System-Inherent Grid Synchronous Stability Boundary and Spontaneous Synchronization

Introduction

Materials and Methods

Figure 1. Schematic diagram of the power system operating coordinate system $(U,{\delta }_{\omega })$.

Figure 1. Schematic diagram of the power system operating coordinate system $(U,{\delta }_{\omega })$.

Data sources and experimental procedures

In this study, the IEEE 39-BUS and IEEE 9-BUS models (Extended Data Figure 4, Figure 5 and Figure 6) were used. The two models are simulated separately using a simulation software package. Here, the fault was set as a three-phase short circuit to ground. The disturbed operating point of each generator was calculated. To observe the movement pattern of the disturbed operating points, no parameters were set for the control elements.

Figure 4. Boundary effects of the 18bus three-phase short circuit to ground fault, with increased fault duration $\Delta t$ from 0.140s to 0.154s ($d(\Delta t)=0.001s$), and the trajectory of the disturbed operating point near the boundary. The arrow shows the direction of increase of $\Delta t$. The simulation result showed instability at $\Delta t$=0.155s.

Figure 4. Boundary effects of the 18bus three-phase short circuit to ground fault, with increased fault duration $\Delta t$ from 0.140s to 0.154s ($d(\Delta t)=0.001s$), and the trajectory of the disturbed operating point near the boundary. The arrow shows the direction of increase of $\Delta t$. The simulation result showed instability at $\Delta t$=0.155s.

In advance, the fault location was fixed, and the fault duration $\Delta t$ was set. This experiment simulated the rotation rate ${{\omega }^{\prime }}_i(t)$ and port bus voltage ${u^{\prime }}_i(t)$ of the ith generator after different disturbances. Then, the angle of rotation rate of the ith generator ${\delta }_i(t)$ was calculated. $\Delta t$ was increased in a fixed step length and ${u^{\prime }}_i(t)$, ${\delta }_i(t)$ were calculated again until the system was destabilised. The faulty position was replaced, and the above steps were repeated.

Subsequently, $({u^{\prime }}_i(t),{{\delta }^{\prime }}_i(t)),i\in (1,2,\dots ,n)$ was arranged and relabelled as $(u_i(t),{\delta }_i(t)),i\in (1,2,\dots ,n)$. This was then averaged as follows:

The mean of $(u_i(t),{\delta }_i(t)),i\in (1,2,\dots ,n)$ over $\left[0,T\right]$ was found: $u_i=\frac{1}{T}{\displaystyle {\int }_0^T{u}_i(\tau )d\tau }$ and ${\delta }_{KL}=\frac{1}{T}{\displaystyle {\int }_0^T{\delta }_{KL}(\tau )d\tau }=\frac{1}{T}{\displaystyle {\int }_0^T\left|{\delta }_K(\tau )-{\delta }_L(\tau )\right|d\tau }=\frac{1}{T}{\displaystyle {\int }_0^T{\delta }_K(\tau )-{\delta }_L(\tau )d\tau }={\delta }_K-{\delta }_L$.

The mean of $(u_i(t),{\delta }_i(t)),i\in (1,2,\dots ,n)$ over $\left[T_m,T_m+dT\right]$ was also found: ${\delta }_i(t_m)=\frac{1}{dT}{\displaystyle {\int }_{T_m}^{T_m+dT}{\delta }_i(\tau )}d\tau ,U_i(t_m)=\frac{1}{dT}{\displaystyle {\int }_{T_m}^{T_m+dT}{U}_i(\tau )}d\tau ,{\delta }_{kl}(t_m)={\delta }_k(t_m)-{\delta }_l(t_m)$. There are several definitions of mean, and the simplest, i.e., the arithmetic mean, was used here.

This work added adjacent meta-generator data $(u_K,u_L,{\delta }_{KL})$ and $(u_K(t_m),u_L(t_m),{\delta }_{KL}(t_m))$ to the coordinate system $u_K-u_L-{\delta }_{KL}$ to assess the system stability (Figure 2.b) and time intervals of instability $\left(T_m,T_m+dT\right)$ (Figure 2.d).

An expression was fitted with $\Delta t$ as the independent variable and $u_K,u_L,{\delta }_K,{\delta }_L,{\delta }_{KL}$ as the dependent variable (Figure 3). CCT and UEP were then calculated.

Near the boundary, $\sigma \left({\delta }_i\right),{\delta }_i,u_i$ was calculated at a finer scale.

Result and discuss

Stability boundary

a. Visualisation of the stable boundary. The stability boundary (blue surface) and $\left(0,u_l,{\delta }_{kl}\right),\left(u_k,0,{\delta }_{kl}\right),\left(u_k,u_l,0\right)$ (pink surface) together enclose the stability domain. The boundary equation is $\left|u_k\right|=\left|u_l\right|\cup \frac{\left|u_l\right|}{\left|u_k\right|}=2\cos {\delta }_{kl}-1$ (see Methods for details: Derivation of the boundary equations). For visualisation purposes, the radian system is used for ${\delta }_{kl}$ in the figure.

b and c are the results of the 12BUS three-phase short circuit to ground simulation, respectively. The system is stable at $\Delta t=0.28s$, unstable at $\Delta t=0.3s$, and unstable in $\left(6s,7s\right)$.

d. Boundary stability results for the 12BUS three-phase short circuit to ground, where $\Delta t$ is the duration of the fault. The positions of all operating points at $\Delta t=0.28s$ and $\Delta t=0.3s$ are indicated by cyan and magenta dots, respectively. The plane $\left|u_l\right|=\left|u_k\right|$ perpendicular to $u_k-u_l$ is not shown. At $\Delta t=0.28s$, all the operating points are clustered together. However, $\Delta t=0.3s$, the operating points are far away from the other points and outside the boundary.

e. The instability process of Multiswing for the 12BUS three-phase ground fault, where $\Delta t=0.3s$. Numbers 1 to 9 indicate period $T_m$ after the fault, and $dT=1s$ (see Methods for details). The system is destabilised in the time interval $\left(6s,7s\right)$ (Figure 2). The diagram shows the operating points of the meta-generators 1 and 2. The operating point crosses the boundary at the time of instability, and ${\delta }_{kl}$ rapidly increases after the voltage changes.

Regarding the synchronization stability, one different assumption,which appears parallel to the Kuramoto model⁸, than before ¹⁸: the system requires the synchronous power $P_{u^{\prime }}=\frac{{\left|u^{\prime }\right|}^2}{\left|Z_{KL}\right|}$ to maintain synchronous stability, and the synchronous power is provided by the coupling power $P_{\Delta u}=\frac{{\left|\Delta u\right|}^2}{\left|Z_{KL}\right|}$ within the system, $Z_{KL}$ is the impedance between the Kth and Lth meta-generators. (see Supplementary Materials for details).

(1)

Eq.1 is the analytical equation for the synchronous stability boundary. Specifically, when ${\delta }_{KL}>0$, the set where $\left|u_L\right|=\left|u_K\right|$ is the isolated stability domain. There has been previous research on the isolated domain². Since it is very difficult to always maintain $\left|u_L\right|=\left|u_K\right|$ after a disturbance of the system, this situation is not discussed in this paper.

In contrast to previous reports^2,3,20, here, The synchronous stability boundary described by Eq.1 is independent of the network topology, system parameters, perturbations and the number of the subsystem. It is an inherent property of the grid systems. Regardless of the changes in the network structure and component parameters, it is only the operating points of the system that change rather than the stability boundary. This feature increases the applicability of the boundary equation to different grid systems and allows the stability of multimachine systems to be analyzed even in scenarios where the topology and parameters are not clear²², such as in a black box situation. Thus, the synchronization stability of the grid can be analyzed in a uniform way. This has been validated on different standard arithmetic models (Figure 2 and Extended Data Figure 4). Eq.1 is also suitable for identifying the stability of a power system with multiple swings²³ and showing the time interval of instability(as in Figure 2c and e).

Eq.1 has a variant as follows:

(2)

When $\left|u_K\right|,\left|u_L\right|$ are sufficiently close²⁴, the stability margin of the Kth and Lth generator angle rate difference ${\delta }_{KL}^{cr}$ also tends to 0. This indicates that the system may already be in a critical state during normal operation. In this case, even if the difference in the values between ${\delta }_K$ and ${\delta }_L$ is small, it takes only a very small perturbation to make the system unstable^25,26. This may explain why some systems with very small speed differences are more likely to lose synchronization than those with much larger speed differences.

Placing the calculated or measured operating points in the coordinate system $u_K-u_L-{\delta }_{KL}$ can help directly determine whether the system is stable. A system is considered unstable as long as at least one point $(u_K,u_L,{\delta }_{KL})$ is outside the boundary (Figure 2b) and far from the other operating points (${\sigma }_{\Delta t}\left({\delta }_i\right)>>{\sigma }_{\Delta t-\epsilon }\left({\delta }_i\right),0<\epsilon <\Delta t$, where $\Delta t$ is the fault duration) (Figure 2b and Figure 4a). To determine the stability of a power system of n generators, only n pairs of variables $u_i,{\delta }_i,i\in (1,2,\dots ,n)$ are needed, which are physically meaningful and easily obtainable.

