The paper considers the computation of the probability of feasible load constellations in a stationary gas
network with uncertain demand. More precisely, a network with a single entry and several
exits with uncertain loads is studied. Feasibility of a load constellation is understood in the sense of
an existing flow meeting these loads along with given pressure bounds in the pipes.
In a first step, feasibility of deterministic exit loads is characterized algebraically and these general
conditions are specified to networks involving at most one cycle.
This prerequisite is essential for determining probabilities in a stochastic setting when exit loads
are assumed to follow some (joint) Gaussian distribution when modeling uncertain customer demand.
The key of our approach is the application of the spheric-radial decomposition of Gaussian random
vectors coupled with Quasi Monte-Carlo sampling. This approach requires an efficient algorithmic
treatment of the mentioned algebraic relations moreover depending on a scalar parameter. Numerical
results are illustrated for different network examples and demonstrate a clear superiority in terms of
precision over simple generic Monte-Carlo sampling. They lead to fairly accurate probability values
even for moderate sample size.
We consider multistage stochastic linear optimization problems combining joint dynamic probabilistic constraints with hard constraints. We develop a method for projecting decision rules onto
hard constraints of wait-and-see type. We establish the relation between the original (infinite
dimensional) problem and approximating problems working with projections from different subclasses of decision policies. Considering the subclass of linear decision rules and a generalized
linear model for the underlying stochastic process with noises that are Gaussian or truncated
Gaussian, we show that the value and gradient of the objective and constraint functions of the
approximating problems can be computed analytically.
On probabilistic constraints with multivariate truncated Gaussian and lognormal distributions
(2016)
Many engineering problems with uncertain data, notably arising
in power management, can be formulated as optimization problems subject to
probabilistic constraints. While dealing with such constraints under continuous distributions of the underlying random parameter remains a difficult task
in general both from the numerical and theoretical point of view, quite some
progress has been made in the special case of multivariate Gaussian distributions. These are not perfectly adequate, however, in many circumstances, in
particular not, when modeling uncertain inflows to hydro reservoirs or uncertain demands in gas networks. Interesting alternatives are offered by truncations of multivariate Gaussian distributions to polyhedra or by multivariate
lognormal distributions. The paper discusses the applicability of such distributions in the context of a simple joint linear probabilistic constraint putting
the emphasis on the numerical approximation of probabilities and their gradients (w.r.t. decisions to be optimized) as well as on the convexity of the set
of feasible decisions.