Synchronization stability is one of central problems in power systems, and it is becoming much more complicated with the high penetration of renewable energy and power electronics devices. In this paper, we review recent work by several nonlinear models for renewable-dominated power systems in terms of multiple timescales, in particular, grid-tied converters within the DC voltage timescale. For the simplest model, a second-order differential equations called the generalized swing equation by considering only the phase-locked loop (PLL) is obtained, which shows a similar form with the well-known swing equation for a synchronous generator in the traditional power systems. With more outer controllers included, fourth-order and fifth-order models can be obtained. The fourth-order model is called the extended generalized swing equation, exhibiting the combined function of grid synchronization and active power balance on the DC capacitor. In addition, a nonlinear model for a two coupled converter system is given. Based on these studies, we find that the PLL plays a key role in synchronization stability. In summary, the value of this paper is to clarify the key concept of the synchronization stability in renewable-dominated power systems based on different nonlinear models, which still lacks systematic studies and is controversial in the field of electrical power engineering. Meanwhile, it clearly uncovers that the synchronization stability of converters has its root in the phase synchronization concept in nonlinear sciences.
REFERENCES
1.
S. H.
Strogatz
, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering
(Westview Press
, 2015
).2.
A. S.
Pikovsky
, M. G.
Rosenblum
, and J.
Kurths
, Synchronization: A Universal Concept in Nonlinear Science
(Cambridge University Press
, 2001
).3.
4.
P. M.
Anderson
and A. A.
Fouad
, Power System Control and Stability
(The Iowa State University Press
, 1994
).5.
F.
Rodrigues
, T.
Peron
, P.
Ji
, and J.
Kurths
, “The Kuramoto model in complex networks
,” Phys. Rep.
610
, 1
–98
(2016
). 6.
D.
Witthaut
, F.
Hellmann
, J.
Kurths
, S.
Kettemann
, H.
Meyer-Ortmanns
, and M.
Timme
, “Collective nonlinear dynamics and self-organization in decentralized power grids
,” Rev. Mod. Phys.
94
, 015005
(2022
). 7.
P.
Kundu
, C.
Hens
, B.
Barzel
, and P.
Pal
, “Perfect synchronization in networks of phase-frustrated oscillators
,” Europhys. Lett.
120
, 40002
(2017
). 8.
Y.
Xue
, T.
Van Cutsem
, and M.
Ribbens-Pavella
, “A simple direct method for fast transient stability assessment of large power systems
,” IEEE Trans. Power Syst.
3
, 400
–412
(1988
). 9.
H.-D.
Chiang
, C.-C.
Chu
, and G.
Cauley
, “Direct stability analysis of electric power systems using energy functions: Theory, applications, and perspective
,” Proc. IEEE
83
, 1497
–1529
(1995
).10.
Y.
Sun
, J.
Ma
, J.
Kurths
, and M.
Zhan
, “Equal-area criterion in power systems revisited
,” Proc. R. Soc. A
474
, 20170733
(2018
). 11.
J.
Ma
, Y.
Sun
, X.
Yuan
, J.
Kurths
, and M.
Zhan
, “Dynamics and collapse in a power system model with voltage variation: The damping effect
,” PLoS One
11
, e0165943
(2016
).12.
D.
Skubov
, A.
Lukin
, and I.
Popov
, “Bifurcation curves for synchronous electrical machine
,” Nonlinear Dyn.
83
, 2323
–2329
(2016
). 13.
Q.
Qiu
, R.
Ma
, J.
Kurths
, and M.
Zhan
, “Swing equation in power systems: Approximate analytical solution and bifurcation curve estimate
,” Chaos
30
, 013110
(2020
). 14.
M.
Sarkar
and S.
Gupta
, “Synchronization in the Kuramoto model in presence of stochastic resetting
,” Chaos
32
, 073109
(2022
). 15.
B.
Schafer
and G.
Yalcin
, “Dynamical modeling of cascading failures in the Turkish power grid
,” Chaos
29
, 093134
(2019
).16.
A.
Motter
, S.
Myers
, M.
Anghel
, and T.
Nishikawa
, “Spontaneous synchrony in power-grid networks
,” Nat. Phys.
9
, 191
–197
(2013
). 17.
C.
Wang
, C.
Grebogi
, and M.
Baptista
, “Control and prediction for blackouts caused by frequency collapse in smart grids
,” Chaos
26
, 093119
(2016
). 18.
J.
Grzybowski
, E.
Macau
, and T.
Yoneyama
, “On synchronization in power-grids modelled as networks of second-order Kuramoto oscillators
,” Chaos
26
, 113113
(2016
). 19.
