# Berlin 2015 – wissenschaftliches Programm

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# SOE: Fachverband Physik sozio-ökonomischer Systeme

## SOE 20: Energy Systems (joint session DY/ AK Energy / SOE)

### SOE 20.5: Vortrag

### Donnerstag, 19. März 2015, 10:30–10:45, BH-N 243

**Large-deviation study of the maximum-disturbance stability of power grids** — •Alexander K. Hartmann^{1}, Timo Dewenter^{1}, Wiebke Heins^{2}, and Benjamin Werther^{2} — ^{1}Institut of Physics, University of Oldenburg — ^{2}Institut for Electrical Energy Technology, Technical University of Clausthal

We study numerically the distribution
of “maximum-disturbance” stability of power grids. The model is
based on networks of oscillators. Here, we consider
different ensembles of random networks,
like standard Erdös-Renyi and two dimensional spacial networks.
To access the distribution down to very small probabilities,
we use specific large deviation techniques [1]. The stability is given
by a conservative estimation of an asymptotic
stability boundary, which is well known in stability theory [2,3].
The starting point is the matrix ** A** defined by

*J*^{T}

**+**

*A*

*A***=**

*J***,**

*E***being the Jacobean Matrix. By calculating the maximum disturbance of**

*J***, which results in the quadratic form**

*x**V*=

*x*^{T}

**=є(**

*A**x***) not being a Lyapunov-function of the system any longer, the boundaries for the stability can be found.**

*x*For comparsion, for the given networks also simple stability measures beased on shortest paths [4], on the eigenvalues of the Jacobi matrix and on a linearized power-flow model [5] are obtained.

[1] A.K. Hartmann, Eur. Phys. J. B **84**,
627-634 (2011)

[2] R. Unbehauen, Systemtheorie (Vol. 2), Oldenbourg, Munich (1998)

[3] E.J. Davison and E.M. Kurak, Automatica **7**, 627-636 (1971)

[4] A.K. Hartmann, Eur. Phys. J. B **87**, 114 (2014)

[5] T. Dewenter and A.K. Hartmann, preprint arXiv:1411.5233 (2014)