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.