As one of the concepts closely related to stability, intentional isolation is an effective way of avoiding widespread blackouts following instability²⁷. Identifying coherent generators is a prerequisite for building intentional islands²⁸. Information regarding the coherence meta-generator groups can be obtained directly from the graph. That is, the partial synchronization of multiple generators, such as chimeric states^29–31, can also be identified with a stable boundary. m operating points outside the boundary indicate that the n meta-generators are sequentially divided into m+1 coherent groups.

Disturbed trajectory and parameters

To understand the behavior of the power system after a disturbance and to calculate the CCT and UEP, it is necessary to study the perturbation trajectories consisting of running points.

a. The projection of the disturbed trajectory in the ${\delta }_1-\Delta t$ plane, fitted using the equation.

b. The projection of the disturbed trajectory in the $u_1-\Delta t$ plane, fitted using the equation.

c. The projection of the disturbed trajectory in the ${\delta }_{1,2}-\Delta t$ plane, fitted using the equation. Notably, ${\delta }_{1,2}$ descended at $\Delta t=0.28s$.

d. The projection of the disturbed trajectory in the ${\delta }_1-{\delta }_2$ plane, fitted using the equation, where ${\delta }_k>{\delta }_l,a_{\delta }>1$.

e. The projection of the disturbed trajectory in the $u_1-u_2$ plane, fitted using the equation.

Figure 3.a and Figure 3.b show that the operating points move with a uniformly variable speed before the system becomes unstable.

Contrary to intuition³², ${\delta }_{12}$ suddenly dropped at $\Delta t=0.28s$ (Figure 3c). This anomaly suggested that the system appears to have a tendency to maintain its own stability.

Figure 3.d and E show that the perturbed trajectories of the subsystems of the coupled system are linearly correlated in the stability domain. This indicates that for a determined power grid, each perturbed trajectory has the same ${T^{\prime }}_i$, ${T^{\prime }}_i$ being the global invariant of the system. The effects of perturbations are global, reflecting the challenges of controlling the stability of complex systems^33–35.

When a high degree of accuracy of the results is not needed, $a_{KL}=0$. The following expression can be derived:

(3)

This can be used to easily and quickly check the stability margin of the system after a disturbance³⁶. By approximating $\Delta T^{\prime }$ as the CCT^19,36, the coordinates of the critical stable operating point $(u_K^{cr},u_L^{cr},{\delta }_{KL}^{cr})$ and critical rate of the meta-generator $l$ in the current system can also be estimated³⁷.

(4)

In summary, the CCT and UEP can be calculated using only information about the rotation rate¹², but considering only a single information source may result in more errors. Theoretically, using $\Delta T^{\prime }$ directly as the CCT would also lead to a conservative result. (Extended Data Table 1 can be referenced for more details)

The current power system is receiving an increasing number of renewable energy sources. These sources are connected to the power grid via inverters, which may change the inertia of the system^38–40, complicating the coefficients. This issue should be further studied.

On a finer scale, the operating points near the stability boundary exhibit unusual behavior (Figure 4).

