We consider optimization problems with a joint probabilistic constraint under
normally distributed uncertain parameters. The parametric constraints
are replaced by one constraint stating that the probability of being feasible
shall exceed or be equal to a prescribed threshold. In order to apply the
concept to gas network optimization under uncertain boundary flows, which
corresponds to the demand of customers, we derive an analytic gradient formula.
The integral corresponding to the probability can be parameterized by
spherical radial decomposition. For this parameterization gradient formulas
are known under convexity assumptions of the parametric constraints in the
parameter. For the application in gas networks that we have in mind, the
convexity assumption of the parametric constraints is not satisfied. Therefore,
we weaken it to convexity of the region of feasible parameters for a fixed
optimization variable. We proceed to show that the assumptions needed for
the gradient formula are met in the gas network optimization problem on
a tree. For the numerical implementation we propose a multilevel sampling
algorithm that uses a coarse approximation of the chance constraint to generate
a warm start for the expensive approximation with fine sampling. The
numerical results illustrate that this approach significantly reduces the computation
time.
We study the transient optimization of gas transport networks including both discrete controls due to switching of controllable elements and nonlinear fluid dynamics described by the system of isothermal Euler equations, which are partial differential equations in time and 1-dimensional space. This combination leads to mixed-integer optimization problems subject to nonlinear hyperbolic partial differential equations on a graph. We propose an instantaneous control approach in which suitable Euler discretizations yield systems of ordinary differential equations on a graph. This networked system of ordinary differential equations is shown to be well-posed and affine-linear solutions of these systems are derived analytically. As a consequence, finite-dimensional mixed-integer linear optimization problems are obtained for every time step that can be solved to global optimality using general-purpose solvers. We illustrate our approach in practice by presenting numerical results on a realistic gas transport network.
Joint model of probabilistic/robust (probust) constraints applied to gas network optimization
(2017)
Optimization tasks under uncertain conditions abound in many
real-life applications. Whereas solution approaches for probabilistic constraints
are often developed in case the uncertainties can be assumed to follow a
certain probability distribution, robust approaches are usually used in case
solutions are sought that are feasible for all realizations of uncertainties within
some pre-defined uncertainty set. As many applications contain different types
of uncertainties that require robust as well as probabilistic treatments, we deal with a class of joint probabilistic/robust constraints as its appears in
optimization problems under uncertainty. Focusing on complex uncertain gas
network optimization problems, we show the relevance of this class of problems
for the task of maximizing free booked capacities in an algebraic model for a
stationary gas network. We furthermore present approaches for their solution.
Finally, we study the problem of controlling a transient system that is governed
by the wave equation. The task consists in determining controls such that a
certain robustness measure remains below some given upper bound, with high
probability.