Small-Signal Stability of Hybrid Inverters with Grid-Following and Grid-Forming Controls

Abstract

In the modern power grid, characterized by the increased penetration of power electronics and extensive utilization of renewable energy, inverter-based power plants play a pivotal role as the principal interface of renewable energy sources (RESs) and the grid. Considering the stability characteristics of grid-following (GFL) inverters when the grid is relatively weak, the application of grid-forming (GFM) controls becomes imperative in enhancing the stability of the entire power plant. Thus, there is an urgent need for suitable and effective models to study the interaction and stability of the paralleled inverters employing GFL and GFM controls. Thus, the small-signal modeling with full-order state-space model and eigenvalues analysis are presented in this paper. First, the small-signal state-space model of the individual GFL and GFM inverters is obtained, considering the control loop, interaction, reference frame, transmissions, and time delays. Then, the models of the individual inverter are extended to the hybrid inverters to study the effects of the GFM inverters on the small-signal stability of the entire system. And the impacts of the inertia and damping are analyzed by the eigenvalues of the state-transition matrix. A case comprising three parallel GFL inverters and two GFL inverters with one GFM inverter, respectively, are studied to examine the effectiveness and accuracy of the model. Finally, the stability margin obtained from the eigenvalue analysis of the entire system is verified by time-domain simulations.

1. Introduction

To address environmental issues such as climate change and carbon neutrality, electricity generation has transitioned to renewable energy, e.g., photovoltaic and wind. This change leads to the traditional grid acceleration, i.e., from being dominated by synchronous generators (SGs) to the grid characterized by a wide-scale application of renewable energy sources (RESs) and high penetration of power electronic converters. The control schemes of RES are typically categorized as grid-following (GFL) and grid-forming (GFM), with the GFL inverters being utilized worldwide nowadays [1]. As shown in Figure 1a, the GFL inverters usually behave like a current source to inject power to the grid with high quality, where the output current is controlled by an inner current controller.

Compared with the traditional SG with a stiff internal voltage, the converters with GFL control have reduced the total inertia and damping of the entire power system, which manifests that the stability and robustness of the frequency and voltage are reduced [2]. Therefore, the inverter-based power system is susceptible to delayed responses or potential instability in the face of grid events, such as disturbances of the frequency and voltage. The European Network of Transmission System Operator for Electricity (ENTSO-E) reports both the decrease of traditional SGs and the increase of GFL inverters as the cause of the reduction in the stability of the power system [3]. Accordingly, extensive efforts have been dedicated to the advancement of GFM control, which can provide inertia and damping to the grid [4]. The inverter with a GFM control usually performs as a voltage source at the point of common coupling (PCC), as shown in Figure 1b. Among various GFM control schemes, the virtual synchronous generator (VSG) is commonly applied to emulate the external characteristics of an SG, which can support the frequency and voltage of the power system in accordance with its droop characteristics ($P-f$ and $Q-V$) [5,6]. Benefited by the control flexibility of the grid-connected inverters, the damping and inertia of the VSG can be designed in response to specific requirements through the modification of parameters within the power control loop. Clearly, to design the GFM control, modeling the inverters is the foremost task.

There have been many publications focusing on the modeling and stability analysis of the individual grid-connected inverter. Modeling approaches of the grid-connected inverters are typically categorized into three methods: electromagnetic transient (EMT) modeling, impedance modeling, and state-space modeling. Based on the detailed model of every component within the entire power system, the EMT can replicate the grid disturbances and provide time-domain simulations for an intuitive analysis of the complete system [7]. The EMT model retains the detailed structure of the system, even including dynamics of nonlinear elements, and thus provides an approximation closely reflecting actual variations. However, EMT modeling lacks theoretical support to unveil the mechanism behind the complex interaction and instability within the power system, and the EMT models for large-scale power systems demand significant computational burdens [8].

The impedance modeling approach focuses on the external characteristics of the grid-connected inverter by taking the inverter and the AC grid as two separated subsystems or two independent “black boxes”. Considering the widely adopted current vector-control of the grid-connected inverter is based on a synchronous reference ($d-q$) frame, ref. [9] studied the impacts of the phase-lock loop (PLL) bandwidth in the $d-q$ frame using a 2 × 2 impedance matrix. The small-signal impedance model of the GFL inverter is derived to demonstrate its unique feature, i.e., the $q-q$ impedance being seen as a negative resistor. The magnitude of the negative resistor is determined by the rated power of the inverter, while the frequency range of this resistor is relevant to the bandwidth of the PLL [10]. Based on the $d-q$ impedance model, ref. [11] analyzes the stability of the GFM inverter under different operating points, manifesting variations in stability margins within the low-frequency range. However, the decoupling control of the $d-q$ rotational frame cannot entirely eliminate the coupling effect between the d and q axis, namely, the presence of the $dq$ and $qd$ channel impedance. If the impact of the coupling impedance is taken into consideration, the impedance of the inverter is a multiple-input multiple-output (MIMO) small-signal model, represented as a 2 × 2 matrix transfer function in the frequency domain. The stability margin of the MIMO system can be evaluated by the generalized Nyquist stability criterion [12] or the generalized MIMO Bode criterion [13]. However, it is found in [14] that the generalized Nyquist stability criterion may lead to inaccurate stability analysis results under a specific grid configuration. The Nyquist generalized stability criterion focuses on the external characteristic of the inverter output port; it is challenging to clarify the interaction mechanism between the inner control parameters and small-signal stability margin. Moreover, achieving the desired optimal parameter design also becomes particularly complex with the impedance model.

