(183c) Global Optimization of Large-Scale Extended Pooling Problems with the EPA Complex Emissions Model

The storage of intermediate refinery streams under limited tankage conditions conditions can be formulated using the pooling problem, an optimization challenge which maximizes profit subject product availability, storage capacity, demand, and product specification constraints [1]. We extend the pooling problem by explicitly incorporating the Environmental Protection Agency (EPA) Title 40 Code of Federal Regulations Part 80.45: Complex Emissions Model [2] into the constraint set.

The EPA Model, which was developed in response to the Clean Air Act of 1990, codifies and legally certifies a mathematical model of reformulated gasoline (RFG) emissions based on eleven measurable fuel components. The nonconvex, piecewise-defined equations defining the volatile organic, NO_X, and toxics emissions depend on the time of year, region of the country, and type of vehicle. Although the EPA model has been previously developed into a mixed-integer nonlinear program [3], we show that some of the ostensibly nonlinear functions in the EPA model are actually piecewise-linear. Additionally, we reduce the number of required variables through effective integration into the pooling problem framework. Because Reid Vapor Pressure (RVP), one of the components monitored by the EPA Complex Emissions Model, blends nonlinearly [4], we also consider nonlinear blending in our model.

In keeping with the large-scale nature of industrially relevant pooling problems, we introduce, underestimate, and globally optimize several large-scale examples that incorporate the EPA model. To relax the standard pooling problem backbone, we employ the relaxation-linearization technique-based 'PQ'-formulation [5, 6] and tightly underestimate the bilinear terms via piecewise relaxation [7, 8, 9, 10]. To underestimate the equations in the EPA Complex Emissions model, we employ piecewise linear techniques [11], outer approximation, and the edge-concave method [12]. We demonstrate that these recently developed methods substantially tighten the problem relaxation. For the extended pooling problem examples, we balance the opposing goals of tight relaxation and problem size.

Finally, we present detailed computational studies on global optimization strategies that take advantage of special structures in the problem. We exploit the implicit bounds in the piecewise-defined Complex Emissions Model to generate good upper bounds on the problem. We also study sensitivity of the EPA model to its parameters and propose algorithms to expedite solution time.

References

[1] C. A. Floudas and A. Aggarwal. A decomposition strategy for global optimum search in the pooling problem. ORSA J. on Comput., 2, 1990.

[2] 40CFR80.45. Code of Federal Regulations: Complex emissions model, 2007. http://ecfr.gpoaccess.gov/cgi/t/text/text-idx?c=ecfr&sid=acf6e89d79c01d2...

[3] K. C. Furman and I. P. Androulakis. A novel MINLP-based representation of the original complex model for predicting gasoline emissions. Comput. & Chem. Eng., 32:2857?2876, 2008.

[4] V. Visweswaran. MINLP: Applications in blending and pooling. In C. A. Floudas and P. M. Pardalos, editors, Encyclopedia of Optimization, pages 2114?2121. Springer Science, 2 edition, 2009.

[5] I. Quesada and I. E. Grossmann. Global optimization of bilinear process networks with multicomponent flows. Comput. & Chem. Eng., 19:1219?1242, 1995.

[6] M. Tawarmalani and N. V. Sahinidis. Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Applications, Software, and Applications. Nonconvex Optimization and Its Applications. Kluwer Academic Publishers, Norwell, MA, USA, 2002.

[7] C. A. Meyer and C. A. Floudas. Global optimization of a combinatorially complex generalized pooling problem. AIChE J., 52(3):1027 ? 1037, 2006.

[8] R. Karuppiah and I.E. Grossmann. Global optimization for the synthesis of integrated water systems in chemical processes. Comput. & Chem. Eng., 30:650?673, 2006.

[9] D. S. Wicaksono and I. A. Karimi. Piecewise MILP under-and overestimators for global optimization of bilinear programs. AIChE J., 54(4):991?1008, 2008.

[10] C. E. Gounaris, R. Misener, and C.A. Floudas. Computational comparison of piecewise-linear relaxations for pooling problems. 2009. Submitted for Publication.

[11] C. A. Floudas. Nonlinear and Mixed-Integer Optimization: Fundamentals and Applications. Oxford University Press, New York, NY, 1995.

[12] C. A. Meyer and C. A. Floudas. Convex envelopes for edge-concave functions. Math. Program., 103 (2):207?224, 2005.