We derive a small-signal transfer function for a system comprising a virtual synchronous generator (VSG), a synchronous generator (SG), and a load, capturing voltage and frequency dynamics. Using this model, we analyze the sensitivity of SG dynamics to VSG parameters, highlighting trade-offs in choosing virtual inertia and governor lag, the limited effect of damper-winding emulation, and several others.
As power electronic interfaces become more prevalent, particularly with the renewable integration, the industry increasingly requires advanced grid-forming (GFM) control. These controllers can form voltage, emulate inertia, and realize droop behavior. For such controllers, small-signal methods are widely used especially in single grid-connected converter systems, e.g., with passivity or Nyquist criteria [1]. In contrast, this paper studies the small-signal properties of an interconnected system comprising a converter operated as a virtual synchronous generator (VSG) 1 , a synchronous generator (SG), and a load. Unlike the related studies in [2]- [5], the current control and the network dynamics are neglected to focus only on slower dynamics. Such simplified models can reveal behaviors that more detailed models may obscure. This same motivation led to the foundational work in [6], which deliberately overlooked network dynamics to derive a parallel VSG-SG model, unveiling new facets of VSG parametrization. However, that study omits voltage and reactive power dynamics, does not analyze zero locations or their impact, and primarily highlights the benefits of higher inertia/damping and reduced governor delay. These limitations motivate our work, and we address each explicitly.
Among more detailed models, [2] considers simulation studies for a nine-bus system, including the dc-link and current control dynamics, highlighting the interaction between fast GFM control and slow SG dynamics. A small-signal study of parallel VSGs in [3] determines their damping coefficients. In [4], a small-signal model of a VSG-SG interconnection with network dynamics is derived to study low-frequency oscillations using pole plots. They state that increasing the governor lag and inductance could weaken the influence of the VSG.
Related simplified models are found in [7]- [11]. For example, [7] analyzes instability with grid-following (GFL) penetration; [8] shows damping versus penetration levels; [9] derives a VSG small-signal model highlighting the coupling of QV and Pω dynamics; [10] studies a GFL converter connected to a VSG; and [11] examines inertia and damping effects in different time intervals during a frequency transient.
The main contribution of this paper is the derivation of a small-signal transfer function mapping load power to internal frequencies and voltages, which we use to analyze the interconnected system with respect to VSG parametrization, in particular, to study the sensitivity of SG dynamics. Compared with [6], the transfer function also captures voltage and reactive power dynamics, and we give specific attention to the locations of its zeros. Section IV shows that these zero locations can significantly affect system behavior. In our analysis, we vary VSG parameters-inertia, damper winding constant, governor lag time constant, QV-droop gain and filter time constant, stator inductance, and XR-ratio-one at a time while keeping others at nominal values, to provide insights. These insights either confirm or extend prior findings. For example, we confirm the results of [4] (discussed above), further distinguish oscillations on two distinct time scales, and examine the influence of zeros in both active and reactive power dynamics. Compared with [10], we also confirm the benefits of increased inertia, but we further highlight the advantages of inertia matching. Finally, we validate the transfer function and the main findings through simulation case studies.
Parameters are denoted with upper-case, e.g., K. Bold capitals, e.g., K, refer to parameter matrices, and K ij describes the entry in the ith row and jth column of K. The identity is I. Functions in upper-case refer to transfer functions, e.g., G(s), while transfer function matrices are bold, e.g., G(s). Here, s is to be interpreted as the derivative operator. Scalar variables are expressed by lower-case, such as v, and their vector versions are denoted bold, e.g., v. Phasors are marked by an arrow, e.g., ⃗ v = v • e jφ = v ̸ φ. Subscripts v , s , b are used to distinguish quantities belonging to the VSG, the SG, and the bus, respectively. Similarly, references are indicated by the subscript r . For example, the reference frequency is ω r . For brevity, we often define quantities only for the VSG or only for the SG. Next, we describe the dynamics and the assumptions, and present the interconnected system.
The VSG dynamics follow [6], with the difference that we include voltage dynamics (see Fig. 1). As in [6], the SG has the same structure as the VSG, with different parametrization. The VSG is modeled as a three-phase voltage source with magnitude v v and instantaneous phase angle φ v (i.e., with frequency ω v = sφ v ) in series with a virtual stator impedance Z v . In contrast to the pure inductances in [6], we have resistive-inductive impedances, i.e.,
Small-signal changes in the reactance are similarly ignored. The VSG is connected to a bus with magnitude v
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