A Novel Approach to Dynamic Optimization of ODE and DAE Systems as High-Index Problems

Solution of many problems in plant operations requires determination of optimal control profi les subject to state constraints for systems modeled by ordinary differential equations (ODEs) or dif ferential-algebraic equations (DAEs). For example, optimal temperature and/or feed rate profiles are important for the oper ation of many batch reactions. Similar observations apply to reflux policies for batch distillation, and feedstock changeover in oil refineries. Currently there are two different classes of methods for determining optimal control profiles for DAEs. Control parameterization techniques rely on the discretization of the control variables to reduce the optimal control problem to an NLP. These methods require repeated integration of the DAEs and some variational equations, which effectively discretizes the state variables within the numerical integrator. Path constraints are typically handled by the master NLP solver, and can force the NLP solver to call for a large number of DAE integrations. Collocation methods reduce the optimal control problem to an NLP by the discretization of both control and state variables. This has the advantage that expensive numerical integrations can be avoided, but stiffness of the DAEs can result in ill-conditioned NLPs. Collocation methods also result in large highly nonlinear NLPs, which are not easily solved using current NLP methods. In this paper we show that dynamic optimization problems with equality path constraints on state variables give rise to high-index DAEs. This has implications for both the collocation and control parameterization methods. In control parameterization, the requirement that the DAEs integrated have index ≤ 1 means that other constraints must be handled as penalties in the master NLP, leading to more degrees of freedom in the NLP than truly exist in the problem. In collocation, proofs that discretization formulations are correct exist only for linear DAEs and index ≤ 2 nonlinear DAEs. Recently, the method of dummy derivatives was described for solving high-index DAEs. Using this method, we may derive symbolically an equivalent index-1 DAE that has the same solution set as the high-index DAE and may be solved with standard DAE codes. We describe our improvements to the dummy derivative method and show how the method may be applied to improve the solution of dynamic optimization problems using both the control parameterization and collocation methods.