(701a) A More Efficient Formulation for the Multiperiod Blending Problem

Castro, P. M., Faculdade de Ciências, Universidade de Lisboa

Finding the most profitable blends of different distilled fractions so as to meet technical and environmental regulations is a problem of great importance in petroleum refineries. It become known as the pooling problem1, where the pools represent blending tanks whose content, generated from the mixing of fuels with known properties, is going to be determined by the optimization.

A few MINLP formulations have been proposed for the pooling problem1-4. The p-formulation1 is perhaps the most intuitive, featuring total flows and compositions as model variables.  The more recent tp-formulation by Alfaki and Haugland4 replaces the pool composition variables with the proportion of the pool being sent to each final product, which add up to one.

The original problem definition has been generalized5 and extended6 but is aimed at steady state operation. In reality, however, supply and demand vary with time, resulting in a multiperiod blending problem7. Models entities gain a time index, and tank capacity and logistic constraints (concerning the movement of materials into and out of tanks) need to be enforced.

The crude oil unloading and inventory management problem8-13 is closely related but more detailed in the sense that the duration of the time intervals is going to be determined by the optimization. In recent work13, we have shown that a Resource-Task Network continuous-time formulation can effectively capture the most important system constraints and be solved very efficiently by commercial global optimization solvers. The material resources are the different crudes and their location as they move from marine vessels to distillation columns, with the contents of a tank consisting of multiple crudes rather than a single resource of unknown composition. The challenge of enforcing that a tank’s outlet stream has the exact same composition of the blend inside the tank is then overcome through bilinear constraints featuring split fraction variables. These are similar to those used in the tp-formulation4, the difference being that the sum of the split fraction over all possible outlet streams does not add to one since some inventory may be left in the tank to be used in subsequent time periods, which does not apply to the steady-state pooling problem.

The main novelty of this work is to define the liquid fuels initially present in the system as new model entities and propose a new formulation with split fraction variables that keeps track of their location through time. Due to the discrete-time nature of the multiperiod blending problem (with static time periods), we can move away from the complex nomenclature of the RTN and present a model that is easier to understand.

Overall, the new formulation leads to a different set of non-convex bilinear terms compared to the original work of Kolodziej et al.7. Through the solution of a set of seven test problems from the literature (freely available online at www.minlp.org), we show that these are better handled by decomposition algorithms that divide9 the problem into MILP and NLP components, as well as by commercial solvers. In fact, BARON14 14.0 and GloMIQO15 2.3 can solve to global optimality all problems resulting from the new formulation. A tailored global optimization algorithm working with a tight mixed-integer relaxation from multiparametric disaggregation16-17 achieves a similar performance.

Acknowledgments: Financial support from Fundação para a Ciência e Tecnologia (FCT) through the Investigador FCT 2013 program and project UID/MAT/04561/2013.


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