Optimal power flow: an introduction to predictive, distributed and stochastic control challenges

Abstract

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.

Toward economic NMPC for multistage AC optimal power flow

“…This paper aims at transferring the dissipativity-based framework for economic nmpc summarized in Theorem 1 to generator dispatch problems (or multistage opf problems) arising in power systems. To this end and similar to the work of Faulwasser et al, 19(Section 2) we consider balanced electrical ac grids at steady state modeled by ( , , , Y ), where  = {1, … , N} is the set of buses (nodes),  ⊆  is the nonempty set of generators,  ⊆  is the set of storages/batteries, and Y = G + jB ∈ ℂ N×N is the bus admittance matrix. 16 The off-diagonal entries of Y can be written as −y l m = g l m + jb l m , whereby g l m is the conductance for the line l m, respectively, b l m is the line susceptance.…”

“…This follows from the structure of the power flow equations (7), which implies a physical upper limit for the power that can be transmitted over a line connecting two buses. Excluding such pathological cases, and given the reachability properties documented in Lemma 2, it is evident that one has three main options to enforce that (19) holds: (a) either one ensures that the set is sufficiently "small" (ie, the disturbances cannot vary too much); or (b) one supposes that the sequence d(·) varies slowly (ie, the reachability properties holds for two subsequent values of d(·)); or (c) one ensures that the input constraint is sufficiently "large" (ie, nonrestrictive), respectively, the state constraint is relaxed by adding more storage. Considering the last option, it deserves to be noted that besides the transmission capacity limit of the grid, the state constraints (which include active and reactive power injections by generators) put a limit on the "size" of the disturbance d in terms of power demand.…”

“…* In multistage opf, the aim is to minimize the (monetary) cost of active power generation while satisfying the physical laws of the underlying grid (power flow equations) and operational constraints like generator limits, voltage bounds, and line limits, see other works. [16][17][18][19] The main challenge of opf problems is twofold: practically relevant grid models can easily comprise several hundreds to thousands of nodes and the power flow equations constitute a set of nonlinear equality constraints. In the past, most opf problems (sometimes also called economic dispatch problems in case of simplified grid models) have been considered decoupled in time (ie, single stage) as the influence of available storage has been negligible.…”

Optimal Power Flow Subject to Non‐Gaussian Stochastic Uncertainties: An L2‐based Approach

“…It is a steady-state optimization problem in terms of the power generation set points that minimizes (monetary) costs whilst satisfying the physics-imposed nonlinear and nonconvex network equations-the so-called power flow equations-and engineering limits (e.g. voltage limits, generation limits, line flow flimits) [1,2]. Mathematically the OPF problem is formulated as a nonlinear program that is parameterized in terms of uncontrollable active and reactive power demand/injections, i.e.…”

“…consumer loads and or uncontrollable injection stemming from solar panels. 1 Traditionally, fixed values are assumed for those uncontrollable powers, leading to deterministic optimal values for the generation set points (assuming the solution to the OPF problem exists). However, the massive installation of renewable energy sources in combination with load fluctuations necessitates to consider uncertain uncontrollable powers.…”