Utilizing the modern control theory, the state-space model of the inverter is derived from the nonlinear differential equations with n state variables, which are linearized around the operating equilibrium point of the inverter. Based on the state-space model, it is easy to calculate the eigenvalues and analyze the stability of the model. The analysis of eigenvalues ${\lambda }_i,i\in \mathbf{Z}\left[1,n\right]$ can be adopted to identify the stability characteristic of the inverter system. The real part of the eigenvalues, $\mathrm{Real}\left({\lambda }_i\right)$, represents the damping of the whole model, which is equivalent to the small-signal stability margin, and $max\left(\mathrm{Real}\left({\lambda }_i\right)\right)<0,i\in \mathbf{Z}\left[1,n\right]$ is the stability criterion of the state-space model. And, the imaginary part of the eigenvalues, $\mathrm{Imag}\left({\lambda }_i\right)$, represents the oscillation characteristic of the inverter. Another advantage of the state-space model lies in the utilization of the participation factors and sensitivity analysis methods. Participation factors facilitate the analysis of the mapping relationship between system state variables and small-signal stability margins [15], while sensitivity analysis enables the examination of the relationship between system control parameters and stability characteristics [16]. The eigenvalue analysis method is highly valuable due to its accurate results. Thus, this paper adopts the state-space method for system modeling.

Most of the work above mainly focuses on the modeling and stability analysis of the individual inverter with a GFL or GFM control, which is connected to an infinite AC bus [17]. However, in industrial applications, especially in RES power plants, the inverters are commonly connected in parallel to the PCC through line impedance, as depicted in Figure 2. Therefore, to analyze the interactive coupling characteristics between parallel inverters, it is important to derive a detailed model of the entire inverter-based system. Many studies focus on the parallel inverters under the same control scheme, e.g., the GFL control [18,19], the droop control [20], the virtual oscillator control (VOC) [21], and the GFM control [22]. The neglect of parallel inverters under different controls declines the operation flexibility of the RES power plant [23]. Based on the impedance model, the work in [24] quantitatively assessed the impact of the increasing ratio of the GFM inverters on the oscillation frequency and damping, which reveals that the oscillation damping can be significantly improved with a higher penetration of GFM inverters. However, this study is based on the impedance model of the two-inverters system, which cannot reflect the relationship between the GFM inverter parameters and system stability margin directly. Therefore, there is still a need to derive the full-order state-space model of the hybrid inverter system, and analyze the effects of the GFM inverters on the stability margin of the system quantitatively seen from the perspectives of damping and frequency of the oscillatory modes.

With the above, this paper presents the small-signal model of parallel inverters, aiming to reveal the stability characteristics of multiple RES systems. More specifically, this paper quantitatively analyzes the stabilizing effect of the inverter with the GFM control on the hybrid system using the eigenvalues loci and the small-signal model. The inertia and damping characteristics of the GFM inverter enable the hybrid inverters system to keep stable when connected to an extremely weak grid, as revealed through the stability analysis. The rest of this paper is organized as follows. Section 2 shows the control schemes of the GFL and GFM, and the accurate full-order state-space models of the individual inverters are formulated. Section 3 models the inverter-based system of GFL inverters in parallel, and then the hybrid inverter-based system with GFL and GFM controls is modeled in detail. Section 4 studies a comparative case of the different characteristics of three parallel GFL inverter systems and the hybrid parallel inverter system (two GFL inverters with a GFM inverter). Section 5 provides concluding remarks.

2. Control and Modeling of the Individual Inverter

The control of power inverters can generally be divided into GFL and GFM control strategies, with which the stability characteristics of the inverter exhibit significant differences. The GFL inverters are controlled as current sources to inject the desired current into the grid with high quality. To achieve the synchronization with the grid voltage, a PLL is utilized to trace the q-axis component of the PCC’s voltage vector $v_{PCC}^{abc}$. Considering the negative impedance of the PLL in a certain frequency range in the weak grid, the GFL inverter takes risks of instability when connected to a weak grid with a low short-circuit ratio (SCR) [25]. On the contrary, the inverter operating in the GFM mode demonstrates enhanced stability in the weak grid [11], and thus, the GFM control is commonly applied to improve the stability of a low-inertia grid [26]. For the most commonly utilized GFM control, the VSG control synchronizes with the grid by the outer power loop. Given the extensive studies and application of the VSG control, this paper similarly employs it as a representative GFM control for modeling and analysis.

2.1. Grid-Following Control

The inverters with GFL control are widely applied as the interface between the RESs and the power system, and this paper adopts the active–reactive power control (PQ control) as the outer power loop of the GFL control. The PLL unit generates the angle signal ${\theta }_{pll}$ for the transformation from the $abc$-axis to $dq$-axis by tracking the PCC’s voltage $v_{PCC}^{abc}$. The current vector controller adjusts the output currents $i_{PCC}^{abc}$ according to current references $i_{dq}^{ref}$ generated by the power loop.

The control block diagram of a grid-connected inverter with the PQ GFL control is shown in Figure 3, and the control is derived in the $dq$ reference frame with the angle ${\theta }_{PLL}$. $V_{DC}$ is the DC-link voltage with a constant value by assuming the voltage is supplied by a stiff DC source; $L_f$ and $R_f$ are the filter inductor and resistance; $C_f$ and $R_d$ are the filter capacitor and its series damping resistor; $L_g$ and $R_g$ are the equivalent inductor and resistor of the grid, and ${\widehat{v}}_{g\_abc}^s$ is the grid voltage. Moreover, ${\widehat{v}}_{L\_abc}^s$ and ${\widehat{i}}_{L\_abc}^s$ are the output voltages and currents of the inverter, while ${\widehat{v}}_{PCC\_abc}^s$ and ${\widehat{i}}_{PCC\_abc}^s$ are the voltages and currents of the PCC, respectively. By utilizing the Park transformation in Equation (1), the AC signals in the $abc$-axis can be converted to $dq$-signals:

$$\left[\begin{array}{c}{x}_d\\ x_q\end{array}\right]=\left[\begin{array}{ccc}cos\left(\theta \right)& cos\left(\theta -\frac{2\pi }{3}\right)& cos\left(\theta +\frac{2\pi }{3}\right)\\ -sin\left(\theta \right)& -sin\left(\theta -\frac{2\pi }{3}\right)& -sin\left(\theta +\frac{2\pi }{3}\right)\end{array}\right]\left[\begin{array}{c}{x}_a\\ x_b\\ x_c\end{array}\right]$$

(1)

where the subscripts d and q denote the d-axis and q-axis components of the corresponding variable x, respectively, and $\theta$ is the phase angle. The parameters of the grid-connected inverter are shown in

Table 1. Circuit parameters of a 4-kW grid-connected inverter.

