Operations Research Proceedings 2012 pp 135-140 | Cite as

# How Does Network Topology Determine the Synchronization Threshold in a Network of Oscillators?

## Abstract

The reliable functioning of an electrical power grid is dependent on the proper interaction between many of its elements. What is critically important is its ability to keep the frequency across the entire system stable. Considering a simple mathematical model, representing the network of coupled oscillators, we study the stability of frequency synchronization. This model can be interpreted as the dynamical representation of frequency synchronization between the power producing and power consuming units. Assuming a uniform network, we analytically derive the formula estimating the relation between the minimum coupling strength required to ensure the frequency synchronization and the network parameters. This minimum value can be efficiently found by solving a binary optimization problem, using universal solver XPRESS, even for large networks,. We validate the accuracy of the analytical estimation by comparing it with numerical simulations on the realistic network describing the European interconnected high-voltage electricity system, finding good agreement. Moreover, by repeatedly solving the binary optimization problem, we can test the stability of the frequency synchronization with respect to link removals. As the threshold value changes only in few cases, we conclude that the network is resilient in this regard. Since the synchronization threshold depends on the network partition representing the synchronization bottleneck, we also evaluate which network areas become critical for the synchronization when removing single links.

## Notes

### Acknowledgments

L.B. was supported by projects VEGA 1/0296/12 and APVV-0760-11.

## References

- 1.Filatrela, G., Nielsen, A., Pedersen, N.: Analysis of a power grid using a Kuramoto-like model. Eur. Phys. J. B.
**61**(4), 485–491 (2008)CrossRefGoogle Scholar - 2.Kuramoto, Y.: Chemical Oscillations, Waves, and Turbulence. Springer-Verlag, New York (1984)CrossRefGoogle Scholar
- 3.Buzna, L., Lozano, S., Díaz-Guilera, A.: Synchronization in symmetric bipolar population networks. Phys. Rev. E
**80**(6), 066120 (2009)Google Scholar - 4.Lozano, S., Buzna, L., Díaz-Guilera, A.: Role of network topology in the synchronization of power systems. Eur. Phys. J. B.
**85**(7), 231–238 (2012)CrossRefGoogle Scholar - 5.Zhou, Q., Bialek, J.: Approximate model of European interconnected system as a benchmark system to study effects of cross-border trades. IEEE Trans. Power Syst.
**20**(3), 782–788 (2005)CrossRefGoogle Scholar - 6.
- 7.