Optimal control of an aluminum casting furnace: Part I. The control model

Abstract

This is a series of two articles on the control of an aluminum casting furnace to bring a mass of liquid aluminum from a known initial temperature to a desired final temperature in a given time with minimal fuel cost. An analytic model of the furnace already exists but is too complex for control purposes. Here in Part I, a simplified nonlinear control model is derived from the analytic model. In Part II, an optimization of the fuel flow is performed on the control model using Pontryagin’s maximum principle. The first article shows that despite the complexity of the analytic model, a tenth-order nonlinear control model with good representativity can be obtained. The second article shows that the maximum principle applied to this problem leads to a solution with optimal fuel cost. If modeling industrial processes is a complex problem in itself, obtaining a control model therefrom is just as delicate. This series of articles proposes an approach that works for the casting furnace and is indeed applicable to other industrial processes as well. In the existing analytic model, the casting furnace is treated as two one-dimensional conducting media (metal and refractories), while its chamber is seen as a well-stirred reactor. In this article, a control model is derived therefrom by a statistical method. The analytic model is run several times to obtain a set of predicted data on which a least-squares approximation is performed to determine the best parameter values to be used for the control model equations. The conduction equations in the two media are linear. The expressions for heat generation in the chamber and for radiative-convective heat transfer from the chamber to the two media are nonlinear and are kept to ensure maximum representativity. The result is a highly representative tenth-order control model, the degree of representativity being assessed by comparing the temperature outputs and the energy balances obtained from the analytic model with those obtained from the control model.