September  2020, 15(3): 543-553. doi: 10.3934/nhm.2020030

Synchronization of a Kuramoto-like model for power grids with frustration

1. 

School of Mathematics, Harbin Institute of Technology, Harbin 150001, China

2. 

School of Mathematics and Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin 150001, China

* Corresponding author: Zhuchun Li

Received  July 2019 Revised  September 2019 Published  September 2020 Early access  September 2020

Fund Project: This work was supported by NSF of China grant 11671109. The work of Z. Li was also supported in part by Heilongjiang Provincial Natural Science Foundation of China (grant LH2019A012)

We discuss the complete synchronization for a Kuramoto-like model for power grids with frustration. For identical oscillators without frustration, it will converge to complete phase and frequency synchronization exponentially fast if the initial phases are distributed in a half circle. For nonidentical oscillators with frustration, we present a framework leading to complete frequency synchronization where the initial phase configurations are located inside the half of a circle. Our estimates are based on the monotonicity arguments of extremal phase and frequency.

Citation: Xiaoxue Zhao, Zhuchun Li. Synchronization of a Kuramoto-like model for power grids with frustration. Networks and Heterogeneous Media, 2020, 15 (3) : 543-553. doi: 10.3934/nhm.2020030
References:
[1]

J. A. Acebrón, L. L. Bonilla, C. J. Pérez Vicente, F. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005). doi: 10.1103/RevModPhys.77.137.

[2]

J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562-564.  doi: 10.1038/211562a0.

[3]

Y.-P. ChoiS.-Y. HaS. Jung and Y. Kim, Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model, Phys. D, 241 (2012), 735-754.  doi: 10.1016/j.physd.2011.11.011.

[4]

Y.-P. Choi and Z. Li, Synchronization of nonuniform Kuramoto oscillators for power grids with general connectivity and dampings, Nonlinearity, 32 (2019), 559–583. doi: 10.1088/1361-6544/aaec94.

[5]

S.-H. ChoiI. Ryoo and B. I. Hong, Complete position synchronization in the power grid system, Appl. Math. Lett., 84 (2018), 19-25.  doi: 10.1016/j.aml.2018.04.004.

[6]

F. Dörfler and F. Bullo, Synchronization in complex networks of phase oscillators: A survey, Automatica J. IFAC, 50 (2014), 1539-1564.  doi: 10.1016/j.automatica.2014.04.012.

[7]

F. Dörfler and F. Bullo, Synchronization and transient stability in power networks and nonuniform Kuramoto oscillators, SIAM J. Control Optim., 50 (2012), 1616-1642.  doi: 10.1137/110851584.

[8]

B. Ermentrout, An adaptive model for synchrony in the firefly Pteroptyx malaccae, J. Math. Biol., 29 (1991), 571-585.  doi: 10.1007/BF00164052.

[9]

G. FilatrellaA. H. Nielsen and N. F. Pedersen, Analysis of a power grid using a Kuramoto-like model, Eur. Phys. J. B, 61 (2008), 485-491.  doi: 10.1140/epjb/e2008-00098-8.

[10]

S.-Y. HaT. Ha and J.-H. Kim, On the complete synchronization of the Kuramoto phase model, Phys. D, 239 (2010), 1692-1700.  doi: 10.1016/j.physd.2010.05.003.

[11]

S.-Y. HaY. Kim and Z. Li, Asymptotic synchronous behavior of Kuramoto type models with frustrations, Netw. Heterog. Media, 9 (2014), 33-64.  doi: 10.3934/nhm.2014.9.33.

[12]

S.-Y. HaD. Ko and Y. Zhang, Emergence of phase-locking in the Kuramoto model for identical oscillators with frustration, SIAM J. Appl. Dyn. Syst., 17 (2018), 581-625.  doi: 10.1137/17M1112959.

[13]

Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, in International Symposium on Mathematical Problems in Theoretical Physics, Lecture Notes in Phys., 39, Springer, Berlin, 1975,420–422. doi: 10.1007/BFb0013365.