Circuit Parameters	Symbol	Value
DC-side voltage	$V_{dc}$	700 V
AC-side voltage (RMS)	$\left|\right|V_g{\left|\right|}_2$	220 V
Rated grid frequency	$f_0$	50 Hz
Switching frequency	$f_s$	20 kHz
Rated active power	$P_N$	10 kW
Rated reactive power	$Q_N$	0 kVar
Grid inductance	$L_g$	2 mH
Grid resistance	$R_g$	0.2 Ω
Filter inductance	$L_f$	5 mH
Filter resistance	$R_f$	0.5 Ω
Filter capacitance	$C_f$	500 μF
Damping resistance	$R_d$	3 Ω
Sampling time	$T_{sample}$	50 μs

.

It is worth mentioning that the reference frame is divided into system and controller frames, denoted by the superscript “s” and “c”, respectively, as depicted in Figure 4, to assess the impact of transient synchronization dynamics [9]. The angle difference between the two frames is defined as ${\theta }_0$, which is equal to 0 in a steady-state operation. Moreover, the small-signal perturbations of variables are denoted with $\Delta$, and the steady-state values of variables are represented with subscript “₀”. In the rest of the paper, the variables are similarly named according to these rules. The system frame $dq\left(s\right)$ is determined by the PCC voltage ${\widehat{v}}_{PCC\_abc}^s$, while the controller frame $dq\left(c\right)$ is determined by the angle signal ${\theta }_{PLL}$ from the PLL. To assess the small-signal stability of the inverter system, a small perturbation is added to the angle signal, and the transformation matrix $T_{cs}$ and $T_{sc}$ between the two frames are given in Appendix A [27].

2.1.1. Phase-Locked Loop

The PLL enables the inverter to synchronize with the grid by orienting the voltage vector $\overrightarrow{V_{PCC}^c}=v_{PCC\_d}^c+j{v}_{PCC\_q}^c$ to the d-axis of the controller rotating reference frame accurately [25]. As shown in Figure 3, the transfer function between $v_{PCC\_q}^c$ and ${\theta }_{PLL}$ can be illustrated as:

$${\theta }_{PLL}=\left(\frac{1}{s}\right)\left(k_{pll\_p}+\frac{k_{pll\_i}}{s}\right)v_{PCC\_q}^c$$

(2)

where $k_{pll\_p}$ and $k_{pll\_i}$ are the proportional and integral parameters of the PLL, respectively. Applying the transformation matrix $T_{cs}$ and $T_{sc}$ and the small-signal linearization method, Equation (2) can be rewritten in the time-domain as:

$$\left\{\begin{array}{c}p\Delta {\xi }_{\mathit{vq}}=\Delta v_{PCC\_q}^c=\Delta v_{PCC\_q}^s-V_{PCC\_d0}\Delta {\theta }_{PLL}\hfill \\ p\Delta {\theta }_{PLL}=k_{pll\_p}\Delta v_{PCC\_q}^c+k_{pll\_i}\Delta {\xi }_{\mathit{vq}}\hfill \end{array}\right.$$

(3)

in which the p denotes the differential operator, i.e., $p=d/dt$.

2.1.2. Power Control Loop

The output active power $P_e$ and reactive power $Q_e$ of the inverter can be calculated as:

$$\left\{\begin{array}{c}{P}_e=\mathrm{Re}\left\{1.5\left(\overrightarrow{V_{PCC}^c}·\overrightarrow{I_{PCC}^c}\right)\right\}\\ Q_e=\mathrm{Im}\left\{1.5\left(\overrightarrow{V_{PCC}^c}·\overrightarrow{I_{PCC}^c}\right)\right\}\end{array}\right.$$

(4)

The power control loop generates the current references $i_{ref\_d}$ and $i_{ref\_q}$ for the inner current controller responding to the operating set-points $P_{ref}$ and $Q_{ref}$. The power control loop commonly consists of a proportional–integral (PI) controller with slow dynamics, and the current reference equations can be given as:

$$\left\{\begin{array}{c}{i}_{ref\_d}^c=\frac{P_{ref}·v_{pcc\_d}^c+Q_{ref}·v_{pcc\_q}^c}{{v_{pcc\_d}^c}^2+{v_{pcc\_q}^c}^2}\\ i_{ref\_q}^c=\frac{P_{ref}·v_{pcc\_q}^c-Q_{ref}·v_{pcc\_d}^c}{{v_{pcc\_d}^c}^2+{v_{pcc\_q}^c}^2}\end{array}\right.$$

