Global optimisation of large-scale quadratic programs: application to short-term planning of industrial refinery-petrochemical complexes

This thesis is driven by an industrial problem arising in the short-term planning of an integrated refinery-petrochemical complex (IRPC) in Colombia. The IRPC of interest is composed of 60 industrial plants and a tank farm for crude mixing and fuel blending consisting of 30 additional units. It considers both domestic and imported crude oil supply, as well as refined product imports such as low sulphur diesel and alkylate. This gives rise to a large-scale mixed-integer quadratically constrained quadratic program (MIQCQP) comprising about 7,000 equality constraints with over 35,000 bilinear terms and 280 binary variables describing operating modes for the process units. Four realistic planning scenarios are recreated to study the performance of the algorithms developed through the thesis and compare them to commercial solvers. Local solvers such as SBB and DICOPT cannot reliably solve such large-scale MIQCQPs. Usually, it is challenging to even reach a feasible solution with these solvers, and a heuristic procedure is required to initialize the search. On the other hand, global solvers such as ANTIGONE and BARON determine a feasible solution for all the scenarios analysed, but they are unable to close the relaxation gap to less than 40% on average after 10h of CPU runtime. Overall, this industrial-size problem is thus intractable to global optimality in a monolithic way. The first main contribution of the thesis is a deterministic global optimisation algorithm based on cluster decomposition (CL) that divides the network into groups of process units according to their functionality. The algorithm runs through the sequences of clusters and proceeds by alternating between: (i) the (global) solution of a mixed-integer linear program (MILP), obtained by relaxing the bilinear terms based on their piecewise McCormick envelopes and a dynamic partition of their variable ranges, in order to determine an upper bound on the maximal profit; and (ii) the local solution of a quadratically-constrained quadratic program (QCQP), after fixing the binary variables and initializing the continuous variables to the relaxed MILP solution point, in order to determine a feasible solution (lower bound on the maximal profit). Applied to the base case scenario, the CL approach reaches a best solution of 2.964 MMUSD/day and a relaxation gap of 7.5%, a remarkable result for such challenging MIQCQP problem. The CL approach also vastly outperforms both ANTIGONE (2.634 MMUSD/day, 32% optimality gap) and BARON (2.687 MMUSD/day, 40% optimality gap). The second main contribution is a spatial Lagrangean decomposition, which entails decomposing the IRPC short-term planning problem into a collection of smaller subproblems that can be solved independently to determine an upper bound on the maximal profit. One advantage of this strategy is that each sub-problem can be solved to global optimality, potentially providing good initial points for the monolithic problem itself. It furthermore creates a virtual market for trading crude blends and intermediate refined–petrochemical streams and seeks an optimal trade-off in such a market, with the Lagrange multipliers acting as transfer prices. A decomposition over two to four is considered, which matches the crude management, refinery, petrochemical operations, and fuel blending sections of the IRPC. An optimality gap below 4% is achieved in all four scenarios considered, which is a significant improvement over the cluster decomposition algorithm.Open Acces

A branch-and-price algorith, for a compressor scheduling problem

This work presents the study and application of a branch-and-price algorithm for solving a compressor scheduling problem. The problem is related to oil production and consists of defining a set of compressors to be activated, supplying the gas-lift demand of a set of wells and minimizing the associated costs. The problem has a non-convex objective function, to which a piecewise-linear formulation has been proposed. This dissertation proposes a column generation approach based on the Dantzig-Wolfe decomposition, which achieves tighter lower bounds than the straightforward linear relaxation of the piecewise-linear formulation. The column generation was embedded in a branch-and-price algorithm and further compared with CPLEX, obtaining optimal solutions in lesser time for a set of instances. Further, the branch-and-price algorithm can find better feasible solutions for large instances under a limited processing time.O presente trabalho realiza o estudo e aplicação de um algoritmo de branch-and-price para a resolução de um problema de escalonamento de compressores. O problema é ligado à produção petrolífera, o qual consiste em definir um conjunto de compressores a serem ativados para fornecer gas de elevação a um conjunto de poços, atendendo toda demanda e minimizando os custos envolvidos. O problema é caracterizado por uma função objetivo não-convexa que é linearizada por partes de forma a ser formulada como um problema de programação inteira mista. A abordagem de geração de colunas é baseada na decomposição de Dantzig-Wolfe e apresenta melhores limitantes inferiores em relação à relaxação linear da formulação compacta. O branch-and-price é comparado ao solver CPLEX, sendo capaz de encontrar a solução ótima em menor tempo para um conjunto de instâncias, bem como melhores soluções factíveis para instâncias maiores em um período de tempo limitado