[14]

E. Oh, C. Choi, B. Kahng and D. Kim, Modular synchronization in complex networks with a gauge Kuramoto model, Europhys. Lett. (EPL), 83 (2008). doi: 10.1209/0295-5075/83/68003.

[15]

C. S. Peskin, Mathematical Aspects of Heart Physiology, Courant Institute of Mathematical Sciences, New York University, New York, 1975.

[16]

H. Sakaguchi and Y. Kuramoto, A soluble active rotator model showing phase transitions via mutual entrainment, Progr. Theoret. Phys., 76 (1986), 576-581.  doi: 10.1143/PTP.76.576.

show all references

References:
[1]

J. A. Acebrón, L. L. Bonilla, C. J. Pérez Vicente, F. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005). doi: 10.1103/RevModPhys.77.137.

[2]

J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562-564.  doi: 10.1038/211562a0.

[3]

Y.-P. ChoiS.-Y. HaS. Jung and Y. Kim, Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model, Phys. D, 241 (2012), 735-754.  doi: 10.1016/j.physd.2011.11.011.

[4]

Y.-P. Choi and Z. Li, Synchronization of nonuniform Kuramoto oscillators for power grids with general connectivity and dampings, Nonlinearity, 32 (2019), 559–583. doi: 10.1088/1361-6544/aaec94.

[5]

S.-H. ChoiI. Ryoo and B. I. Hong, Complete position synchronization in the power grid system, Appl. Math. Lett., 84 (2018), 19-25.  doi: 10.1016/j.aml.2018.04.004.

[6]

F. Dörfler and F. Bullo, Synchronization in complex networks of phase oscillators: A survey, Automatica J. IFAC, 50 (2014), 1539-1564.  doi: 10.1016/j.automatica.2014.04.012.

[7]

F. Dörfler and F. Bullo, Synchronization and transient stability in power networks and nonuniform Kuramoto oscillators, SIAM J. Control Optim., 50 (2012), 1616-1642.  doi: 10.1137/110851584.

[8]

B. Ermentrout, An adaptive model for synchrony in the firefly Pteroptyx malaccae, J. Math. Biol., 29 (1991), 571-585.  doi: 10.1007/BF00164052.

[9]

G. FilatrellaA. H. Nielsen and N. F. Pedersen, Analysis of a power grid using a Kuramoto-like model, Eur. Phys. J. B, 61 (2008), 485-491.  doi: 10.1140/epjb/e2008-00098-8.

[10]

S.-Y. HaT. Ha and J.-H. Kim, On the complete synchronization of the Kuramoto phase model, Phys. D, 239 (2010), 1692-1700.  doi: 10.1016/j.physd.2010.05.003.

[11]

S.-Y. HaY. Kim and Z. Li, Asymptotic synchronous behavior of Kuramoto type models with frustrations, Netw. Heterog. Media, 9 (2014), 33-64.  doi: 10.3934/nhm.2014.9.33.

[12]

S.-Y. HaD. Ko and Y. Zhang, Emergence of phase-locking in the Kuramoto model for identical oscillators with frustration, SIAM J. Appl. Dyn. Syst., 17 (2018), 581-625.  doi: 10.1137/17M1112959.

[13]

Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, in International Symposium on Mathematical Problems in Theoretical Physics, Lecture Notes in Phys., 39, Springer, Berlin, 1975,420–422. doi: 10.1007/BFb0013365.

[14]

E. Oh, C. Choi, B. Kahng and D. Kim, Modular synchronization in complex networks with a gauge Kuramoto model, Europhys. Lett. (EPL), 83 (2008). doi: 10.1209/0295-5075/83/68003.

[15]

C. S. Peskin, Mathematical Aspects of Heart Physiology, Courant Institute of Mathematical Sciences, New York University, New York, 1975.

[16]

H. Sakaguchi and Y. Kuramoto, A soluble active rotator model showing phase transitions via mutual entrainment, Progr. Theoret. Phys., 76 (1986), 576-581.  doi: 10.1143/PTP.76.576.

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