John W. Simpson-Porco

Synchronization and power sharing for droop-controlled inverters in islanded microgrids

Motivated by the recent and growing interest in smart grid technology, we study the operation of DC/AC inverters in an inductive microgrid. We show that a network of loads and DC/AC inverters equipped with power-frequency droop controllers can be cast as a Kuramoto model of phase-coupled oscillators. This novel description, together with results from the theory of coupled oscillators, allows us to characterize the behavior of the network of inverters and loads. Specifically, we provide a necessary and sufficient condition for the existence of a synchronized solution that is unique and locally exponentially stable. We present a selection of controller gains leading to a desirable sharing of power among the inverters, and specify the set of loads which can be serviced without violating given actuation constraints. Moreover, we propose a distributed integral controller based on averaging algorithms, which dynamically regulates the system frequency in the presence of a time-varying load. Remarkably, this distributed-averaging integral controller has the additional property that it preserves the power sharing properties of the primary droop controller. Our results hold for any acyclic network topology, and hold without assumptions on identical line admittances or voltage magnitudes.

Breaking the Hierarchy: Distributed Control and Economic Optimality in Microgrids

Modeled after the hierarchical control architecture of power transmission systems, a layering of primary, secondary, and tertiary control has become the standard operation paradigm for islanded microgrids. Despite this superficial similarity, the control objectives in microgrids across these three layers are varied and ambitious, and they must be achieved while allowing for robust plug-and-play operation and maximal flexibility, without hierarchical decision making and time-scale separations. In this paper, we explore control strategies for these three layers and illuminate some possibly unexpected connections and dependencies among them. Building from a first-principle analysis of decentralized primary droop control, we study centralized, decentralized, and distributed architectures for secondary frequency regulation. We find that averaging-based distributed controllers using communication among the generation units offer the best combination of flexibility and performance. We further leverage these results to study constrained ac economic dispatch in a tertiary control layer. Surprisingly, we show that the minimizers of the economic dispatch problem are in one-to-one correspondence with the set of steady states reachable by droop control. In other words, the adoption of droop control is necessary and sufficient to achieve economic optimization. This equivalence results in simple guidelines to select the droop coefficients, which include the known criteria for power sharing. We illustrate the performance and robustness of our designs through simulations.

Secondary Frequency and Voltage Control of Islanded Microgrids via Distributed Averaging

In this paper, 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 tradeoff 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, eliminating 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.

Voltage stabilization in microgrids via quadratic droop control

Motivated by the growing interest in energy technology and smart grid architectures, we consider the problem of voltage stability and reactive power balancing in low-voltage electrical networks equipped with DC/AC inverters (“microgrids”). It is generally believed that high-voltage equilibria of such networks are stable, but the locations of these equilibria are unknown, as is the critical network load where stability is lost. Inspired by the “control by interconnection” paradigm developed for port-Hamiltonian systems, we propose a novel droop-like inverter controller which is quadratic in the local voltage magnitude. Remarkably, under this controller the closed-loop network is again a well-posed electrical circuit. We find that the equilibria of the quadratic droop-controlled network are in exact correspondence with the solutions of a reduced power flow equation. For general network topologies, we study some simple yet insightful solutions of this equation, and for the frequently-encountered case of a parallel microgrid, we present a concise and closed-form condition for the existence of an exponentially stable high-voltage network equilibrium. Our condition establishes the existence of a critical inductive load for the network, which depends only on the network topology, admittances, and controller gains. We compare and contrast our design with the conventional droop controller, investigate the relationship between the two, and validate the robustness of our design through simulation.

Voltage collapse in complex power grids.

A large-scale power grids ability to transfer energy from producers to consumers is constrained by both the network structure and the nonlinear physics of power flow. Violations of these constraints have been observed to result in voltage collapse blackouts, where nodal voltages slowly decline before precipitously falling. However, methods to test for voltage collapse are dominantly simulation-based, offering little theoretical insight into how grid structure influences stability margins. For a simplified power flow model, here we derive a closed-form condition under which a power network is safe from voltage collapse. The condition combines the complex structure of the network with the reactive power demands of loads to produce a node-by-node measure of grid stress, a prediction of the largest nodal voltage deviation, and an estimate of the distance to collapse. We extensively test our predictions on large-scale systems, highlighting how our condition can be leveraged to increase grid stability margins.

