The inevitable presence of uncertain parameters in critical applications of process optimization can lead to undesirable or infeasible solutions. For this reason, optimization under parametric uncertainty was, and continues to be a core area of research within Process Systems Engineering. Multiparametric programming is a strategy that offers a holistic perspective for the solution of this class of mathematical programming problems. Specifically, multiparametric programming theory enables the derivation of the optimal solution as a function of the uncertain parameters, explicitly revealing the impact of uncertainty in optimal decision-making. By taking advantage of such a relationship, new breakthroughs in the solution of challenging formulations with uncertainty have been created. Apart from that, researchers have utilized multiparametric programming techniques to solve deterministic classes of problems, by treating specific elements of the optimization program as uncertain parameters. In the past years, there has been a significant number of publications in the literature involving multiparametric programming. The present review article covers recent theoretical, algorithmic, and application developments in multiparametric programming. Additionally, several areas for potential contributions in this field are discussed, highlighting the benefits of multiparametric programming in future research efforts.
In this paper, we develop an energy-focused model of an industrial roller hearth heat treating furnace. The model represents radiation heat transfer with nonparticipating gas and convective heat transfer. The model computes the exit temperature profile of the treated steel parts and the energy consumption and efficiency of the furnace. We propose a dual iterative numerical scheme to solve the conservation equations and validate its efficacy by simulating the dynamics of the furnace during startup, as well as for steady-state operation. A case study investigates energy consumption within the furnace under temperature control. We first consider a heuristic control strategy using simple linear controllers. A response surface approach is then used to find the optimal set points that minimize energy consumption while ensuring desired part temperature properties are met when processing is complete. With optimized set points, 4.8% less energy per part is required versus the heuristic set points.
Quench hardening is the process of strengthening and hardening ferrous metals and alloys by heating the material to a specific temperature to form austenite (austenitization), followed by rapid cooling (quenching) in water, brine or oil to introduce a hardened phase called martensite. The material is then often tempered to increase toughness, as it may decrease from the quench hardening process. The austenitization process is highly energy-intensive and many of the industrial austenitization furnaces were built and equipped prior to the advent of advanced control strategies and thus use large, sub-optimal amounts of energy. The model computes the energy usage of the furnace and the part temperature profile as a function of time and position within the furnace under temperature feedback control. In this paper, the aforementioned model is used to simulate the furnace for a batch of forty parts under heuristic temperature set points suggested by the operators of the plant. A model predictive control (MPC) system is then developed and deployed to control the the part temperature at the furnace exit thereby preventing the parts from overheating. An energy efficiency gain of 5.3% was obtained under model predictive control compared to operation under heuristic temperature set points tracked by a regulatory control layer.
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