(562d) Optimal Crude Blending in Floating, Storage and Offloading Platforms (FSOPs)

Decisions based on crude transportation, crude handling, crude blending and crude delivery provide optimization opportunities to the FSOP to minimize huge expenses which could otherwise be worth millions of dollars per day. FSOPs can facilitate crude storage, transportation and blending in waters where either pipeline infrastructure is not cost effective or Ultra Large Crude Carriers (ULCCs) and Very Large Crude Carriers (VLCCs) are unable to enter due to shallow draughts. Scheduling in the Oil and Gas Industry using FSOP as the central procurement unit is typically order driven. Crude schedule is determined according to the forecasted (weekly or monthly) demand from the refineries as well as the current availability of crude in the tanks of the FSOP. Economic and operability benefits associated with the delivery of right blend of crude at the right time are numerous. Crude blends delivered by the FSOP play a very important role in the functioning of the distillation units and the downstream units at the refineries. Blends with extreme crude properties may disrupt these units' operations and further cause heavy economic losses. Thus, the idea is to ensure that only right quality of crude reaches the refineries from the FSOP as per refineries' specifications. The refineries lay out crude quality specifications based on 15 crude properties viz. specific gravity, sulphur, Reid Vapour Pressure, aromatics, naphthas, viscosity etc. As quoted by Li et al, 2004 there exists a linearly additive blending index for almost every crude property involving highly non mixing rules. In this paper, we discuss crude blending in FSOP as an important sub problem of the bigger Crude Oil Scheduling problem using FSOP as an offshore technology. While the volume based crude properties introduce bilinearity, the weight based properties give rise to trilinearity. This on the whole, reduces the problem into a Mixed Integer Trilinear Programming (MITLP) problem. As most local solvers like DICOPT in GAMS fail to solve the MITLP, there is a need to develop algorithms which can solve this crude scheduling problem in a robust and efficient manner.

Solution Strategy

We employ an MILP-NLP iterative scheme to solve the MITLP problem. This scheme consists of converting the original MITLP problem into a relaxed MILP first, with cost minimization as the objective. We achieve the Relaxed MILP (MILP0) by linearizing and convexifying the bilinear domain through the use of convex and concave envelopes developed by McCormick (1973). For the trilinear domain, we apply Recursive Arithmetic Intervals (RAI) methodology of generating convex and concave envelopes by Ryoo and Sahanidis (2001). Another technique for convexifying the positive trilinear domain is the use of convex and concave envelopes developed by Meyer and Floudas (2003). We name this technique as MF.

MILP0 yields a lower bound on cost based on the discrete decisions of parcel transfer and delivery from the FSOP. However, MILP0 fails to determine the inventory levels that should be kept in tanks to ensure all properties fall within specified bounds. In order to make such continuous decisions, we find it mandatory to solve an NLP problem which has binary variables fixed from MILP0. We call this NLP as NLP0 with its objective now as the minimization of sum of cost and summation of slack variables. NLP0 consists of all the constraints of the original MITLP except that the binary variables in it are fixed from MILP0. Introducing slack variables in NLP0 counters infeasibilities possibly generated as a result of any of the crude properties falling out of their specified bounds and ensures that the NLP0 is always feasible. We continue solving this series of MILP-NLP iteratively until summation of slack variables in NLP0 is below tolerance, all crude properties are within bounds and we get the lowest possible cost.

Results

As an illustration, we consider an FSOP with 3 tanks. 4 crude parcels arrive at different times during the scheduling horizon of 90 hours. The FSOP tanks deliver 4 crude orders to the refineries within this scheduling horizon. We consider only 3 volume based properties and 1 weight based property of crude in this illustration. The solutions were obtained on a Windows XP PC with Intel Pentium 4 processor using GAMS 22.8 with CPLEX (11) and CONOPT (3) as the MILP and NLP solvers respectively. Table 1 compares the key aspects of the results obtained by the aforementioned methods.

Table 1 Key Aspects of the Solutions

Serial No. Technique used Cost ($) Solution Time (secs)

1 RAI 16644.540 5.88

2 MF 16644.540 3.35

We obtain locally optimal solutions by both RAI and MF. Note that the costs obtained by both the methods are same. MF solves faster than RAI for the aforementioned case. We also observe that all crude properties in both the methods are within bounds and our algorithm solves the MITLP problem with relative ease giving this result in the first iteration itself.

Conclusions

We employed some existing methodologies to counter trilinearity arising from crude blending in our larger crude scheduling problem using FSOP. As we observe, our algorithm gives relatively fast results with both the methodologies used in this paper. Future work includes extending these methods to bigger case studies that include all crude properties.

References

(1) Li, J.; Li, W.; Karimi, I. A.; Srinivasan, R. (2007) ?Improving the robustness and efficiency of crude scheduling algorithms,? AIChE J, 53 (10), 2659?2680.

(2) Ryoo, Hong Seo and Sahanidis, Nikolaos V. (2001), ?Analysis of Bounds for Multilinear Functions,? Journal of Global Optimization, pp.403-424.

(3) Meyer, C.A., Floudas, C.A., (2003) ?Trilinear Monomials with Positive or Negative Domains: Facets of the Convex and Concave Envelopes,? Frontiers in Global Optimization, pp.327-351.

(4) Lee, T., Ryu, J., Lee, I.B., Lee, H., (2009) ?A Synchronized Feed Scheduling of Petrochemical Industries Simultaneously Considering Vessel Scheduling and Storage Tank Management?, IECR, DOI: 10.1021/ie800741m.