Friedrich-Alexander-Universität Erlangen-Nürnberg
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.
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.
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.
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.