# Small-Signal Analysis of Grid-Forming Vector Current Control

Ognjen Stanojev, Orcun Karaca, Mario Schweizer

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Ognjen Stanojev, Orcun Karaca, Mario Schweizer. Small-Signal Analysis of Grid-Forming Vector Current Control. The 26th European Conference on Power Electronics and Applications, GDR SEEDS France & EPE Association, Mar 2025, Paris, France. 10.34746/epe2025-0072. hal-05081863

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Submitted on 23 May 2025

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<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Small-Signal Analysis of Grid-Forming Vector Current Control

Ognjen Stanojev, Orcun Karaca, Mario Schweizer

ABB Corporate Research Center, Switzerland

Segelhofstrasse 1K

Baden-Daettwil, Switzerland

E-Mail: {ognjen.stanojev, orcun.karaca, mario.schweizer}@ch.abb.com

***Index Terms***—voltage source converters, grid-forming control, modeling, small-signal stability.

***Abstract***—Grid-Forming Vector Current Control (GFVCC) is a novel control strategy that enables grid-connected converters to provide stiff voltage regulation, inertia emulation, and power sharing. It enhances conventional Phase-Locked Loop (PLL)-based vector current control by incorporating a special PLL tuning strategy and virtual admittance, thus enabling the grid-forming functionalities. Although previous studies have validated its technical feasibility, further analysis of its behavior and its tuning are necessary. This work first develops a comprehensive Differential-Algebraic Equation (DAE) model of GFVCC in the dq-domain. Subsequently, the DAEs are numerically linearized to perform small-signal analysis and uncover additional properties. Our findings are established through modal and parameter sensitivity analysis in a single machine, infinite bus setup.

## I. INTRODUCTION

With the proliferation of power electronic interfaces in power systems, there is an increasing need in the industry for grid-forming control strategies. Key features of grid-forming control schemes include voltage forming, inertia emulation, and power sharing. The most widely discussed control methods in the literature that can provide the above-mentioned grid services are droop control [1] and virtual synchronous generator control [2]. Despite the demonstrated efficacy of these methods in various studies, they exhibit several limitations. These limitations include high complexity and challenges in parameter tuning, attributed to the large number of control parameters, thereby requiring considerable commissioning efforts.

Recently, a novel grid-forming strategy known as Grid-Forming Vector Current Control (GFVCC) has been proposed to mitigate the aforementioned limitations [3]. This strategy emulates the combined effect

of a virtual synchronous condenser and a virtual current source in parallel. Specifically, the virtual current source adheres to the standard vector current control with a Phase-Locked Loop (PLL), while the virtual synchronous condenser emulation is responsible for virtual inertia provision and voltage forming. The main advantages of GFVCC over previously established grid-forming schemes lie in a reduced number of control tuning parameters, inherent overcurrent handling capabilities, and a modular design. Nevertheless, given the recent introduction of the GFVCC concept, comprehensive small-signal stability analysis has not been performed thus far. Such an analysis could yield valuable insights into the operational characteristics of GFVCC.

Small-signal characteristics of standard grid-forming techniques have been extensively studied in the literature [4]–[8]. These studies first validate the proposed small-signal models of individual inverter control schemes and subsequently provide insights into their stability characteristics by analyzing modes and eigenvalue sensitivities to relevant system parameters. In particular, [4] and [5] investigate the stability properties of inverters operating as virtual synchronous generators. Meanwhile, [6] compares the dynamic characteristics of droop-based grid-following and grid-forming converters, and [7] extends these analyses to various types of grid-forming and grid-following control schemes. Finally, the small-signal stability characteristics of large-scale networks with grid-forming and grid-following converters have been examined in [8], [9].

