Assessment of Power System Stability Considering Multiple Time-Scale Dynamics: Insights Into Saddle-Node and Hopf Bifurcations

Real power systems exhibit dynamics that evolve over various time scales. Some dynamics are exceptionally fast, such as electromagnetic transients, which occur on the order of microseconds. Others are notably slow, like load restoration processes and the action of Over-Excitation Limiters (OELs), which can span tens of seconds or even minutes. Between these extremes lie electromechanical transients, characterized by dynamics ranging from a few milliseconds to several seconds. Traditionally, power engineers categorize stability phenomena and related studies based on these time scales. For example, when assessing fast dynamics involving fast inverter controls (e.g., inner current controls), it is common to ignore slower dynamics like OELs that operate over several seconds or minutes. Conversely, when evaluating slower phenomena such as long-term voltage stability, faster dynamics (e.g., fast inverter controls) are often disregarded. While these assumptions may be valid in many cases, they imply that certain dynamics are inherently stable by completely neglecting them. This time-scale decomposition is effectively achieved using a technique called singular perturbation, which is so ingrained in the power engineering community that engineers rarely question it or may not even recognize its implications. Nevertheless, real networks do not perform time-scale decomposition. On the contrary, multi-time scale dynamics are intertwined and can have a significant impact on each other. In fact, well-known literature, such as [1], refers to this dependency between dynamics that evolve over different time scales as S-LT1, S-LT2, and S-LT3. In simple terms, they explain the different mechanisms by which slow dynamics can act in a way that forces the fast dynamics to lose equilibrium. These connections between different time-scale dynamics are not explicitly addressed in the well-known IEEE classification of power system stability. Typical results of such interactions include: 1) Saddle-Node Bifurcation of the fast dynamics induced by slow dynamics, which typically manifests as a sudden system collapse when a generator loses synchronism, and 2) a Hopf Bifurcation of fast dynamics (see Fig. 1) induced by slow dynamics, which typically manifests as oscillatory instabilities when complex-conjugate eigenvalues are pushed into the right semi-plane. This paper addresses the importance of assessing system stability considering the intrinsic interactions between different dynamics. It shows that if time-scale decomposition is always performed, instabilities that may occur in a real physical system can be missed in simulations. This paper builds on works previously published by the author [2], [3]. It advances the previous investigations by mathematically formulating the types of instabilities in terms of their bifurcation types and showing the consequences of each. The paper uses the well-known IEEE test system for voltage stability assessment (known as the Nordic test system). Since addressing new system technologies is essential in state-of-the-art research, this work analyzes the performance of grid-following and grid-forming inverters, highlighting potential issues and advantages of each technology within the scope of multi-time scale dynamics.