Friedrich-Alexander-Universität Erlangen-Nürnberg
Refine
Year of publication
Keywords
- Bilevel optimization (8)
- Optimal control (7)
- Gas networks (6)
- Uniqueness (6)
- Convergence (5)
- Networks (5)
- Robust optimization (5)
- robust optimization (5)
- Branch-and-cut (4)
- Mixed-Integer Nonlinear Optimization (4)
- Mixed-Integer Nonlinear Programming (4)
- Mixed-integer optimization (4)
- Mixed-integer programming (4)
- Multilevel Optimization (4)
- optimal control (4)
- A posteriori error estimates (3)
- Alternating direction methods (3)
- Gas Networks (3)
- Linear complementarity problems (3)
- Mixed-integer nonlinear optimization (3)
- Perfect competition (3)
- Pricing (3)
- Robust Optimization (3)
- Time-domain decomposition (3)
- Turnpike (3)
- Turnpike phenomenon (3)
- Adjustable Robustness (2)
- Big-M (2)
- Booking (2)
- Bookings (2)
- Combinatorial optimization (2)
- Complementarity constraints (2)
- Computational complexity (2)
- Electricity market equilibrium models (2)
- European entry-exit market (2)
- Existence (2)
- Gas Market (2)
- Gas transport networks (2)
- Global Optimization (2)
- Graph Partitioning (2)
- Lipschitz Optimization (2)
- Market Equilibria (2)
- Mixed-integer linear optimization (2)
- Multilevel optimization (2)
- Multiplicity (2)
- Network design (2)
- Nodal Control (2)
- Penalty methods (2)
- Piecewise Linearization (2)
- Potential networks (2)
- Potential-based flows (2)
- Probabilistic Constraints (2)
- Semilinear hyperbolic systems (2)
- Space- and time-domain decomposition (2)
- chance constrained programming (2)
- chance constraints (2)
- gas networks (2)
- isothermal Euler equations (2)
- linear hyperbolic systems (2)
- optimal nodal control (2)
- optimal power flow (2)
- p-Laplace problem on a graph (2)
- 1-d wave equations (1)
- 28 (1)
- AC Optimal Power Flow (1)
- Active elements (1)
- Adaptive error control (1)
- Aggregation (1)
- Applications (1)
- Averaged Control (1)
- Averaged controllability, averaged observability, observability, random heat equation (1)
- BV-coefficients (1)
- Benders Decomposition (1)
- Bilevel Optimization (1)
- Black-box Optimization (1)
- Black-box constraints (1)
- Boundary Control (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)
- Bounding polyhedra (1)
- Branch-and-bound (1)
- Chance constraints (1)
- Classical Solutions (1)
- Clique Problem (1)
- Competitive equilibrium (1)
- Complementarity Constraints (1)
- Complete description (1)
- Computational Analysis (1)
- Computational Complexity (1)
- Computational Equilibrium Models; Electricity Markets; Investment Incentives; Distribution Network Expansion Planning; Storage Investment and Operation; Renewable Energy Production; (Self-) Consumption (1)
- Computational analysis (1)
- Connectivity (1)
- Continuous Optimization (1)
- Control Constraint (1)
- Convex Envelope (1)
- Convex mixed-integer nonlinear optimization (1)
- Convexity (1)
- Cut-set inequalities (1)
- Cutting planes (1)
- Cycled Networks (1)
- DC Power Flow (1)
- Decomposition (1)
- Decomposition Method (1)
- Decomposition Methods (1)
- Demand side flexibility Load management Multi market bidding Stochastic programming Production planning Demand response (1)
- Derivative-free Optimization (1)
- Differential-algebraic equations (1)
- Direct methods (1)
- Dirichlet boundary control (1)
- Discrete Decisions (1)
- Discrete arc sizing (1)
- District heating networks (1)
- Domain decomposition (1)
- Dynamic Network Flows (1)
- Dynamic Robust Flow (1)
- Electric fuels, Hydrogen Utilization, Hydrogen Import, LOHC, Mobility (1)
- Electricity Market Design} (1)
