Håkon Dahl-Olsen, Sridharakumar Narasimhanand 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

yh(x)

minu(t ),t f

J(x(t f ))

x f (x,u)c(x,u)0u(t)U[0,T ]x(t f )X

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

H (t)T f (x,u)

L(t)H (t) H *(t)

Maximum gain rule

L(t)Huu

0

1

2uTHuuu

Need to relate variations in inputs to variations in outputs

yhxT

xuT

u Gu

L(t)1

2yTG THuuG

1y

Maximum gain rule for dynamic optimization

•Assume the model is scaled such that δymax≈1

min G THuuG 1

2

2

t0

t f

dt 1/2Select 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

G hxT

K

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

BioreactorX

O2

S X+P

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.

Example

X < 3.7

Biomass concentration

SSubstrate

concentration

PProduct

concentration

V Volume

u < 1Substrate feedrate

BioreactorX

O2

S X+P

u [L/h]Sin=200 g/L

ControlleControllerr

Measurements

Example

X < 3.7

Biomass concentration

SSubstrate

concentration

PProduct

concentration

V Volume

u < 1Substrate feedrate

X (S)X u

VX

S (S)XYX

XYP

u

VSin S

P X u

VP

V u

BioreactorX

O2

S X+P

Biomass growth rate

0

0.005

0.01

0.015

0.02

0 0.5 1

Substrate (g/L)

BioreactorX

O2

S X+P

(S)MS

KM S S2 / Ki

Consider gain magnitude

•Transformation of input: ξ=√u

•Hξξ is constant

•Minimizing of 1/K2 corresponds to maximizing K2

BioreactorX

O2

S X+P

S

Comparison of gains

XPV

1.0E-03

1.0E+02

1.0E+07

0 20 40 60 80

Time

K

BioreactorX

O2

S X+P

Simulation results

Summary

•Maximum gain rule extended to dynamics

•Variational gain from neighboring optimal control theory

•Method works well for a small case study