(5)

with $v_{pcc\_d}^c$ and $v_{pcc\_q}^c$ being the $dq$ components of the voltage ${\widehat{v}}_{PCC\_abc}^s$ in the controller axis, and $i_{ref\_d}^c$ and $i_{ref\_q}^c$ being the current references in the $dq$-axis. In this paper, these are applied as the power loop of the GFL inverters. By adding a small-signal perturbation to Equation (5), the following can be obtained:

$$\left[\begin{array}{c}\Delta i_{ref\_d}^c\\ \Delta i_{ref\_q}^c\end{array}\right]=K\left[\begin{array}{cccc}{k}_{dd}& k_{dq}& V_{pcc\_d0}& V_{pcc\_q0}\\ k_{qq}& k_{qq}& V_{pcc\_q0}& -V_{pcc\_d0}\end{array}\right]{\left[\begin{array}{cccc}\Delta v_{pcc\_d}^c& \Delta v_{pcc\_q}^c& \Delta P_{ref}& \Delta Q_{ref}\end{array}\right]}^T$$

(6)

where

$$\left\{\begin{array}{c}K=\frac{2}{3}{\left(\frac{1}{{\left(V_{pcc\_d0}\right)}^2+{\left(V_{pcc\_q0}\right)}^2}\right)}^2\hfill \\ k_{dd}=P_{ref}\left({V_{pcc\_d0}}^2+{V_{pcc\_q0}}^2\right)-2V_{pcc\_d0}\left(P_{ref}{V}_{pcc\_d0}+Q_{ref}{V}_{pcc\_q0}\right)\hfill \\ k_{dq}=Q_{ref}\left({V_{pcc\_d0}}^2+{V_{pcc\_q0}}^2\right)-2V_{pcc\_q0}\left(P_{ref}{V}_{pcc\_d0}+Q_{ref}{V}_{pcc\_q0}\right)\hfill \\ k_{qd}=-Q_{ref}\left({V_{pcc\_d0}}^2+{V_{pcc\_q0}}^2\right)-2V_{pcc\_d0}\left(-Q_{ref}{V}_{pcc\_d0}+P_{ref}{V}_{pcc\_q0}\right)\hfill \\ k_{qq}=P_{ref}\left({V_{pcc\_d0}}^2+{V_{pcc\_q0}}^2\right)-2V_{pcc\_q0}\left(-Q_{ref}{V}_{pcc\_d0}+P_{ref}{V}_{pcc\_q0}\right)\hfill \end{array}\right.$$

(7)

Here, the $P_{ref}$ and $Q_{ref}$ are the set-points of the inverter.

2.1.3. Current Controller Loop

The current controllers are applied to track the current references from the power loop with the PI controller, and the decoupling parts are added to mitigate the coupling effect of the $dq$ axis. The grid voltage feedforward control is also applied to suppress the effect of the harmonics of the grid voltage and to enhance the synchronization stability with the voltage feedforward coefficients $F_i$ [28]. The current controllers are given as:

$$\left\{\begin{array}{c}{v}_d^{*}=\left(k_{i\_p}+\frac{k_{i\_i}}{s}\right)\left(i_{ref\_d}^c-i_{pcc\_d}^c\right)-{\omega }_g{i}_{pcc\_d}^c+F_i{v}_{pcc\_d}^c\\ v_q^{*}=\left(k_{i\_p}+\frac{k_{i\_i}}{s}\right)\left(i_{ref\_q}^c-i_{pcc\_q}^c\right)+{\omega }_g{i}_{pcc\_q}^c+F_i{v}_{pcc\_q}^c\end{array}\right.$$

(8)

where $\omega$ is the rated angular velocity of the grid; $k_{i\_p}$ and $k_{i\_i}$ are the proportional and integral parameters of the current controller, respectively; $v_d^{*}$ and $v_q^{*}$ are the modulating voltages.

Combing the matrices $T_{cs}$ and $T_{sc}$, Equation (8) can be linearized as follows:

$$\left\{\begin{array}{c}\Delta v_d^{*}=-{\omega }_g{L}_f·\Delta i_{L\_q}^s+F_i·\Delta v_{pcc\_d}^s+k_{p\_i}·\Delta i_{d\_ref}^c-k_{p\_i}·\Delta i_{L\_d}^s+k_{i\_i}·\Delta {\psi }_{id}+G_{vd}^{\theta }·\Delta \theta \hfill \\ \Delta v_q^{*}={\omega }_g{L}_f·\Delta i_{L\_d}^s+F_i·\Delta v_{pcc\_q}^s+k_{p\_i}·\Delta i_{q\_ref}^c-k_{p\_i}·\Delta i_{L\_q}^s+k_{i\_i}·\Delta {\psi }_{iq}+G_{vq}^{\theta }·\Delta \theta \hfill \end{array}\right.$$

(9)

where

$$\left\{\begin{array}{c}{G}_{vd}^{\theta }=I_{L\_d0}^c·{\omega }_g{L}_f+V_{pcc\_q0}^c·F_i-I_{L\_q0}^c·k_{p\_i}\hfill \\ G_{vq}^{\theta }=I_{L\_q}^c·{\omega }_g{L}_f-V_{pcc\_d0}^c·F_i+I_{L\_d0}^c·k_{p\_i}\hfill \\ p\Delta {\psi }_{id}=\Delta i_{d\_ref}^c-\Delta i_{L\_d}^c\hfill \\ p\Delta {\psi }_{iq}=\Delta i_{q\_ref}^c-\Delta i_{L\_q}^c\hfill \end{array}\right.$$

(10)

in which $\Delta {\psi }_{id}$ and $\Delta {\psi }_{iq}$ are the outputs of the integral part of the current controllers; $G_{vd}^{\theta }$ and $G_{vq}^{\theta }$ represent the effect of the synchronization disturbance on the current loop.

