Disturbance Decoupling in Grid-Forming Inverters for Enhanced Dynamic Response

ABSTRACT

This paper presents a control strategy for grid-forming inverters, utilizing a cascaded dual-control scheme that integrates current and voltage controllers, along with an outer loop for regulating the injected active and reactive power. The proposed approach enhances system performance by addressing key challenges faced by grid-forming inverters. One of the highlights of this strategy is the improved disturbance rejection capability of the current controller, achieved through capacitor voltage decoupling. This modification effectively reduces the impact of disturbances on system performance. Additionally, the voltage controller incorporates the concept of disturbance input decoupling, which further improves the inverter’s dynamic response under varying operating conditions. Notably, the inclusion of disturbance input decoupling enhances the loop gain of the voltage controller in the low-frequency range, leading to superior performance of the inverter. Experimental results validate the efficacy of the proposed control strategy in managing real-word scenarios such as voltage sags, load changes and variations in power references. These findings highlight the robustness of the proposed methodology, demonstrating its potential for enhancing the operation of grid-forming inverters in modern power systems.

KEYWORDS
control strategy; grid-forming inverters; disturbance rejection capability; disturbance input decoupling

I. INTRODUCTION

Power electronic converters play a crucial role in integrating renewable energy sources (RESs) into modern power systems, which are increasingly influenced by power electronics [¹], [²]. With the rising adoption of inverter-based resources (IBRs), particularly wind and solar photovoltaic systems, as well as battery energy storage systems (BESSs), the structure of the electrical grid is undergoing significant changes. This evolution is shifting the balance between the energy demands and the services provided by the grid. As synchronous machines (SMs) are gradually replaced by IBRs, the essential services that SMs traditionally offered—such as voltage and frequency regulation—are becoming less prevalent. Consequently, it is imperative for IBRs to take on these critical functions to ensure grid stability and reliability [³]. Conventional IBRs control strategies have typically employed the grid-following (GFL) approach, and in this method, the inverter operates by synchronizing its output to the grid, fundamentally following the grid’s voltage and frequency. To accomplish this, these parameters are measured at the point of common coupling (PCC) using a phase-locked loop (PLL). However, the PLL can negatively impact system stability, particularly when the grid is weak [⁴], [⁵]. Basically, the GFL IBRs are configured to operate at the rated output power and inject it into a powered grid. Power control loops are used to regulate active and reactive power ($P$ and $Q$) and to generate current references for the current control loop.

Another control strategy for IBRs is the grid-forming (GFM) approach, and according to NERC (North American Electric Reliability Corporation) the GFM IBR control can be defined as “an internal voltage phasor that is constant or nearly constant in the sub-transient to transient time frame” [⁶]. Although the term GFM has been widely used in recent years, its concept, which is based on regulating voltage and frequency, was proposed many years ago [⁷].

Among the GFM strategies, those that stand out the most are the virtual synchronous machine (VSM) [⁸]–[¹¹], that emulates the behavior of a synchronous machine, adding inertia and damping of a traditional synchronous machine to IBRs; droop control [¹²]–[¹⁴], that adjusts $P/Q$ power output based on local frequency/voltage deviations (P-f and Q-V droops), enabling decentralized load-sharing among distributed resources without communication; power synchronization control (PSC) [¹⁵]–[¹⁷], that enables IBRs to synchronize their output power with the grid by adjusting voltage, frequency, and phase; Virtual Oscillator Control (VOC) [¹⁸]–[²⁰], that leverages nonlinear oscillator dynamics to self-synchronize IBRs with the grid autonomously, enabling decentralized operation without PLL; Synchronverter [²¹]–[²⁴], that replicates exactly the mathematical model of a synchronous machine (electromechanical equations, flux dynamics, and excitation control) in IBRs and Synchronous Power Controller (SPC) [²⁵]–[²⁷], that combines droop characteristics with virtual inertia and damping, blending static power-frequency regulation. The strategies mentioned previously have the characteristic of work either isolated or connected to the grid, and unlike the GFL inverter, the GFM inverter typically does not rely on a PLL to maintain synchronization with the grid. Instead, most reported GFM inverters use the power synchronization mechanism of synchronous machines to stay synchronized with the grid [²⁸]. However, when the GFM inverter works connected to the grid, it is more susceptible to instability as the short circuit ratio (SCR) of the grid increases, as well as the increase in the X/R ratio of the feeders.