The contribution of this work is two-fold. Firstly, we present a detailed Differential-Algebraic Equation (DAE) model of GFVCC, in contrast to the transfer function-based frequency domain model presented in [3]. The proposed DAE model is numerically linearized around relevant operating points to derive small-signal models. Furthermore, it can also serve as a basis for implementing GFVCC in standard electromagnetic transient simulation tools. Secondly, we investigate the stability
<figure><img src="image_2.png" /><figcaption>Fig. 1: General configuration of the considered system, including a GFVCC inverter connected to an infinite bus.</figcaption></figure>

margins of a GFVCC inverter connected to an infinite bus using modal analysis and various parameter sensitivity studies.

The remainder of the paper is structured as follows. In Section II, DAE models of individual building blocks of GFVCC in dq-domain are presented. Subsequently, in Section III, the complete DAE system is formulated and linearized to obtain a small-signal model. Section IV showcases the stability analysis results, whereas Section V concludes the paper.

## II. DIFFERENTIAL-ALGEBRAIC EQUATION MODEL OF GRID-FORMING VECTOR CURRENT CONTROL

### A. Notation

We denote the set of real numbers by $\mathbb{R}$ and the set of complex numbers by $\mathbb{C}$. The real part of a complex number $z \in \mathbb{C}$ is denoted by $\mathfrak{R}(z)$ and the imaginary part by $\mathfrak{I}(z)$. For column vectors $x \in \mathbb{C}^n$ and $y \in \mathbb{C}^m$, we use $(x, y) := [x^\top, y^\top]^\top \in \mathbb{C}^{n+m}$ to denote a stacked vector. Given an angle $\theta \in [-\pi, \pi)$, the 2-D rotation matrix is defined by

$$ \mathcal{R}(\theta) := \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}, \quad \mathcal{J} := \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}, $$

where $\mathcal{J} := \mathcal{R}(\pi/2)$ is the commonly used 90° rotation matrix. Furthermore, $\|x\|$ denotes the Euclidean norm.

### B. General Configuration

An overview of the considered system is depicted in Fig. 1, where an inverter is connected to an infinite bus through an RLC filter. The converter is controlled using the GFVCC scheme, which consists of system-level controls and a current controller. The system-level control generates a current reference based on the given setpoints $u$ and the emulation of the combined effect of a virtual synchronous condenser and a virtual current

<figure><img src="image_3.png" /><figcaption>Fig. 2: Illustration of the GFVCC concept.</figcaption></figure>

source, as illustrated in Fig. 2. This reference current signal is then tracked by the current control loop. In the following, we derive the model in DAE form based on the frequency domain model of GFVCC presented in [3]. For a detailed derivation of the control scheme, readers are referred to the original paper.

### C. AC Side Circuit Model

The power electronic system considered in this work consists of an ideal DC voltage source, a lossless DC/AC converter, and an RLC filter on the converter AC side. The DC-side dynamics are thus neglected, and a separate control loop is assumed to stabilize the DC-link voltage. Modeling and control of the converter system are established in a synchronous reference frame and a per unit system, with the mathematical model defined in dq-vector form, i.e., $x := (x_d, x_q) \in \mathbb{R}^2$.

The RLC filter is defined by the respective parameters $(r_f, \ell_f, c_f) \in \mathbb{R}^3$ and can be described in dq-coordinates as

$$ \frac{d}{dt}i_f = \frac{\omega_b}{\ell_f}(v_m - v_f) - \left(\frac{r_f}{\ell_f}\omega_b + \mathcal{J}\omega_b\omega_g\right)i_f, \quad (1a) $$

$$ \frac{d}{dt}v_f = \frac{\omega_b}{c_f}(i_f - i_g) - \mathcal{J}\omega_b\omega_g v_f, \quad (1b) $$

where $v_m \in \mathbb{R}^2$ is the converter output voltage, $i_f \in \mathbb{R}^2$ and $v_f \in \mathbb{R}^2$ are respectively the filter current and voltage, and $i_g \in \mathbb{R}^2$ is the terminal current, i.e., current injected into the grid; the system base angular frequency is represented by $\omega_b \in \mathbb{R}$ and $\omega_g \in \mathbb{R}$ is the normalized grid (reference) angular frequency.

