Håkon Dahl-Olsen, Sridharakumar Narasimhan and Sigurd Skogestad Optimal output selection for batch processes

Outline Batch optimization Implementation schemes Variable selection method Reactor case study Summary

Batch optimization Minimum time to given specification Maximum product in fixed time

Dynamic optimization

Implementation Online optimization: measurements used to update model Self-optimizing control: good outputs give near optimal performance

Variable selection Unconstrained degrees of freedom Based on Pontryagin’s minimum principle Look for variables which give small deviation from optimality when controlled at fixed reference, even under disturbances

Maximum gain rule Loss is defined in terms of value of the Hamiltonian The loss is time-varying

Maximum gain rule Need to relate variations in inputs to variations in outputs

Maximum gain rule for dynamic optimization Assume the model is scaled such that δy max ≈1 Select y’s to minimize the following expression along the nominal trajectory

How to obtain G How does variations in inputs map to the states? Neighboring optimal control gives δu(t) = K(t)δx(t) We estimate G by

Example Maximize production of product P in a fed-batch bioreactor with fixed final time of 150 hours Reaction is driven by the presence of a substrate S, which is consumed in the biomass generation The biomass concentration is constrained Bioreactor Srinivasan, B., D. Bonvin, et al. (2002). "Dynamic optimization of batch processes II. Role of measurements in handling uncertainty." Comp. chem. eng. 27:

Example X < 3.7 Biomass concentration S Substrate concentration P Product concentration VVolume u < 1 Substrate feedrate Bioreactor u [L/h] S in =200 g/L Controller Measurements

Example X < 3.7 Biomass concentration S Substrate concentration P Product concentration VVolume u < 1 Substrate feedrate Bioreactor

Biomass growth rate Bioreactor

Consider gain magnitude Transformation of input: ξ=√u H ξξ is constant Minimizing of 1/K 2 corresponds to maximizing K 2 Bioreactor

S Comparison of gains X P V Bioreactor

Simulation results

Summary Maximum gain rule extended to dynamics Variational gain from neighboring optimal control theory Method works well for a small case study