Many studies have been conducted with the goal of enhancing the dynamic response of GFM inverters, especially regarding reducing power oscillation. In [²⁹], an active-damping control method based on self- and mutual-damping controllers to attenuate both self- and mutually induced low-frequency power oscillations was proposed. In [³⁰], it was proposed a virtual inductance control strategy to mitigate the unstable oscillation of frequency and powers, which differs from conventional virtual inductance strategies in that it does not dependent on a dual-loop control architecture. In [³¹], it was implemented virtual reactance instead of using virtual resistance to dampen the low-frequency resonance of the GFM inverter. The work presented in [³²] decided to use a distinct solution based on artificial intelligence to predict oscillation modes and enhance damping of electromechanical inter-areas oscillations. Basically, the strategy presented in [³³] aims to predict and adaptively tune a dedicated loop of the SPC to damp oscillations and enhance system stability. In [³⁴], it was proposed a control strategy using virtual damping for enhancing the system damping and virtual reactance to suppress oscillations without altering the synchronverters’ fundamental characteristics. In [³⁵], it was proposed a strategy based on virtual damper winding applied to GFM inverter to decrease the low-frequency oscillations, utilizing existing state variables without requiring a PLL. In [³⁶], it was proposed a generic voltage control scheme for GFM inverters that enhances voltage tracking and power regulation in both grid-tied and stand-alone modes, addressing conflicts in power loop dynamics and improving overall system stability and performance.

In these works, the investigation was carried out almost entirely on the power control loops, without a deep analysis of the inner loops to improve the dynamic response of the GFM inverter. In light of the above, this paper aims to develop a control strategy for GFM inverters, focusing on the inner loops design (current and voltage control loops) to improve the dynamic response of the inverter. The main contributions of this work are:

1. A current control loop with high-bandwidth and disturbance decoupling, which allows enhanced disturbance rejection capability, in situations such as voltage sags.

2. A voltage control loop with disturbance input decoupling, which reduces oscillations in the inverter's electrical parameters, specifically improving active power, reactive power, and current.

This paper is an extension of the conference paper in [³⁷] by the same authors. In this extended version, the mathematical development is presented in greater detail, and experimental results are included to complement the theoretical analysis. The paper is organized as follows: Section II outlines the system description, highlighting each of the parameters that are part of the model. Section III presents a detailed development of the modified inner loops (current and voltage loops). Section IV presents experimental results to validate theoretical development. Finally, Section V concludes this paper.

II. SYSTEM DESCRIPTION

The topology of a GFM inverter for grid-connected operation, incorporating an LCL filter and inner control loops for current and voltage, is depicted in Fig. 1. The DC-link voltage is represented by $V_{dc}$, grid voltage by $v_g$, capacitor voltages by $v_c$, grid-side currents by $i_g$, capacitor currents by $i_c$ and voltages at the PCC by $v_{pcc}$. The filter parameters are as follows: $L_1$ is the converter-side inductor, $C$ is the filter capacitor, $L_2$ is the grid-side inductor, and $r_1$ and $r_2$ are the equivalent series resistance (ESR) of $L_1$ and $L_2$, respectively. The grid impedance is represented by $Z_g$.

FIGURE 1
Topology of a three-phase GFM inverter.