Conclusions

References

1. Motter, A.E.; Myers, S.A.; Anghel, M.; Nishikawa, T. Spontaneous synchrony in power-grid networks. Nat. Phys. 2013, 9, 191–197. [Google Scholar] [CrossRef]
2. Yu, Y.; Liu, Y.; Qin, C.; Yang, T. Theory and Method of Power System Integrated Security Region Irrelevant to Operation States: An Introduction. Engineering 2020, 6, 754–777. [Google Scholar] [CrossRef]
3. Yang, P.; Liu, F.; Wei, W.; Wang, Z. Approaching the Transient Stability Boundary of a Power System: Theory and Applications. IEEE Trans. Autom. Sci. Eng. 2022, 1–12. [Google Scholar] [CrossRef]
4. Molnar, F.; Nishikawa, T.; Motter, A.E. Asymmetry underlies stability in power grids. Nat. Commun. 2021, 12, 1–9. [Google Scholar] [CrossRef] [PubMed]
5. Martínez, I.; Messina, A.R.; Vittal, V. Normal form analysis of complex system models: A structure-preserving approach. IEEE Trans. Power Syst. 2007, 22, 1908–1915. [Google Scholar] [CrossRef]
6. Bhui, P.; Senroy, N. Real-Time Prediction and Control of Transient Stability Using Transient Energy Function. IEEE Trans. Power Syst. 2017, 32, 923–934. [Google Scholar] [CrossRef]
7. Zhu, L.; Hill, D.J. Synchronization of Kuramoto Oscillators: A Regional Stability Framework. IEEE Trans. Automat. Contr. 2020, 65, 5070–5082. [Google Scholar] [CrossRef]
8. Koronovskii, A. A., Moskalenko, O. I. & Hramov, A. E. synchronization in complex networks. Tech. Phys. Lett. 2012, 38, 924–927.
9. Casals, M. R. Knowing power grids and understanding complexity science. Int. J. Crit. Infrastructures 2015, 11, 4. [Google Scholar] [CrossRef]
10. Gurrala, G.; Dimitrovski, A.; Pannala, S.; Simunovic, S.; Starke, M. Parareal in Time for Fast Power System Dynamic Simulations. IEEE Trans. Power Syst. 2016, 31, 1820–1830. [Google Scholar] [CrossRef]
11. Gurrala, G. Large Multi-Machine Power System Simulations Using Multi-Stage Adomian Decomposition. IEEE Trans. Power Syst. 2017, 32, 3594–3606. [Google Scholar] [CrossRef]
12. Wang, B.; Fang, B.; Wang, Y.; Liu, H.; Liu, Y. Power System Transient Stability Assessment Based on Big Data and the Core Vector Machine. IEEE Trans. Smart Grid 2016, 7, 2561–2570. [Google Scholar] [CrossRef]
13. Al-Ammar, E.A.; El-Kady, M.A. Application of operating security regions in power systems. IEEE PES Transm. Distrib. Conf. Expo. Smart Solut. a Chang. World 2010. [Google Scholar] [CrossRef]
14. Kundur, P. Definition and classification of power system stability. IEEE Trans. Power Syst. 2004, 19, 1387–1401. [Google Scholar]
15. Student Member, B.B.; Senior Member, G.A. On the nature of unstable equilibrium points in power systems. IEEE Trans. Power Syst. 1993, 8, 738–745. [Google Scholar]
16. Chiang, H.D.; Wu, F.F.; Varaiya, P.P. A BCU Method for Direct Analysis of Power System Transient Stability. IEEE Trans. Power Syst. 1994, 9, 1194–1208. [Google Scholar] [CrossRef]
17. Shubhanga, K.N.; Kulkarni, A.M. Application of Structure Preserving Energy Margin Sensitivity to Determime the Effectiveness of Shunt and Serles FACTS Devices. IEEE Power Eng. Rev. 2002, 22, 57. [Google Scholar] [CrossRef]
18. Al Marhoon, H.H.; Leevongwat, I.; Rastgoufard, P. A fast search algorithm for Critical Clearing Time for power systems transient stability analysis. 2014 Clemson Univ. Power Syst. Conf. PSC 2014, 2014. [Google Scholar] [CrossRef]
19. Rimorov, D.; Wang, X.; Kamwa, I.; Joos, G. An approach to constructing analytical energy function for synchronous generator models with subtransient dynamics. IEEE Trans. Power Syst. 2018, 33, 5958–5967. [Google Scholar] [CrossRef]
20. Dörfler, F.; Chertkov, M.; Bullo, F. Synchronization in complex oscillator networks and smart grids. Proc. Natl. Acad. Sci. U. S. A. 2013, 110, 2005–2010. [Google Scholar] [CrossRef]
21. Cuadra, L.; Salcedo-Sanz, S.; Del Ser, J.; Jiménez-Fernández, S.; Geem, Z.W. A critical review of robustness in power grids using complex networks concepts. Energies 2015, 8, 9211–9265. [Google Scholar] [CrossRef]
22. Zhou, J. Large-Scale Power System Robust Stability Analysis Based on Value Set Approach. IEEE Trans. Power Syst. 2017, 32, 4012–4023. [Google Scholar] [CrossRef]
