In this thesis we analyze stationary and dynamic gas flow with uncertain
boundary data in networks of pipelines. The gas flow in pipeline networks
is modeled by the isothermal Euler equations. The uncertain boundary data is
modeled by probability distributions, they represent the a priori unknown gas
demand of the consumers. The aim of this work is the analysis of optimization
problems with probabilistic constraints in the context of gas transport.
For computing the probability that an uncertain gas demand is feasible
we use both, a kernel density estimator approach and the spheric radial decomposition.
Feasible in this context means, that the demanded gas can be
transported through the network, s.t. bounds for the pressure at the nodes are
satisfied. Moreover we discuss advantages and disadvantages of both methods.
In the stationary case we extend our model by compressor control and
bounds for the pressure at the entry nodes, and we also compute the probability
for an uncertain gas demand to be feasible. In the dynamic setting the
uncertain gas demand is time dependent, which is modeled by randomized
Fourier series.
Further we analyze certain optimization problems with probabilistic constraints,
in which the probabilistic constraints are approximated by the kernel
density estimator approach. On the one hand we show the existence of optimal
solutions for both, the exact and the approximated problems, and on the
other hand we show that if the approximation is sufficiently accurate, then the
optimal solutions of the approximated problems are close to the solutions of
the exact problems. With the approximation of the probabilistic constraints
via the kernel density estimator we are able to compute derivatives of the
approximated optimization constraints, which allows us to derive necessary
optimality conditions for the approximated optimization problems with probabilistic
constraints.
In real life decision problems, one almost always is confronted with uncertainty and risk. For practical optimization problems this is manifested by unknown parameters within the input data, or, an inexact knowledge about the system description itself. In case the uncertain problem data is governed by a known probability distribution, stochastic programming offers a variety of models hedging against uncertainty and risk. Most widely employed are two-stage models, who admit a recourse structure: The first-stage decisions are taken before the random event occurs. After its outcome, a recourse (second-stage) action is made, often but not always understood as some "compensation".
In the present thesis, the optimization problems that involve parameters which are not known with certainty are semidefinite programming problems. The constraint sets of these optimization problems are given by intersections of the cone of symmetric, positive semidefinite matrices with either affine or more general equations. Objective functions, formally, may be fairly general, although they often are linear as in the present thesis.
We consider risk neutral and risk averse two-stage stochastic semidefinite programs with continuous and mixed-integer recourse, respectively. For these stochastic optimization problems we analyze their structure, derive solution methods relying on decomposition, and finally apply our results to unit commitment in alternating current (AC) power systems.
Furthermore, deterministic unit commitment in AC power transmission systems is addressed. Beside traditional unit commitment constraints, the physics of power flow are included. To gain globally optimal solutions a recent semidefinite programming (SDP) approach is used which leads to large-scale semidefinite programs with discrete variables on top. As even the SDP relaxation of these programs is too large for being handled in an all-at-once manner by general SDP solvers, it requires an efficient and reliable method to tackle them. To this end, an algorithm based on Benders decomposition is proposed.
With power demand (load) and in-feed from renewables serving as sources of uncertainty, two-stage stochastic programs are set up heading for unit commitment schedules which are both cost-effective and robust with respect to data perturbations. The impact of different, risk neutral and risk averse, stochastic criteria on the shapes of the optimal stochastic solutions will be examined. To tackle the resulting two-stage programs, we propose to approximate AC power flow by semidefinite relaxations. This leads to two-stage stochastic mixed-integer semidefinite programs having a special structure. To solve the latter, the L-shaped method and dual decomposition have been applied and compared.
