We present an adaptive grid refinement algorithm to solve probabilistic optimization problems with infinitely many random constraints. Using a bilevel approach, we iteratively aggregate inequalities that provide most information not in a geometric but in a probabilistic sense. This conceptual idea, for which a convergence proof is provided, is then adapted to an implementable algorithm. The efficiency of our approach when compared to naive methods based on uniform grid refinement is illustrated for a numerical test example as well as for a water reservoir problem with joint probabilistic filling level constraints.
We present a novel mathematical algorithm to assist gas network operators in managing uncertainty,
while increasing reliability of transmission and supply. As a result, we solve an optimization problem
with a joint probabilistic constraint over an infinite system of random inequalities. Such models arise
in the presence of uncertain parameters having partially stochastic and partially non-stochastic character.
The application that drives this new approach is a stationary network with uncertain demand
(which are stochastic due to the possibility of fitting statistical distributions based on historical measurements)
and with uncertain roughness coefficients in the pipes (which are uncertain but non-stochastic due to a lack of
attainable measurements).
We study the sensitivity of local uncertainties in the roughness coefficients and their impact on a highly reliable
network operation. In particular, we are going to answer the question, what is the maximum uncertainty that is
allowed (shaping a 'maximal' uncertainty set) around nominal roughness coefficients, such that random demands in
a stationary gas network can be satisfied at given high probability level for no matter which realization of
true roughness coefficients within the uncertainty set.
One ends up with a constraint, which is probabilistic with respect to the load of gas
and robust with respect to the roughness coefficients. We demonstrate how such constraints can be dealt with in
the framework of the so-called spheric-radial decomposition of multivariate Gaussian distributions.
The numerical solution of a corresponding optimization problem is illustrated.
The results might assist the network operator with the implementation
of cost-intensive roughness measurements.
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.
Quasi-Monte Carlo algorithms are studied for designing discrete approximations of two-stage linear stochastic programs with random right-hand side and
continuous probability distribution. The latter should allow for a transformation to a
distribution with independent marginals. The two-stage integrands are piecewise linear,
but neither smooth nor lie in the function spaces considered for QMC error analysis.
We show that under some weak geometric condition on the two-stage model all terms
of their ANOVA decomposition, except the one of highest order, are continuously differentiable and that first and second order ANOVA terms have mixed first order partial
derivatives and belong to L2 . Hence, randomly shifted lattice rules (SLR) may achieve
the optimal rate of convergence O(n−1+δ ) with δ ∈ (0, 12 ] and a constant not depending
on the dimension if the effective superposition dimension is at most two. We discuss
effective dimensions and dimension reduction for two-stage integrands. The geometric
condition is shown to be satisfied almost everywhere if the underlying probability distribution is normal and principal component analysis (PCA) is used for transforming
the covariance matrix. Numerical experiments for a large scale two-stage stochastic
production planning model with normal demand show that indeed convergence rates
close to the optimal are achieved when using SLR and randomly scrambled Sobol’ point
sets accompanied with PCA for dimension reduction.
In this paper, we deal with a renewable-powered mini-grid, connected to an unreliable main grid, in a Joint Chance Constrained (JCC) programming setting.
In many countries with low energy access rates, grid-connected mini-grid system operators contend with four different types of uncertainties: stochastic solar
power and demand forecast errors; absolute uncertain national grid outage onset
times; and outages duration subjected to statistical analysis. These uncertainties
pose new challenges to the classical power system’s operation tasks. Two alternatives to the JCC problem are presented. In particular, we present an Individual
Chance Constraint (ICC) and a purely deterministic dispatch model. The JCC
model has the capability to address all four uncertainties, while the ICC covers only three of them, overlooking the uncertainty about the outage duration.
In contrast, the purely deterministic model completely ignores any uncertain
parameters. We illustrate the three models through a comparison of outcomes
attained from a real mini-grid in Lake Victoria, Tanzania. Results show how the
dispatch is modified across the models to plan the battery and diesel reserves in
the chance-constrained models, with the reserves in the JCC being larger than
in the ICC model. In comparison between all models, we prove that the JCC model offers the most robust results, since it can handle uncertainties about forecasting errors, on the one hand, and grid outages, on the other. The results also
show that the decrease in profits due to the hedging with reserves kept in the
MG is significantly small compared to the high level of reliability reached and
the potential load shedding that could be avoided in the case of an outage.
In this paper we consider systems that are governed by linear time-discrete dynamics with an initial condition and a terminal condition for the expected values. We study optimal control problems where in the objective function a term of tracking type for the expected values and a control cost appear. In addition, the feasible states have to satisfy a conservative probabilistic constraint that requires that the probability that the trajectories remain in a given set F is greater than or equal to a given lower bound. An application are
optimal control problems related to storage management systems with uncertain in- and output. We give suffcient conditions that imply that the optimal expected trajectories remain close to a certain state that can be characterized as the solution of an optimal control problem without prescribed initial- and terminal condition. Hence we contribute to the study of the turnpike phenomenon that is well-known in mathematical economics.
The spherical cap discrepancy is a widely used measure for how uniformly a sample of points on the sphere is distributed. Being hard to compute, this discrepancy measure is typically replaced by some lower or upper estimates when designing optimal sampling schemes for the uniform distribution on the sphere. In this paper, we provide a fully explicit, easy to implement enumerative formula for the spherical cap discrepancy. Not surprisingly, this formula is of combinatorial nature and, thus, its application is limited to spheres of small dimension and moderate sample sizes. Nonetheless, it may serve as a useful calibrating tool for testing the efficiency of sampling schemes and its explicit character might be useful also to establish necessary optimality conditions when minimizing the discrepancy with respect to a sample of given size.
In optimal control problems, often initial data are required
that are not known exactly in practice.
In order to take into account this uncertainty,
we consider optimal control problems for a system with an uncertain initial
state. A finite terminal time is given. On account of the uncertainty of the
initial state, it is not possible to prescribe an exact terminal state.
Instead, we are looking for controls that steer the system into a given
neighborhood of the desired terminal state with sufficiently high
probability. This neighborhood is described in terms of an inequality for
the terminal energy. The probabilistic constraint in the considered optimal
control problem leads to optimal controls that are robust against the
inevitable uncertainties of the initial state.
We show the existence of such optimal controls.
Numerical examples with
optimal Neumann control of the wave equation are presented.
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.
Assuming a pipe-wise constant structure of the friction coefficient in the modeling of natural gas transport through a passive network of pipes via semilinear systems of balance laws with associated linear coupling and boundary conditions, uncertainty in this parameter is quantified by a Markov chain Monte Carlo method. Information on the prior distribution is obtained from practitioners. The results are applied to the problem of validating technical feasibility under random exit demand in gas transport networks. The impact of quantified uncertainty to the probability level of technical feasible exit demand situations is studied by two example networks of small and medium size. The gas transport of the network is modeled by stationary solutions that are steady states of the time dependent semilinear problems.
