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Humboldt-Universität zu Berlin

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Institut für Mathematik / Projects / C-SE3 Stability analysis of power networks and power network models

## C-SE3 Stability analysis of power networks and power network models

 Project heads Christian Mehl, Volker Mehrmann, Caren Tischendorf Staff Philipp Pade, Punit Scharma, Barbara Scherlein Duration June 2014 - May 2017

This research is carried out in the framework of Matheon supported by Einstein Foundation Berlin.

### Project Background and Challenges

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.

### Results

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.

### Publications

 [1] L. Batzke. Sign characteristics of regular Hermitian matrix pencils under generic rank-1 and rank-2 perturbations. Electron. J. Linear Algebra, Volume 30, pp. 760--794, 2015. [DOI] [2] L. Batzke, C. Mehl, A. Ran and L. Rodman. Generic rank-$k$ perturbations of structured matrices. Technical Report 1078, Research Center Matheon, TU Berlin, 2015. To appear in Oper. Theory Adv. Appl. [preprint] [3] S. Bora, M. Karow, C. Mehl and P. Sharma. Structured Eigenvalue Backward Errors of Matrix Pencils and Polynomials with Palindromic Structures. SIAM Journal on Matrix Analysis and Applications, Volume 36:2, pp. 393--416, 2015. [DOI] [4] J.D. Hart, J.P. Pade, T. Pereira, T.E. Murphy and R. Roy. Adding connections can hinder network synchronization of time-delayed oscillators. Physical Review E , Volume 92:2, 2015. [DOI] [5] L. Jansen and C. Tischendorf. A Unified (P)DAE Modeling Approach for Flow Networks. Progress in DAEs, Springer book, pp. 127--151, 2014. [DOI] [6] C. Mehl, V. Mehrmann, A.C.M. Ran and L. Rodman. Eigenvalue perturbation theory of structured real matrices and their sign characteristics under generic structured rank-one perturbations. Linear and Multilinear Algebra, Volume 64:3, pp. 527--556, 2016. [DOI] [7] L. Batzke. Generic rank-two perturbations of structured regular matrix pencils. Operators and Matrices, Volume 10:1, pp. 83--112, 2016 [pdf] [8] J.P. Pade and T. Pereira. Improving the network structure can lead to functional failures. Nature Scientific Reports, Volume 5, 2015 [pdf]

### Proceedings

 [9] C. Tischendorf. Regularization of Electrical Circuits. 8th Vienna International Conferenceon Mathematical Modelling, IFAC-PapersOnLine, Volume 48, Issue 1, pp. 312-313, 2015. [DOI]

### Submitted articles

 [10] C. Mehl, V. Mehrmann and M. Wojtylak. Parameter-dependent rank-one perturbations of singular Hermitian or symmetric pencils. Technical Report 1371, Research Center Matheon, TU Berlin, 2016 [preprint] [11] C. Mehl, V. Mehrmann, and P. Sharma. Structured distances to instability for linear Hamiltonian systems with dissipation Technical Report 1377, Research Center Matheon, TU Berlin, 2016 [preprint]

### PhD Theses

 [12] L. Batzke. Generic low rank perturbations of structured matrix pencils. Institute of Mathematics, TU Berlin, 2015 [DOI] [13] L. Jansen. A Dissection concept for DAEs - structural decoupling, unique solvability, convergence theory and half-explicit methods. Institute of Mathematics, HU Berlin, 2015 [pdf] [14] J.P. Pade. Synchrony and Bifurcations in Coupled Dynamical Systems and Effects of Time Delay. Institute of Mathematics, HU Berlin, 2015 [pdf]