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Neural Processing Letters
Published in: Neural Processing Letters 4/2021

22-04-2021

The Kuramoto Model: The Stability Conditions in the Presence of Phase Shift

Authors: Reza Farhangi, Mohammad Taghi Hamidi Beheshti

Published in: Neural Processing Letters | Issue 4/2021

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Abstract

A set of coupled Kuramoto oscillators is the main applied model for harmonization study of oscillating phenomena in physical, biological and engineering networks. In line with the previous studies and to bring the analytical results into conformity with further realistic models, in present paper the synchronization of Kuramoto oscillators has been investigated and the necessary and sufficient conditions for the frequency synchronization and phase cohesiveness have been introduced using the contraction property. The novelty of this paper lies in the following: (I) we consider the heterogeneous second-order model with non-uniformity in coupling topology; (II) we apply a non-zero and non-uniform phase shift in coupling function; (III) we introduce a new Lyapunov-based stability analysis technique. We demonstrate how the heterogeneity in network and the phase shift in coupling function are the key factors in network synchronization. The synchronization conditions are presented on the basis of network graph-theoretical characteristics and the oscillators’ parameters. Investigation of the analytical results reveals that an increase in the phase shift and heterogeneity of oscillators will complicate the synchronization conditions. The validity of the main theoretical results has been confirmed through the numerical simulations.
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Literature
1.
Kuramoto Y (1984) Chemical oscillations, waves and turbulence. Springer, BerlinCrossRef
2.
Wu CW, Chua LO (1994) A unified framework for synchronization and control of dynamical systems. Int J Bifurc Chaos 4:979–998MathSciNetCrossRef
3.
Strogatz SH (2000) From Kuramoto to crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D 143:1–20MathSciNetCrossRef
4.
Dörfler F, Bullo F (2014) Synchronization in complex networks of phase oscillators: a survey. Automatica 50:1539–1564MathSciNetCrossRef
5.
Acebrón JA, Bonilla LL (2005) The Kuramoto model: a simple paradigm for synchronization phenomena. Rev Mod Phys 77:137CrossRef
6.
7.
Mureau L (2003) Leaderless coordination via bidirectional and unidirectional time-dependent communication. In: 42nd IEEE international conference on decision and control
8.
Jadbabaie A, Motee N, Barahona M (2005) On the stability of the Kuramoto model of coupled nonlinear oscillators. In: Proceedings of the American control conference, pp 4296–4301
9.
Chopra N, Spong MW (2005) On synchronization of Kuramoto oscillators. In: Proceedings of the 44th IEEE conference on decision and control and the European control conference, Seville, Spain, pp 3916–3922
10.
Chopra N, Spong MW (2009) On exponential synchronization of Kuramoto oscillators. IEEE Trans Autom Control 54:353–357MathSciNetCrossRef
11.
Choi YP, Ha SY, Jung S, Kim Y (2012) Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model. Physica D 241:735–754MathSciNetCrossRef
12.
Ha S-Y, Ha T, Kim J-H (2010) On the complete synchronization of the Kuramoto phase model. Physica D 239:1692–1700MathSciNetCrossRef
13.
Ha SY, Kim Y, Li Z (2014) Asymptotic synchronous behavior of kuramoto type models with frustrations. Netw Heterog Med 9:33–64MathSciNetCrossRef
14.
Fortuna L, Frasca M, Fiore AS (2011) Analysis of the Italian power grid based on Kuramoto-like model. In: 5th international conference on physics and control, Leon, Spain (2011)
15.
Filatrella G, Nielsen AH, Pedersen NF (2008) Analysis of a power grid using a Kuramoto-like model. Eur Phys J B 61:485–491CrossRef
16.
van Hemmen JL, Wreszinski WF (1993) Lyapunov function for the Kuramoto model of nonlinearly coupled oscillators. J Stat Phys 72:145–166CrossRef
17.
Rogge JA, Aeyels D (2004) Existence of partial entrainment and stability of phase locking behavior of coupled oscillators. Prog Theor Phys 112:921–942CrossRef
18.
Canale E, Monzon P (2008) Almost global synchronization of symmetric Kuramoto coupled oscillators. In: Systems structure and control. InTech Education and Publishing, pp 167–190
19.
Moreau L (2005) Stability of multiagent systems with time-dependent communication links. IEEE Trans Autom Control 50:169–182MathSciNetCrossRef
20.
Lin Z, Francis B, Maggiore M (2007) State agreement for continuous-time coupled nonlinear systems. SIAM J Control Optim 46:288–307MathSciNetCrossRef
21.
Schmidt GS, Münz U, Allgöwer F (2009) Multi-agent speed consensus via delayed position feedback with application to Kuramoto oscillators. In: European control conference, Budapest, Hungary, pp 2464–2469
22.
Dörfler F, Bullo F (2010) Synchronization and transient stability in power networks and non-uniform Kuramoto oscillators. In: Proceedings of the American control conference, pp 930–937
