Friedrich-Alexander-Universität Erlangen-Nürnberg
Refine
Has Fulltext
- no (25) (remove)
Keywords
- Branch-and-cut (2)
- Combinatorial optimization (2)
- Networks (2)
- Aggregation (1)
- Boundary feedback control, feedback stabilization, exponential stability, isothermal Euler equations, second-order quasilinear equation, Lyapunov function, stationary state, non-stationary state, gas pipeline. (1)
- Complete description (1)
- Cut-set inequalities (1)
- Decomposition Method (1)
- Dynamic Network Flows (1)
- Dynamic Robust Flow (1)
The economics of global green ammonia trade – "Shipping Australian wind and sunshine to Germany"
(2023)
This paper contributes to understanding the transformation of global energy trade to green energy carriers, focusing on green ammonia as the foreseeable first green hydrogen carrier. We provide a comprehensive overview of today's ammonia trade and assess scaling options for the trade of green ammonia. To that aim, we develop an optimization model for the integrated assessment of the green ammonia value chain that covers all steps from green ammonia production in an exporting country, up to delivery to a harbor in an importing country. The model endogenously chooses among different technology options and determines cost minimal operation. In a case study, we apply the model to the large-scale import of ammonia from Australia to Germany in a scenario for 2030. The results show that green ammonia can reach cost parity with gray ammonia even for moderate gas prices (but not necessarily with blue ammonia) if CO2 prices are high enough. We also provide a sensitivity analysis with respect to the interest rate and other key technical and economic parameters and show that cracking ammonia to provide pure hydrogen comes at a 45 % cost markup per MWh at the destination.
We investigate the long-time behaviour of solutions of quasilinear hyperbolic systems with transparent boundary conditions when small source terms are incorporated in the system. Even if the finite-time stability of the system is not preserved, it is shown here that an exponential convergence towards the steady state still holds with a decay rate which is proportional to the logarithm of the amplitude of the source term. The result is stated for a system with dynamical boundary conditions in order to deal with initial data that are free of any compatibility condition. The proof of the existence and uniqueness of a solution defined for all positive times is also provided in this paper.
We consider optimal control problems for gas flow in pipeline networks. The equations of motion are taken to be represented by a first-order system of hyperbolic semilinear equations derived from the fully nonlinear isothermal Euler gas equations. We formulate an optimal control problem on a network and introduce a tailored time discretization thereof. In order to further reduce the complexity, we consider an instantaneous control strategy. The main part of the paper is concerned with a nonoverlapping domain decomposition of the optimal control problem on the graph into local problems on smaller sub-graphs - ultimately on single edges. We prove convergence of the domain decomposition method on networks and study the wellposedness of the corresponding time-discrete optimal control problems. The point of the paper is that we establish virtual control problems on the decomposed subgraphs such that the corresponding optimality systems are in fact equal to the systems obtained via the domain decomposition of the entire optimality system.
We discuss coupling conditions for the p-system in case of a transition from supersonic states to subsonic states. A single junction with adjacent pipes is considered where on each pipe the gas ow is governed by a general p-system. By extending the notion of demand and supply known from traffic fiow analysis we obtain a constructive existence result of solutions compatible with the introduced conditions.
Neumann boundary feedback stabilization for a nonlinear wave equation: A strict H2-Lyapunov function
(2017)
For a system that is governed by the isothermal Euler equations with friction for ideal gas, the corresponding field of characteristic curves is determined by the velocity of the flow. This velocity is determined by a second-order quasilinear hyperbolic equation. For the corresponding initial-boundary value problem with Neumann-boundary feedback, we consider non-stationary solutions locally around a stationary state on a finite time interval and discuss the well-posedness of this kind of problem. We introduce a strict H2-Lyapunov function and show that the boundary feedback constant can be chosen such that the H2-Lyapunov function and hence also the H2-norm of the difference between the non-stationary and the stationary state decays exponentially with time.
The development of mathematical simulation and optimization models and algorithms for solving gas transport problems is an active field of research. In order to test and compare these models and algorithms, gas network instances together with demand data are needed. The goal of GasLib is to provide a set of publicly available gas network instances that can be used by researchers in the field of gas transport. The advantages are that researchers save time by using these instances and that different models and algorithms can be compared on the same specified test sets. The library instances are encoded in an XML format. In this paper, we explain this format and present the instances that are available in the library.
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.
We study optimal control problems for linear systems with prescribed initial and terminal states. We analyze the exact penalization of the terminal constraints. We show that for systems that are exactly controllable, the norm-minimal exact control can be computed as the solution of an optimization problem without terminal constraint but with a nonsmooth penalization of the end conditions in the objective function, if the penalty parameter is sufficiently large. We describe the application of the method for hyperbolic and parabolic systems of partial differential equations, considering the wave and heat equations as particular examples. Copyright © 2016 John Wiley & Sons, Ltd.
We consider a vibrating string that is fixed at one end with Neumann control action at the other end. We investigate the optimal control problem of steering this system from given initial data to rest, in time TT, by minimizing an objective functional that is the convex sum of the L2L2-norm of the control and of a boundary Neumann tracking term.
We provide an explicit solution of this optimal control problem, showing that if the weight of the tracking term is positive, then the optimal control action is concentrated at the beginning and at the end of the time interval, and in-between it decays exponentially. We show that the optimal control can actually be written in that case as the sum of an exponentially decaying term and of an exponentially increasing term. This implies that, if the time TT is large, then the optimal trajectory approximately consists of three arcs, where the first and the third short-time arcs are transient arcs, and in the middle arc the optimal control and the corresponding state are exponentially close to 00. This is an example of a turnpike phenomenon for a problem of optimal boundary control. If T=+∞T=+∞ (infinite time horizon problem), then only the exponentially decaying component of the control remains, and the norms of the optimal control action and of the optimal state decay exponentially in time. In contrast to this situation, if the weight of the tracking term is zero and only the control cost is minimized, then the optimal control is distributed uniformly along the whole interval [0,T][0,T] and coincides with the control given by the Hilbert Uniqueness Method.
