Chance Constrained Programming : with applications in Energy Management
In optimization problems involving uncertainty, probabilistic constraints are an important tool for defining safety of decisions. In Energy management, many optimization problems have some underlying uncertainty. In particular this is the case of unit commitment problems. In this Thesis, we will investigate probabilistic constraints from a theoretical, algorithmic and applicative point of view. We provide new insights on differentiability of probabilistic constraints and on convexity results of feasible sets. New variants of bundle methods, both of proximal and level type, specially tailored for convex optimization under probabilistic constraints, are given and convergence shown. Both methods explicitly deal with evaluation errors in both the gradient and value of the probabilistic constraint. We also look at two applications from energy management: cascaded reservoir management with uncertainty on inflows and unit commitment with uncertainty on customer load. In both applications uncertainty is dealt with through the use of probabilistic constraints. The presented numerical results seem to indicate the feasibility of solving an optimization problem with a joint probabilistic constraint on a system having up to 200 constraints. This is roughly the order of magnitude needed in the applications. The differentiability results involve probabilistic constraints on uncertain linear and nonlinear inequality systems. In the latter case a convexity structure in the underlying uncertainty vector is required. The uncertainty vector is assumed to have a multivariate Gaussian or Student law. The provided gradient formulae allow for efficient numerical sampling schemes. For probabilistic constraints that can be rewritten through the use of Copulae, we provide new insights on convexity of the feasible set. These results require a generalized concavity structure of the Copulae, the marginal distribution functions of the underlying random vector and of the underlying inequality system. These generalized concavity properties may hold only on specific sets.