(581a) Designing Subsea Production Facilities for the Worst Case Using Semi-Infinite Programming

Diminishing petroleum supply from traditional sources along
with the desire for reduced dependence on petroleum from foreign sources has
motivated exploration into increasingly more hostile environments.  One
promising frontier is ultra deep-sea oil and gas in the Gulf of Mexico.  The
application of traditional floating platform technologies to ultra deep-sea
production has been considered to be a very risky choice.  Alternatively,
remote compact subsea production facilities are considered to be the enabling
technology for ultra deep-sea oil and gas production.  However, as recent events
have demonstrated, operational failures in this environment are extremely
costly in terms of economic and environmental damages.  In this case, the cost
associated with ?over-designing? the process does not outweigh the cost
associated with operational failure.  Therefore, remote compact subsea production
facilities must be designed for the worst-case scenario, however improbable. 

In order to assess the robustness of a proposed design, a
rigorous model-based approach must be taken.  In such an approach, uncertainty
in the input disturbances to the process as well as uncertainties that are
inherent in the modeling parameters must be considered.  The question that must
be answered is ?for every realization of uncertainty, does there exist a
control setting such that all performance and safety specifications are never
violated??

In [3], the authors attempt to address the robustness
problem by formulating it as a bilevel optimization problem with the model
equations taken as equality constraints.  Since there are no algorithms currently
available for solving equality constrained bilevel optimization problems in
general, this technique is not applicable to the complex case of modeling a subsea
production facility.  Alternatively, in [5], the authors propose the idea of
reformulating the bilevel optimization program into an equivalent semi-infinite
program (SIP).  They suggest that, due to developments in globally solving SIPs
[1,2], the SIP reformulation is more tractable.  However, there is one major
caveat with this reformulation: an implicit function is embedded in the
semi-infinite constraint from solving the equality constraints (model
equations) for the state variables as an implicit function of the uncertain
variables and the controls [5].  Therefore, it does not have a known closed
form and can only be approximated using a procedure such as a fixed-point
iteration [5].

New developments in the construction of convex
underestimators and concave overestimators (or relaxations) of nonconvex
functions [4,6] have enabled the authors of this paper to extend the algorithm
of [1,2] to SIPs with embedded implicit functions.  Utilizing the most recent
developments [6], the SIP reformulation can be solved globally in the general
case. 

In this paper the authors discuss a ?zeroth-order? model of
a remote compact subsea production facility.  The novel ideas of relaxing
implicit functions [6] were applied within the new SIP algorithm to solve the
SIP reformulation and answer the question of robust feasibility.  The result is
a working proof-of-concept that the robust SIP formulation is tractable for
nontrivial examples.

[1]         
Bhattacharjee, B., Green Jr. W. H., and P. I. Barton. Interval Methods for
Semi-Infinite Programs. Computational Optimization and
Applications, 30:63-93, 2005.

[2]         
Bhattacharjee, B., Lemonidis, P., Green Jr. W. H., and P. I. Barton. Global
Solution of Semi-Infinite Programs. Math. Program., Ser. B
103:283-307, 2005.

[3]          Halemane, K.P. and Grossmann, I.E. Optimal
Process Design Under Uncertainty.  AIChE Journal, 29(3):425-433, 1983.

[4]          Scott, J.K., Stuber, M.D., and P.I. Barton.
Generalized McCormick Relaxations. Journal of Global Optimization, DOI:
10.1007/x10898-011-9664-7, In press 2011.

[5]          Stuber, M.D. and P.I. Barton. Robust Simulation
and Design Using Semi-Infinite Programs with Implicit Functions. Int. J. of
Reliability and Safety, In press 2011.

[6]          Stuber, M.D., Scott, J.K., and P.I. Barton.
Global Optimization of Implicit Functions. In preparation 2011.