2.1.4. Physical Circuit Model

Based on Kirchhoff’s law, the linearized equations of the physical circuit can be given as shown in Equation (11), which reflects the dynamics of the filter of the inverter and the impedance of the grid. This paper especially considers the existence of the damping resistance $R_d$, which exhibits a noticeable effect on the stability of the inverter in the high-frequency range.

$$\left\{\begin{array}{c}p\Delta v_{PCC\_d}^s=\left(p{R}_d-\frac{1}{C_f}\right)\left(\Delta i_{L\_d}^s-\Delta i_{pcc\_d}^s\right)-{\omega }_g{R}_d\Delta i_{L\_q}^s+{\omega }_g{R}_d\Delta i_{pcc\_q}^s+{\omega }_g\Delta v_{pcc\_q}^s\hfill \\ p\Delta v_{PCC\_q}^s=\left(p{R}_d-\frac{1}{C_f}\right)\left(\Delta i_{L\_q}^s-\Delta i_{pcc\_q}^s\right)+{\omega }_g{R}_d\Delta i_{L\_d}^s-{\omega }_g{R}_d\Delta i_{pcc\_d}^s-{\omega }_g\Delta v_{pcc\_d}^s\hfill \\ p\Delta i_{L\_d}^s=\frac{1}{L_f}\left(\Delta v_{L\_d\_d}^s+{\omega }_g{L}_f·\Delta i_{L\_q}^s-R_f·\Delta i_{L\_d}^s-\Delta v_{pcc\_d}^s\right)\hfill \\ p\Delta i_{L\_q}^s=\frac{1}{L_f}\left(\Delta v_{L\_q\_d}^s-{\omega }_g{L}_f·\Delta i_{L\_d}^s-R_f·\Delta i_{L\_q}^s-\Delta v_{pcc\_q}^s\right)\hfill \\ p\Delta i_{pcc\_d}^s=\frac{1}{L_g}\left(\Delta v_{pcc\_d}^s+{\omega }_g{L}_g·\Delta i_{pcc\_q}^s-R_g·\Delta i_{pcc\_d}^s-\Delta u_{g\_d}^s\right)\hfill \\ p\Delta i_{pcc\_q}^s=\frac{1}{L_g}\left(\Delta v_{pcc\_q}^s-{\omega }_n{L}_g·\Delta i_{pcc\_d}^s-R_g·\Delta i_{pcc\_q}^s-\Delta u_{g\_q}^s\right)\hfill \end{array}\right.$$

(11)

where the $u_{g\_d}^s$ and $u_{g\_q}^s$ represent the disturbance of the grid; $v_{L\_d\_d}^s$ and $v_{L\_q\_d}^s$ denote the result of the modulating voltages $v_d^{*}$ and $v_q^{*}$ being effected by a time delay, and the approximation of the time delay will be discussed later.

2.1.5. Effect of Time Delay

The digital delay of the control and communication is considered as the factors resulting in the instability of grid-connected inverters [29]. Time delays of the inverter will bring a phase lag to the control loop, which will decrease the phase margin and control bandwidth, thus degrading the performance of the inverter [30]. The effect of a time delay can be evaluated by the Padé approximation effectively, and the fitting frequency range and accuracy can be designed by the approximation orders $n_{delay}$. This paper adopts a 3-order Padé approximation, which can be given as:

$$\left\{\begin{array}{c}p\left[\begin{array}{c}{x}_1^{del}\\ x_2^{del}\\ x_3^{del}\end{array}\right]=\left[\begin{array}{ccc}\frac{-12}{T_d}& \frac{-60}{T_d^2}& \frac{-120}{T_d^3}\\ 1& 0& 0\\ 0& 1& 0\end{array}\right]\left[\begin{array}{c}{x}_1^{del}\\ x_2^{del}\\ x_3^{del}\end{array}\right]+\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]v\hfill \\ v_{delay}=\left[\begin{array}{ccc}\frac{24}{T_d}& 0& \frac{240}{T_d^3}\end{array}\right]\left[\begin{array}{c}{x}_1^{del}\\ x_2^{del}\\ x_3^{del}\end{array}\right]-v\hfill \end{array}\right.$$

(12)

where v represents the original signal, and $v_{delay}$ is the corresponding signal considering the time delay. $T_d$ is the time delay and its value commonly considered to be 1.5 times the switching period, i.e., $T_d=1.5T_s$ [30].

2.1.6. State-Space Model of the GFL Inverter

By considering Equations (1), (3), (6), (7) and (9)–(12), the state-space model of the PQ-GFL inverter can be shown as:

$$p\Delta x_{GFL}={\mathbf{A}}_{\mathbf{GFL}}\Delta x_{GFL}+{\mathbf{B}}_{\mathbf{GFL}}\Delta u_{GFL}$$

(13)

where $x_{GFL}$ is a $16\times 1$ vector of state variables, ${\mathbf{A}}_{\mathbf{GFL}}$ is a $16\times 1$ matrix, and ${\mathbf{B}}_{\mathbf{GFL}}$ is a $16\times 4$ matrix. $\Delta u_{GFL}$ is a $4\times 1$ vector of input variables, given as:

$$\Delta u_{GFL}={\left[\begin{array}{cccc}\Delta u_{gd}& \Delta u_{gq}& \Delta P_{ref}& \Delta Q_{ref}\end{array}\right]}^T$$

(14)

in which the $\Delta P_{ref}$ and $\Delta Q_{ref}$ are the disturbance of the set points.

The control parameters of a grid-connected GFL inverter are listed in

Table 2. Control parameters of the 4-kW GFL inverter.