The converter system is connected to an infinite bus through a resistance $r_g \in \mathbb{R}$ and an inductance $\ell_g \in \mathbb{R}$, and the grid current dynamics are given by

$$ \frac{d}{dt}i_g = \frac{\omega_b}{\ell_g}(v_f - v_g) - \left(\frac{r_g}{\ell_g}\omega_b + \mathcal{J}\omega_b\omega_g\right)i_g, \quad (2) $$

where $v_g \in \mathbb{R}^2$ is a constant voltage of the infinite bus.
## D. Current Control

At the heart of the GFVCC concept is its functional separation into the virtual synchronous condenser and a virtual current source connected in parallel. The two parts create separate current references, which are combined into the total reference $i_f^* \in \mathbb{R}^2$ and followed by the current control loop:

$$ \frac{d}{dt}\eta = i_f^* - i_f^c, \quad (3a) $$

$$ v_m^c = K_P^i(i_f^* - i_f^c) + K_I^i\eta + K_F^i v_f^c + \mathcal{J}\omega_g l_f i_f^c, \quad (3b) $$

where $\eta \in \mathbb{R}^2$ is the integrator state, and $(K_P^i, K_I^i, K_F^i) \in \mathbb{R}^3$ are the respective proportional, integral and feed-forward gains. The superscript c on voltages and currents is used to indicate that these quantities are represented in the converter reference frame, which is related to the corresponding physical quantities (no superscript) in (3) via a rotation given below

$$ i_f^c = -\mathcal{R}(\theta_c - \theta_g) i_f, \quad v_f^c = -\mathcal{R}(\theta_c - \theta_g) v_f, \quad (4) $$

where $\theta_c - \theta_g := \Delta\theta_c \in [-\pi, \pi)$, is the difference between the the PLL angle $\theta_c$ and the network reference frame angle $\theta_g$. Considering that the DC-link voltage is constant and normalized, the voltage reference generated in (3b) is translated into the output voltage via a reverse rotation, as follows:

$$ v_m = \mathcal{R}(\theta_c - \theta_g) v_m^c. \quad (5) $$

## E. Virtual Synchronous Condenser

The virtual synchronous condenser part of the control scheme has two roles: (i) synchronization and provision of virtual inertia to the system and (ii) voltage forming. These roles can be achieved via the joint effect of PLL and virtual impedance. The standard type-2 PLL implementation is used, which estimates the phase angle change $\Delta\theta_c \in [-\pi, \pi)$ of the voltage $v_f$ and the grid frequency $\omega_g$ using a PI controller that diminishes the q-component of the rotated voltage $v_f^c$, as

$$ \frac{d}{dt}\varepsilon = v_{f,q}^c, \quad (6a) $$

$$ \omega_c = \omega_g + K_P^s v_{f,q}^c + K_I^s \varepsilon, \quad (6b) $$

$$ \frac{d}{dt}\Delta\theta_c = \omega_b(\omega_c - \omega_g), \quad (6c) $$

where $(K_P^s, K_I^s) \in \mathbb{R}^2$ are the proportional and integral gains of the PLL, and $\varepsilon \in \mathbb{R}$ is the integrator state. It has been demonstrated in [3] that the equivalence between PLL and the swing equation is achieved with the following PLL tuning:

$$ K_P^s = \frac{K_f}{M\|v_f\|}, \quad K_I^s = \frac{\|v_c^*\|}{Mx_v}, $$

where $M \in \mathbb{R}$ denotes the virtual inertia constant, $K_f \in \mathbb{R}$ the virtual damper winding constant, and $v_c^* := (V_v, 0)$ is the back EMF of the virtual synchronous condenser with magnitude $V_v \in \mathbb{R}$ and aligned with the d-axis of the converter reference frame. We note that a steady-state constant value of the filter voltage is typically used instead of the time-varying magnitude $\|v_f\|$. Furthermore, virtual stator impedance of resistance $r_v \in \mathbb{R}$ and reactance $x_v := \omega_g l_v \in \mathbb{R}$ is used to explicitly couple the output power of the synchronous condenser and angle difference between $v_c^*$ and $v_f^c$ via a current reference:

$$ \frac{d}{dt}i_v = \frac{\omega_b}{l_v}(v_c^* - v_f^c) - \left(\frac{r_v}{l_v}\omega_b + \mathcal{J}\omega_b\omega_g\right)i_v. \quad (7) $$

