Electric power systems are undergoing significant changes with the deployment of large-scale wind and solar plants connected to the transmission system and small-scale Distributed Energy Resources (DERs) and microgrids connected to the distribution system, making the latter an active system. A microgrid is a small-scale power system that interconnects renewable and non-renewable generating units such as solar photo-voltaic panels and micro-turbines, storage devices such as batteries and fly wheels, and loads. Typically, it is connected to the distribution feeders via power electronic converters with fast control responses within the micro-seconds. These new developments have prompted growing research activities in stability analysis and control of the transmission and the distribution systems. Unfortunately, these systems are treated as separated entities, limiting the scope of the applicability of the proposed methods to real systems. It is worth stressing that the transmission and distribution systems are interconnected via HV/MV transformers and therefore, are interacting dynamically in a complex way. In this research work, we overcome this problem by investigating the dynamics of the transmission and distribution systems with parallel microgrids as an integrated system . Specifically, we develop a generic model of a microgrid that consists of a DC voltage source connected to an inverter with real and reactive power control and voltage control. We analyze the small-signal stability of the two-area four-machine system with four parallel microgrids connected to the distribution feeders though different impedances. We show that the conventional PQ control of the inverters is insufficient to stabilize the voltage at the point-of-common coupling when the feeder impedances have highly unequal values. To ensure the existence of a stable equilibrium point associated with a sufficient stability margin of the system, we propose a new voltage control implemented as an additional feedback control loop of the conventional inner and outer current control schemes of the inverter. Furthermore, we carry out a modal analysis of the four-machine system with microgrids using Koopman mode analysis. We reveal the existence of local modes of oscillation of a microgrid against the rest of the system and between parallel microgrids at frequencies that range between 0.1 and 3 Hz. When the control of the microgrid becomes unstable, the frequencies of the oscillation are about 20 Hz. Recall that the Koopman mode analysis is a new technique developed in fluid dynamics and recently introduced in power systems by Suzuki and Mezic. It allows us to carry out small signal and transient stability analysis by processing only measurements, without resorting to any model and without assuming any linearization. / Ph. D.

In this thesis I investigate a number of problems in the nonlinear stability of density stratified plane Couette flow. I begin by describing the history of transient growth phenomena, and in particular the recent application of adjoint based optimisation to find nonlinear optimal perturbations and associated minimal seeds for turbulence, the smallest amplitude perturbations that are able to trigger transition to turbulence. I extend the work of Rabin et al. (2012) in unstratified plane Couette flow to find minimal seeds in both vertically and horizontally sheared stratified plane Couette flow. I find that the coherent states visited by such minimal seed trajectories are significantly altered by the stratification, and so proceed to investigate these states both with generalised Koopman mode analysis and by stratifying the self-sustaining process described by Waleffe (1997). I conclude with an introductory problem I considered that investigates the linear Taylor instability of layered stratified plane Couette flow, and show that the nonlinear evolution of the primary Taylor instability is not coupled to the form of the linearly unstable mode, in contrast to the Kelvin-Helmholtz instability, for example. I also include an appendix in which I describe joint work conducted with Professor Neil Balmforth of UBC during the 2015 WHOI Geophysical Fluid Dynamics summer programme, investigating stochastic homoclinic bifurcations.

To view the electricity supply in our society as just sockets mountedin our walls with a constant voltage output is far from the truth. Inreality, the power system supplying the electricity or the grid, is themost complex man-made dynamical system there is. It demands severecontrol and safety measures to ensure a reliable supply of electric power.Throughout the world, incidents of widespread power grid failures havebeen continuously reported. The state where electricity delivery to customersis terminated by a disturbance is called a blackout. From a stateof seemingly stable operating conditions, the grid can fast derail intoan uncontrollable state due to cascading failures. Transmission linesbecome automatically disconnected due to power flow redirections andparts of the grid become isolated and islands are formed. An islandedsub-grid incapable of maintaining safe operation conditions experiencesa blackout. A widespread blackout is a rare, but an extremely costlyand hazardous event for society.During recent years, many methods to prevent these kinds of eventshave been suggested. Controlled islanding has been a commonly suggestedstrategy to save the entire grid or parts of the grid from a blackout.Controlled islanding is a strategy of emergency control of a powergrid, in which the grid is intentionally split into a set of islanded subgridsfor avoiding an entire collapse. The key point in the strategy is todetermine appropriate separation boundaries, i.e. the set of transmissionlines separating the grid into two or more isolated parts.The power grid exhibits highly nonlinear response in the case oflarge failures. Therefore, this thesis proposes a new controlled islandingmethod for power grids based on the nonlinear Koopman Mode Analysis(KMA). The KMA is a new analyzing technique of nonlinear dynamicsbased on the so-called Koopman operator. Based on sampled data followinga disturbance, KMA is used to identify suitable partitions of thegrid.The KMA-based islanding method is numerically investigated withtwo well-known test systems proposed by the Institute of Electrical andElectronics Engineers (IEEE). By simulations of controlled islanding inthe test system, it is demonstrated that the grid’s response following afault can be improved with the proposed method.The proposed method is compared to a method of partitioning powergrids based on spectral graph theory which captures the structural propertiesof a network. It is shown that the intrinsic structural propertiesof a grid characterized by spectral graph theory are also captured by theKMA. This is shown both by numerical simulations and a theoreticalanalysis.