- Electricity Markets (1)
- Electricity Markets, Network Expansion, Generation Expansion, Investment Incentives, Computational Equilibrium Models (1)
- Electricity Markets, Network Expansion, Generation Expansion, Investment Incentives, Market Design, Congestion Management, Computational Equilibrium Models (1)
- Electricity Markets, Redispatch, Congestion Management, Computational Equilibrium Models (1)
- Electricity Networks (1)
- Electricity System (1)
- Electricity market design (1)
- Electricity market modeling (1)
- Electricity tariffs (1)
- Ellipsoidal uncertainty sets (1)
- Energy Markets (1)
- Energy Networks (1)
- Energy markets (1)
- Energy storage, Consumer segmentation, Adoption decision, Innovation diffusion (1)
- Entry-Exit Gas Market (1)
- Entry-Exit System (1)
- Entry-exit system (1)
- Equilibrium Problems (1)
- Equilibrium computation (1)
- Euler and St. Venant equations (1)
- Euler equations (1)
- European Entry-Exit Gas Market (1)
- European gas market (1)
- Exact control (1)
- Exact controllability (1)
- Extended Incremental Method (1)
- First-Discretize-Then-Optimize (1)
- Flow Networks (1)
- Flow models (1)
- Fundamental Welfare Theorems (1)
- Galerkin projection (1)
- Game Theory (1)
- Game theory (1)
- Gas Dynamics (1)
- Gas Market Design (1)
- Gas Network Optimization (1)
- Gas Pipelines (1)
- Gas Transport (1)
- Gas Transport Networks (1)
- Gas markets (1)
- Gas network optimization (1)
- GasLib (1)
- Generalized Benders Decomposition (1)
- Global classical solutions Ideal gas Bi-directional flow Transsonic flow (1)
- Graph decomposition (1)
- Graph partitioning (1)
- Hardness (1)
- Heat equations with memory (1)
- Hyperbolic Balance Laws, Stabilization, Exact Controllability, Modeling of Gas Flow, Finite-Volume Schemes, Optimal control, Uncertainty (1)
- Illiquid investments (1)
- Infinite-time admissibility (1)
- Instantaneous control (1)
- Inverse parametric quadratic programming (1)
- Investment Incentives (1)
- Isothermal Euler Equations (1)
- Isothermal stationary Euler equations (1)
- Kernel Density Estimator (1)
- LQ optimal control problem (1)
- Linear Bilevel Optimization (1)
- Linear Complementarity Problems (1)
- Linear prices (1)
- Lipschitz continuity (1)
- MINLP (1)
- MPCC (1)
- MPEC (1)
- Market power (1)
- Marking strategy (1)
- Mathematical programs with complementarity constraints (MPCC) (1)
- Max-cut (1)
- Maximum flow problem (1)
- Measure Turnpike (1)
- Measure turnpike (1)
- Mixed-Integer (Non)Linear Programming (1)
- Mixed-Integer Nonlinear Optimization, Global Optimization, Lipschitz Optimization, Gas Networks (1)
- Mixed-Integer Nonlinear Optimization, Lipschitz Optimization, Inexact Function Evaluations, Decomposition Methods, Gas Networks (1)
- Mixed-Integer Optimization (1)
- Mixed-integer bilevel optimization (1)
- Mixed-integer nonlinear programming (1)
- Mixed-integer optimal control (1)
- Mixed-integer optimal control problems (1)
- Mixing (1)
- Model hierarchy (1)
- Multi-leader multi-follower game (1)
- Multilevel Modeling (1)
- Multilevel programming (1)
- Multiple-Choice Constraints (1)
- Multistage stochastic optimization (1)
- NP-Hardness (1)
- Nash Games (1)
- Nash-Cournot competition (1)
- Nash-Cournot equilibria (1)
- Natural Gas Markets (1)
- Natural Gas System (1)
- Neighboring Markets (1)
- Network Expansion (1)
- Network expansion (1)
- Network flow problems (1)
- Network reduction (1)
- Neumann boundary control (1)
- Neural ODEs (1)
- Nodal Stabilization (1)
- Nomination (1)
- Nonconvex Equilibrium Models (1)
- Nonconvex games (1)
- Nonlinear optimization (1)
- OR in energy (1)
