(635g) Robust Optimization for Chemical Process Design and Applications to Carbon Capture Technology

Robust Optimization for Chemical
Process Design and Applications to Carbon Capture Technology

Natalie M. Isenberg^a, Paul
Akula^b, Debangsu Bhattacharya^b, David C. Miller^c,
Chrysanthos E. Gounaris^a

^a Dept. of Chemical Engineering,
Carnegie Mellon University

^b Dept. of Chemical and Biomedical Engineering,
West Virginia University

^c The National Energy Technology
Laboratory

Data in mathematical optimization models is
often subject to some level of uncertainty. This uncertainty originates from uncertain
system properties (e.g., kinetics, heat/mass transfer), uncertain operating
conditions (e.g., variability in feedstocks) or market stochasticity (e.g.,
product prices). In the context of process systems engineering, where critical
design and control decisions are made via solving models that are subject to
such uncertainties, it is especially important to understand the effects of parametric
uncertainties on the performance of the chosen solutions, and if significant,
to mitigate uncertainty during the optimization phase. Robust optimization (RO)
is one approach in the mathematical programming literature for formulating and
solving risk-averse models.

There is a breadth of literature
investigating robust optimization in the context of linear and convex models [1,2].
However, chemical processes typically possess complex nonlinearities and a
large amount of non-convexity originating from physical and chemical state
equations. This means that traditional reformulation methods in the RO
literature may lead to either overly conservative solutions (e.g., due to
duality gaps) or robust infeasible solutions (e.g., due to convergence to KKT
points that are only locally optimal) [3]. To address limitations of this
nature, a local search algorithm has been proposed in the literature for
identifying robust feasible solutions to uncertain optimization problems with
non-convex inequality constraints [4]. Additionally, methods for formulating
robust counterparts of nonlinear process models that induce robust feasibility
within a small neighborhood of uncertain parameter realizations have also been
proposed [5]. However, there remains a need to develop general approaches that
can identify robust solutions in nonlinear, non-convex process models that
consist mostly of equality constraints or state equations, as is the case for chemical
process systems.

Here, we extend the robust cutting-set proposed
by Mutapcic and Boyd, which sequentially hedges against realizations of
parametric uncertainties [6]. We note that the original cutting-set method does
not consider optimization models containing both nonlinear inequality and
equality constraints. Therefore, our proposed generalized robust cutting-set
method handles the case where an optimization model consists of mostly state
equations that cannot be readily simplified or solved out of the formulation,
and it is also valid for models with nonlinearities and non-convexities in both
the decision variables and uncertain parameters.

The original robust cutting-set algorithm proposes
to maintain copies of state variables and equations for each iteration to
ensure the feasibility of the master problem. However, since copying state
variables and constraints causes model sizes to grow quickly, this approach can
rapidly become intractable for all but the smallest of process systems
engineering models. In order to counteract this, the generalized robust
cutting-set method is coupled with a new approach for handling state variables
via non-linear decision rules, as seen in the area of adjustable robust
optimization [7]. First, a general, non-linear decision rule is postulated for
all (or some) of the state variables, and the latter are replaced within the
optimization model with their non-linear decision rule dependence. After
completing a coefficient-matching procedure and solving the set of resulting
equations for the homogeneous solution, the resulting set of state equations features
significantly fewer equations referencing uncertain parameters. Therefore, the
generalized robust cutting-set algorithm requires significantly fewer
iterations to converge, improving its overall tractability.

We test the performance of the proposed
approach in the context of a highly complex, equation oriented post-combustion
carbon capture (PCC) process design model. The process represents a
monoethanolamine solvent-based CO₂ capture system. The main
components of the process model pertain to the absorber and regenerator column
units. These columns are modeled with dynamic mass and energy balances,
embedding detailed mass and heat transfer relations as state equations. These
include a two-film theory-based mass transfer between phases, as well as
complex solvent equilibria. The models of chemical and physical properties in
this system are known to contain significant uncertainties in mass transfer,
interfacial area, and viscosity parameters [8,9]. Therefore, there is a need to
identify optimal solutions to the PCC process design problem that are robust
against the range possible uncertainties.

References

[1] Ben-Tal, A.,
El Ghaoui, L. and Nemirovski, A. (2009). Robust optimization (Vol. 28).
Princeton University Press.

[2] Bertsimas, D.,
Brown, D.B., Caramanis, C. (2011). Theory and applications of robust
optimization. SIAM review, 53(3), 464-501.

[3] Leyffer, S.,
Menickelly, M., Munson, T., Vanaret, C., Wild, S.M. (2018). Nonlinear Robust
Optimization. Argonne National Laboratory, preprint.

[4] Bertsimas, D.,
Nohadani, O. and Teo, K.M., (2010). Nonconvex robust optimization for problems
with constraints. INFORMS journal on computing, 22(1), 44-58.

[5] Yuan, Y., Li,
Z. and Huang, B., (2018). Nonlinear robust optimization for process design.
AIChE Journal, 64(2), 481-494.

[6] Mutapcic, A.,
Boyd, S. (2009). Cutting-set methods for robust convex optimization with
pessimizing oracles. Optimization Methods & Software, 24.3, 381-406.

[7] Ben-Tal, A.,
Goryashko, A., Guslitzer, E. and Nemirovski, A., (2004). Adjustable robust
solutions of uncertain linear programs. Mathematical Programming, 99(2), 351-376.

[8] Morgan, J.C.,
Bhattacharyya, D., Tong, C., Miller, D.C. (2015). Uncertainty quantification of
property models: Methodology and its application to CO2‐loaded aqueous MEA
solutions. AIChE Journal, 61(6), 1822-1839.

[9] Soares Chinen,
A., Morgan, J.C., Omell, B., Bhattacharyya, D., Tong, C. and Miller, D.C., (2018).
Development of a Rigorous Modeling Framework for Solvent-Based CO2 Capture. 1.
Hydraulic and Mass Transfer Models and Their Uncertainty Quantification.
Industrial & Engineering Chemistry Research, 57(31), 10448-10463.