The outer loops are represented by the active power controller (APC) and the reactive power controller (RPC). The former aims to control the active power ($P$), with the active power reference represented by $P_{ref}$ and generates the phase reference at the output, represented by $\theta$. The latter aims to control the reactive power ($Q$), with the reactive power reference represented by $Q_{ref}$ and generates the voltage magnitude reference at the output, represented by $E$. The parameters $\theta$ and $E$ together determine the reference voltage in the $\alpha \beta$ coordinate system, $v_{c\alpha \beta }$, for the capacitor voltage controller $C_v(z)$, which is a PR controller. The transfer function $G_{ff}(z)$ is incorporated into the voltage loop to enhance the dynamic performance of the GFM inverter, and its details will be discussed in Section III-B.

The injected current $i_g$ is regulated by a $P$ controller $C_i(z)$, utilizing capacitor current feedback for active damping and capacitor voltage decoupling. The capacitor current $i_c$ is fed back through the active damping transfer function $G_{ad}(z)$ to suppress the effects of the resonance frequency $f_r$, while $v_c$ is decoupled using the transfer function $G_{dec}(z)$ to enhance the disturbance rejection capabilities of the current controller. The system parameters depicted in Fig. 1 are detailed in Table I. Fig. 2 presents the control block diagram of the closed-loop system for the proposed GFM inverter, excluding the power control loops. The computational delay is one sampling period ($T_s$), and it is modeled as $z^{-1}$. The PWM delay is represented by the zero-order hold (ZOH).

Table I
System Parameters

Parameter	Value
DC-link voltage ($v_{dc}$)	650 $V$
Grid line voltage (RMS) ($v_g$)	380 $V$
Grid frequency ($f_g$)	60 $Hz$
Inductor ($L_1$)	1 $mH$
Inductor ($L_2$)	300 $\mu H$
ESR ($R_1$)	0.1 $\mathrm{\Omega }$
ESR ($R_2$)	0.1 $\mathrm{\Omega }$
Capacitor ($C$)	15 $\mu F$
Sample period ${(T}_s)$	100 $\mu s$
Switching frequency ${(f}_{sw})$	10 $kHz$

III. INNER LOOP CONTROLLERS DESIGN

This section will discuss the design of the current and voltage controllers, as well as the enhancements they will bring to the operation of the GFM inverter.

FIGURE 2
GFM inverter control block diagram.

A. DESIGN OF THE CURRENT CONTROLLER

The injected current is controlled by a lead compensator, whose transfer function is given by (1). $R_a$ is the proportional gain, and $k_L$ [³⁸] is the lead gain designed to reduce the effects of computational delay.

$$C_i(z)=\frac{R_a}{1+k_L{z}^{-1}}$$ (1)

The complete transfer function of the LCL filter, which relates $i_g$ to the terminal voltage $v_{inv}$ of the inverter, is given by (2). However, to simplify the tuning of $R_a$, the filter is approximated by an equivalent $L$ filter, where the total inductance $L_T$ is the sum of $L_1$ and $L_2$, and the total ESR $r_T$ is the sum of $r_1$ and $r_2$ [³⁹], [⁴⁰]. The simplified transfer function is given by (3).

$$\frac{i_g(s)}{v_{inv}(s)}=\frac{1}{\begin{array}{c}\hfill \left(L_1{L}_{2}C\right)s^3+(L_1{r}_2+L_2{r}_1)C{s}^2+\\ \hfill (L_1+L_2+r_1{r}_{2}C)s+r_1+r_2\end{array}}$$ (2)

$$\frac{i_{g_s}(s)}{v_{in{v}_s}(s)}=\frac{1}{L_{T}s+r_T}$$ (3)

To validate this approximation, Fig. 3 shows the frequency response of (2) and (3). It can be noticed that up to a frequency of approximately 2 kHz, the models are equivalent. Therefore, the bandwidth of the current controller will be fixed at 2 kHz.

FIGURE 3
Open-loop frequency response of the LCL filter and the equivalent L filter.

The closed-loop block diagram used to control $i_g$ is shown in Fig. 4. The model of the plant, $i_{g_s}(z)/{v_{inv}}_s(z)$, is given by (4), where $a=e^{-\left(\frac{r_T}{L_T}\right)T_s}$ and $b=(1-a)/r_T$.

