Voltage Stabilization in Microgrids via Quadratic Droop Control

We consider the problem of voltage stability and reactive power balancing in islanded small-scale electrical networks outfitted with DC/AC inverters ("microgrids"). A droop-like voltage feedback controller is proposed which is quadratic in the local voltage magnitude, allowing for the application of circuit-theoretic analysis techniques to the closed-loop system. The operating points of the closed-loop microgrid are in exact correspondence with the solutions of a reduced power flow equation, and we provide explicit solutions and small-signal stability analyses under several static and dynamic load models. Controller optimality is characterized as follows: we show a one-to-one correspondence between the high-voltage equilibrium of the microgrid under quadratic droop control, and the solution of an optimization problem which minimizes a trade-off between reactive power dissipation and voltage deviations. Power sharing performance of the controller is characterized as a function of the controller gains, network topology, and parameters. Perhaps surprisingly, proportional sharing of the total load between inverters is achieved in the low-gain limit, independent of the circuit topology or reactances. All results hold for arbitrary grid topologies, with arbitrary numbers of inverters and loads. Numerical results confirm the robustness of the controller to unmodeled dynamics.Comment: 14 pages, 8 figure

Secondary Frequency and Voltage Control of Islanded Microgrids via Distributed Averaging

In this work we present new distributed controllers for secondary frequency and voltage control in islanded microgrids. Inspired by techniques from cooperative control, the proposed controllers use localized information and nearest-neighbor communication to collectively perform secondary control actions. The frequency controller rapidly regulates the microgrid frequency to its nominal value while maintaining active power sharing among the distributed generators. Tuning of the voltage controller provides a simple and intuitive trade-off between the conflicting goals of voltage regulation and reactive power sharing. Our designs require no knowledge of the microgrid topology, impedances or loads. The distributed architecture allows for flexibility and redundancy, and eliminates the need for a central microgrid controller. We provide a voltage stability analysis and present extensive experimental results validating our designs, verifying robust performance under communication failure and during plug-and-play operation.Comment: Accepted for publication in IEEE Transactions on Industrial Electronic

Input-Output Performance of Linear-Quadratic Saddle-Point Algorithms With Application to Distributed Resource Allocation Problems

Saddle-point or primal-dual methods have recently attracted renewed interest as a systematic technique to design distributed algorithms, which solve convex optimization problems. When implemented online for streaming data or as dynamic feedback controllers, these algorithms become subject to disturbances and noise; convergence rates provide incomplete performance information, and quantifying input-output performance becomes more important. We analyze the input-output performance of the continuous-time saddle-point method applied to linearly constrained quadratic programs, providing explicit expressions for the saddle-point $\mathcal {H}_2$ norm under a relevant input-output configuration. We then proceed to derive analogous results for regularized and augmented versions of the saddle-point algorithm. We observe some rather peculiar effects-a modest amount of regularization significantly improves the transient performance, while augmentation does not necessarily offer improvement. We then propose a distributed dual version of the algorithm, which overcomes some of the performance limitations imposed by augmentation. Finally, we apply our results to a resource allocation problem to compare the input-output performance of various centralized and distributed saddle-point implementations and show that distributed algorithms may perform as well as their centralized counterparts

Synchronization of coupled limit cycles

A unified approach for analyzing synchronization in coupled systems of autonomous differential equations is presented in this work. Through a careful analysis of the variational equation of the coupled system we establish a sufficient condition for synchronization in terms of the geometric properties of the local limit cycles and the coupling operator. This result applies to a large class of differential equation models in physics and biology. The stability analysis is complemented with a discussion of numerical simulations of a compartmental model of a neuron.Comment: Journal of Nonlinear Science, accepte

Subject specific demands of teaching: Implications for out-of-field teachers

This chapter provides a framework for thinking about the subject-specific nature of teaching in terms of the knowledge, modes of inquiry and discursive practices that delineate one subject from another in the traditional school curriculum. The chapter will explore how these disciplinary traits are translated into teaching as curriculum, knowledge and pedagogy, and how this subject-specificity of teaching is juxtaposed against the more generic aspects of teaching. The chapter explores the idea that if a teacher’s expertise can be situated within a field, then they can also be positioned out-of-field. Implications for teaching out-of-field are discussed in terms of the subject-specific knowledge, processes and skills, and the difficulties associated with teacher practice. English and Australian illustrations of teacher practices from in-field and out-of-field situations are provided, in particular highlighting the demands of moving across subject boundaries. Cross-fertilisation is especially evident when subjects are integrated, therefore, the issues associated with integrated curriculum are discussed where the traditional subject boundaries are being challenged as schools are reorganised to integrate subjects through, for example, STEM teaching, or holistic curriculum designs

Discrete gradient flows for general curvature energies

We consider the numerical approximation of theL2–gradient flow of general curvatureenergies∫G(|~κ|) for a curve inRd,d≥2. Here the curve can be either closed, or it can be open andclamped at the end points. These general curvature energies, and the considered boundary conditions,appear in the modelling of the power loss within an optical fibre. We present two alternative finiteelement approximations, both of which admit a discrete gradient flow structure. Apart from beingstable, in addition, one of the methods satisfies an equidistribution property. Numerical resultsdemonstrate the robustness and the accuracy of the proposedmethods