Droop-Controlled Inverters Are Kuramoto Oscillators

Abstract Motivated by the recent interest in smart grid technology and by the push towards distributed and renewable energy, we study the parallel operation of DC/AC inverters in a lossless microgrid. We show that the parallel interconnection of DC/AC inverters equipped with conventional droop controllers is precisely described by the Kuramoto model of coupled phase oscillators. This novel description, together with results from the theory of coupled oscillators, allows us to characterize the behavior of the network of inverters. Specifically, we provide a necessary and sufficient condition for the existence of a synchronized solution that is unique and exponentially stable. Remarkably, we find that the existence of such a synchronized solution does not depend on the selection of droop coefficients. We prove that the inverters share the network power demand in proportion to their power ratings if and only if the droop coefficients are selected proportionally, and we characterize the set of feasible loads which can be serviced. These results hold without assumptions on identical line characteristics or voltage magnitudes.

Further results on distributed secondary control in microgrids

This work presents several analysis and design results for primary droop control and secondary control in inductive microgrids. Building on our recent work, we study the problem of set point design in droop-controlled microgrids, and provide a choice of droop coefficients leading to the desired power flows. We then compare and contrast two distributed secondary controllers, and extend them to the case where only a fraction of inverters cooperate in regulating the network frequency. We show that both these secondary-control schemes achieve frequency regulation in the presence of resistive losses.

Contraction theory on Riemannian manifolds

Abstract Contraction theory is a methodology for assessing the stability of trajectories of a dynamical system with respect to one another. In this work, we present the fundamental results of contraction theory in an intrinsic, coordinate-free setting, with the presentation highlighting the underlying geometric foundation of contraction theory and the resulting stability properties. We provide coordinate-free proofs of the main results for autonomous vector fields, and clarify the assumptions under which the results hold. We state and prove several interesting corollaries to the main result, study cascade and feedback interconnections of contracting systems, study some simple examples, and highlight how contraction theory has arisen independently in other scientific disciplines. We conclude by illustrating the developed theory for the case of gradient dynamics.

On reactive power flow and voltage stability in microgrids

This paper focuses on reactive power flow and voltage stability in electrical grids. We provide novel analytical understanding of the solutions to the classic nonlinear polynomial equations describing the decoupled reactive power flow. As of today, solutions to these equations can be found only via numerical methods. Yet an analytical understanding would be beneficial to the rigorous design of future electrical grids. This paper has two main contributions. First, for sufficiently high reference voltages, we guarantee the existence of a high-voltage solution for the reactive power flow equations and provide its approximate analytical expression. We bound the approximation error in terms of network topology and parameters. Second, we consider a recently proposed droop control strategy for voltage stabilization in a microgrid equipped with inverters. For sufficiently high reference voltages, we prove the existence and the exponential stability of a high-voltage fixed point of the closed-loop dynamics. We provide an approximate expression for this fixed point and bound the approximation error in terms of the network topology and parameters. Finally, we validate the accuracy of our approximations through numerical simulation of the IEEE 37 standard test case.

On Resistive Networks of Constant-Power Devices

This brief examines the behavior of DC circuits comprised of resistively interconnected constant-power devices (CPDs), as may arise in dc microgrids containing microsources and constant-power loads. We derive a sufficient condition for all operating points of the circuit to lie in a desirable set, where the average nodal voltage level is high, and nodal voltages are tightly clustered near one another. Our condition has the elegant physical interpretation that the ratio of resistive losses to total injected power should be small compared with a measure of network heterogeneity, as quantified by a ratio of conductance matrix eigenvalues. Perhaps surprisingly, the interplay between the circuit topology, branch conductances, and CPDs implicitly defines a nominal voltage level for the circuit, despite the explicit absence of voltage-regulated nodes.