FIGURE 4
Closed-loop block diagram used for controlling ${\mathbf{i}}_{\mathbf{g}}$.

$$\frac{i_{g_s}(z)}{{v_{inv}}_s(z)}=(1-z^{-1})𝒵\left[\frac{i_{g_s}(s)}{{{s•v}_{inv}}_s(s)}\right]=\frac{b{z}^{-1}}{1-a{z}^{-1}}$$ (4)

The closed-loop transfer function, as illustrated in Fig. 4, is presented in (5), with its poles required to satisfy the condition outlined in (6) [⁴¹].

$$\frac{i_{g\alpha \beta }(z)}{i_{g\alpha \beta }^{*}(z)}=\frac{R_{a}b}{(z+k_L)(z-a)+R_{a}b}$$ (5)

$$(z-p_1)(z-p_2)=(z+k_L)(z-a)+R_{a}b\to z^2-(p_1+p_2)z+p_1{p}_2=z^2+(k_L-a)z-k_{L}a+R_{a}b$$ (6)

The closed-loop dominant poles are $p_1$ and $p_2$. By equating both sides of the equation, the parameters $k_L$ and $R_a$ can be determined as shown in (7) and (8), respectively.

$$k_L=a-(p_1+p_2)$$ (7)

$$R_a=\frac{(p_1{p}_2+k_{L}a)}{b}$$ (8)

The desired locations of $p_{1,2}$ are specified in (9), while the damped natural frequency, ${\omega }_d$, is defined in (10). The damping factor and the natural frequency are represented by $\xi$ and ${\omega }_n$, respectively.

$$p_{1,2}=e^{-\xi {\omega }_n{T}_s}[\cos \left({\omega }_d{T}_s\right)\pm j\sin \left({\omega }_d{T}_s\right)]$$ (9)

$${\omega }_d={\omega }_n\sqrt{1-{\xi }^2}$$ (10)

The current controller bandwidth was set to 2 kHz, with $\xi =0.9$ and ${\omega }_n=2\pi •2000$ rad/s. The current controller gains are listed in Table II.

Table II
Current and Voltage controller gains

Gain	Value
Proportional gain ($R_a$) – current	4.86
Lead gain ($k_L$)	0.22
Proportional gain ($K_{pv}$) – voltage	0.04
Resonant gain ($K_{rv}$)	40

1. ACTIVE DAMPING DESIGN

As previously mentioned, the GFM inverter is connected to the grid through an LCL-type filter, and therefore, the resonance effect inherent to it must be mitigated. To accomplish this, a first-order lead compensator is used to increase the region of positive resistance. The function $G_{ad}(s)$ is presented in (11) [⁴²].

$$G_{ad}(s)=\frac{1+{\tau }_{L}s}{1+{\alpha \tau }_{L}s}$$ (11)

The parameter ${\tau }_L$ is tuned based on (12), and the main goal of $G_{ad}(s)$ is to provide the greatest phase lead around the filter’s resonance frequency ${\omega }_{res}$. Therefore, the value of ${\omega }_{res}$ is chosen based on the filter parameters presented in Table I. The value assigned to $\alpha$ is 0.1 [⁴³]. The calculated value of ${\tau }_L$ is 1.86 ∙ 10⁻⁴.

$${\tau }_L=\frac{1}{{\omega }_{res}\sqrt{\alpha }}$$ (12)

To obtain the transfer function $G_{ad}(z)$, $G_{ad}(s)$ was discretized using the Tustin method to achieve the discrete-time implementation, as this method provided the best approximation between the continuous and discrete time domains.

2. CAPACITOR VOLTAGE DECOUPLING DESIGN

Before moving forward with the $G_{dec}(z)$ design, the expression for the GFM inverter output impedance $Z_o(z)$ must be derived. This can be achieved through several simplifications of the block diagram presented in Fig. 5. This diagram is essentially the same as the one presented in Fig. 2, but without the voltage loop.

