Optimal output selection for batch processes

1
Optimal output selection for batch processes
Håkon Dahl-Olsen, Sridharakumar Narasimhan and
Sigurd Skogestad
2
Outline

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

3
Batch optimization

• Minimum time to given specification
• Maximum product in fixed time

4
Dynamic optimization
5
Implementation

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

6
Variable selection

• Unconstrained degrees of freedom
• Based on Pontryagins minimum principle
• Look for variables which give small deviation
  from optimality when controlled at fixed
  reference, even under disturbances

7
Maximum gain rule

• Loss is defined in terms of value of the
  Hamiltonian
• The loss is time-varying

8
Maximum gain rule
Need to relate variations in inputs to variations
in outputs
9
Maximum gain rule for dynamic optimization

• Assume the model is scaled such that dymax1

Select ys to minimize the following expression
along the nominal trajectory
10
How to obtain G

• How does variations in inputs map to the states?
• Neighboring optimal control gives du(t)K(t)dx(t)
• We estimate G by

11
Example
Bioreactor

• 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

Srinivasan, B., D. Bonvin, et al. (2002).
"Dynamic optimization of batch processes II. Role
of measurements in handling uncertainty." Comp.
chem. eng. 27 27-44.
12
Example
Bioreactor
u L/h Sin200 g/L
X lt 3.7 Biomass concentration
S Substrate concentration
P Product concentration
V Volume
u lt 1 Substrate feedrate
Controller
Measurements
13
Example
Bioreactor
X lt 3.7 Biomass concentration
S Substrate concentration
P Product concentration
V Volume
u lt 1 Substrate feedrate
14
Biomass growth rate
Bioreactor
15
Consider gain magnitude
Bioreactor

• Transformation of input ?vu
• H?? is constant
• Minimizing of 1/K2 corresponds to maximizing K2

16
Comparison of gains
Bioreactor
S
X
P
V
17
Simulation results
18
Summary

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