P. J.
Menck
, J.
Heitzig
, J.
Kurths
, and H.
Joachim Schellnhuber
, “How dead ends undermine power grid stability
,” Nat. Commun.
5
, 3969
(2014
). 20.
P.
Schultz
, J.
Heitzig
, and J.
Kurths
, “Detours around basin stability in power networks
,” New J. Phys.
16
, 125001
(2014
). 21.
Y.
Yang
, T.
Nishikawa
, and A. E.
Motter
, “Small vulnerable sets determine large network cascades in power grids
,” Science
358
, eaan3184
(2017
). 22.
M.
Rohden
, A.
Sorge
, M.
Timme
, and D.
Witthaut
, “Self-organized synchronization in decentralized power grids
,” Phys. Rev. Lett.
109
, 064101
(2012
). 23.
F.
Dörfler
, M.
Chertkov
, and F.
Bullo
, “Synchronization in complex oscillator networks and smart grids
,” Proc. Natl. Acad. Sci. U.S.A.
110
, 2005
–2010
(2013
).24.
Z.
Yang
, R.
Ma
, S.
Cheng
, and M.
Zhan
, “Problems and challenges of power-electronic-based power system stability: A case study of transient stability comparison
,” Acta Phys. Sin.
69
, 103
–116
(2020
) (in Chinese).25.
J.
Yu
, Z.
Yang
, J.
Kurths
, and M.
Zhan
, “Small-signal stability of multi-converter infeed power grids with symmetry
,” Symmetry
13
, 1
–25
(2021
).26.
X.
He
, H.
Geng
, and G.
Mu
, “Modeling of wind turbine generators for power system stability studies: A review
,” Renew. Sust. Energy Rev.
143
, 110865
(2021
). 27.
Joint NERC and WECC Staff
, “900 mw fault induced solar photovoltaic resource interruption disturbance report,” Technical Report, 2018.28.
Australian Energy Market Operator, “Black system South Australia,” Technical Report, 2016.
29.
National Grid ESO, “Technical report on the events of 9 August 2019,” Technical Report, 2019.
30.
Project Group of Global Engineering Fronts of Chinese Academy of Engineering
, Engineering Fronts
(Higher Education Press
, 2021
), p. 115
.31.
H.
Pico
and B.
Johnson
, “Transient stability assessment of multi-machine multi-converter power systems
,” IEEE Trans. Power Syst.
34
, 3504
–3514
(2019
). 32.
Y.
Zhang
, C.
Zhang
, and X.
Cai
, “Large-signal grid-synchronization stability analysis of pll-based VSCs using Lyapunov’s direct method
,” IEEE Trans. Power Syst.
37
, 788
–791
(2022
). 33.
Q.
Hu
, L.
Fu
, F.
Ma
, and F.
Ji
, “Large signal synchronizing instability of PLL-based VSC connected to weak AC grid
,” IEEE Trans. Power Syst.
34
, 3220
–3229
(2019
). 34.
X.
Fu
, J.
Sun
, M.
Huang
, Z.
Tian
, H.
Yan
, H. H.-C.
Iu
, P.
Hu
, and X.
Zha
, “Large-signal stability of grid-forming and grid-following controls in voltage source converter: A comparative study
,” IEEE Trans. Power Electron.
36
, 7832
–7840
(2021
). 35.
R.
Ma
, J.
Li
, J.
Kurths
, S.
Cheng
, and M.
Zhan
, “Generalized swing equation and transient synchronous stability with PLL-based VSC
,” IEEE Trans. Energy Convers.
37
, 1428
–1441
(2022
). 36.
C.
Zhang
, X.
Cai
, A.
Rygg
, and M.
Molinas
, “Modeling and analysis of grid-synchronizing stability of a type-IV wind turbine under grid faults
,” Int. J. Electr. Power Energy Syst.
117
, 105544
(2020
). 37.
M.
Huang
, Y.
Peng
, C.
Tse
, Y.
Liu
, J.
Sun
, and X.
Zha
, “Bifurcation and large-signal stability analysis of three-phase voltage source converter under grid voltage dips
,” IEEE Trans. Power Electron.
32
, 8868
–8879
(2017
). 38.
Z.
Yang
, R.
Ma
, S.
Cheng
, and M.
Zhan
, “Nonlinear modeling and analysis of grid-connected voltage-source converters under voltage dips
,” IEEE J. Emerg. Sel. Top. Power Electron.
8
, 3281
–3292
(2020
). 39.
R.
Ma
, Z.
Yang
, S.
Cheng
, and M.