GFL Control Parameters	Symbol	Value
Proportional coefficient of the PLL	$k_{pll\_p}$	$0.38$
Integral coefficient of the PLL	$k_{pll\_i}$	$50.8$
Proportional coefficient of the current controller	$k_{i\_p}$	$12.5$
Integral coefficient of the current controller	$k_{i\_i}$	800
Voltage-forward coefficient	$F_i$	0
Active power reference	$P_{ref}$	4 kW
Reactive power reference	$Q_{ref}$	0 Var
Delay time	$T_{delay}$	75 μs
Delay orders	$n_{delay}$	3

, and an EMT model in SIMULINK is compared to verify the accuracy of the state-space model of the GFL inverter. As shown in Figure 5, the active power reference $P_{ref}$ steps from 4 kW (1 pu) to 4.8 kW (1.2 pu) at 0.5 s. The blue line is the response from the state-space (SS) model, while the red curve represents that from the EMT model. As depicted in Figure 5, the active power response of the SS model matches well with the EMT model during transition while presenting negligible ripples during a steady-state operation.

Based on the SS model, the eigenvalue analysis can be applied to quantitatively assess the impact of system parameters on the stability. As the traces of eigenvalues responding to the sweeps of the PLL parameters $k_{pll\_p}$ and $k_{pll\_i}$, grid impedance $L_g$ and $R_g$ are depicted in Figure 6. Here, $L_g$ and $R_g$ vary in the range [0, 35] mH and [0, 3] Ω, respectively, to represent various grid strengths with different SCRs, and the SCR can be calculated as:

$$SCR=\frac{1.5{\left({∥V_g∥}_2\right)}^2}{P_{ref}}\frac{1}{\sqrt{R_g^2+{\left({\omega }_g{L}_g\right)}^2}}$$

(15)

It can be observed in Figure 6a,b that the GFL inverter suffers from instability when the grid is relatively weak. With the decrease of the SCR, i.e., the increase of $L_g$ and $R_g$, the locations of the eigenvalues are moving to the right half of the plane, meaning instability of the system. With the increase of $L_g$, the eigenvalues of the GFL inverter will move to the right half of the plane with $max\left\{\mathrm{Real}\left({\lambda }_i\right)\right\}>0$, which means instability occurs, as depicted in Figure 6a. However, with the increase of the grid resistance $R_g$, the eigenvalues move to the right, but they still stay in the left half of the plane with $max\left\{\mathrm{Real}\left({\lambda }_i\right)\right\}<-60$, as shown in Figure 6b. Therefore, $L_g$ has more significance compared to $R_g$, considering the inductor will affect the phase of the PCC voltage ${\widehat{v}}_{PCC\_abc}^s$. According to the bandwidth ${\omega }_c$ of the PLL defined in [25], Figure 6c,d shows the eigenvalue changes with respect to ${\omega }_c$ in the range of [1, 300] Hz under different grid strengths. When the grid is relatively weak, the GFL inverter has a smaller stability margin, and with a large ${\omega }_c$, the GFL inverter will go to instability, as demonstrated in Figure 6c. Figure 6d shows a similar result. That is, the eigenvalues of the GFL inverter will move closer to the right half of the plane with the increase of ${\omega }_c$. Meanwhile, with the same ${\omega }_c$, the real parts of the eigenvalues in Figure 6c are larger than those in Figure 6b, which means the GFL inverter has a smaller stability margin when the grid is relatively weak.

2.2. Grid-Forming Control

This paper adopts the commonly used VSG control as the GFM control, and the VSG control can provide inertia and voltage support to the grid by emulating the external characteristic of an SG. The control scheme of the VSG GFM inverter is depicted in Figure 7. The main differences between the GFL and GFM control lie in the synchronization mechanism and the voltage controller. The GFM can achieve synchronization with the grid by the power synchronization loop (PSL) without detecting the phase of the grid voltage. The inertia characteristic of the PSL enables the GFM inverter to respond to frequency events with a transient power supply. The modeling of the current controller, the physical circuit and the time delay for the GFM inverter are the same as those for the GFL inverter.

2.2.1. Power Loop Control

The active power loop generates the control angle for the inverter by mimicking the mechanical part of an SG, while the reactive power loop determines the excitation electromotive force of the inverter following the stator voltage equation of an SG. The small-signal equations of the power loop can be given as:

$$\left\{\begin{array}{c}p\Delta {\omega }_{vsg}=\frac{1}{J}\left(-D_p\Delta {\omega }_{vsg}+\frac{1}{{\omega }_n}\left(\Delta P_{ref}-\Delta P_e\right)\right)\hfill \\ p\Delta \theta =\Delta {\omega }_{vsg}\hfill \\ p\Delta E_m=\frac{1}{K_Q}\left(\Delta Q_{ref}-\Delta Q_e\right)\hfill \end{array}\right.$$

(16)

where J is the virtual inertia coefficient; $D_p$ is the damping coefficient; $K_Q$ is the integral coefficient of the reactive power loop. Without considering the virtual impedance, the output of the reactive power loop $E_m$ is applied as the d-axis component of the voltage reference, while the q-axis component of the voltage reference is set to zero.

2.2.2. Voltage Control Loop

The voltage controller is applied to trace the voltage references from the power loop, and the output current feedforward control is applied with the gain of $F_v$. The equations of the voltage controller can be given as:

$$\left\{\begin{array}{c}\Delta i_{d\_ref}^c=\left(\Delta E_m-\Delta v_{pcc\_d}^c\right)·\left(k_{p\_v}+\frac{k_{i\_v}}{s}\right)-{\omega }_n{C}_f·\Delta v_{pcc\_q}^c+F_v·\Delta i_{pcc\_d}^c\hfill \\ \Delta i_{q\_ref}^c=\left(-\Delta v_{pcc\_q}^c\right)·\left(k_{p\_v}+\frac{k_{i\_v}}{s}\right)+{\omega }_n{C}_f·\Delta v_{pcc\_d}^c+F_v·\Delta i_{pcc\_q}^c\hfill \end{array}\right.$$

(17)

where $k_{v\_p}$ and $k_{v\_i}$ are the proportional and integral parameters of the voltage controller, respectively.

