Series Preface * Preface to the Third Edition * 1 Linear Systems * 2 Nonlinear Systems: Local Theory * 3 Nonlinear Systems: Global Theory * 4 Nonlinear Systems: Bifurcation Theory * References * Index

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Differential Equations and Dynamical Systems

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L2 Gain And Passivity Techniques In Nonlinear Control

The electric power system is currently undergoing a period of unprecedented changes. Environmental and sustainability concerns lead to replacement of a significant share of conventional fossil fuel-based power plants with renewable energy resources. This transition involves the major challenge of substituting synchronous machines and their well-known dynamics and controllers with power electronics-interfaced generation whose regulation and interaction with the rest of the system is yet to be fully understood. In this article, we review the challenges of such low-inertia power systems, and survey the solutions that have been put forward thus far. We strive to concisely summarize the laid-out scientific foundations as well as the practical experiences of industrial and academic demonstration projects. We touch upon the topics of power system stability, modeling, and control, and we particularly focus on the role of frequency, inertia, as well as control of power converters and from the demand-side.

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Foundations and Challenges of Low-Inertia Systems (Invited Paper)

A recent article ‘Burden of proof: A comprehensive review of the feasibility of 100% renewable-electricity systems’ claims that many studies of 100% renewable electricity systems do not demonstrate sufficient technical feasibility, according to the criteria of the article's authors (henceforth ‘the authors’). Here we analyse the authors’ methodology and find it problematic. The feasibility criteria chosen by the authors are important, but are also easily addressed at low economic cost, while not affecting the main conclusions of the reviewed studies and certainly not affecting their technical feasibility. A more thorough review reveals that all of the issues have already been addressed in the engineering and modelling literature. Nuclear power, which the authors have evaluated positively elsewhere, faces other, genuine feasibility problems, such as the finiteness of uranium resources and a reliance on unproven technologies in the medium- to long-term. Energy systems based on renewables, on the other hand, are not only feasible, but already economically viable and decreasing in cost every year.

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Response to 'Burden of proof': A comprehensive review of the feasibility of 100% renewable-electricity systems'

The electric power system is witnessing a shift in the technology of generation. Conventional thermal generation based on synchronous machines is gradually being replaced by power electronics interfaced renewable generation. This new mode of generation, however, lacks the natural inertia and governor damping, which are quintessential features of synchronous machines. The loss of these features results in increasing frequency excursions and, ultimately, system instability. Among the numerous studies on mitigating these undesirable effects, the main approach involves virtual inertia (VI) emulation to mimic the behavior of synchronous machines. In this paper, explicit models of grid-following and grid-forming VI devices are developed for inertia emulation and fast frequency response in low-inertia systems. An optimization problem is formulated to optimize the parameters and location of these devices in a power system to increase its resilience. Finally, a case study based on a high-fidelity model of the South-East Australian system is used to illustrate the effectiveness of such devices.

Placement and Implementation of Grid-Forming and Grid-Following Virtual Inertia and Fast Frequency Response

Grid-forming control for power converters based on matching of synchronous machines

We consider the problem of grid-forming control of power converters in low-inertia power systems. Starting from an average-switch three-phase inverter model, we draw parallels to a synchronous machine (SM) model and propose a novel grid-forming converter control strategy which dwells upon the main characteristic of a SM: the presence of an internal rotating magnetic field. In particular, we augment the converter system with a virtual oscillator whose frequency is driven by the DC-side voltage measurement and which sets the converter pulse-width-modulation signal, thereby achieving exact matching between the converter in closed-loop and the SM dynamics. We then provide a sufficient condition assuring existence, uniqueness, and global asymptotic stability of equilibria in a coordinate frame attached to the virtual oscillator angle. By actuating the DC-side input of the converter we are able to enforce this sufficient condition. In the same setting, we highlight strict incremental passivity, droop, and power-sharing properties of the proposed framework, which are compatible with conventional requirements of power system operation. We subsequently adopt disturbance decoupling techniques to design additional control loops that regulate the DC-side voltage, as well as AC-side frequency and amplitude, while in the end validating them with numerical experiments.

Grid-forming Control for Power Converters based on Matching of Synchronous Machines

Grid-Friendly Matching of Synchronous Machines by Tapping into the DC Storage*

We study local (also referred to as small-signal) stability of a network of identical DC/AC converters having a rotating degree of freedom. We develop a stability theory for a class of partitioned linear systems with symmetries that has natural links to classical stability theories of interconnected systems. We find stability conditions descending from a particular Lyapunov function involving an oblique projection onto the complement of the synchronous steady state set and enjoying insightful structural properties. Our sufficient and explicit stability conditions can be evaluated in a fully decentralized fashion, reflect a parametric dependence on the converter's steady-state variables, and can be one-to-one generalized to other types of systems exhibiting the same behavior, such as synchronous machines. Our conditions demand for sufficient reactive power support and resistive damping. These requirements are well aligned with practitioners' insights.

Parametric local stability condition of a multi-converter system

We address the problem of local synchronization of two DC/AC converters connected via a RL line in closed-loop with machine matching control by pursuing a novel approach for the assessment of small-signal stability. Our method is defined using an oblique projection. We use this Lyapunov function to obtain a local stability certificate in the neighborhood of a well-defined set of equilibria characterized by frequency synchronization. To this end, we first reduce the fifth-order DC/AC converter model to a third-order one. Then, we investigate local stability based on Lyapunov's method performed on the linearized and reduced-order model. Moreover, we certify stability via sufficient and fully decentralized conditions that can be satisfied by choosing an appropriate operating point and control gains. Finally, we demonstrate our results using numerical simulations.

Taouba Jouini

Papers

Grid-forming control for power converters based on matching of synchronous machines

Grid-forming Control for Power Converters based on Matching of Synchronous Machines

Grid-Friendly Matching of Synchronous Machines by Tapping into the DC Storage*

Parametric local stability condition of a multi-converter system

Local Synchronization of Two DC/AC Converters Via Matching Control