(545c) Bounding the Solutions of Chemical Kinetics Models Using ODEs with Linear Programs Embedded

A method is presented for the computation of rigorous bounds enclosing all solutions of chemical kinetics models with uncertain parameters and/or controls. In many chemical processes, there are uncertain parameters that can lead to a wide range of possible process behaviors. Given a range of possible parameter or control values, one often wishes to determine the range of possible states that results, called the design envelope. It is similar in nature to related concepts such as the "reachable set". Rigorously bounding the design envelope is an important task in safety verification and global dynamic optimization of these processes.

This work will focus on bounding the design envelope of a batch reactor subject to uncertainty. The main assumption is that the parameters or controls have known upper and lower bounds. Mathematically, this problem reduces to bounding the set of possible solutions of an initial value problem in parametric ordinary differential equations (ODEs). Recent work has shown that tight component-wise upper and lower bounds can be computed by formulating a related ODE system which enforces physical information that might not be explicit in the model. Based on arguments involving differential inequalities, the resulting ODE system depends on the solutions of "embedded" parametric optimization problems. The physical information that is enforced takes the form of an invariant set, or a set which the states cannot leave at any time or for any parameter/control value. The invariant set restricts the feasible set of the optimization problems.

In chemical kinetics models, the invariant set is typically a convex polyhedron. This is because mass conservation arguments typically furnish upper and lower bounds that must hold for all time and parameter values. Meanwhile the states must obey affine relations due to stoichiometry. By taking affine relaxations of the objective functions of the embedded optimization problems, one obtains an ODE with linear programs (LPs) embedded.

A previously developed algorithm has been extended to handle further parametric dependence that occurs in the embedded LPs. This algorithm reformulates the system as differential algebraic equations, which leads to a robust and efficient way to solve numerically the resulting system of ODEs with LPs embedded. The resulting bounds are tight.