Implementing the virtual impedance with the full dynamic model ensures passivity and damping of high-frequency resonances. Contrary to the standard virtual synchronous machine approaches, the above equations emulate a spinning wheel without a governor. Governor functionality is added separately, as explained in the following subsection.

## F. Virtual Current Source

The governor functionality of maintaining the power balance in steady-state is implemented via the droop control loop by computing a d-axis current reference

$$ i_{r,d} = \|\tilde{v}_f\|^{-1}(p_c^* - K_g(\tilde{\omega}_c - \omega_g)), \quad (8) $$

where $\|\tilde{v}_f\| \in \mathbb{R}$ is the low-pass filtered converter terminal voltage magnitude, $K_g \in \mathbb{R}$ is the droop gain, $\tilde{\omega}_c \in \mathbb{R}$ is converter frequency defined in (6) processed by a second-order low-pass filter, and $p_c^* \in \mathbb{R}$ is the active power setpoint. An additional q-component current reference is introduced to impose voltage magnitude regulation and reactive power setpoint tracking, computed by the following expression

$$ \frac{d}{dt}i_{r,q} = -T_a^{-1}(q_c^* - K_a(\|v_f^c\| - \|v_c^*\|)), \quad (9) $$

with $T_a \in \mathbb{R}$ denoting the AVR time constant, $K_a \in \mathbb{R}$ being the voltage droop gain and $q_c^* \in \mathbb{R}$ representing the reactive power setpoint. Furthermore, a virtual termination resistor can be included in the scheme for stabilization via an additional current reference

$$ i_{ad} = r_{ad}^{-1}\tilde{v}_f, \quad (10) $$

where $\tilde{v}_f \in \mathbb{R}^2$ is the converter terminal voltage filtered by a second-order high-pass filter, and $r_{ad} \in \mathbb{R}$ is the virtual resistance. Finally, the total current reference is
obtained by summing individual references of both the virtual synchronous condenser and the virtual current source, as

$$i_f^* = i_v + i_r + i_{ad}, \qquad (11)$$

which is then tracked by the current controller (3b).

### G. Filter Modeling

In this section, we discuss modeling of the previously used high-pass and low-pass filters. The terminal voltage magnitude in (8) is filtered by a first-order low-pass filter of the form

$$\frac{d}{dt} \|\tilde{v}_f\| = \omega_b \omega_z (\|v_f^c\| - \|\tilde{v}_f\|), \qquad (12)$$

where $\omega_z \in \mathbb{R}$ is the filter cut-off frequency. Additionally, a second-order low-pass filter is employed in the same equation to set the bandwidth of the governor and is modeled by

$$\frac{d}{dt}\alpha_1 = \alpha_2, \qquad (13a)$$

$$\frac{d}{dt}\alpha_2 = \omega_c - 2\zeta_{zlp}\omega_b\omega_{zlp}\alpha_2 - \omega_b^2\omega_{zlp}^2\alpha_1, \qquad (13b)$$

where $\zeta_{zlp} \in \mathbb{R}$ is the filter damping factor, $\omega_{zlp} \in \mathbb{R}$ is the cut-off frequency, and $(\alpha_1, \alpha_2) \in \mathbb{R}^2$ are the filter states. The filtered converter frequency is obtained as $\tilde{\omega}_c = \omega_b^2\omega_{zlp}^2\alpha_1$. The second-order high-pass filter used for active damping (10) takes a similar form

$$\frac{d}{dt}\beta_1 = \beta_2, \qquad (14a)$$

$$\frac{d}{dt}\beta_2 = v_f^c - 2\zeta_{zhp}\omega_b\omega_{zhp}\beta_2 - \omega_b^2\omega_{zhp}^2\beta_1, \qquad (14b)$$

with analogous notation and the high-pass filtered voltage computed as $\tilde{v}_f = v_f^c - 2\zeta\omega_b\omega_{zhp}\beta_2 - \omega_b^2\omega_{zhp}^2\beta_1$.

