(215f) Active Constraint Regions for Economically Optimal Operation of Distillation Columns

Control and operation of distillation columns has been widely studied, as
illustrated by for example Skogestad (1993). Normally, it is assumed that the
task is to control the product compositions at given setpoints (”two-point control”).
However, in many cases it may be economically optimal to overpurify
one of the products, and it is then no longer clear that two-point control is the
best.
Somewhat surprisingly, the issue of optimal economic operation of distillation
columns has been studied relatively little. The issue of active constraints
for economically optimal operation was addressed about 40 years ago by (Maarleveld
and Rijnsdorp, 1970), but they considered the extra degrees of freedom
related to column pressure and preheating, and assumed two-point composition
control. (Gordon, 1986) studied economical optimal operation and how
it relates to product purity control. He introduced the avoid product giveaway
rule, which states that it is optimal to maximize the flow of the more expensive
product by keeping the amount of impurities in that stream at the maximum
allowed (Gordon, 1986). The implication is that that one should always control
the purity of the expensive product (active constraint rule). In this paper, we
discuss this rule and point out its possible limitations.
When seeking to operate any plant in an economically optimal way, it is
often critical to know which operational constraints are optimally active. The
main focus of the paper is on showing how to identify active constraint regions
as function of disturbances, through:
• Using knowledge of the optimization problem and of the behavior of distillation
columns.
• Finding points in the disturbance space where individual constraint functions
change between active and inactive, by finding where the sum of
each constraint and its corresponding Lagrange multiplier is zero.
The second is used as an alternative to optimizing the model at a large
number of points all over the disturbance space, thus decreasing the need for
repeated optimizations.
We have identified how the active constraint set varies with the disturbances
(feed flow rate and energy cost) for a single two-product column (with fixed
product prices, case Ia) and with purity-dependent distillate price (case Ib))
and for two distillation columns in sequence (case II).
The results of this work serve to illustrate the following:
1. The complexity of the active constraint regions even for a relatively simple
problem
2. How process knowledge can be used to reduce the need for (potentially
time-consuming) optimization
3. Show when the avoid product giveaway rule is valid
For case Ia, we found that we had three distinct regions with different active
sets. In case Ib we identified five regions, and in case II we identified eight.
Only in case Ib did we find a region in which no constraints were active, which
was in keeping with the avoid product giveaway rule. We showed that the
borders between regions where only purity constraints are optimally active, are
independent of feed flow rate.
We have also explained how the avoid product giveaway rule depends on
two assumptions, of which one (one gets paid for the entire product stream, not
just the desired component) is often valid and the second (overpurification costs
energy) is always valid for realistic distillation cases.

References
Gordon, L., 1986. Simple optimization for dual composition control. Hydrocarbon
Process.;(United States) 65 (6).
Maarleveld, A., Rijnsdorp, J., 1970. Constraint control on distillation columns.
Automatica 6 (1), 51 – 58.
Skogestad, S., 1993. Dynamics and control of distillation columns – a critical
survey:. Control Engineering Practice 1 (3), 564 – 564.