Prediction of optimal operating parameters for continuous casting of billets

Process optimization determines process parameters that maximize or minimize (optimize) so aspect of a process (the obieciive function), while ensuring that the process operates wr established limits. In this work a mathematical model that simulates heat flow and solidification continuously cast steel strand is coupled with mathematical optimization techniques to predict opt process parameters lor several aspects ol the continuous casting process The optimizations constrained so that representative process constraints are enlorced A description ol the model optimization method, and the means ol coupling are presented The tormulaiion ol obieciive tune and constraints tor continuous casting ol billets and the predictions resulting iioni optimizing t lormuiattons is also discussed Introduction The continuous casting process currently accounts for more than 50% of total world crude steel production 1 Many applications require steel of a quality level only obtainable through continuous casting The productivity of a continuous casting operation and the quality of the resulting product are largely dependent on the casting parameters used during the casting process. The operating parameters for the continuous casting process need to be chosen so that a predetermined balance between productivity, product quality and operating costs is optimized. The selection of optimal operating parameters becomes even more important as the use of direct charging of hot strands to rolling operations becomes more prevalent. The temperature distribution and total heat content in the strand must be closely controlled in order to roll high quality products. The problem of selecting continuous casting process parameters that optimize some function ol the caster operating state falls within the Iramework of problems known as constrained optimization problems. We desire to optimize (maximize or minimize) an objective function (a function used to determine if one operating state is more or less desirable than another), while ensuring that constraints that represent physical limits on the process are obeyed. The casting process is represented by a mathematical model, for reasons of cost and convenience and to allow the optimization process to proceed to the optimal point by paths that may include infeasible states (operating states where one or more of the process constraints are violated). In this work, only heat flow aspects of the continuous casting process are considered, although extensions to stress/strain and other aspects can certainly be made within the Iramework presented here. All of the objective and constraint functions are therefore stated in terms of temperature fields and thermal behavior. The relationships between the objective function, constraint values and process variables are available only through the use of a numerical heat transfer simulation for a continuous caster. These relationships are nonlinear, hence the optimization problem is a Non-Linear Program (NLP). An optimization technique known as Successive Quadratic Programming(SOP) has been used in this study to solve the constrained NLP problems. This technique was chosen because it typically requires fewer function evaluations than other NLP methods and function evaluations (model simulations) have been lound to be quite expensive in terms of both real time and computer time. The SOP algorithm can be derived from a Newton-Raphson approach applied to the optimally conditions for the nonlinear programming problem. Numerous applications of this method have been made to chemical process optimization problems (see the paper by Biegler for a review) and currently it is the algorithm of choice for solving moderately sized optimization problems based on computationally intensive models. In addition, SOP has excellent constraint handling features and requires only function and gradient information from the process model. Based on the implementation of Biegler and Cuthrell the optimization algorithm is relatively straightforward to apply to general purpose optimization problems with smooth objective and constraint functions. Previous attempts at applying nonlinear constrained optimization techniques to continuous casting processes have been lew In the work by Larrecq. et.al., a detailed list of process operation and product quality constraints is presented. A gradient method is used to minimize a cost function that represents violated constraints, at constant casting speed, and then the casiing speed is manipulated manually until a maximum casting speed is found. Holappa. et.al.. have used a similar method 8 Neither group has allowed the casting rate to be a variable in the optimization process even though the casting rate has an extremely important effect on the temperature distribution and metallurgical structure of the cast product. The optimization system is comprised mainly of two parts, the model and the optimizer. These are shown schematically in Figure 1 The model is further subdivided into a part that calculates the temperature field in a continuous caster and a part that uses the resulting temperature field to calculate 1056 Figure 1: Schematic Representation ol Optimization System Figure 2: Schematic Representation of Slice Modelling Technique