### H. Complete Differential-Algebraic Equation Model

With all control blocks within the GFVCC scheme and the electrical circuit defined, the formulation of the DAE model is complete. Collecting equations (1)-(14), we obtain a set of nonlinear DAEs of the following form

$$\frac{d}{dt}x = f(x, u, z), \qquad (15a)$$

$$0 = g(x, u, z), \qquad (15b)$$

where $x$ and $z$ are the vectors of differential and algebraic states, and $u$ is the vector of inputs, defined as

$$x = (i_f, v_f, \eta, \varepsilon, \Delta\theta_c, i_v, i_{r,q}, \|\tilde{v}_f\|, \alpha_1, \alpha_2, \beta_1, \beta_2, i_g), \qquad (16a)$$

$$z = (i_f^c, v_f^c, \omega_c, i_{r,d}, i_{ad}, i_f^*, v_m^c, v_m, v_g), \qquad (16b)$$

$$u = (V_v, p_c^*, q_c^*). \qquad (16c)$$

Thus, the resulting DAE model consists of $n_x = 20$ differential states, $n_z = 14$ algebraic states, and $n_u = 3$ control inputs.

### III. SMALL-SIGNAL ANALYSIS

Given the nonlinearity and complexity of the obtained model (15), we aim to derive its small-signal counterpart to enable further analysis using linear system tools. Linearizing (15) around a selected operating point of the system $(x_o, u_o, z_o)$ results in the linear DAE given below

$$\frac{d}{dt}\Delta x = A_{xx}\Delta x + B_{xu}\Delta u + A_{xz}\Delta z + b_x, \qquad (17a)$$

$$0 = A_{zx}\Delta x + B_{zu}\Delta u + A_{zz}\Delta z + b_z, \qquad (17b)$$

where $\Delta x = x - x_o$, $\Delta u = u - u_o$, $\Delta z = z - z_o$ are the appropriate small-signal variables, $A_{xx}$, $B_{xu}$, and $A_{xz}$ are the respective Jacobians of $f$ w.r.t. $x, u, z$, and $A_{zx}$, $B_{zx}$, $A_{zz}$ are the Jacobians of $g$ w.r.t. $x, u, z$, and $b_x$ and $b_z$ are the bias terms. The algebraic part of the model (17b) can be rewritten

$$\Delta z = A_{zz}^{-1}(A_{zx}\Delta x + B_{zu}\Delta u + b_z), \qquad (18)$$

and substituted in (17a) to eliminate the algebraic variables

$$\frac{d}{dt}\Delta x = \underbrace{(A_{xx} + A_{xz}A_{zz}^{-1}A_{zx})}_{=:A}\Delta x + \underbrace{b_x + A_{zz}^{-1}b_z}_{=:b} + \underbrace{(B_{xx} + A_{xz}A_{zz}^{-1}B_{zu})}_{=:B}\Delta u, \qquad (19)$$

which leads to a small-signal ordinary differential equation model of the form

$$\frac{d}{dt}\Delta x = A\Delta x + B\Delta u + b. \qquad (20)$$

The eigenvalues of the system matrix $A$ are denoted as $\lambda_i$ with $i \in \{1, \dots, n_x\}$. The right and left eigenvectors associated to $\lambda_i$ are denoted as $w_i$ and $v_i$, respectively, and defined by

$$Aw_i = w_i\lambda_i, \quad w_i \neq 0, \quad i \in \{1, \dots, n_x\}, \\ v_i^\top A = \lambda_i v_i^\top, \quad v_i \neq 0, \quad i \in \{1, \dots, n_x\}.$$