In addition, we establish a similarity theorem stating that, for every T>0T>0, there exists an appropriate weight λ<1λ<1 for which the optimal solutions of the corresponding finite horizon optimal control problem and of the infinite horizon optimal control problem coincide along the first part of the time interval [0,2][0,2]. We also discuss the turnpike phenomenon from the perspective of a general framework with a strongly continuous semi-group.
In this paper we analyze peak-load pricing in the presence of network constraints. In our setup, firms facing fluctuating demand decide on the size and location of production facilities. They make production decisions constrained by the invested capacities, taking into account that market prices reflect scarce transmission capacities. We state general conditions for existence and uniqueness of the market equilibrium and provide a characterization of equilibrium investment and production. The presented analysis covers the cases of perfect competition and monopoly - the case of strategic firms is approximated by a conjectural variations approach. Our result is a prerequisite for analyzing regulatory policy options with computational multilevel equilibrium models, since uniqueness of the equilibrium at lower levels is of key importance when solving these models. Thus, our paper contributes to an evolving strand of literature that analyzes regulatory policy based on computational multilevel equilibrium models and aims at taking into account individual objectives of various agents, among them not only generators and customers but also, e.g., the regulator deciding on network expansion.
We propose an equilibrium model that allows to analyze the long-run impact of the electricity market design on transmission line expansion by the regulator and investment in generation capacity by private firms in liberalized electricity markets. The model incorporates investment decisions of the transmission system operator and private firms in expectation of an energy-only market and cost-based redispatch. In different specifications we consider the cases of one vs. multiple price zones (market splitting) and analyze different approaches to recover network cost—in particular lump sum, generation capacity based, and energy based fees. In order to compare the outcomes of our multilevel market model with a first best benchmark, we also solve the corresponding integrated planner problem. Using two test networks we illustrate that energy-only markets can lead to suboptimal locational decisions for generation capacity and thus imply excessive network expansion. Market splitting heals these problems only partially. These results are valid for all considered types of network tariffs, although investment slightly differs across those regimes.
Feasibility pumps are highly effective primal heuristics for
mixed-integer linear and nonlinear optimization.
However, despite their success in practice there are only few works
considering their theoretical properties.
We show that feasibility pumps can be seen as alternating
direction methods applied to special reformulations of the original
problem, inheriting the convergence theory of these methods.
Moreover, we propose a novel penalty framework that encompasses
this alternating direction method, which allows us to refrain from random
perturbations that are applied in standard versions of feasibility
pumps in case of failure.
We present a convergence theory for the new penalty based alternating
direction method and compare the new variant of the feasibility
pump with existing versions in an extensive numerical study for
mixed-integer linear and nonlinear problems.
In this work we study polyhedra in the context of network flow problems, where the flow value on each arc lies in one of several predefined intervals. This is motivated by nonlinear
problems on transportation networks, where nonlinearities are handled by piecewise linear approximation or relaxation - a common and established approach in many applications.
Several methods for modeling piecewise linear functions are known which provide a complete description for a single network arc. However, in general this property is lost when considering multiple arcs. We show how to strengthen the formulation for specific substructures consisting of multiple arcs by linear inequalities. For the case of paths of degree-two-nodes we give a complete description of the polyhedron projected to the integer variables. Our model is based on - but not limited to - the multiple choice method; we also show how to transfer our results to a formulation based on the incremental method. Computational results show that a state-of-the-art MIP-solver greatly benefits from using our cutting planes for random and realistic network topologies.
In this work, we present an exact approach for solving network design problems that is based on an iterative graph aggregation procedure. The scheme allows existing preinstalled capacities. Starting with an initial aggregation, we solve a sequence of network design master problems over increasingly fine-grained representations of the original network. In each step, a subproblem is solved that either proves optimality of the solution or gives a directive where to refine the representation of the network in the subsequent iteration. The algorithm terminates with a globally optimal solution to the original problem. Our implementation uses a standard integer programming solver for solving the master problems as well as the subproblems. The computational results on random and realistic instances confirm the profitable use of the iterative aggregation technique. The computing time often reduces drastically when our method is compared to solving the original problem from scratch.
We study dynamic network flows with uncertain input data under a robust optimization perspective. In the dynamic maximum flow problem, the goal is to maximize the flow reaching the sink within a given time horizon T, while flow requires a certain travel time to traverse an arc. In our setting, we account for uncertain travel times of flow. We investigate maximum flows over time under the assumption that at most Γ travel times may be prolonged simultaneously due to delay. We develop and study a mathematical model for this problem. As the dynamic robust flow problem generalizes the static version, it is NP-hard to compute an optimal flow. However, our dynamic version is considerably more complex than the static version. We show that it is NP-hard to verify feasibility of a given candidate solution. Furthermore, we investigate temporally repeated flows and show that in contrast to the non-robust case (i.e., without uncertainties) they no longer provide optimal solutions for the robust problem, but rather yield a worst case optimality gap of at least T. We finally show that for infinite delays, the optimality gap is at most O(k log T), where k is a newly introduced instance characteristic. The results obtained in this paper yield a first step towards understanding robust dynamic flow problems with uncertain travel times.