- Optimal Transport (1)
- Optimal control, Gas networks, Euler's equation, Semilinear PDE, Nonoverlapping domain decomposition (1)
- Optimal shape design (1)
- Optimal value function (1)
- Optimality system (1)
- Optimization (1)
- Optimization under uncertainty (1)
- Optimization with Differential Equations (1)
- Optimization with differential equations (1)
- Outer Approximation (1)
- Outer approximation (1)
- PDE constrained optimization (1)
- PDE constrained optimization, probabilistic constraints, uncertain initial data (1)
- PDEs on graphs (1)
- Parabolic equation (1)
- Parametric optimization (1)
- Partial differential equations on graphs (1)
- Peak-Load Pricing (1)
- Peak-load pricing (1)
- Penalty method (1)
- Perfect Competition (1)
- Piecewise Linear Relaxation (1)
- Piecewise linear functions (1)
- Pilot projects, Organizational setup, Adoption decision, Quadruple Helix, Innovation diffusion (1)
- Policy risk (1)
- Polyhedral demand uncertainty (1)
- Portfolio optimization (1)
- Potential-Based Flows (1)
- Potential-based flows, Gas networks, Computational complexity, European entry-exit market, Bookings (1)
- Power-to-Gas (1)
- Presolve (1)
- Price Zones (1)
- Primal Heuristic (1)
- Probabilistic Robustness (1)
- Problem Instances (1)
- Quadratic programming (1)
- R-vine copula (1)
- Random Boundary Data (1)
- Reformulations (1)
- Regional Pricing (1)
- Renewables (1)
- Riccati equations (1)
- Risk Aversion (1)
- Robust Optimization, Mixed-integer programming, Uncertainty sets, Robust feasibility (1)
- Robust market equilibria (1)
- Robust network design (1)
- SOS-1 (1)
- Scenario generation (1)
- Scheduling (1)
- Second-Order Cone Programming (1)
- Sector Coupling (1)
- Sector Coupling, Eletricity Markets, Electric fuels, Hydrogen Utilization, LOHC, Mobility (1)
- Semidefinite Programming (1)
- Separation under uncertainty (1)
- Series-parallel graphs (1)
- Short- and Long-Run Market Equilibrium (1)
- Simulation Based Optimization (1)
- Simultaneous Convexification (1)
- Spheric Radial Decomposition (1)
- State constraint (1)
- Stationary points (1)
- Stochastic Programming (1)
- Storage Capacity Maximization (1)
- Strict dissipativity (1)
- Supply chain (1)
- Tariff design (1)
- Technical Capacities (1)
- Total Unimodularity (1)
- Total unimodularity (1)
- Tractable counterparts (1)
- Traffic equilibrium problems (1)
- Traffic flow (1)
- Transient Gas Transport Optimization (1)
- Transmission and generation investment (1)
- Transmission management (1)
- Transport Costs (1)
- Transport Equation (1)
- Trees (1)
- Uncertain Travel Times (1)
- Unit Commitment with AC Power Flow (1)
- Valid inequalities (1)
- Validation of nominations, Stationary states, Isothermal Euler equations, Gas networks, Convexity, Star-Shapedness, Spheric-radial decomposition (1)
- Validation of nominations, Stationary states, Isothermal Euler equations, Ideal Gas, Gas network, Gas transport, Compressor control, Uncertainty, Chance Constraints, Spheric-radial decomposition (1)
- ammonia trade (1)
- boundary control (1)
- case study Australia to Germany (1)
- chance constraint (1)
- conditional uncertainty set (1)
- conservation laws on network (1)
- coupling conditions (1)
- data classification (1)
- decision trees (1)
- decision-dependent uncertainty (1)
- decomposition of the flow (1)
- deep learning (1)
- delay (1)
- descriptor systems (1)
- distributed parameter system (1)
- distributionally robust optimization (1)
- dynamic control (1)
- energy decay (1)
- exact control (1)
- exponential stability (1)
- finite time blow-up (1)
- gas network (1)
- gas pipeline flow (1)