Combing the matrices $T_{cs}$ and $T_{sc}$, Equation (17) can be linearized as:

$$\left\{\begin{array}{c}\Delta i_{d\_ref}^c=-{\omega }_n{C}_f·\Delta v_{pcc\_q}^s+F_v·\Delta i_{pcc\_d}^s+k_{p\_v}·\Delta E_m-k_{v\_p}·\Delta v_{pcc\_d}^s+k_{v\_i}·\Delta {\xi }_{vd}+G_{id}^{\theta }·\Delta \theta \hfill \\ \Delta i_{q\_ref}^c={\omega }_n{C}_f·\Delta v_{pcc\_d}^s+F_v·\Delta i_{pcc\_q}^s-k_{v\_p}·\Delta v_{pcc\_q}^s-k_{v\_i}·\Delta {\xi }_{vq}++G_{iq}^{\theta }·\Delta \theta \hfill \end{array}\right.$$

(18)

with

$$\left\{\begin{array}{c}{G}_{id}^{\theta }=V_{pcc\_d}^c·{\omega }_n{C}_f+I_{pcc\_q}^c·F_v-V_{pcc\_q}^c·k_{v\_p}\hfill \\ G_{iq}^{\theta }=V_{pcc\_q}^c·{\omega }_n{C}_f-I_{pcc\_d}^c·F_v+V_{pcc\_d}^c·k_{v\_p}\hfill \\ p\Delta {\xi }_{vd}=\Delta i_{d\_ref}^c-\Delta i_{L\_d}^c\hfill \\ p\Delta {\xi }_{vq}=\Delta i_{q\_ref}^c-\Delta i_{L\_q}^c\hfill \end{array}\right.$$

(19)

where $\Delta {\xi }_{vd}$ and $\Delta {\xi }_{vq}$ are the outputs of the integral part of the voltage controllers; $G_{id}^{\theta }$ and $G_{iq}^{\theta }$ represent the effect of the synchronization disturbance on the voltage control loop, d-axis and q-axis, respectively.

2.2.3. State-Space Model of the GFM Inverter

Similar to the GFL inverter, the state-space model of the VSG-GFM inverter can be given as:

$$p\Delta x_{GFM}={\mathbf{A}}_{\mathbf{GFM}}\Delta x_{GFM}+{\mathbf{B}}_{\mathbf{GFM}}\Delta u_{GFM}$$

(20)

where $x_{GFM}$ is a $19\times 1$ vector of state variables, ${\mathbf{A}}_{\mathbf{GFM}}$ is a $19\times 1$ matrix, and ${\mathbf{B}}_{\mathbf{GFM}}$ is a $19\times 4$ matrix. $\Delta u_{GFM}$ is a $4\times 1$ vector of input variables that is the same as Equation (14).

The control parameters of the 4 kW GFM inverter are listed in

Table 3. Control parameters of the 4 kW GFM inverter.

GFM Control Parameters	Symbol	Value
Inertia coefficient	J	$0.005$
Damping coefficient	$D_P$	$1.7$
Integral coefficient of the Q-V loop	$K_Q$	$5.5$
Proportional coefficient of the voltage controller	$k_{v\_p}$	$0.12$
Integral coefficient of the voltage controller	$k_{v\_i}$	$0.08$
Proportional coefficient of the current controller	$k_{i\_p}$	28
Integral coefficients of the current controller	$k_{i\_i}$	600
Current-forward coefficient	$F_v$	0
Voltage-forward coefficient	$F_i$	0
Active power reference	$P_{ref}$	4 kW
Reactive power reference	$Q_{ref}$	0 Var
Delay time	$T_{delay}$	75 μs
Delay orders	$n_{delay}$	3

, and the comparison between the EMT model and SS model is shown in Figure 8, where the active power reference $P_{ref}$ steps from 4 kW (1 pu) to 4.4 kW (1.1) pu at 2.5 s. It can be observed that the blue line (SS model) matches well with the red line (EMT model), which shows the effectiveness of the SS model.

In Figure 9, the traces of eigenvalues are presented to demonstrate the effect of the GFM parameters on the system’s stability. Here, J and $D_p$ of the active power loop sweep in the range [1 × 10⁻³, 1] and [0.1, 10], respectively, in Figure 9a,b, to exhibit the influence of different inertia and damping characteristics. Moreover, $K_Q$ of the reactive power loop varies in the range [0.1, 10] in Figure 9c, while the grid impedance $L_g$ changes from 0 mH to 35 mH in Figure 9d. With the variation of active power parameters, the eigenvalue loci in Figure 9a show that an unreasonable large value of system inertia J produces a higher risk of instability, manifesting as a slow frequency oscillation. Similarly, Figure 9b shows the necessity for the GFM inverter to properly set the damping parameter $D_P$. Inadequate or excessive damping can lead to the dominant pole of the system locating quite near the imaginary axis, which even makes the inverter potentially unstable. Furthermore, Figure 9c shows the eigenvalues’ loci with the droop gain $K_Q$ of the reactive power loop; it is clear that the increase of $K_Q$ results in a narrower stable operation range of the GFM inverter and even causes oscillation in the low frequency.

2.3. Small-Signal Modeling of the Power Plant

Without loss of generality, the configuration of the renewable power plant is simplified as depicted in Figure 2, and an example with three inverters in parallel considering the transmission line dynamics is shown in Figure 10.

