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.