Simulation of differential-algebraic equation systems with optimization criteria embedded in Modelica

Abstract

Differential-algebraic equations with embedded optimization criteria (DAEO) are a class of mathematical models for underdetermined differential-algebraic equation (DAE) systems with less algebraic equations than algebraic variables. The algebraic variables may be calculated as the solution of an embedded (non)linear program, yielding a DAEO system. An example for DAEOs is the dynamic flux balance analysis (DFBA) approach, where the formulation of metabolic reaction networks leads to an underdetermined equation system for the intracellular fluxes that are assumed to behave optimally with respect to some cell-specific optimization criterion.We present a toolbox that allows formulation of DAEOs in the object-oriented Modelica modeling language. The solution method is based on substituting the embedded optimization problem with its first-order Karush-Kuhn-Tucker conditions to obtain a nonsmooth DAE system that can be simulated by a root-finding DAE solver. One nonlinear example and two examples based on DFBA demonstrate the performance of the toolbox.