The small-signal model of the transmission line can be derived as follows:

$$\left\{\begin{array}{c}p\Delta i_{o\_d}^s=\frac{1}{L_g}\left(\Delta v_{BSU\_d}^s+{\omega }_g{L}_g·\Delta i_{o\_q}^s-R_g·\Delta i_{o\_d}^s-\Delta u_{g\_d}^s\right)\hfill \\ p\Delta i_{o\_q}^s=\frac{1}{L_g}\left(\Delta v_{BUS\_q}^s-{\omega }_n{L}_g·\Delta i_{o\_d}^s-R_g·\Delta i_{o\_q}^s-\Delta u_{g\_q}^s\right)\hfill \end{array}\right.$$

(21)

where $i_{o\_d}^s$ and $i_{o\_q}^s$ are the state variables of the transmission line; $v_{BSU\_d}^s$ and $v_{BUS\_q}^s$ are the voltage of the connected bus.

3. Small-Signal Stability Analysis of Parallel Inverters

To quantitatively analyze the effect of the GFM inverter on the stability of the power plant, a case study involving parallel GFL inverters (three GFL inverters) and hybrid inverters (two GFL inverters and one GFM inverter) are built. The diagram of the system is depicted in Figure 10. As observed in Figure 10, the inverters are connected to the same AC bus through the line impedance of $L_i$ and $R_i$, while the AC bus injects power to the grid through the transmission line with the impedance being $L_g$ and $R_g$. For a more general discussion, the act power operation points of the inverters are set differently in the sequence of 3, 4, and 5 kW, and the other circuit parameters, GFL control parameters, and GFM control parameters are listed in Table 1, Table 2 and Table 3, respectively.

With the above models, the small-signal model of the power plant can be given as:

$$p\left[\begin{array}{c}\Delta x_{GFL1}\\ ⋮\\ \Delta x_{GFLi}\\ ⋮\\ \Delta x_{GFM1}\\ ⋮\\ \Delta x_{GFMi}\\ ⋮\\ \Delta x_{line}\end{array}\right]=A_{sys}{\left[\begin{array}{c}\Delta x_{GFL1}\\ ⋮\\ \Delta x_{GFLi}\\ ⋮\\ \Delta x_{GFM1}\\ ⋮\\ \Delta x_{GFMi}\\ ⋮\\ \Delta x_{line}\end{array}\right]}^T+B_{sys}\left[\begin{array}{c}\Delta u_{gd}\\ \Delta u_{gq}\\ \Delta u_{GFL1}\\ ⋮\\ \Delta u_{GFM1}\\ ⋮\end{array}\right]$$

(22)

where $\Delta x_{line}$ is the current of the transmission line; $A_{sys}$ and $B_{sys}$ are the system matrix and output vector, which can be easily derived according to the aforementioned equations.

The SS model of three GFL inverters in parallel is established to represent a power plant with 100% GFL inverters, and Figure 11 depicts the comparison results between the EMT model and SS model, which validate the accuracy of the SS model. Similarly, when one of the GFL inverters (Inverter 3, see Figure 10) is controlled in GFM mode, the SS model is verified by the EMT model, as shown in Figure 12.

As shown in Figure 13a, parts of the eigenvalues of the system will move to the left plane, and the system cannot maintain stability anymore when the grid impedance $L_g$ varies from 0 mH to 35 mH. That means the renewable power plant with 100% GFL inverters when the grid is relatively weak undermines the applicability of the power plant, especially in certain remote regions. This characteristic of the parallel GFL inverters is similar to the individual GFL inverter, as the synchronization with ${\widehat{v}}_{PCC\_abc}^s$ is necessary for all GFL inverters. The eigenvalues’ loci of the hybrid inverter system is depicted in Figure 13b, which shows that the GFM inverter will enhance system stability, especially in weak grid conditions. The mechanism behind this can be explained by: as the GFM inverter participates in the establishment of the ${\widehat{v}}_{PCC\_abc}^s$, the equivalent grid is strengthened and improved. Consequently, the stability is enhanced.

4. Simulation Validation

In order to verify the stability analysis in Section 3, the EMT model of the case study system is established in Matlab/Simulink 2023a.

The control and circuit of an individual inverter are masked into a subsystem, and the details of the control schemes can be found in Figure 3 and Figure 7. The power set-points are considered as the inputs of the subsystem. And Figure 14a shows three GFL inverters in parallel, while one of the GFL inverters is changed to the GFM control in Figure 14b.

The simulation results are shown in Figure 15. As depicted in Figure 15, the parallel GFL inverters suffer from instability when the connected grid is relatively low with $L_g$ = 20 mH, which will hazard the operation of the entire system. However, when the control of one inverter is changed from the GFL to GFM mode, the hybrid inverter system keeps a stable operation under the same grid condition as the locations of the eigenvalues are improved and moved to the left half of the plane, which is beneficial for the application of the GFM, especially when considering the effect of grid conditions. Comparing the results of Figure 15a,b, the efficacy of the stability analysis and the models are verified.

5. Conclusions

This paper focused on the small-signal modeling and analysis of an inverter with GFL and GFM control, and the effect of the selected parameters on the inverter’s stability was studied by analyzing the eigenvalues’ loci. First, the state-space model of the GFL and GFM inverter was derived step by step, and the detailed state-space models for GFL and GFM inverters can be used to quantify the influence of transmission lines, PLL dynamics, line impedance, and power loop parameters. Then, the accuracy of the small-signal models was assessed by the corresponding EMT simulations. Consequently, the state-space modeling of the renewable power plant was derived to quantitatively analyze the improvement of the GFM inverter on the stability margin of the power plant. The comparison of the stability analysis of a case consisted of three parallel GFL inverters and two GFL inverters with one GFM inverter. The latter is superior in stabilizing the hybrid system due to its inherent inertia and damping characteristics, and this was further verified by the time-domain simulations. The quantitative analysis of the application of the GFM inverter on the hybrid inverter system’s stability can help the system operator in determining a proper ratio and placement of the GFM inverter to improve the stable margin of the entire system.