Martin Mevissen

Optimal Power Flow as a Polynomial Optimization Problem

Formulating the alternating current optimal power flow (ACOPF) as a polynomial optimization problem makes it possible to solve large instances in practice and to guarantee asymptotic convergence in theory. We formulate the ACOPF as a degree-two polynomial program and study two approaches to solving it via convexifications. In the first approach, we tighten the first-order relaxation of the nonconvex quadratic program by adding valid inequalities. In the second approach, we exploit the structure of the polynomial program by using a sparse variant of Lasserres hierarchy. This allows us to solve instances of up to 39 buses to global optimality and to provide strong bounds for the Polish network within an hour.

Fast non-linear optimization for design problems on water networks 1

As water infrastructure ages and repair costs increase, optimization techniques are increasingly used for the design and operation of water networks. A key challenge for optimization on water systems is the fast and accurate simulation of hydraulic equations. Conventional simulation tools such as Epanet are fast but cannot perform optimization alone and so must be coupled to an optimization engine, typically a metaheuristic such as a genetic algorithm. In contrast, mathematical optimization methods take into account hydraulic equations as constraints. The energy equation for pipe ow is a challenging constraint because it is non-linear and given by an explicit function with a rational exponent (Hazen-Williams) or an implicit function (Colebrook-White). This paper uses a quadratic approximation for pipe head loss that provides very good accuracy. The approximation is applied to pose and solve a mixed integer non-linear program (MINLP) for placing and setting pressure reducing valves. The problem is addressed using both local and global solvers. Computational results show accuracy comparable to Epanet and signicant potential to reduce non revenue water by deploy

MINLP in transmission expansion planning

Transmission expansion planning requires forecasts of demand for electric power and a model of the underlying physics, i.e., power flows. We present three approaches to deriving exact solutions to the transmission expansion planning problem in the alternating-current model, for a given load.

Polynomial optimization for water networks: Global solutions for the valve setting problem

This paper explores polynomial optimization techniques for two formulations of the energy conservation constraint for the valve setting problem in water networks. The sparse hierarchy of semidefinite programing relaxations is used to derive globally optimal bounds for an existing cubic and a new quadratic problem formulation. Both formulations use an approximation for friction loss that has an accuracy consistent with the experimental error of the classical equations. Solutions using the proposed approach are reported on four water networks ranging in size from 4 to 2000 nodes and are compared against a local solver, Ipopt and a global solver, Couenne. Computational results found global solutions using both formulations with the quadratic formulation having better time efficiency due to the reduced degree of the polynomial optimization problem and the sparsity of the constraint matrix. The approaches presented in this paper may also allow global solutions to other water network steady-state optimization problems formulated with continuous variables.

Robustness to Time Discretization Errors in Water Network Optimization

Water network optimization problems require modeling the progression of flow and pressure over time. The time discretization step for the resulting differential algebraic equation must be chosen carefully; a large time step can result in a solution that bears little relevance to the physical system, and small time steps impact a problem’s tractability. We show that a large time step can result in meaningless results and we construct an upper bound on the error in the tank pressures when using a forward Euler scheme. We provide an optimization formulation that is robust to this discretization error; robustness to model uncertainty is novel in water network optimization.