Blume Steffen O P, Sansavini Giovanni
Future Resilient Systems, Singapore-ETH Centre, 1 Create Way, CREATE Tower, #06-01, Singapore 138602.
Reliability and Risk Engineering Laboratory, Department of Mechanical and Process Engineering, ETH Zürich, Leonhardstrasse 21, 8092 Zürich, Switzerland.
Chaos. 2017 Dec;27(12):121101. doi: 10.1063/1.5008315.
Complex dynamical systems face abrupt transitions into unstable and catastrophic regimes. These critical transitions are triggered by gradual modifications in stressors, which push the dynamical system towards unstable regimes. Bifurcation analysis can characterize such critical thresholds, beyond which systems become unstable. Moreover, the stochasticity of the external stressors causes small-scale fluctuations in the system response. In some systems, the decomposition of these signal fluctuations into precursor signals can reveal early warning signs prior to the critical transition. Here, we present a dynamical analysis of a power system subjected to an increasing load level and small-scale stochastic load perturbations. We show that the auto- and cross-correlations of bus voltage magnitudes increase, leading up to a Hopf bifurcation point, and further grow until the system collapses. This evidences a gradual transition into a state of "critical coupling," which is complementary to the established concept of "critical slowing down." Furthermore, we analyze the effects of the type of load perturbation and load characteristics on early warning signs and find that gradient changes in the autocorrelation provide early warning signs of the imminent critical transition under white-noise but not for auto-correlated load perturbations. Furthermore, the cross-correlation between all voltage magnitude pairs generally increases prior to and beyond the Hopf bifurcation point, indicating "critical coupling," but cannot provide early warning indications. Finally, we show that the established early warning indicators are oblivious to limit-induced bifurcations and, in the case of the power system model considered here, only react to an approaching Hopf bifurcation.
复杂动力系统会突然转变为不稳定和灾难性状态。这些临界转变是由压力源的逐渐变化引发的，压力源的变化会将动力系统推向不稳定状态。分岔分析可以表征此类临界阈值，超过该阈值系统就会变得不稳定。此外，外部压力源的随机性会导致系统响应出现小尺度波动。在某些系统中，将这些信号波动分解为前兆信号可以揭示临界转变之前的早期预警信号。在此，我们对一个承受不断增加的负载水平和小尺度随机负载扰动的电力系统进行了动力学分析。我们表明，母线电压幅值的自相关和互相关会增加，直至达到霍普夫分岔点，并进一步增大，直到系统崩溃。这证明了向“临界耦合”状态的逐渐转变，这与已确立的“临界减速”概念相辅相成。此外，我们分析了负载扰动类型和负载特性对早期预警信号的影响，发现自相关的梯度变化在白噪声情况下能提供即将发生临界转变的早期预警信号，但对于自相关负载扰动则不然。此外，所有电压幅值对之间的互相关在霍普夫分岔点之前和之后通常都会增加，表明存在“临界耦合”，但无法提供早期预警指示。最后，我们表明，已确立的早期预警指标对极限诱导分岔不敏感，在此处考虑的电力系统模型中，仅对即将到来的霍普夫分岔做出反应。