Zhan
, “Sustained oscillations and bifurcations in three-phase voltage source converters tied to AC grid
,” IET Renew. Power Gener.
14
, 3770
–3781
(2020
). 40.
H.
Wu
and X.
Wang
, “Design-oriented transient stability analysis of PLL-synchronized voltage-source converters
,” IEEE Trans. Power Electron.
35
, 3573
–3589
(2020
). 41.
Z.
Zhang
, R.
Schuerhuber
, L.
Fickert
, K.
Friedl
, G.
Chen
, and Y.
Zhang
, “Domain of attraction’s estimation for grid connected converters with phase-locked loop
,” IEEE Trans. Power Syst.
37
, 1351
–1362
(2022
). 42.
C.
Zhang
, M.
Molinas
, Z.
Li
, and X.
Cai
, “Synchronizing stability analysis and region of attraction estimation of grid-feeding VSCs using sum-of-squares programming
,” Front. Energy Res.
8
, 1
–12
(2020
).43.
J.
Hu
, X.
Yuan
, and S.
Cheng
, “Multi-time scale transients in power-electronized power systems considering multi-time scale switching control schemes of power electronics apparatus
,” Proc. CSEE
39
, 5457
–5467
(2019
) (in Chinese).44.
J.
Hu
, B.
Wang
, W.
Wang
, H.
Tang
, Y.
Chi
, and Q.
Hu
, “Small signal dynamics of dfig-based wind turbines during riding through symmetrical faults in weak AC grid
,” IEEE Trans. Energy Convers.
32
, 720
–730
(2017
). 45.
A.
Sajadi
, R.
Kenyon
, and B.-M.
Hodge
, “Synchronization in electric power networks with inherent heterogeneity up to 100% inverter-based renewable generation
,” Nat. Commun.
13
, 2490
(2022
). 46.
Z.
Yang
, M.
Zhan
, D.
Liu
, C.
Ye
, K.
Cao
, and S.
Cheng
, “Small-signal synchronous stability of a new-generation power system with 100% renewable energy
,” IEEE Trans. Power Syst.
(published online 2022
).47.
X.
Zhao
, P.
Thakurta
, and D.
Flynn
, “Grid-forming requirements based on stability assessment for 100% converter-based Irish power system
,” IET Renew. Power Gener.
16
, 447
–458
(2022
). 48.
C.
Breyer
, S.
Khalili
, D.
Bogdanov
, M.
Ram
, A. S.
Oyewo
, A.
Aghahosseini
, A.
Gulagi
, A. A.
Solomon
, D.
Keiner
, G.
Lopez
, P. A.
Ostergaard
, H.
Lund
, B. V.
Mathiesen
, M. Z.
Jacobson
, M.
Victoria
, S.
Teske
, T.
Pregger
, V.
Fthenakis
, M.
Raugei
, H.
Holttinen
, U.
Bardi
, A.
Hoekstra
, and B. K.
Sovacool
, “On the history and future of 100% renewable energy systems research
,” IEEE Access
10
, 78176
–78218
(2022
). 49.
X.
Wang
, M. G.
Taul
, H.
Wu
, Y.
Liao
, F.
Blaabjerg
, and L.
Harnefors
, “Grid-synchronization stability of converter-based resources—An overview
,” IEEE Open J. Ind. Appl.
1
, 115
–134
(2020
). 50.
R.
Rosso
, X.
Wang
, M.
Liserre
, X.
Lu
, and S.
Engelken
, “Grid-forming converters: Control approaches, grid-synchronization, and future trends—A review
,” IEEE Open J. Ind. Appl.
2
, 93
–109
(2021
). 51.
Q.
Jiang
and C.
Zhao
, “Electromagnetic transient synchronization stability issue of grid-connected inverters
,” J. Tsinghua Univ.
21
, 1
–14
(2021
) (in Chinese).52.
H.
Geng
, C.
He
, Y.
Liu
, X.
He
, and M.
Li
, “Overview on transient synchronization stability of renewable-rich power systems
,” High Voltage Eng.
48
, 3367
–3383
(2022
) (in Chinese).53.
Y.
Zhang
, X.
Cai
, C.
Zhang
, J.
Lyu
, and Y.
Li
, “Transient synchronization stability analysis of voltage source converters: A review
,” Proc. CSEE
41
, 1687
–1701
(2021
) (in Chinese).54.
Y.
Gu
and T. C.
Green
, “Power system stability with a high penetration of inverter-based resources
,” Proc. IEEE
(published online, 2022
). 55.
Q.
Zhong
and G.
