Tillmann Mühlpfordt

Alexander Engelmann, Yuning Jiang, Tillmann Mühlpfordt et al.

The present paper discusses the application of the recently proposed Augmented Lagrangian Alternating Direction Inexact Newton (ALADIN) method to non-convex AC Optimal Power Flow Problems (OPF) in a distributed fashion. In contrast to the often used Alternating Direction of Multipliers Method (ADMM), ALADIN guarantees locally quadratic convergence for AC OPF. Numerical results for 5 to 300 bus test cases indicate that ALADIN is able to outperform ADMM and to reduce the number of iterations by about one order of magnitude. We compare ALADIN to numerical results for ADMM documented in the literature. The improved convergence speed comes at the cost of increasing the communication effort per iteration. Therefore, we propose a variant of ALADIN that uses inexact Hessians to reduce communication. Additionally, we provide a detailed comparison of these ALADIN variants to ADMM from an algorithmic and communication perspective. Moreover, we prove that ALADIN converges locally at quadratic rate even for the relevant case of suboptimally solved local NLPs.

Tillmann Mühlpfordt, Rolf Findeisen, Veit Hagenmeyer et al.

Methods based on polynomial chaos expansion allow to approximate the behavior of systems with uncertain parameters by deterministic dynamics. These methods are used in a wide range of applications, spanning from simulation of uncertain systems to estimation and control. For practical purposes the exploited spectral series expansion is typically truncated to allow for efficient computation, which leads to approximation errors. Despite the Hilbert space nature of polynomial chaos, there are only a few results in the literature that explicitly discuss and quantify these approximation errors. This work derives error bounds for polynomial chaos approximations of polynomial and non-polynomial mappings. Sufficient conditions are established, which allow investigating the question whether zero truncation errors can be achieved and which series order is required to achieve this. Furthermore, convex quadratic programs, whose argmin operator is a special case of a piecewise polynomial mapping, are studied due to their relevance in predictive control. Several simulation examples illustrate our findings.

Tillmann Mühlpfordt, Timm Faulwasser, Veit Hagenmeyer

Deregulated energy markets, demand forecasting, and the continuously increasing share of renewable energy sources call---among others---for a structured consideration of uncertainties in optimal power flow problems. The main challenge is to guarantee power balance while maintaining economic and secure operation. In the presence of Gaussian uncertainties affine feedback policies are known to be viable options for this task. The present paper advocates a general framework for chance-constrained OPF problems in terms of continuous random variables. It is shown that, irrespective of the type of distribution, the random-variable minimizers lead to affine feedback policies. Introducing a three-step methodology that exploits polynomial chaos expansion, the present paper provides a constructive approach to chance-constrained optimal power flow problems that does not assume a specific distribution, e.g. Gaussian, for the uncertainties. We illustrate our findings by means of a tutorial example and a 300-bus test case.

Timm Faulwasser, Alexander Engelmann, Tillmann Mühlpfordt et al.

The Energiewende is a paradigm change that can be witnessed at latest since the political decision to step out of nuclear energy. Moreover, despite common roots in Electrical Engineering, the control community and the power systems community face a lack of common vocabulary. In this context, this paper aims at providing a systems-and-control specific introduction to optimal power flow problems which are pivotal in the operation of energy systems. Based on a concise problem statement, we introduce a common description of optimal power flow variants including multi-stage-problems and predictive control, stochastic uncertainties, and issues of distributed optimization. Moreover, we sketch open questions that might be of interest for the systems and control community.

Tillmann Mühlpfordt, Veit Hagenmeyer, Timm Faulwasser

The operation of power systems has become more challenging due to feed-in of volatile renewable energy sources. Chance-constrained optimal power flow (ccOPF) is one possibility to explicitly consider volatility via probabilistic uncertainties resulting in mean-optimal feedback policies. These policies are computed before knowledge of the realization of the uncertainty is available. On the other hand, the hypothetical case of computing the power injections knowing every realization beforehand---called in-hindsight OPF(hOPF)---cannot be outperformed w.r.t. costs and constraint satisfaction. In this paper, we investigate how ccOPF feedback relates to the full-information hOPF. To this end, we introduce different dimensions of the price of uncertainty. Using mild assumptions on the uncertainty we present sufficient conditions when ccOPF is identical to hOPF. We suggest using the total variational distance of probability densities to quantify the performance gap of hOPF and ccOPF. Finally, we draw upon a tutorial example to illustrate our results.