Parametric local stability condition of a multi-converter system Taouba Jouini 1 and Florian D¨ orfler 2 Abstract—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 sta- bility 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. I. I NTRODUCTION The major shift in power generation from conventional syn- chronous machines to renewables led to the study of problems of network stability composed mainly of DC/AC converters mimicking the electro-mechanical interaction of synchronous machines with the grid. Power system stability amounts to the ability of an electric power system, for a given initial operating condition, to regain state of desired operation, after being subjected to a disturbance [2]. Synchronous machines embody mechanical systems having a rotational degree of freedom. As a result, power system dynamics admit trigonometric nonlin- earities, diffusive coupling between electrical and mechanical angles and a rotational symmetry. Converters controlled to emulate synchronous machines [3]–[8] endow the closed loop with a virtual rotating angle and thus inherit these dynamics, which are challenging for many stability analysis approaches. In this context, different power system stability conditions have been proposed: In [6], [9], [10] sufficient stability con- ditions are obtained for a single-machine/converter case. Typ- ically, the individual generators and converters are designed and controlled to be stable in isolation. Hence, the bulk of the literature focuses on the network case. In [11] a link is drawn between the synchronization of power systems and Kuramoto oscillators resulting in conditions on the system topology and parameters. Although these conditions give qualitative insights into the sensitivities influencing stability, they usually require strong (and often unrealistic) assumptions. Also the conditions in [11] and elsewhere [3], [4], [11], [12] cannot be assessed without an omniscient knowledge of 1 Taouba Jouini is with the Department of Automatic Control at LTH, Lund University. 2 Florian D¨ orfler is with Automatic Control Laboratory, ETH Zurich, Switzerland. Emails: taouba.jouini@control.lth.se, doerfler@ethz.ch. A preliminary version of part of the results in this paper was presented at the European Control Conference 2019 [1]. This work was funded by the European Union’s Horizon 2020 research and innovation program under grant agreement N ◦ 691800 and ETH Z¨ urich funds. system parameters and the operating point. In general, explicit stability conditions require strong assumptions (e.g., strong mechanical or electrical damping) [7]–[11], whereas implicit conditions are based on semi-definite programs and thus not very insightful [13], [14]. Some conditions are valid only in radial networks [12]. Classical energy function analysis [15] and its modern extensions [16] consider low-order machine models and quasi-stationary lossless line models implying a weak coupling between active versus reactive power as well as voltage angle versus magnitude. These assumptions make the stability analysis tractable but in real power systems rotor angle stability and voltage stability are inevitably coupled, low-order models are not always a truthful representation, and lines have resistive losses and non-negligible dynamics — especially on the shorter time scales and on lower volt- age levels relevant for converters [17]. Quite opposing these assumptions, experimental and theoretical findings underscore that sufficient (virtual) resistive damping and reactive power support are necessary for power system stability [18], [19]. In this paper, we consider a higher-order multi-source power system model consisting of identical DC/AC converters interconnected in a general topology through lossy lines with uniform inductive-resistive dynamics. The converters are con- trolled through matching control [6]–[8] which renders them structurally equivalent to synchronous machines. Thus, our analysis approach also extends to synchronous machines. Our model exhibits a rotational symmetry corresponding to the absence of an absolute angle. We pursue a parametric linear stability analysis at a synchronous and rotationally invariant steady state. Towards this, we develop a novel analysis ap- proach for a class of partitioned linear systems characterized by a stable subsystem and a one-dimensional invariant sub- space. We propose a class of Lyapunov functions characterized by an oblique projection onto the complement of the invariant subspace, following the direction of a matrix to be chosen as solution to Lyapunov and Ricatti equations. Our approach has natural cross-links with analysis concepts for interconnected systems, e.g., passive systems, though our assumptions are less restrictive. For the multi-source power system model, we arrive at explicit stability condition that depend only on the converter’s parameters and steady-state values and can thus be evaluated in a fully decentralized fashion. Unlike other works, our conditions do not require the restrictive assumption of overly strong mechanical (respectively DC-side) damping but rather sufficient reactive power support and AC-side resistance. II. MODELING AND SETUP A. DC/AC Converter Model Consider a balanced three-phase DC/AC converter model, as in [6] and as illustrated in Figure 1. The dynamics, formulated arXiv:1904.11288v3 [math.OC] 12 Oct 2020