Dynamic Parallel Optimization For Secondary Cooling Water Flow In Continuous Casting Under Change Of Casting Speed Based On Transient Nonlinear Convection Heat Transfer Equation

The physical processes of heat conduction are widely found in the metallurgical industry,the chemical industry,and the manufacturing process of manufacturing,for example,slab heating,slab solidification,fluidized bed reaction,glass cooling,etc.The quality requirements and process constraints for products containing heat transfer processes constitute the objective function and constraints for the optimization problem.Unlike the traditional optimization problem,the thermal conduction physical process is described by the parabolic partial differential equation,which leads to the optimization problem has the partial differential equation constraints.Taking the solidification process of the slab in the continuous casting process as an example,the slab is passed through the mold and the supporting device of secondary cooling zone.Meanwhile,cooling water is performed on the surface of the slab to finally completely solidify the liquid molten steel into a slab.The temperature field dynamics of the slab can be described by a parabolic partial differential equation.The cooling process of the slab is realized by the secondary cooling zone control system.The secondary cooling zone water control system is composed of a water flow setting subsystem and a secondary cooling zone water flow control subsystem.The water flow setting subsystem gives the set value of the water flow distribution in each section of the secondary cooling zone,and then the secondary cooling water flow control subsystem adjusts the water flow with reference to the set value,and then the cooling water spray nozzles jet the cooling water on the slab.Only when the water flow setting subsystem gives a reasonable water flow setting value,the secondary cooling water flow control subsystem can achieve better tracking control of the set value.At present,most of the methods for setting the water flow belong to the parameter water distribution method,which can provide a reasonable water flow setting value,when the casting speed is not changed.However,when casting speed is changed,the water flow setting value given by the parameter water distribution method has a large variation of the surface temperature of the slab,which may result in the occurrence of quality defects such as cracks.How to give a reasonable water set value when the casting speed changes is an urgent problem to be solved in the current production of slab.In this paper,a dynamic optimization method for water flow in continuous casting secondary cooling zone based on partial differential equation constrained optimization problem is proposed.This method can give a reasonable water set value when the casting speed changes.The work of this paper is summarized as follows:First,nonlinear 3D partial differential equation can describe the heat transfer process of a continuous casting billet with latent heat of solidification.For this reason,3D nonlinear partial differential equations constrained dynamic optimization problem is studied.According to the metallurgical guidelines,this paper establishes a model for the dynamic optimization problem.This dynamic optimization problem has the following difficulties:(1)The coefficients of the boundary conditions of the nonlinear partial differential equations are inaccurate and need to be identified by measurement temperature data.The measured temperature data is affected by the oxide of slab,which results in the measurement temperature data containing outliers with non-white noise.The traditional least squares method only works well when the measurement data contains white noise data.How to make full use of these data with outliers for continuous casting heat transfer parameter identification is an urgent problem to be solved.(2)Nonlinear 3D partial differential equation constrained dynamic optimization problem cannot obtain explicit expressions of the objective functional derivatives.Therefore,the Jacobian matrix is approximated by numerical derivatives,which causes the large computational task.The variation of the casting speed causes the change of the temperature in about 1 minute,which leads to the traditional method does not meet the real-time requirement for the change of the casting speed during the continuous casting production process.How to reduce its calculation time is an urgent problem to be solved?To this end,the following work is carried out:(1)Based on weighted least squares,a parameter identification model is established.The credibility of each measurement data is given by kernel function estimation method,and then the weights are determined according to the credibility of the data.The impact of outliers on parameter identification accuracy can be reduced.