(269b) Robust Design of Subsea Production Facilities

The depletion of petroleum resources from traditional sources
has led to exploration into increasingly more hostile environments.  One new
frontier for the petroleum industry is deep-sea oil and gas production.  It has
been suggested that remote compact subsea production facilities are the
enabling technology for deep-sea oil and gas production.  However, because of
the extreme nature of deep-sea environments, the costs associated with
operational failures of such units are extraordinarily high and therefore the
risk of such failures must be accounted for at the design stage.  By taking
into account uncertainty in the input disturbances as well as inherent
uncertainties in the process model parameters, the robustness of a process can
be assessed at the design stage.

In [3], the authors propose addressing the robustness
question by formulating the problem as a bilevel optimization program.  The steady-state
model equations are represented as equality constraints in the inner program,
and the performance and safety specifications as the objective function of the
inner program.  The inner (minimization) program answers the question ?for some
realization of uncertainty, does there exist a feasible control such that the
constraints are not violated??  The outer (maximization) program essentially
isolates the worst-case realization of uncertainty.  Upon solving this bilevel
program, one obtains an answer to the question: ?for all realizations of
uncertainty, does there exist a control such that the constraints are not
violated??  However, due to the complexity of the problem, this bilevel program
is often extremely difficult or impossible to solve. 

In this paper it is proposed that the bilevel optimization
program be reformulated into an equivalent semi-infinite program (SIP); a
nonlinear optimization problem that has finitely many decision variables and an
infinite number of constraints.  In order to reformulate the bilevel program as
an SIP, the model equations must be solved for the process state variables, x,
as implicit functions of the disturbances, d, parametric uncertainty, p,
and the controls, u.  Due to the complexity of the model equations, the
functional form of x(u,d,p) is not known
explicitly, but can be approximated using a number of numerical methods, such
as Newton's method.  By applying this technique, the performance and safety
constraint becomes an implicit function itself.  The resulting program is an
SIP whose semi-infinite constraint is an implicit function.

In [1,2] the authors present new theory and an algorithm
that allows for the global solution of SIPs with an explicit semi-infinite
constraint.  The algorithm relies on interval arithmetic and McCormick's
convex/concave relaxations [4].  New theoretical developments in
McCormick-based relaxations as well as in interval Newton methods have enabled
this algorithm to be extended to solve SIPs with implicit semi-infinite
constraints, as encountered here.  The developments in interval Newton methods [6]
are used to calculate inner-approximations of the SIP feasible set.  Solving
the resulting problem on the restricted set yields a valid lower bound on the
SIP.  Similarly, using the newly developed interval Newton methods and newly
developed Generalized McCormick Relaxations [5], a valid outer-approximation of
the SIP feasible set can be generated, as well as valid convex/concave
relaxations of the implicit semi-infinite function.  Solving the resulting
convex problem yields a valid upper bound on a global solution of the SIP.  Armed
with this information, the branch-and-bound framework, and the existence of a
Slater point arbitrarily close to a solution, can guarantee finite ε-optimal
convergence to a global solution of the SIP.  The value of the global solution
provides a rigorous ?yes/no? answer to the robustness question.  With this
tool, design engineers will be able to guarantee, with certainty, their
proposed design will not fail under any realization of uncertainty considered.

[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 Grossman, I. E. Optimal Process Design Under Uncertainty. AIChE
Journal, 29(3):425-433, 1983.

[4]          McCormick,
Garth P. Computability of Global Solutions to Factorable Nonconvex Programs:
Part I ? Convex Underestimating Problems. Math. Program., 10:147-175,
1976.

[5]          Scott,
J. K., Stuber, M. D., and P. I. Barton. Generalized McCormick Relaxations. submitted
2009.

[6]          Stuber,
M. D. and P. I. Barton. Parametric Interval Newton Methods. In preparation
2010.