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