In addition to analyzing modes of the system (20) and their sensitivity to GFVCC parameterization, we will perform participation factor analysis to evaluate the contribution of individual states to these modes. To this end, the participation matrix $P \in \mathbb{R}^{n_x \times n_x}$ associated to $A$ is defined as

$$P_{ki} = [v_i]_k[w_i]_k, \qquad (21)$$

where $[v]_k$ denotes the $k^{th}$ entry of vector $v$. As discussed in [10], the $(k, i)$-entry of the participation matrix $P$ measures the relative contribution of the $k^{th}$ state in the time response associated with the $i^{th}$ mode.
# IV. RESULTS

## A. Simulation Setup

The proposed DAE model (15) evaluation and the small-signal stability analysis are conducted on the system depicted in Fig. 1, where a GFVCC-controlled inverter is connected to an infinite bus through a grid impedance. The system parameters are closely aligned with the infinite bus case studies presented in [3], with the specific system and control parameters detailed in Table I. Note that the transformer and line equivalents used in the previous work are included in the grid impedance with SCR of 50. The considered test system configuration is implemented in MATLAB using CasADi [11] for symbolic modeling, and the odel5s variable time-step solver in MATLAB is employed to solve the DAEs. The calculation of Jacobians of *f* and *g* required to obtain (19) is performed using CasADi's automatic differentiation.

## B. Model Validation

In order to validate the DAE model (15) and the corresponding small-signal model (20), time domain simulations are performed and the obtained state trajectories are compared against the trajectories generated with the original MATLAB Simulink model. The considered event is an active power setpoint change from 0 p.u. to 1 p.u. Trajectories of converter voltage, active and reactive power, and PLL frequency for all the three considered models are presented in Fig. 3.

As illustrated in the figure, the response of the DAE model closely matches that of the original Simulink implementation, thereby validating the proposed mathematical formulation. However, minor transient and steady-state discrepancies can be observed in the voltage and reactive power responses of the small-signal model. Interestingly, no significant errors are observed in the active power and frequency responses. This can be attributed to the linear behavior of the power-angle curve under stiff-grid conditions, where angle deviations are small. In contrast, the linearization of voltage magnitude is valid

<figure><img src="image_4.png" /><figcaption>Fig. 3: Transient response of DAE, linear ordinary differential equation (small-signal), and Simulink models to a step change in active power setpoint from 0 p.u. to 1 p.u.</figcaption></figure>

over a smaller region due to the nonlinear properties of vector magnitude calculation. Nevertheless, considering that the investigated event drives the system through a full range of operating points, the observed errors are acceptable, and the model's validity is satisfactory.

## C. Modal Analysis

In this subsection, the eigenvalues of the small-signal model (20) are calculated for an operating point defined by $(V_v, p_c^*, q_c^*) = (1, 1, 0)$. All $n_x = 20$ eigenvalues are reported in Table II both as complex numbers in rectangular coordinates and through the associated damping and natural angular frequency in rad/s. Additionally, the participation factor analysis is performed to identify the states with the highest participation in the observed modes. The results are reported in the same table, first listing the symbols of the highest participation states and then the corresponding total participation factor of these states, calculated based on the participation factor matrix in (21).

As shown in the table, a broad spectrum of modes has been identified. The slowest eigenvalues are associated

