constraint condition about optimization problem.
Step 3. Check for similarity solutions.This step uses Algorithm 2 to test the diversity of population.
Step 4. Make differential evolutionary computation in reference to the actual individuals of concentration.Step 5. Associate population-update strategy with go-beyond strategy.If the quality of child  child that is generated with combination method outperforms its parent   , then gobeyond strategy is used to further exploit solution intensification.Otherwise, go-beyond strategy is not performed.
Step 6. Escape from suboptimal solution.If the parent   is replaced by the child, then the number of consecutive iterations   is reset as 0; otherwise   =   + 1, and estimate whether   ≥  change .If the result is affirmative,   will be substituted by the random one of the remaining  size −  members.
Step 7. Repeatedly perform Step 4-Step 6 until all members of population come through this process.
Step 8. Until stopping criterion is met, go into Step 3.
In order to clarify this algorithm procedure, a flow chart was shown in Figure 2.   The process of DEACOP algorithm optimization for load distribution of the hot rolling is shown as follows.
Step 1.The load distribution model considering flatness is established based on the actual production process parameters.
Step 2. First of all, thickness value ℎ   can be obtained according to experiential load distribution (6).Variables optimized are determined as Δℎ  , so exit thickness optimized is denoted as ℎ  = ℎ   + Δℎ  .
Step 3. Use DEACOP to optimize mathematic model of load distribution.According to constraint condition of modeling details, the process parameters which are calculated by optimizing variable must satisfy actual production requirements.
Step 4. Until stopping criterion is met, go back to Step 3.

Simulation and Discussion
. In this part, we have considered three methods for optimizing load distribution, including, experience distribution, EACOP, and DEACOP.
Meanwhile, the results generated through those algorithms were compared and analyzed.Under constraints conduction and objective function, as we can notice that top three stands' reduction must be as large as possible.For the desire of rolling force, the first stand rolling force  1 is expected as 90% of the second stand rolling force  2 , and  2 should be equal to the third stand rolling force  3 .In addition, the purpose of optimization should guarantee integrated performance of shape and gauge control system, so the object function desires that rolling force of the last four stands should be descended one by one, and relative crown remains consistent as far as possible.The calculation results about rolling force distribution as well as relative crown of each stand were simulated separately by Matlab Platform, and the results are shown in Tables 2 and 3.
According to reference with the constraints conduction and objective function in actual rolling process, Figure 3 shows optimization effect of rolling force.For top three stands, all the results optimized by DEACOP are greater than those optimized by EACOP and classic optimization.As for the empirical distribution, the conclusion cannot meet the characteristics of objective optimization function because  1 and  2 are not equal.Meanwhile, the last four rolling forces which empirical distribution configure cannot be in accord with objective function neither, while rolling force allocated by DEACOP and EACOP is in line with in turn reduced law.Besides, as we can see from Table 4, the relative crown of the last four mill stands almost maintains the same by DEACOP algorithm, which perfectly meets the demand on rolling schedule that the relative crown of the last four stands should be equal.
The thickness distribution of every stand is shown in Table 4. Since the values of DEACOP optimization fully meet the requirements of objective function based on gauge control system, a similar experiment was made to verify its better results in strip shape optimization curve; the data of DEACOP in Table 4 was curved with the consideration of Shohet discriminant criterion, which is useful to determine whether shape has met requirements.As shown in Figure 4, the relative crown difference between entrances and exits which is optimized by DEACOP does not exceed the scope   of moderate wave and edge wave and has a larger margin.Conclusions show that Shohet discriminant verifies the reliability of the experimental results.

Conclusions
In this paper, The DEACOP which has a flexible frame structure embedding in various submethods has been introduced.This algorithm was presented to optimize the rolling schedule and show its superior ability of global searching.Moreover, it can not only escape from suboptimal solutions, but also advance the search efficiency.
According to the experimental results within actual data in hot rolling process, the DEACOP still can get feasible and better mathematical solution and validate the real-time application even by fewer adjustable parameters, which is more suitable for the actual load distribution problems.With this algorithm, the optimized rolling schedule can make full use of the upstream finishing mill equipment which controls top three stands' reduction and improves the total rolling consumption.The rolling force of the last four stands which control exit thickness can be used as an important means of shape control.Therefore, the improvement of efficiency in plate-shaped regulating by DEACOP is recommended as an important issue for further investigation.
for the Central Universities (FRF-TP-12-104A, FRF-SD-12-008B).The authors heartily appreciate the data support from Ansteel Company in Liaoning Province, China.Meanwhile, great thanks also go to former researchers for their excellent work, which gives great help for our academic study.

for 𝑖 = 1
to b for  =  to b if  ( ⃗   , ⃗   ) =      ⃗   − ⃗        ≤  then one of two solutions will be replaced with random solution within the search space.else until the end of the loop endif endfor endfor Algorithm 1: Check for solutions similarity algorithm.

Figure 3 :
Figure 3: Comparison of rolling force distribution.

Table 1 :
Set parameters for optimization.Set Initial Parameters and Optimization Steps.In order to validate the effectiveness of DEACOP optimization for load distribution of the hot rolling, Q235 Steel was used for simulation experiments.The parameters load distribution was listed in Table1, including the width of strip steel B, initial thickness of workpiece  0 , finish product thickness h  , exit temperature of roughing mill  RC , objective crown CR  , and the number of stands .Moreover, parameters in DEACOP include the number of population b (including the number of high-quality solutions 1 and the number of random solutions 2), the size of initial set  size , the number of dropping into suboptimal region  change , radius of suboptimal region r, and iteration times of optimization ITTM.

Table 2 :
Comparison of rolling force distribution.

Table 3 :
Relative crown of each stand.

Table 4 :
Thickness distribution of each stand.