23. Ajala, O.; Dominguez-Garcia, A.; Sauer, P.; Liberzon, D. A Second-Order Synchronous Machine Model for Multi-swing Stability Analysis. 51st North Am. Power Symp. NAPS 2019, 2019. [Google Scholar] [CrossRef]
24. Karatekin, C.Z.; Uçak, C. Sensitivity analysis based on transmission line susceptances for congestion management. Electr. Power Syst. Res. 2008, 78, 1485–1493. [Google Scholar] [CrossRef]
25. Mei, S.; Ni, Y.; Wang, G.; Wu, S. A study of self-organized criticality of power system under cascading failures based on AC-OPF with voltage stability margin. IEEE Trans. Power Syst. 2008, 23, 1719–1726. [Google Scholar]
26. Dobson, I.; Carreras, B.; Lynch, V.; Newman, D. An initial model for complex dynamics in electric power system blackouts. Proc. Hawaii Int. Conf. Syst. Sci. 2001, 51. [Google Scholar] [CrossRef]
27. Ding, L.; Gonzalez-Longatt, F.M.; Wall, P.; Terzija, V. Two-step spectral clustering controlled islanding algorithm. IEEE Trans. Power Syst. 2013, 28, 75–84. [Google Scholar] [CrossRef]
28. Znidi, F.; Davarikia, H.; Rathore, H. Power Systems Transient Stability Indices: Hierarchical Clustering Based Detection of Coherent Groups Of Generators. 2021.
29. Kuramoto, Y.; Battogtokh, D. Coexistence of Coherence and Incoherence in Nonlocally Coupled Phase Oscillators. Physics (College. Park. Md). 2002, 4, 380–385. [Google Scholar]
30. Martens, E.A.; Thutupalli, S.; Fourrière, A.; Hallatschek, O. Chimera states in mechanical oscillator networks. Proc. Natl. Acad. Sci. USA 2013, 110, 10563–10567. [Google Scholar] [CrossRef]
31. Panaggio, M.J.; Abrams, D.M. Chimera states: Coexistence of coherence and incoherence in networks of coupled oscillators. Nonlinearity 2015, 28, R67–R87. [Google Scholar] [CrossRef]
32. Amirthalingam, K.M.; Ramachandran, R.P. Improvement of transient stability of power system using solid state circuit breaker. Am. J. Appl. Sci. 2013, 10, 563–569. [Google Scholar] [CrossRef]
33. Liu, X.; Shahidehpour, M.; Cao, Y.; Li, Z.; Tian, W. Risk assessment in extreme events considering the reliability of protection systems. IEEE Trans. Smart Grid 2015, 6, 1073–1081. [Google Scholar] [CrossRef]
34. Huang, R. Learning and Fast Adaptation for Grid Emergency Control via Deep Meta Reinforcement Learning. IEEE Trans. Power Syst. 2022, 37, 4168–4178. [Google Scholar] [CrossRef]
35. Guo, M.; Xu, D.; Liu, L. Design of Cooperative Output Regulators for Heterogeneous Uncertain Nonlinear Multiagent Systems. IEEE Trans. Cybern. 2022, 52, 5174–5183. [Google Scholar] [CrossRef] [PubMed]
36. Roberts LG, W.; Champneys, A.R.; Bell KR, W.; Di Bernardo, M. Analytical Approximations of Critical Clearing Time for Parametric Analysis of Power System Transient Stability. IEEE J. Emerg. Sel. Top. Circuits Syst. 2015, 5, 465–476. [Google Scholar] [CrossRef]
37. Owusu-Mireku, R.; Chiang, H.D.; Hin, M. A Dynamic Theory-Based Method for Computing Unstable Equilibrium Points of Power Systems. IEEE Trans. Power Syst. 2020, 35, 1946–1955. [Google Scholar] [CrossRef]
38. Sajadi, A.; Kenyon, R.W.; Hodge, B.M. Synchronization in electric power networks with inherent heterogeneity up to 100% inverter-based renewable generation. Nat. Commun. 2022, 13, 1–12. [Google Scholar]
39. Sun, M.; et al. On-line power system inertia calculation using wide area measurements. Int. J. Electr. Power Energy Syst. 2019, 109, 325–331. [Google Scholar] [CrossRef]
40. Zhang, Y.; Bank, J.; Muljadi, E.; Wan, Y.H.; Corbus, D. Angle instability detection in power systems with high-wind penetration using synchrophasor measurements. IEEE J. Emerg. Sel. Top. Power Electron. 2013, 1, 306–314. [Google Scholar] [CrossRef]
41. Dörfler, F.; Bullo, F. Synchronization in complex networks of phase oscillators: A survey. Automatica 2014, 50, 1539–1564. [Google Scholar] [CrossRef]
42. Chen, G. Searching for Best Network Topologies with Optimal Synchronizability: A Brief Review. IEEE/CAA J. Autom. Sin. 2022, 9, 573–577. [Google Scholar] [CrossRef]
43. Li, X.; Wei, W.; Zheng, Z. Promoting synchrony of power grids by restructuring network topologies. Chaos An Interdiscip. J. Nonlinear Sci. 2023, 33, 63149. [Google Scholar] [CrossRef] [PubMed]
44. Zhang, Y.; Motter, A.E. Symmetry-Independent Stability Analysis of Synchronization Patterns. SIAM Rev. 2020, 62, 817–836. [Google Scholar] [CrossRef]