- gradient descent (1)
- green ammonia (1)
- here-and-now decision (1)
- hierarchy of models (1)
- hybrid parabolic-hyperbolic behavior (1)
- hyperbolic differential equation (1)
- hyperbolic system (1)
- instantaneous control (1)
- learning over time (1)
- linear systems (1)
- long time behavior (1)
- machine learning (1)
- mixed-integer programming (1)
- multilevel (1)
- network (1)
- network modelling (1)
- networks (1)
- nodal control (1)
- nodal profile con-trol (1)
- nonlinear stochastic optimization (1)
- nonlocal conservation laws (1)
- nonoverlapping omain decomposition, optimal control of semi-linear ellioptic systems on netowrks (1)
- optimization model (1)
- p-parabolic problems (1)
- p-system (1)
- partial differential-algebraic equations (1)
- pipes (1)
- polynomial optimization (1)
- port-Hamiltonian systems (1)
- power networks (1)
- quasilinear hyperbolic problem (1)
- random demand (1)
- random forests (1)
- real gas (1)
- relaxations (1)
- renewable energy auction, combinatorial auction, market design (1)
- shape optimization (1)
- simultaneous control (1)
- spheric-radial decomposition (1)
- spherical radial decomposition (1)
- state constraints (1)
- stationary gas transport (1)
- stochastic gradient method (1)
- systems of hyperbolic pdes (1)
- topological derivative (1)
- traveling waves (1)
- turnpike property (1)
- wave equation (1)
- Γ-robustness (1)
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.
Pareto efficiency for robust linear programs was introduced by
Iancu and Trichakis in [9]. We generalize their approach and theoretical results
to robust optimization problems in Euclidean spaces with linear uncertainty.
Additionally, we demonstrate the value of this approach in an exemplary
manner in the area of robust semidefinite programming (SDP). In particular,
we prove that computing a Pareto robustly optimal solution for a robust SDP
is tractable and illustrate the benefit of such solutions at the example of the
maximal eigenvalue problem. Furthermore, we modify the famous algorithm of
Goemans and Williamson [8] in order to compute cuts for the robust max cut
problem that yield an improved approximation guarantee in non-worst-case
scenarios.
We present a solution framework for general alternating current optimal power flow (AC OPF) problems that include discrete decisions.
The latter occur, for instance, in the context of the curtailment of renewables or the
switching of power generation units and transmission lines.
Our approach delivers globally optimal solutions and is provably convergent.
We model AC OPF problems with discrete decisions as mixed-integer nonlinear programs.
The solution method starts from a known framework that uses piecewise linear relaxations.
These relaxations are modeled as as mixed-integer linear programs and adaptively refined until some termination criterion is fulfilled.
In this work, we extend and complement this approach by problem-specific as well as very general algorithmic enhancements.
In particular, these are mixed-integer second-order cone programs as well as primal and dual cutting planes.
For example objective cuts and no-good-cuts help to compute good feasible solutions as where outer approximation constraints tighten the relaxations.
We present extensive numerical results for various AC OPF problems where discrete decisions play a major role.
Even for hard instances with a large proportion of discrete decisions, the method is able
to generate high quality solutions efficiently.
Furthermore, we compare our approach with state-of-the-art MINLP.
Our method outperforms all other algorithms.
