#### For citation:

Arinushkin P. A., Anishchenko V. S. The influence of the output power of the generators on the frequency characteristics of the grid in a ring topology. *Izvestiya VUZ. Applied Nonlinear Dynamics*, 2019, vol. 27, iss. 6, pp. 25-38. DOI: 10.18500/0869-6632-2019-27-6-25-38

# The influence of the output power of the generators on the frequency characteristics of the grid in a ring topology

Great interest in the field of dynamic systems and nonlinear processes is caused by research in the field of energy networks. A power grid is a complex network of coupled oscillators that demonstrates collective behavior by synchronizing

network elements at the base frequency of a power grid.

The purpose of the work is to study the dynamic stability of the synchronous state of the grid. The behavior of a network with homogeneous characteristics and a ring-shaped topology is investigated. An idealized energy network consisting of ten generators and ten consumers is considered. The influence of the generator output power and the inertia coefficient on the network synchronization is considered.

Method. An efficient network model (Nishikawa T., Motter A.E. Comparative analysis of existing models for powergrid synchronization) is studied, which excludes changes in consumer power from consideration, assuming that the power of all consumers is constant. This model allows us to consider the generators of the power system as coupled oscillators, the dimensionless parameters of which are determined by a large set of real physical parameters.

Results. The results showed that the natural frequency of the oscillator depends on the output power of the generator and does not always coincide with the synchronization frequency of the oscillators. With increasing generator power, the natural frequency is much higher than the synchronization frequency. At a critical value of the output power, oscillator oscillations become unstable, the generator frequency becomes equal to its natural frequency. The value of inertia plays a significant role in the stability of the generators; it is shown in the work that small values of the coefficient of inertia generators can produce energy in large ranges of output power without loss of synchronous state and regardless of the set of initial conditions.

Discussion. From obtained results it became known that the stable functioning of the generators with increase in output power is possible at small values of inertia coefficient. At large values of inertia coefficient, the synchronism of one or several generators is disrupted, leading the output generator to work at its own frequency, the value of which significantly exceeds the frequency of the network standard 50 Hz. The frequency perturbation depends on the location of one or another generator with respect to unstable generator.