<table><caption>TABLE I: Nominal system and control parameters in p.u. with following bases: $P_b = 100$ kW, $V_b = 400$ V, and $f_b = 50$ Hz.</caption><thead><tr><th>Parameter</th><th>Value</th><th>Parameter</th><th>Value</th></tr></thead><tbody><tr><td>(r<sub>f</sub>, l<sub>f</sub>, c<sub>f</sub>)</td><td>(0.002, 0.04, 0.1)</td><td>(K<sub>g</sub>, ω<sub>z</sub>)</td><td>(33.33, 2/ω<sub>b</sub>)</td></tr><tr><td>(r<sub>g</sub>, l<sub>g</sub>)</td><td>(0.005, 0.08)</td><td>(ω<sub>zlp</sub>, ξ<sub>zlp</sub>)</td><td>(0.5, 0.7)</td></tr><tr><td>(K<sup>i</sup><sub>P</sub>, K<sup>i</sup><sub>I</sub>, K<sup>i</sup><sub>F</sub>)</td><td>(0.2, 4, 0)</td><td>(K<sub>a</sub>, T<sub>a</sub>)</td><td>(25, 0.1)</td></tr><tr><td>(K<sup>s</sup><sub>P</sub>, K<sup>s</sup><sub>I</sub>)</td><td>(0.094, 2.778)</td><td>r<sub>ad</sub></td><td>∞</td></tr><tr><td>(r<sub>v</sub>, l<sub>v</sub>)</td><td>(0.045, 0.18)</td><td>(ω<sub>zhp</sub>, ξ<sub>zhp</sub>)</td><td>(0.2, 0.8)</td></tr></tbody></table>
<table><caption>TABLE II: System Eigenvalues.</caption><thead><tr><th>ID</th><th>Mode</th><th>ζ</th><th>ω<sub>n</sub></th><th>States</th><th>PF</th></tr></thead><tbody><tr><td>λ<sub>1</sub></td><td>-2</td><td>1</td><td>2</td><td>i<sub>f</sub>, ||ṽ<sub>f</sub>||</td><td>0.66</td></tr><tr><td>λ<sub>2</sub></td><td>-13</td><td>1</td><td>13</td><td>i<sub>r,d</sub></td><td>0.44</td></tr><tr><td>λ<sub>3</sub>, λ<sub>4</sub></td><td>-17.7 ± j6.96</td><td>0.93</td><td>19</td><td>i<sub>f</sub>, i<sub>r,d</sub></td><td>0.4</td></tr><tr><td>λ<sub>5</sub>, λ<sub>6</sub></td><td>-11.3 ± j21.9</td><td>0.46</td><td>24.7</td><td>i<sub>f</sub>, i<sub>r,d</sub>, i<sub>v</sub></td><td>0.61</td></tr><tr><td>λ<sub>7</sub>, λ<sub>8</sub></td><td>-50.3 ± j37.7</td><td>0.8</td><td>62.8</td><td>α<sub>1</sub>, β<sub>1</sub></td><td>0.98</td></tr><tr><td>λ<sub>9</sub>, λ<sub>10</sub></td><td>-50.3 ± j37.7</td><td>0.8</td><td>62.8</td><td>α<sub>1</sub>, β<sub>1</sub></td><td>0.99</td></tr><tr><td>λ<sub>11</sub>, λ<sub>12</sub></td><td>-103.1 ± j130</td><td>0.62</td><td>166</td><td>i<sub>f</sub>, i<sub>v</sub>, i<sub>g</sub></td><td>0.91</td></tr><tr><td>λ<sub>13</sub>, λ<sub>14</sub></td><td>-59.3 ± j312</td><td>0.12</td><td>318</td><td>i<sub>f</sub>, i<sub>v</sub>, i<sub>g</sub></td><td>0.98</td></tr><tr><td>λ<sub>15</sub>, λ<sub>16</sub></td><td>-779 ± j203</td><td>0.97</td><td>805</td><td>i<sub>f</sub>, i<sub>g</sub></td><td>0.86</td></tr><tr><td>λ<sub>17</sub>, λ<sub>18</sub></td><td>-444 ± j5800</td><td>0.08</td><td>5820</td><td>i<sub>f</sub>, v<sub>f</sub></td><td>0.83</td></tr><tr><td>λ<sub>19</sub>, λ<sub>20</sub></td><td>-389 ± j6240</td><td>0.06</td><td>6260</td><td>i<sub>f</sub>, v<sub>f</sub></td><td>0.83</td></tr></tbody></table>

with the governor and voltage regulator. The employed tuning of virtual admittance results in eigenvalues in the mid-frequency range, while the fastest modes are linked to the filter and grid dynamics. The least damped modes are simultaneously the highest frequency modes, stemming from small values of RLC filter components.