Stochastic Optimization (SO) is a classical approach for optimization under uncertainty that typically requires knowledge about the probability distribution of uncertain parameters. As the latter is often unknown, Distributionally Robust Optimization (DRO) provides a strong alternative that determines the best guaranteed solution over a set of distributions (ambiguity set). In this work, we present an approach for DRO over time that uses online learning and scenario observations arriving as a data stream to learn more about the uncertainty. Our robust solutions adapt over time and reduce the cost of protection with shrinking ambiguity. For various kinds of ambiguity sets, the robust solutions converge to the SO solution. Our algorithm achieves the optimization and learning goals without solving the DRO problem exactly at any step. We also provide a regret bound for the quality of the online strategy which converges at a rate of $ O(\log T / \sqrt{T})$, where $T$ is the number of iterations. Furthermore, we illustrate the effectiveness of our procedure by numerical experiments on mixed-integer optimization instances from popular benchmark libraries and give practical examples stemming from telecommunications and routing. Our algorithm is able to solve the DRO over time problem significantly faster than standard reformulations.
We propose a mathematical optimization model and its solution for joint chance constrained DC Optimal Power Flow. In this application, it is particularly important that there is a high probability of transmission limits being satisfied, even in the case of uncertain or fluctuating feed-in from renewable energy sources. In critical network situations where the network risks overload, renewable energy feed-in has to be curtailed by the transmission system operator (TSO). The TSO can reduce the feed-in in discrete steps at each network node. The proposed optimization model minimizes curtailment while ensuring that there is a high probability of transmission limits being maintained. The latter is modeled via (joint) chance constraints that are computationally challenging. Thus, we propose a solution approach based on the robust safe approximation of these constraints. Hereby, probabilistic constraints are replaced by robust constraints with suitably defined uncertainty sets constructed from historical data. The uncertainty sets are calculated by encompassing randomly drawn scenarios using the scenario approach proposed by Margellos et al. (IEEE Transactions on Automatic Control, 59 (2014)). The ability to discretely control the power feed-in then leads to a robust optimization problem with decision-dependent uncertainties, i.e. the uncertainty sets depend on decision variables. We propose an equivalent mixed-integer linear reformulation for box uncertainties with the exact linearization of bilinear terms. Finally, we present numerical results for different test cases from the Nesta archive, as well as for a real network. We consider the discrete curtailment of solar feed-in, for which we use real-world weather and network data. The experimental tests demonstrate the effectiveness of this method and run times are very fast. Moreover, on average the calculated robust solutions lead only to a small increase in curtailment, when compared to nominal solutions.
Robust DC Optimal Power Flow with Modeling of Solar Power Supply Uncertainty via R-Vine Copulas
(2021)
We present a robust approximation of joint chance constrained DC Optimal Power Flow in combination with a model-based prediction of uncertain power supply via R-vine copulas.
It is applied to optimize the discrete curtailment of solar feed-in in an electrical distribution network and guarantees network stability under fluctuating feed-in.
This is modeled by a two-stage mixed-integer stochastic optimization problem proposed by Aigner et al. (European Journal of Operational Research, (2021)).
The solution approach is based on the approximation of chance constraints via robust constraints using suitable uncertainty sets.
The resulting robust optimization problem has a known equivalent tractable reformulation.
To compute uncertainty sets that lead to an inner approximation of the stochastic problem, an R-vine copula model is fitted to the distribution of the multi-dimensional power forecast error, i.e., the difference between the forecasted solar power and the measured feed-in at several network nodes.
The uncertainty sets are determined by encompassing a sufficient number of samples drawn from the R-vine copula model.
Furthermore, an enhanced algorithm is proposed to fit R-vine copulas which can be used to draw conditional samples for given solar radiation forecasts.
The experimental results obtained for real-world weather and network data demonstrate the effectiveness of the combination of stochastic programming and model-based prediction of uncertainty via copulas.
We improve the outcomes of previous work by showing that the resulting uncertainty sets are much smaller and lead to less conservative solutions while maintaining the same probabilistic guarantees.
Ongoing policy discussions on the reconfiguration of bidding zones in European electricity markets induce uncertainty about the future market design. This paper deals with the question of how this uncertainty affects market participants and their long-run investment decisions in generation and transmission capacity. Generalizing the literature on pro-active network expansion planning, we propose a stochastic multilevel model which incorporates generation capacity investment, network expansion, and market operation, taking into account uncertainty about the future bidding zone configuration. Using a stylized two-node network, we disentangle different effects that uncertainty has on market outcomes. If there is a possibility that future bidding zone configurations provide improved regional price signals, welfare gains materialize even if the change does not actually take place. As a consequence, welfare gains of an actual change of the bidding zone configuration are substantially lower due to those anticipatory effects. Additionally, we show substantial distributional effects in terms of both expected gains and risks, between producers and consumers and between different generation technologies.
In the course of the energy transition, load and supply centers are growing apart in electricity markets worldwide, rendering regional price signals even more important to provide adequate locational investment incentives. This paper focuses on electricity markets that operate under a zonal pricing market design. For a fixed number of zones, we endogenously derive the optimal configuration of price zones and available transfer capacities on a network in order to optimally govern investment and production decisions in the long run. In a multilevel mixed-integer nonlinear model that contains a graph partitioning problem on the first level, we determine welfare-maximizing price zones and available transfer capacities for a given electricity market and analyze their impact on market outcomes. Using a generalized Benders decomposition approach developed in Grimm et al. (2019) and a problem-tailored scenario clustering for reducing the input data size, we are able to solve the model to global optimality even for large instances. We apply the approach to the German electricity market as an example to examine the impact of optimal zoning on key performance indicators such as welfare, generation mix and locations, or electricity prices. It turns out that even for a small number of price zones, an optimal configuration of zones induces a welfare level that almost approaches the first best.
We study gas network problems with compressors and control valves under uncertainty that can be formulated as two-stage robust optimization problems. Uncertain data are present in the physical parameters of the pipes as well as in the overall demand. We show how to exploit the special decomposable structure of the problem in order to reformulate the two-stage robust problem as a standard single-stage optimization problem. Since this structure is present in similar problems on e.g., water or direct current electricity networks, we investigate the consequences of the decomposable structure in an abstract setting: The right-hand side of the single-stage problem can be precomputed by solving a series of optimization problems and multiple elements of the right-hand side can be combined into one optimization task. In order to apply our results to gas network problems, we extend piecewise relaxations and preprocessing techniques to incorporate uncertain input data. The practical feasibility and effectiveness of our approach is demonstrated with benchmarks on realistic gas network instances. We observe large speedups due to the described aggregation method together with the developed preprocessing strategies. Furthermore, we are able to solve even comparably large gas network instances quickly for the price of slightly more conservative solutions.
In this paper we study feasibility and infeasibility of nonlinear two-stage
fully adjustable robust feasibility problems with an empty first stage. This
is equivalent to deciding set containment of a projection of the feasible
region and the uncertainty set. For answering this question, two very general
approaches using methods from polynomial optimization are presented --- one
for showing feasibility and one for showing infeasibility. The developed
methods are approximated through sum of squares polynomials and solved using
semidefinite programs.
Deciding robust feasibility and infeasibility is important for gas network
operations, which is a \nonconvex quadratic problem with absolute values
functions. Concerning the gas network problem, different topologies are
considered. It is shown that a tree structured network can be decided exactly
using linear programming. Furthermore, a method is presented to reduce a tree
network with one additional arc to a single cycle network. In this case,
removing the absolute values and solving the problem can be decided with
linearly many polynomial optimization problems.
Lastly, the effectivity of the methods is tested on a variety of small cyclic
networks. For instances where robust feasibility or infeasibility can be
decided, level~2 or level~3 of the Lasserre relaxation hierarchy is typically
sufficient.