We propose a parallel solver for linear systems of equations arising from the application of Primal Dual Interior Point methods to Dynamic Optimal Power Flow problems. Our solver is based on the Generalised Minimal Residual method in combination with an additive Schwarz domain decomposition method as preconditioner. This preconditioner exploits the structure of Dynamic Optimal Power Flow problems which, after linearization, is given as a matrix with large diagonal blocks and only a few off-diagonal elements. These elements correspond to intertemporal couplings due to ramping and energy storage constraints and are partially neglected in order to solve the problem in parallel. We test our method on a large-scale optimisation problem and show that a parallel speedup can be obtained.

In times of growing integrated electricity markets and needed coordination of large interregional physical power flows, multi-area Optimal Power Flow (OPF), also referred to as distributed OPF, has gained importance in research. However, the conventional OPF is only of limited use since a TSO is strongly interested in N-1 security. Furthermore, time-dependent constraints such as generator ramping or energy storage limits play a growing role. Consequently, a Security-Constrained Dynamic OPF (SC-D-OPF) includes both N-1 security and quasi-stationary dynamics. We present a decoupling approach to compute an SC-D-OPF by coordination among interconnected areas. Privacy is maintained by implementing an Alternating Direction of Multipliers Method (ADMM), where only results of boundary variables are exchanged with a neighbor. We show the functionality of the approach in a small test case, where the distributed result is close to that of a centralized optimization.

The Vlasov–Fokker–Planck (VFP) equation, a variant of the Boltzmann equation, is frequently used to model the dynamics of laboratory and astrophysical plasmas. A common approach in solving the VFP equation in numerical and analytic studies is to expand the momentum part of the distribution function $f$ (i.e. the particle density in phase-space) in terms of known functions. The Cartesian tensor and spherical harmonic expansion have been widely used, leading to the question of how to convert between the different coefficients of the two expansions. This problem is also familiar in multipole expansions of an electrostatic (or gravitational) potential. The coefficients of the Cartesian tensor expansion of the potential are called (Cartesian) multipole moments and the ones of the spherical harmonic expansion are called spherical multipole moments. In this paper, we investigate the relation between the two kinds of multipole moments and provide a general formalism to convert between them. We subsequently apply this formalism to the coefficients of the expansions of the distribution function $f$ . A free, open-source command-line tool which implements this formalism is provided.

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