Open-loop optimal control of batch chromatographic separation processes using direct collocation

This contribution presents a novel model-based methodology for open-loop optimal control of batch high-pressure liquid chromatographic (HPLC) separation processes. The framework allows for simultaneous optimization of target component recovery yield and production rate with respect to a parameterization of the input elution trajectory and fractionating interval endpoints. The proposed methodology implies formulating and solving a large-scale dynamic optimization problem (DOP) constrained by partial differential equations (PDEs) governing the multi-component system dynamics. It is based on a simultaneous method where both the control and state variables are fully discretized in the temporal domain, using direct local collocation on finite elements, and the state variables are discretized in the spatial domain, using an adaptive finite volume weighted essentially non-oscillatory (WENO) scheme. The direct transcription of the DOP described by Modelica, and its extension Optimica, code into a sparse nonlinear programming problem (NLP) is thoroughly presented. The NLP was subsequently solved using CasADi's (Computer algebra system with Automatic Differentiation) interface to the primal-dual interior point method IPOPT. The advantages of the open-loop optimal control strategy are highlighted through the solution of a challenging ternary complex mixture separation problem of human insulin analogs, with the intermediately eluting component as the target, for a hydrophobic interaction chromatography system. Moreover, the high intercorrelation between the shape of the optimal elution trajectories and the fractionation interval endpoints is thoroughly investigated. It is also demonstrated that the direct transcription methodology enabled accurate and efficient computation of optimal cyclic-steady-state solutions, which govern that state and control variables conform to periodicity constraints imposed on column re-generation and re-equilibration. By these means, the generic methods and tools developed here are applicable to continuous chromatographic separation technologies, including the continuous simulated moving bed (SMB) and the multicolumn counter-current solvent gradient purification (MCSGP) process