## D. Bifurcation Analysis

We further conduct a bifurcation analysis to identify critical parameter tuning values at which the system transitions from stable to unstable operation. Our analysis reveals that these parameters are primarily related to the tuning of the current controller and virtual impedance. Figure 4a illustrates the movement of the eigenvalue spectrum as the virtual impedance is reduced from 100% to 20% while Fig. 4b shows the impact of changing the current control bandwidth from the nominal 250 Hz to 50 Hz. In both figures, a bifurcation is observed, indicating that the system becomes unstable at the lowest considered values of these parameters.

Interestingly, the virtual impedance value that leads to instability corresponds to a scenario where $\ell_v \le \ell_f$. This suggests that for a system with a dominantly inductive RLC filter and a sufficiently high current control bandwidth, setting $\ell_v > \ell_f$ is a necessary condition for small-signal stability.

Moreover, our observations indicate that a very low bandwidth of the current controller also leads to instability. Further analysis reveals that in such cases, as indicated by the stability map in Fig. 5, appropriate tuning of the virtual admittance can stabilize the system. These observations highlight the importance of careful parameter tuning of virtual impedance when the current control bandwidth is limited.

<figure><img src="image_5.png" /><figcaption>Fig. 4: Movement of eigenvalues due to the changes in (a) virtual admittance parameters; (b) current control bandwidth.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 5: Stability map on the virtual inductance – current control bandwidth plane, with indicated filter inductance value.</figcaption></figure>

## E. Parameter Sensitivity Analysis

To complete the analysis, eigenvalue spectrum scans were conducted for the remaining control parameters in the GFVCC strategy. Due to space constraints, we focus on the tuning of the governor and AVR, as these control loops significantly impact the slowest modes in the system, as shown in Table II. Figure 6 illustrates the movement of the eigenvalue spectrum when $K_g$ and $K_a$ are varied from 1 p.u. to 100 p.u..

Consistent with observations in [4], increasing the frequency droop gain improves the damping of slow modes $\lambda_5, \lambda_6$ while adversely affecting the damping of fast modes $\lambda_{11}, \lambda_{12}$. Nonetheless, stability is maintained in both scenarios.

Similarly, it can be observed in Fig. 6b that the
<figure><img src="image_7.png" /><figcaption>Fig. 6: Movement of eigenvalues due to the changes in (a) governor gain; (b) automatic voltage regulator gain.</figcaption></figure>

droop gain of AVR does not compromise the small-signal stability for any reasonable tuning. We observe that increasing the AVR gain shifts $λ_2$ further into the left half-plane, enhancing system stability. Additionally, an oscillatory mode corresponding to $λ_3, λ_4$ becomes perfectly damped when the AVR gain is tuned above 90 p.u. Specifically, $λ_3$ moves in the direction of the complex plane origin, whereas $λ_4$ moves further into the left half-plane.

## V. CONCLUSION

This paper analyzed the Grid-Forming Vector Current Control (GFVCC) scheme from a small-signal perspective. To this end, a Differential-Algebraic Equation (DAE) model was developed and numerically linearized. Our results demonstrate that the developed DAE model and the corresponding small-signal model closely match the original Simulink implementation, thereby confirming their validity. We observed that, unlike other grid-forming control schemes, GFVCC exhibits linear behavior over a wide range of operating points. Furthermore, our bifurcation analysis highlights the importance of appropriate virtual admittance tuning, especially when the current control bandwidth is limited. Additionally, we found that the droop gains of the AVR and governor can be freely tuned based on the desired power-sharing

allocation in the system without compromising small-signal stability. Our future work will focus on the small-signal stability of networks of GFVCC inverters as well as the transient stability analysis of GFVCC.

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