- Nishikawa T., Motter A.E. Comparative analysis of existing models for powergrid synchronization // New Journal of Physics. 2015. Vol. 17, no. 1. P. 015012.
- Heagy J.F., Pecora L.M., and Carroll T.L. Short wavelength bifurcations and size instabilities in coupled oscillator systems // Phys Rev. Lett. 1995. Vol. 74, no. 21. P. 4185–4188.
- Barahona M. and Pecora L.M. Synchronization in small-world systems // Phys. Rev. Lett. 2002. Vol. 89, no. 5. 054101.
- Menck P.J., Heitzig J., Marwan N., and Kurths J. How basin stability complements the linearstability paradigm // Nat. Phys. 2013. Vol. 9. P. 89–92.
- Boccaletti S., Hwang D.-U., Chavez M., Amann A., Kurths J., and Pecora L.M. Synchronization in dynamical networks: Evolution along commutative graphs // Phys. Rev. E. 2006. Vol. 74, no. 1. 016102.
- Arinushkin P.A., Anishchenko V.S. Synchronous modes analysis of coupled oscillators in power grids. Izvestiya VUZ. Applied Nonlinear Dynamics, 2018, vol. 26, no. 3, pp. 62–77 (in Russian). DOI: 10.18500/0869-6632-2018-26-3-62-77.
- Anderson P.M. and Fouad A.A. Power System Control and Stability // 2nd ed. (IEEE Press). 2003.
- Mallada E. and Tang A. Improving damping of power networks: Power scheduling and impedance adaptation // IEEE Conference on Decision and Control. 2011. P. 7729–7734.
- Caliskan S. and Tabuada P. Compositional transient stability analysis of multimachine power networks // IEEE T. Contr. Netw. Syst. 2014. Vol. 1. P. 4–14.
- Schmietendorf K., Peinke J., Friedrich R., and Kamps O. Self-organized synchronization and voltage stability in networks of synchronous machines // Eur. Phys. J. 2014. Vol. 223. P. 2577–2592.
- Nagat M., Fujiwara N., Tanaka G., Suzuki H., Kohda E., and Aihara K. Node-wise robustness against fluctuations of power consumption in power grids // Eur. Phys. J. 2014. Vol. 223. P. 2549– 2559.
- Ortega R., van der Schaft A.J., Mareels Y., and Maschke B.M. Putting energy back in control // Control Syst. Mag. 2001. Vol. 21, no. 2. P. 18–33.
- Pearmine R., Song Y.H., Chebbo A. Influence of wind turbine behaviour on the primary frequency control // IET Renewable Power Generation, 2007, Vol. 1, no. 2. P. 142–150.
- Ramtharan G., Jenkins N., Anaya-Lara O. Modelling and control of synchronous generators for wide-range variable-speed wind turbines // Wind Energy. 2007. Vol. 10. P. 231–246.
- Etxegarai A., Eguia P., Torres E., Iturregi A., Valverde V. Review of grid connection requirements for generation assets in weak power grids // Renewable and Sustainable Energy Reviews. 2015. Vol. 41. P. 1501–1514.
- Nishikawa T., Motter A.E. Maximum performance at minimum cost in network synchronization // Physica D: Nonlinear Phenomena. 2006. Vol. 224, no. 1–2. P. 77–89.
- Menck P.J., Heitzig J. How dead ends undermine power grid stability // Nature communications. 2014. Vol. 5. P. 3969.
- Rohden M., Sorge A., Witthaut D. Impact of network topology on synchrony of oscillatory power grids // Chaos: An Interdisciplinary Journal of Nonlinear Science. 2014. Vol.24, no. 1. 013123.
- Changsong Zhou,Motter A.E., and Kurths J. Universality in the synchronization of weighted random networks // Phys. Rev. Lett. 2006. Vol. 96, no. 1. 034101.
- Lozano S., Buzna L., and Dıaz-Guilera A. Role of network topology in the synchronization of power systems // Eur. Phys. J. 2012. Vol. 85. Pp. 1–8.
- Delille G., Francois B. and Malarange G. Dynamic frequency control support by energy storage to reduce the impact of wind and solar generation on isolated power system’s inertia // in IEEE Transactions on Sustainable Energy. 2012. Vol. 3, no. 4. Pp. 931–939.
- Mamis M.S. and Meral M.E. State-space modeling and analysis of fault arcs // Electric Power Systems Research. 2005. Vol. 76. Pp. 46–51.
- Dorfler F., Chertkov M., Bullo F. Synchronization in complex oscillator networks and smart grids // Proceedings of the National Academy of Sciences of the United States of America. 2013. Vol. 110, no. 6. Pp. 2005–2010.
- Dorfler F. and Bullo F. Synchronization and transient stability in power networks and nonuniform Kuramoto oscillators // SIAM Journal on Control and Optimization. 2012. Vol. 50, no. 3. Pp. 1616–1642.
- Motter A.E., Myers S.A., Anghel M. and Nishikawa T. Spontaneous synchrony in power-grid networks // Nature Physics. 2013. Vol. 9(3). P. 191.
- Schiffer J., Ortega R., Astolfi A., Raisch J. and Sezi T. Conditions for stability of droop-controlled inverter-based microgrids // Automatica. 2014. Vol. 50(10). Pp. 2457–2469.
- Schiffer J., Zonetti D., Ortega R., Stankovic A.M., Sezi T. and Raisch J. A survey on modeling of microgrids—From fundamental physics to phasors and voltage sources // Automatica. 2016. Vol. 74. Pp. 135–150.
- Rohden M., Sorge A., Timme M. and Witthaut D. Self-organized synchronization in decentralized power grids // Physical Review Letters. 2012. Vol. 109(6). 064101.
- Dorfler F., Bullo F. Kron reduction of graphs with applications to electrical networks // IEEE Transactions on Circuits and Systems I: Regular Papers. 2013. Vol. 60, no. 1. Pp. 150–163.

- 1525 reads