Computational Experience with Piecewise-Linear Relaxations for Petroleum Refinery Planning

Refinery planning optimization is a challenging problem as regards handling the nonconvex bilinearity mainly due to pooling operations in processes such as crude oil distillation and product blending. This work investigates the performance of several representative piecewise-linear (or piecewise-affine) relaxation schemes (referred to as McCormick, bm, nf5, nf6t, and de (which is a new approach proposed based on eigenvector decomposition) that mainly give rise to mixed-integer optimization programs to convexify a bilinear term using predetermined univariate partitioning for instances of uniform and non-uniform partition sizes. Computational results show that applying these schemes give improved relaxation tightness than only applying convex and concave envelopes as estimators. Uniform partition sizes typically perform better in terms of relaxation solution quality and convergence behavior. It is also seen that there is a limit on the number of partitions that contributes to relaxation tightness, which does not necessarily correspond to a larger number of partitions, while a direct relation between relaxation size and tightness does not always hold for non-uniform partition sizes.

≤ cp  max , ∀ ∈  (A2) where   = load of unit  and cp  max = maximum capacity of u.
CDU outlet flow rate of cut p is given by:  , =     ,  = CDU, ∀ ∈  (A3) where   = weight transfer ratio of p that is determined based on true boiling point data of .
Weight transfer ratio of p sums to unity: Middle-of-point (or midpoint) weight transfer ratio   of fraction  is given by: ), ∀  ∈ \{}.(A5) A.2. Fluid Catalytic Cracking Unit CDU cut of bottom residue is fed to FCC to be converted into more valuable products.FCC outlet flow rate of product fraction f is given by:  FCC, =  FCC   , ∀ ∈  (A6) where  FCC, = flow rate of f from FCC and   = weight transfer ratio of f from FCC.
All weight transfer ratios of  ∈  sums to unity: is determined using the following regression-based relation: where regression coefficients given by   0 ,   1 , and   2 are known constants.
To achieve a desired FCC conversion level, part of its outlet flow of total gas oil (TGO) is recycled (as    ) and mixed with total inlet feed (   ): where  FCC  = total inlet flow rate to FCC,  CDU,BR = flow rate of bottom residue (BR) outlet stream from CDU, and  TGO  = flow rate of TGO recycle stream.
FCC load is equal to its inlet flow rate: TGO recycle stream flow rate are bounded (from above) by the following constraints: Remaining TGO stream (after split for recycle) is sold as heavy oil (FHO): where  FCC,FHO = flow rate of FHO product from FCC.

A.3. Gasoline Blending Unit
Lighter CDU fractions of GO and HN are processed further to improve their for gasoline blending to meet required research octane number (RON) specifications: where  ,  = flow rate of gasoline product grade g , G = set of gasoline product grades with RON of 90 (g90) and 93 (g93), and   = set of CDU fractions for gasoline blending.
To improve product quality, additives (e.g., MTBE) are mixed with blended CDU fractions according to the following relation: where    = flow rate of additive r and    = flow rate of final product p.
FCC gasoline fraction called Fgas is blended to improve its quality.The flow rate of FCC blended fraction equals the sum of flow rates of its respective blended products, as follows: where  ,  is the flow rate of final product fraction  (Fgas),  ,  is the flow rate of intermediate blended product , which is produced by blending flow stream Fgas from FCC.
The gasoline final products g90 and g93 are sold to customers.Their flow rates are calculated using equation (A17 ).   is the flow rate of intermediate product  from the DB, which is produced by blending feed  (LD, HD) from the CDU.  is the cost of crude oil,   is the cost of additive raw materials,   is the operating cost of process unit .
represents flow rate of final product  from DB.A.5.Quality Specifications Octane numbers of light CDU fractions GO and HN and pour points of CDU heavy fractions LD and HD are calculated using property correlations from the literature: