Power networks are typically modeled via large (port-Hamiltonian) systems of delay
differential-algebraic equations (DAEs). An important task in the
analysis of power grids is the analysis of the stability
of the system if it is subjected to small changes in the sense that a small number of
vertices in the network graph is added or removed. In applications, this corresponds to
network extensions or the interruption of power lines by
damage or theft. Major practical questions are the stability of the network under such changes,
the behavior under high loads or under the sudden occurrence of impulsive solutions which
may trigger a cascadic shut-down of the grid.
In the first phase of the project, the focus has been on the perturbation
theory of structured matrices and pencils. Filling an existing gap in the literature, the
effect of real generic structured rank-one perturbations has been investigated
in [6] for the case that the matrices under consideration are
real. Furthermore, in [2] it could be shown for many classes of structured matrices
that a generic structure-preserving low rank perturbation can be interpreted as
a sequence of consecutive rank-one perturbations. With this result, the theory
of generic structure-preserving low rank perturbation of structured matrices is
now well understood.
Generalizations of these results to matrix pencils have been undertaken in
[7] and [1]. Special emphasis was given to
perturbations that perturb only one of the two matrix coefficients as this case
is relevant for the power network stability application. In these cases, it may
also happen that the perturbations make the pencil under consideration singular
which leads to major difficulties in numerical simulation methods. Therefore,
the case of singular pencils was investigated in [10] in order to
understand the behavior of network systems that are close to being singular.
Moreover, structured backward errors for eigenvalues of matrix pencils and
polynomials were developed in [3]. With these results it is
possible to trace the movement of eigenvalues under structure-preserving versus
general perturbations. Currently the local perturbation analysis for
nonlinear non-stationary port-Hamiltonian is investigated.
The distance to instability for port-Hamiltonian systems (arising from
power grids, electrical circuits, or damped mechanical systems) under
structure preserving perturbations has been in analyzed in [11].
In order to develop a more detailed understanding of the dynamics in electrical circuits,
a new decoupling concept was developed in [13]. The dissection concept allows for
a decoupling of nonlinear circuit equations by constant projections based on the network
topology into the inherent dynamic part and the algebraic part.
In the second phase of the project the focus is on relating the dynamical
impact of structural perturbations to the coupling topology of the underlying
network. The stability of synchronization in coupled systems of ODEs is known
for certain classes of networks. First investigations have shown that the well
developed algebraic graph theory can be employed in order to infer general stability results for
dynamic networks. For instance, it was shown that adding vertices in undirected
networks increases the synchronizability. However, in directed networks, one
can observe the counter-intuitive phenomenon of a stability loss upon the
addition of a directed vertex [4,8,14]. So far,
this approach has never been used in the context of power systems. However, our
first results show that it can be extended to this case. The interest of this
strategy is two-fold. First, it involves a generalization of algebraic graph
theory to graphs with different types of vertices. Second, it allows for a
classification of vertices according to their dynamical impact in the
corresponding DAE. This will enable us to improve the design and control of
power-grids.