Claudia D'Ambrosio

An MILP Approach for Short-Term Hydro Scheduling and Unit Commitment With Head-Dependent Reservoir

The paper deals with a unit commitment problem of a generation company whose aim is to find the optimal scheduling of a multiunit pump-storage hydro power station, for a short term period in which the electricity prices are forecasted. The problem has a mixed-integer nonlinear structure, which makes very hard to handle the corresponding mathematical models. However, modern mixed-integer linear programming (MILP) software tools have reached a high efficiency, both in terms of solution accuracy and computing time. Hence we introduce MILP models of increasing complexity, which allow to accurately represent most of the hydroelectric system characteristics, and turn out to be computationally solvable. In particular we present a model that takes into account the head effects on power production through an enhanced linearization technique, and turns out to be more general and efficient than those available in the literature. The practical behavior of the models is analyzed through computational experiments on real-world data.

Piecewise linear approximation of functions of two variables in MILP models

We consider three easy-to-implement methods for the piecewise linear approximation of functions of two variables. We experimentally evaluate their approximation quality, and give a detailed description of how the methods can be embedded in a MILP model. The advantages and drawbacks of the three methods are discussed on numerical examples.

An MINLP solution method for a water network problem

We propose a solution method for a water-network optimization problem using a nonconvex continuous NLP relaxation and an MINLP search. We report successful computational experience using available MINLP software on problems from the literature and on difficult real-world instances.

Valid inequalities for the pooling problem with binary variables

The pooling problem consists of finding the optimal quantity of final products to obtain by blending different compositions of raw materials in pools. Bilinear terms are required to model the quality of products in the pools, making the pooling problem a non-convex continuous optimization problem. In this paper we study a generalization of the standard pooling problem where binary variables are used to model fixed costs associated with using a raw material in a pool. We derive four classes of strong valid inequalities for the problem and demonstrate that the inequalities dominate classic flow cover inequalities. The inequalities can be separated in polynomial time. Computational results are reported that demonstrate the utility of the inequalities when used in a global optimization solver.

On interval-subgradient and no-good cuts

Interval-gradient cuts are (nonlinear) valid inequalities derived from continuously differentiable nonconvex constraints. In this paper we define interval-subgradient cuts, a generalization to nondifferentiable constraints, and show that no-good cuts with 1-norm are a special case of interval-subgradient cuts. We then briefly discuss what happens if other norms are used.

Heuristic algorithms for the general nonlinear separable knapsack problem

We consider the nonlinear knapsack problem with separable nonconvex functions. Depending on the assumption on the integrality of the variables, this problem can be modeled as a nonlinear programming or as a (mixed) integer nonlinear programming problem. In both cases, this class of problems is very difficult to solve, both from a theoretical and a practical viewpoint. We propose a fast heuristic algorithm, and a local search post-optimization procedure. A series of computational comparisons with a heuristic method for general nonconvex mixed integer nonlinear programming and with global optimization methods shows that the proposed algorithms provide high-quality solutions within very short computing times.

Surrogate‐based methods for black‐box optimization

In this paper, we survey methods that are currently used in black-box optimization, i.e. the kind of problems whose objective functions are very expensive to evaluate and no analytical or derivative information are available. We concentrate on a particular family of methods, in which surrogate (or meta) models are iteratively constructed and used to search for global solutions.

The power edge set problem

The automated real time control of an electrical network is achieved through the estimation of its state using phasor measurement units. Given an undirected graph representing the network, we study the problem of finding the minimum number of phasor measurement units to place on the edges such that the graph is fully observed. This problem is also known as the Power Edge Set problem, a variant of the Power Dominating Set problem. It is naturally modeled using an iteration-indexed binary linear program, whose size turns out to be too large for practical purposes. We use a fixed-point argument to remove the iteration indices and obtain a more compact bilevel formulation. We then reformulate the latter to a single-level mixed-integer linear program, which performs better than the natural formulation. Lastly, we provide an algorithm that solves the bilevel program directly and much faster than a commercial solver can solve the previous models. We also discuss robust variants and extensions of the problem.

On a Nonconvex MINLP Formulation of the Euclidean Steiner Tree Problem in n-Space

The Euclidean Steiner Tree Problem in dimension greater than 2 is notoriously difficult. Successful methods for exact solution are not based on mathematical-optimization -- rather, they involve very sophisticated enumeration. There are two types of mathematical-optimization formulations in the literature, and it is an understatement to say that neither scales well enough to be useful. We focus on a known nonconvex MINLP formulation. Our goal is to make some first steps in improving the formulation so that large instances may eventually be amenable to solution by a spatial branch-and-bound algorithm. Along the way, we developed a new feature which we incorporated into the global-optimization solver SCIP and made accessible via the modeling language AMPL, for handling piecewise-smooth univariate functions that are globally concave.

Observing the State of a Smart Grid Using Bilevel Programming

Monitoring an electrical network is an important and challenging task. Phasor measurement units are measurement devices that can be used for a state estimation of this network. In this paper we consider a PMU placement problem without conventional measurements and with zero injection nodes for a full observability of the network. We propose two new approaches to model this problem, which take into account a propagation rule based on Ohms and Kirchoffs law. The natural binary linear programming description models an iterative observability process. We remove the iteration by reformulating its fixed point conditions to a bilevel program, which we then further reformulate to a single-level mixed-integer linear program. We also present a bilevel algorithm to solve directly the proposed bilevel model. We implemented and tested our models and algorithm: the results show that the bilevel algorithm is better in terms of running time and size of instances which can be solved.