42 research outputs found
Matrix Minor Reformulation and SOCP-based Spatial Branch-and-Cut Method for the AC Optimal Power Flow Problem
Alternating current optimal power flow (AC OPF) is one of the most
fundamental optimization problems in electrical power systems. It can be
formulated as a semidefinite program (SDP) with rank constraints. Solving AC
OPF, that is, obtaining near optimal primal solutions as well as high quality
dual bounds for this non-convex program, presents a major computational
challenge to today's power industry for the real-time operation of large-scale
power grids. In this paper, we propose a new technique for reformulation of the
rank constraints using both principal and non-principal 2-by-2 minors of the
involved Hermitian matrix variable and characterize all such minors into three
types. We show the equivalence of these minor constraints to the physical
constraints of voltage angle differences summing to zero over three- and
four-cycles in the power network. We study second-order conic programming
(SOCP) relaxations of this minor reformulation and propose strong cutting
planes, convex envelopes, and bound tightening techniques to strengthen the
resulting SOCP relaxations. We then propose an SOCP-based spatial
branch-and-cut method to obtain the global optimum of AC OPF. Extensive
computational experiments show that the proposed algorithm significantly
outperforms the state-of-the-art SDP-based OPF solver and on a simple personal
computer is able to obtain on average a 0.71% optimality gap in no more than
720 seconds for the most challenging power system instances in the literature
Mixed-integer Nonlinear Optimization: a hatchery for modern mathematics
The second MFO Oberwolfach Workshop on Mixed-Integer Nonlinear Programming (MINLP) took place between 2nd and 8th June 2019. MINLP refers to one of the hardest Mathematical Programming (MP) problem classes, involving both nonlinear functions as well as continuous and integer decision variables. MP is a formal language for describing optimization problems, and is traditionally part of Operations Research (OR), which is itself at the intersection of mathematics, computer science, engineering and econometrics. The scientific program has covered the three announced areas (hierarchies of approximation, mixed-integer nonlinear optimal control, and dealing with uncertainties) with a variety of tutorials, talks, short research announcements, and a special "open problems'' session
Relaxations and discretizations for the pooling problem
The pooling problem is a folklore NP-hard global optimization problem that finds applications in industries such as petrochemical refining, wastewater treatment and mining. This paper assimilates the vast literature on this problem that is dispersed over different areas and gives new insights on prevalent techniques. We also present new ideas for computing dual bounds on the global optimum by solving high-dimensional linear programs. Finally, we propose discretization methods for inner approximating the feasible region and obtaining good primal bounds. Valid inequalities are derived for the discretized models, which are formulated as mixed integer linear programs. The strength of our relaxations and usefulness of our discretizations is empirically validated on random test instances. We report best known primal bounds on some of the large-scale instances
Tight-and-Cheap Conic Relaxation for the AC Optimal Power Flow Problem
The classical alternating current optimal power flow problem is highly
nonconvex and generally hard to solve. Convex relaxations, in particular
semidefinite, second-order cone, convex quadratic, and linear relaxations, have
recently attracted significant interest. The semidefinite relaxation is the
strongest among them and is exact for many cases. However, the computational
efficiency for solving large-scale semidefinite optimization is lower than for
second-order cone optimization. We propose a conic relaxation obtained by
combining semidefinite optimization with the reformulation-linearization
technique, commonly known as RLT. The proposed relaxation is stronger than the
second-order cone relaxation and nearly as tight as the standard semidefinite
relaxation. Computational experiments using standard test cases with up to 6515
buses show that the time to solve the new conic relaxation is up to one order
of magnitude lower than for the chordal relaxation, a semidefinite relaxation
technique that exploits the sparsity of power networks
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