Abstract
Critical infrastructure systems like the power system are in danger of abrupt transitions into unstable and oftentimes catastrophic regimes. These critical transitions occur due to the modification of certain control parameters or an external forcing acting on the dynamical system. Bifurcation analysis can help to characterize the critical threshold beyond which systems become unstable. Moreover, some systems emit early warning signs prior to the critical transition, detectable from small-scale signal fluctuations triggered by the stochasticity of the external forcing. We present here our analysis of a time-domain dynamical power system model subjected to an increasing load level and small-scale stochastic load perturbations. We confirm previous findings from other literature that the autocorrelation of system signals increases, as the load level approaches a critical threshold characterized by a Hopf bifurcation point. Furthermore, we analyze the effects of load homogeneity and load connectivity on early warning signs. We assert that load connectivity does not influence autocorrelation coefficients. In addition, we show that changes in load homogeneity shift the location of the critical threshold.
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Notes
- 1.
In case of the 10-machine 39-bus New England power system model used here, the system of equations consists of 85 differential and 138 algebraic equations: 4 differential and 4 algebraic equations for each of the G generators (\(G = 10\)); 5 differential and 2 algebraic equations for every generator with a turbine governor and voltage regulator (\(G^* = 9\)); 2 algebraic equations for each of the \(N_b = 39\) network buses; and 2 algebraic equations for the synchronous reference frequency and rotor angle; Thus \(m_x = \left( 4G + 5G^* \right) \) differential equations and \(m_y = \left( 4G + 2G^* + 2N_b + 2\right) \) algebraic equations. Stochastic perturbations in both the active and reactive components of each of the \(N = 19\) loads result in \(m_\eta = 2N\).
- 2.
The load level increases by 0.05 within the 270 s window.
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Acknowledgements
We would like to thank the IT Services at ETH Zürich for the provision of computing resources and access of the EULER cluster. This research was conducted under the Future Resilient Systems program at the Singapore-ETH Centre and funded by the National Research Foundation of Singapore.
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Appendix
Appendix
Since this analysis deals with discrete signals, an estimate of the autocorrelation for lag k of a signal with length N is obtained from [29]
where \(\bar{x}\) is the sample mean of the signal and \(s^2\) is the unbiased sample variance.
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Blume, S.O.P., Sansavini, G. (2018). The Influence of Load Characteristics on Early Warning Signs in Power Systems. In: D'Agostino, G., Scala, A. (eds) Critical Information Infrastructures Security. CRITIS 2017. Lecture Notes in Computer Science(), vol 10707. Springer, Cham. https://doi.org/10.1007/978-3-319-99843-5_7
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