Weiss
, “Synchronverters: Inverters that mimic synchronous generators
,” IEEE Trans. Ind. Electron.
58
, 1259
–1267
(2011
). 56.
V.
Gevorgian
and B.
O’Neill
, “Advanced grid-friendly controls demonstration project for utility-scale PV power plants,” Technical Report, National Renewable Energy Laboratory, 2016.57.
E.
Rokrok
, T.
Qoria
, A.
Bruyere
, B.
Francois
, and X.
Guillaud
, “Transient stability assessment and enhancement of grid-forming converters embedding current reference saturation as current limiting strategy
,” IEEE Trans. Power Syst.
37
, 1519
–1531
(2021
). 58.
X.
Zha
, M.
Huang
, Y.
Liu
, and Z.
Tian
, “An overview on safe operation of grid-connected converters from resilience perspective: Analysis and design
,” Int. J. Electr. Power Energy Syst.
143
, 108511
(2022
). 59.
A.
Yazdani
and R.
Iravani
, Voltage-Sourced Converters in Power Systems: Modeling, Control, and Applications
(John Wiley & Sons
, 2010
).60.
X.
Yuan
, S.
Cheng
, and J.
Hu
, “Multi-time scale voltage and power angle dynamics in power electronics dominated large power systems
,” Proc. CSEE
36
, 5145
–5154
(2016
) (in Chinese).61.
R.
Ma
, Q.
Qiu
, J.
Kurths
, and M.
Zhan
, “Fast-slow-scale interaction induced parallel resonance and its suppression in voltage source converters
,” IEEE Access
9
, 90126
–90141
(2021
). 62.
M.
Zarif Mansour
, S.
Me
, S.
Hadavi
, B.
Badrzadeh
, A.
Karimi
, and B.
Bahrani
, “Nonlinear transient stability analysis of phased-locked loop based grid-following voltage source converters using Lyapunov’s direct method
,” IEEE J. Emerg. Sel. Top. Power Electron.
10
, 2699
–2709
(2022
). 63.
Y.
Kang
, X.
Lin
, Y.
Zheng
, X.
Quan
, J.
Hu
, and X.
Yuan
, “The static stable-limit and static stable-working zone for single-machine infinite-bus system of renewable-energy grid-connected converter
,” Proc. CSEE
40
, 4506
–4515
(2020
) (in Chinese).64.
L.
Xiong
, F.
Zhuo
, F.
Wang
, X.
Liu
, Y.
Chen
, M.
Zhu
, and H.
Yi
, “Static synchronous generator model: A new perspective to investigate dynamic characteristics and stability issues of grid-tied PWM inverter
,” IEEE Trans. Power Electron.
31
, 6264
–6280
(2016
). 65.
Z.
Yang
, J.
Yu
, J.
Kurths
, and M.
Zhan
, “Nonlinear modeling of multi-converter systems within DC-link timescale
,” IEEE J. Emerg. Sel. Top. Circuits Syst.
11
, 5
–16
(2021
). 66.
C.
Shen
, Z.
Shuai
, Y.
Shen
, Y.
Peng
, X.
Liu
, Z.
Li
, and Z. J.
Shen
, “Transient stability and current injection design of paralleled current-controlled VSCs and virtual synchronous generators
,” IEEE Trans. Smart Grid
12
, 1118
–1134
(2021
). 67.
X.
He
and H.
Geng
, “Transient stability of power systems integrated with inverter-based generation
,” IEEE Trans. Power Syst.
36
, 553
–556
(2021
). 68.
X.
Fu
, M.
Huang
, S.
Pan
, and X.
Zha
, “Cascading synchronization instability in multi-VSC grid-connected system
,” IEEE Trans. Power Electron.
37
, 7572
–7576
(2022
). 69.
X.
He
and H.
Geng
, “Synchronization stability analysis and enhancement of grid-tied multi-converter systems,” in 2020 IEEE Industry Applications Society Annual Meeting (IEEE, 2020), pp. 1–8.70.
Y.
Ji
, W.
He
, S.
Cheng
, J.
Kurths
, and M.
Zhan
, “Dynamic network characteristics of power-electronics-based power systems
,” Sci. Rep.
10
, 1
–16
(2020
). 71.
X.
Yang
, R.
Ma
, and M.
Zhan
, “Dynamic and static network analysis and power transmission characteristics of power system oscillations,” in Proceedings of CSEE (to be published) (2022) (in Chinese); available at https://kns.cnki.net/kcms/detail/11.2107.TM.20220926.1758.018.html.© 2023 Author(s). Published under an exclusive license by AIP Publishing.
2023
Author(s)
You do not currently have access to this content.