23.
Dörfler F, Bullo F (2012) Exploring synchronization in complex oscillator networks. In: Proceedings of 51th IEEE conference on decision and control (CDC), Maui, HI, USA, pp 7157–7170
24.
Dörfler F, Chertkov M, Bullo F (2013) Synchronization in complex oscillator networks and smart grids. Proc Natl Acad Sci USA 110:2005–2010MathSciNetCrossRef
25.
Ainsworth N, Grijalva S (2013) A structure-preserving model and sufficient condition for frequency synchronization of lossless droop inverter-based ac networks. IEEE Trans Power Syst 28:4310–4319CrossRef
26.
Simpson-Porco J, Dörfler F, Bullo F (2012) Droop-controlled inverters are Kuramoto oscillators. In: Proceedings of the 3rd IFAC workshop distributed estimation and control in networked systems, vol 3
27.
Simpson-Porco JW, Dörfler F, Bullo F (2013) Synchronization and power sharing for droop-controlled inverters in islanded microgrids. Automatica 49:2603–2611MathSciNetCrossRef
28.
Bergen AR, Hill DJ (1981) A structure preserving model for power system stability analysis. IEEE Trans Power Apparat Syst 100:25–35CrossRef
29.
Giraldo J, Mojica-Nava E, Quijano N (2013) Synchronization of dynamical networks with a communication infrastructure: a smart grid application. In: 52nd IEEE conference decision and control, Los Angeles, CA, USA, pp 4638-4643
30.
Giraldo J, Mojica E, Quijano N (2019) Synchronization of heterogeneous Kuramoto oscillators with sampled information and a constant leader. Int J Control 92:2591–2607CrossRef
31.
Motter AE, Myers SA, Anghel M, Nishikawa T (2013) Spontaneous synchrony in power grid networks. Nat Phys 9:191–197CrossRef
32.
Mao Y, Zhang Z (2016) Distributed frequency synchronization and phase-difference tracking for Kuramoto oscillators and its application to islanded microgrids. In: IEEE 55th conference on decision and control (CDC), pp 4364–4369
33.
34.
Choi YP, Ha SY, Yun SB (2011) Complete synchronization of Kuramoto oscillators with finite inertia. Physica D 240:32–44MathSciNetCrossRef
35.
Ha SY, Kim Y, Li Z (2014) Large-time dynamics of Kuramoto oscillators under the effects of inertia and frustration. SIAM J Appl Dyn Syst 13:466–492MathSciNetCrossRef
36.
Choi Y-P, Li Z, Ha S-Y, Xue X, Yun S-B (2014) Complete entrainment of Kuramoto oscillators with inertia on networks via gradient-like flow. J Differ Equ 257:2591–2621MathSciNetCrossRef
37.
Ha S-Y, Noh S-E, Park J (2017) Interplay of inertia and heterogeneous dynamics in an ensemble of Kuramoto oscillators. Anal Appl 15:837–861MathSciNetCrossRef
38.
Choi Y, Li Z (2019) Synchronization of nonuniform Kuramoto oscillators for power grids with general connectivity and dampings. Nonlinearity 32:559MathSciNetCrossRef
39.
Vu TL, Turitsyn K (2014) Lyapunov functions family approach to transient stability assessment. IEEE Trans Power Syst 31:1269–1277CrossRef
40.
Li Z, Xue X, Yu D (2014) Synchronization and transient stability in power grids based on Lojasiewica inequalities. SIAM J Control Optim 52:2482–2511MathSciNetCrossRef
41.
Yang L, Guo Y, Chen N, Qian M, Xue X, Yu D (2019) Power system transient stability analysis via second-order non-uniform Kuramoto model. Adv Mater Res 19:1054–1057
42.
Li H, Chen G, Liao X, Huang T (2017) Attraction region seeking for power grids. IEEE Trans Circuits Syst II Express Briefs 64:201–205CrossRef
43.
Mallada E, Tang A (2011) Improving damping of power networks: power scheduling and impedance adaptation. In: 50th IEEE conference on decision and control and European control conference, pp 7734–7729
44.
45.
Li B, Wong KYM (2017) Optimizing synchronization stability of the Kuramoto model in complex networks and power grids. Phys Rev E 95:012207CrossRef
46.
Hill DJ, Chen G (2006) Power systems as dynamic networks. In: IEEE international symposium on circuits and systems, Greece, pp 722–725
47.
48.
Hill DJ, Bergen AR (1982) Stability analysis of multimachine power networks with linear frequency dependent loads. IEEE Trans Circuits Syst 29:840–848CrossRef
49.
Zhu L, Hill D (2016) Transient stability analysis of microgrids with network-preserving structure. In: 6th IFAC workshop on distributed estimation and control in networked systems, vol 49, pp 339–344
50.
51.
Zhu L, Hill DJ (2018) Stability analysis of power systems: a network synchronization perspective. SIAM J Control Optim 56:1640–1664MathSciNetCrossRef
52.
Long T, Turitsyn K (2015) Synchronization stability of Lossy and uncertain power grids. In: American control conference (ACC), pp 5056–5061
Metadata
Title
The Kuramoto Model: The Stability Conditions in the Presence of Phase Shift
Authors
Reza Farhangi
Mohammad Taghi Hamidi Beheshti
Publication date
22-04-2021
Publisher
Springer US
Published in
Neural Processing Letters / Issue 4/2021
Print ISSN: 1370-4621
Electronic ISSN: 1573-773X
DOI
https://doi.org/10.1007/s11063-021-10510-0

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