(2)This paper proposes an improved stochastic gradient algorithm based on sparse Jacobian matrix.Due to the characteristics of continuous casting production,it can be considered that the amount of water in the i-th section of the second cooling zone affects only the temperature of the second cooling zone of the i-th segment and the temperature of the secondary cooling zone of the i+1 section for a short period of time.Therefore,the main diagonal and the low diagonal(the diagonal below the main diagonal)of the Jacobian matrix are not zero.Using this property,a new perturbation factor is designed.The entire Jacobian matrix can be obtained and the solution times of heat transfer model is from N+1(N is the number of secondary cooling zone)to 3.Simulation results show that the method can reduce the calculation time by about 60%.This study combines the idea of stochastic gradient algorithm with sparse Jacobian matrix computation.In each process of gradient iteration,it is not necessary to calculate the complete Jacobian matrix,and only a part of the elements of the Jacobian matrix are randomly sampled to perform the iterative process of gradient descent.Theoretically,it can be proved that the upper bound of the error of the solution and the true optimal solution can be reduced without increasing the amount of calculation.Second,linear 3D parabolic partial differential equation can describe the continuous heat transfer process without the latent heat of solidification.Since the nonlinear 3D partial differential equations constrained dynamic optimization problem is very sensitive to the initial solution,this paper simplifies the problem into a linear 3D parabolic partial differential equation constrained optimization problem without the latent heat.The solution result is used as the initial solution to the nonlinear dynamic optimization problem of nonlinear 3D partial differential equations.For this purpose,the problem of constrained dynamic optimization for linear 3D parabolic partial differential equation with source and robin boundary conditions is studied.This kind of dynamic optimization problem is essentially an optimal control problem with partial differential equation constraints.The difficulty is that the Lipschitz continuity of the derivative of the objective function is a prerequisite for the convergence of the numerical optimization algorithm,and with the source term and robin boundary condition.The Lipschitz continuity of the objective function derivative of the optimal control problem for 3D parabolic partial differential equation is still difficult to prove theoretically,which makes the application of numerical optimization algorithms on this kind of problem lack of theoretical guarantee.Most of the traditional proofs are for 1D or 2D,and do not contain source terms and robin boundary conditions.To this end,the following work is carried out:(1)Starting from the optimality condition,the extended Cauchy-Schwartz inequality is used to prove Lipschitz continuity with the robin boundary condition and source term situations.(2)Using the variational method to derive the optimality condition of the optimal control problem for linear 3D partial differential equation,then design a conjugate gradient algorithm with sufficient descent to solve the optimal control problem according to the optimality condition.The convergence of the algorithm is proved by the Lipschitz continuity.Third,the continuous casting secondary cooling zone model predictive controller needs to solve with a smaller time granularity(about 10 seconds),while the traditional serial computing based computing framework cannot meet the real-time requirements.The main difficulties are(1)the numerical calculation of 3D nonlinear heat conduction equation,which has millions of temperature variables.The calculation of 3D nonlinear partial differential equations is very huge task.The optimization algorithm needs to calculate the Jacobian matrix to solve the dynamic optimization problem,and the calculation of the Jacobian matrix requires to solve 3D nonlinear partial differential equations,repeatedly,which further leads to slower calculation.(2)The traditional GPU-based model predictive control solution strategy separates the parallelization of the optimization algorithm and the parallelization of the simulation model.It does not consider the parallel level of different computing tasks is different.Because the GPU is used for calculation,data transfer between GPU and CPUs is inevitable,how to reduce the data transfer time between the GPU and the CPU is a key issue.To this end,the following work is carried out:(1)In order to overcome the problem of large computational complexity of 3D nonlinear partial differential equations in continuous casting secondary cooling zone,a parallel solution method for 3D nonlinear partial differential equations is designed.First,the explicit difference is used to discretize the 3D nonlinear partial differential equation,and then the computational task of each node is allocated to each thread.Millions of threads can fully utilize the powerful multi-thread parallel processing capability of the GPU to accelerate the calculation of 3D nonlinear partial differential equations and reduce the computational time of 3D nonlinear partial differential equations from 389.67s to 4.84s.(2)In order to combine the parallel model and the parallel optimization algorithm organically,consider the different level of different tasks.Therefore,this paper proposes a two-level parallel model predictive control solution method based on GPU stream parallelism.The sparse Jacobian calculations task is divided into three streams,which makes the data transfer and calculation time overlap each other.Finally,the GPU-based two level improved stochastic gradient algorithm for solving the secondary cooling water MPC problem can reduce the calculation relative time(the time used for calculation/the time of the actual physical process simulated)to 0.023.The application in the optimization problem provides a strong guarantee.