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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.

Uncertainty often plays an important role in dynamic flow problems. In this paper, we consider both, a stationary and a dynamic flow model with uncertain boundary data on networks. We introduce two different ways how to compute the probability for random boundary data to be feasible, discussing their advantages and disadvantages. In this context, feasible means, that the flow corresponding to the random boundary data meets some box constraints at the network junctions. The first method is the spheric radial decomposition and the second method is a kernel density estimation.
In both settings, we consider certain optimization problems and we compute derivatives of the probabilistic constraint using the kernel density estimator. Moreover, we derive necessary optimality conditions for the stationary and the dynamic case.
Throughout the paper, we use numerical examples to illustrate our results by comparing them with a classical Monte Carlo approach to compute the desired probability.

In this paper, problems of optimal control are considered where in the objective function, in addition to the control cost, there is a tracking term that measures the distance to a desired stationary state. The tracking term is given by some norm, and therefore it is in general not differentiable. In the optimal control problem, the initial state is prescribed. We assume that the system is either exactly controllable in the classical sense or nodal profile controllable. We show that both for systems that are governed by ordinary differential equations and for infinite-dimensional systems, for example, for boundary control systems governed by the wave equation, under certain assumptions, the optimal system state is steered exactly to the desired state after finite time.

We deal with a stationary model for flow through a
network. The flows are determined by the values at the boundary nodes of the
network that we call the loads of the network. In the applications , the feasible loads
must satisfy some box constraints. We analyze the structure of the set of feasible
loads. Our analysis is motivated by gas pipeline flows, where the box constraints are
pressure bounds.
We present sufficient conditions to show, that the feasible set is star-shaped with
respect to special points. For stronger conditions, we prove the convexity of the
set of feasible loads. All the results are given for active and passive networks, i.e.
networks with and without inner control.
This analysis is motivated by the aim to use the spheric-radial decomposition for
stochastic boundary data in this model. This paper can be used for simplifying the
algorithmic use of the spheric-radial decomposition.

We introduce a stationary model for gas flow based on simplified isothermal Euler equations in a non-cycled pipeline network.
Especially the problem of the feasibility of a random load vector is analyzed. Feasibility in this context means the existence of a
flow vectormeeting these loads, which satisfies the physical conservation laws with box constraints for the pressure. An important
aspect of the model is the support of compressor stations, which counteract the pressure loss caused by friction in the pipes.The
network is assumed to have only one influx node; all other nodes are efflux nodes.With these assumptions the set of feasible loads
can be characterized analytically. In addition we show the existence of optimal solutions for some optimization problems with
probabilistic constraints. A numerical example based on real data completes this paper.

The European gas market is governed by rules that are agreed on by the European Union. We present a mathematical market model that
takes into account this structure, where the technical system operator (TSO)
offers certain transportation capacities that can be booked and later nominated within the previously chosen bookings. The TSO also fixes booking fees and defines an operational control of the gas pipeline system in order to deliver the gas according to the nominations. Since the gas
flow is governed by a system of partial differential equations, to realize this control structure partial differential equations (PDEs) should be involved in the model.
While the four level gas market model has been discussed previously, in this
paper we take into account the
flow model by PDEs in the discussion of the model and in the reduction to a single level problem, where we also state the corresponding necessary optimality conditions.

In this paper we analyze the turnpike phenomenon for optimal boundary control problems with a linear transport equation with source term. The convex objective function depends on
the boundary traces of the transport equation and is strictly convex with respect to the boundary control. We show an integral turnpike result for an optimal Dirichlet boundary control problem in the sense that if the time horizon goes to infinity, then the dynamic optimal control converges to
the corresponding steady state optimal control.
The novelty of this work is two-sided. On the one hand, even if turnpike results for this kind of optimal boundary control problem already exist, we present a new direct proof without using adjoint calculus that leads to sharper estimates. On the other hand we consider uncertainty in
the initial data and/or in the source term. We show that the integral turnpike result also holds considering uncertainty. Throughout the paper we use numerical examples to illustrate the results.

In the transition to renewable energy sources, hydrogen will potentially play an important role for energy storage. The efficient transport of this gas is possible via pipelines. An understanding of the possibilities to control the gas flow in pipelines is one of the main building blocks towards the optimal use of gas.
For the operation of gas transport networks it is important to take into account the randomness of the consumers’ demand, where often information on the probability distribution is available.
Hence in an efficient optimal control model the corresponding probability should be included and the optimal control should be such that the state that is generated by the optimal control satisfies given state constraints with large probability. We comment on the modelling of gas pipeline flow and the problems of optimal nodal control with random demand, where the aim of the optimization is to determine controls that generate states that satisfy given pressure bounds with large probability. We include the H2 norm of the control as control cost, since this avoids large pressure fluctuations which are harmful in the transport of hydrogen since they can cause
embrittlement of the pipeline metal.

In the operation of pipeline networks, compressors play a crucial role in ensuring the network’s functionality for various scenarios. In this contribution we address the important question of finding the optimal location of the compressors. This problem is of a novel structure, since it is related
with the gas dynamics that governs the network flow. That results in non-convex mixed integer stochastic optimization problems with probabilistic constraints.
Using a steady state model for the gas flow in pipeline networks including compressor control and uncertain loads given by certain probability distributions, the problem of finding the optimal location for the
control on the network, s.t. the control cost is minimal and the gas pressure stays within given bounds, is considered.
In the deterministic setting, explicit bounds for the pipe length and the inlet pressure, s.t. a unique optimal compressor location with minimal control cost exists, are presented. In the probabilistic setting,
an existence result for the optimal compressor location is presented and the uniqueness of the solution is discussed depending on the probability distribution. For Gaussian distributed loads a uniqueness result
for the optimal compressor location is presented.
Further the problem of finding the optimal compressor locations on networks including the number of compressor stations as variable is considered. Results for the existence of optimal locations on a graph in
both, the deterministic and the probabilistic setting, are presented and the uniqueness of the solutions is discussed depending on probability distributions and graph topology. The paper concludes with an illustrative example demonstrating that the compressor locations determined using a steady state approach are also admissible in transient settings.

On the Convergence of Optimization Problems with Kernel Density Estimated Probabilistic Constraints
(2024)

Uncertainty plays a significant role in applied mathematics and probabilistic constraints are widely used to model uncertainty in various fields, even if probabilistic constraints often demand computational challenges. Kernel density estimation (KDE) provides a data-driven approach for properly estimating probability density functions and efficiently evaluate corresponding probabilities.
In this paper, we investigate optimization problems with probabilistic constraints, where the probabilities are approximated using a KDE approach. We establish sufficient conditions under which the solution of the KDE approximated optimization problem converges to the solution of the original problem as the sample size goes to infinity.
The main results of this paper include three theorems: (1) For sufficiently large sample sizes, the solution of the original problem is also a solution of the approximated problem, if the probabilistic constraint is passive; (2) The limit of a convergent sequence of solutions of the approximated problems is a solution of the original problem, if the KDE uniformly converges; (3) We provide sufficient conditions for the existence of a convergent sequence of solutions of the approximated problems.