Small-signal modeling and parametric sensitivity of a virtual synchronous machine in islanded operation Salvatore D’Arco a , Jon Are Suul a,b,⇑ , Olav B. Fosso b a SINTEF Energy Research, 7465 Trondheim, Norway b Department of Electric Power Engineering, Norwegian University of Science and Technology, 7495 Trondheim, Norway article info Article history: Received 1 February 2015 Accepted 16 February 2015 Available online 29 March 2015 Keywords: Power electronic control Small-signal stability Stand-alone operation Virtual synchronous machine abstract The concept of Virtual Synchronous Machines (VSMs) is emerging as a flexible approach for controlling power electronic converters in grid-connected as well as in stand-alone or microgrid applications. Several VSM implementations have been proposed, with the emulation of inertia and damping of a tradi- tional Synchronous Machine (SM) as their common feature. This paper investigates a VSM imple- mentation based on a Voltage Source Converter (VSC), where a virtual swing equation provides the phase orientation of cascaded voltage and current controllers in a synchronous reference frame. The con- trol system also includes a virtual impedance and an outer loop frequency droop controller which is func- tionally equivalent to the governor of a traditional SM. The inherent capability of the investigated VSM implementation to operate in both grid-connected and islanded mode is demonstrated by numerical sim- ulations. Then, a linearized small-signal model of the VSM operated in islanded mode while feeding a local load is developed and verified by comparing its dynamic response to the time-domain simulation of a nonlinear system model. Finally, this small-signal model is applied to identify the dominant modes of the system and to investigate their parametric sensitivity. Ó 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Introduction Virtual Synchronous Machines (VSMs) have recently been pro- posed as a suitable concept for controlling power electronic con- verters in power system applications [1–4]. In the context of large-scale power systems, VSMs can provide a flexible approach for introducing additional damping and virtual inertia as an inher- ent part of the control system of grid integrated Voltage Source Converters (VSCs) [2,5,6]. A few proposed implementations of the VSM concept can also allow for stand-alone and parallel-connected operation in Microgrids or other isolated system configurations with similar performance and flexibility as traditional Synchronous Machines (SMs) [4,7–9]. The VSM concept is still in an early stage of development and many possible implementations, targeted for various types of applications, have been proposed, as reviewed in [4,10]. Thus, most publications until now have been mainly concerned with the development of particular VSM implementations and the presentation of case studies demonstrating the corresponding operational features. A systematic small-signal analysis of a speci- fic VSM implementation was first presented in [11], intended for controller tuning and stability improvement by utilizing the sensitivities of the system eigenvalues with respect to the con- troller parameters. The VSC control system investigated in [11] included only the VSM swing equation for damping and inertia emulation, a droop- based reactive power controller according to [12,13] and cascaded voltage and current control loops. However, there was no external power control included in the model, and the implementation of the damping of the VSM did not automatically take into account variations in the steady-state grid frequency. Thus, the applicabil- ity of the studied control system was limited to either stand-alone operation for feeding a local load or the operation in a strong grid with a known, fixed, frequency. An extension of the VSM control system design described in [11] was presented in [14]. To achieve full flexibility in allowable operating conditions, the resulting con- trol system included an outer loop frequency droop controller with functionality equivalent to the steady-state control characteristics of traditional SMs [16]. A Phase Locked Loop (PLL) [17,18] was also introduced for tracking the actual grid frequency needed for imple- menting the VSM inertial damping under deviations from the nominal grid frequency. Furthermore, a virtual impedance, similar to the implementations proposed in [19,20], was included in the http://dx.doi.org/10.1016/j.ijepes.2015.02.005 0142-0615/Ó 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). ⇑ Corresponding author at: SINTEF Energy Research, Postboks 4761 Sluppen, 7465 Trondheim, Norway. Tel.: +47 95 91 09 13. E-mail addresses: salvatore.darco@sintef.no (S. D’Arco), Jon.A.Suul@sintef.no, jon.are.suul@ntnu.no (J.A. Suul), olav.fosso@ntnu.no (O.B. Fosso). Electrical Power and Energy Systems 72 (2015) 3–15 Contents lists available at ScienceDirect Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes