Multiparametric Programming in Process Systems Engineering: Recent Developments and Path Forward

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.

Fault Detection in Wastewater Treatment Systems Using Multiparametric Programming

In this work, a methodology for fault detection in wastewater treatment systems, based on parameter estimation, using multiparametric programming is presented. The main idea is to detect faults by estimating model parameters, and monitoring the changes in residuals of model parameters. In the proposed methodology, a nonlinear dynamic model of wastewater treatment was discretized to algebraic equations using Euler’s method. A parameter estimation problem was then formulated and transformed into a square system of parametric nonlinear algebraic equations by writing the optimality conditions. The parametric nonlinear algebraic equations were then solved symbolically to obtain the concentration of substrate in the inflow, , inhibition coefficient, , and specific growth rate, , as an explicit function of state variables (concentration of biomass, ; concentration of organic matter, ; concentration of dissolved oxygen, ; and volume, ). The estimated model parameter values were compared with values from the normal operation. If the residual of model parameters exceeds a certain threshold value, a fault is detected. The application demonstrates the viability of the approach, and highlights its ability to detect faults in wastewater treatment systems by providing quick and accurate parameter estimates using the evaluation of explicit parametric functions.