FIGURE 5
Closed-loop block diagram for the current loop.

By applying the ZOH discretization to the model in Fig. 5, the discrete model can be obtained as shown in Fig. 6 where $G_{vc}(z)$, $G_{ic}(z)$, $v_c(s)/v_{inv}(s)$ and $i_c(s)/v_{inv}(s)$ are defined by (13) - (16), respectively.

FIGURE 6
Discrete model for the closed-loop current loop block diagram.

$$G_{vc}(z)=\frac{v_c(z)}{v_{inv}(z)}=𝒵\{ZOH•\frac{v_C(s)}{v_{inv}(s)}\}$$ (13)

$$G_{ic}(z)=\frac{i_c(z)}{v_{inv}(z)}=𝒵\{ZOH•\frac{i_C(s)}{v_{inv}(s)}\}$$ (14)

$$\frac{v_C(s)}{v_{inv}(s)}=\frac{L_{2}s+R_2}{\begin{array}{c}\hfill s^3{L}_1{L}_{2}C+s^{2}C(L_1{R}_2+L_2{R}_1)+\\ \hfill +s(L_1+L_2+C{R}_1{R}_2)+R_1+R_2\end{array}}$$ (15)

$$\frac{i_C(s)}{v_{inv}(s)}=\frac{s^2{L}_{2}C+s{R}_{2}C}{\begin{array}{c}\hfill s^3{L}_1{L}_{2}C+s^{2}C(L_1{R}_2+L_2{R}_1)+\\ \hfill +s(L_1+L_2+C{R}_1{R}_2)+R_1+R_2\end{array}}$$ (16)

The transfer function $i_g(z)/v_{l2}(z)$ is defined by (17), where $i_g(z)/v_{inv}(z)$ is obtained by applying the ZOH discretization in (2), and $v_{l2}(z)/v_{inv}(z)$ is obtained by applying the ZOH discretization in (18).

$$G_{ig}(z)=\frac{i_g(z)}{v_{l2}(z)}=\frac{i_g(z)}{v_{in}(z)}{\left[\frac{v_{l2}(z)}{v_{in}(z)}\right]}^{-1}$$ (17)

$$\frac{v_{l2}(z)}{v_{inv}(z)}\mathcal{=}𝒵\{ZOH[(L_{2}s+R_2)•\frac{i_g(s)}{v_{inv}(s)}]\}$$ (18)

Fig. 7 shows a simplified block diagram obtained after several modifications to the diagram presented in Fig. 6, where $\beta$ is given by (19).

FIGURE 7
Simplified block diagram for the current loop.

$$\beta =\frac{C_i(z)G_{vc}(z)z^{-1}}{1+[G_{ad}(z)G_{ic}(z)-G_{dec}(z)G_{vc}(z)]z^{-1}}$$ (19)

Thus, based on (17) and (19), $Z_o(z)$ is represented by the unnumbered equation shown at the bottom of this page.

The decoupling function $G_{dec}$ was designed to enhance the disturbance rejection capability of the current controller. It incorporates a lead-lag compensation, which consisted of a low-pass Butterworth filter in series with a lead compensator, as shown in (20).

$$G_{dec}(s)=\frac{{\omega }_c}{s+{\omega }_c}•\frac{1+{\tau }_{z}s}{1+{\tau }_{p}s}$$ (20)

The cutoff frequency, ${\omega }_c$, of the filter was set to $2\pi •1500rad/s$, and it was chosen as a compromise between stability and enhanced disturbance rejection capability. The parameters ${\tau }_z$ and ${\tau }_p$ were tuned with the help of MATLAB to compensate for the delay at the fundamental frequency (60 Hz). The values obtained were ${\tau }_z=1.8041\times {10}^{-4}$ and ${\tau }_p=3.4354\times {10}^{-5}$ [⁴²]. Fig. 8 presents the frequency response of $Z_o(z)$ for the inverter, and it shows that the disturbance rejection capability of the inverter is improved when capacitor voltage decoupling is used. To obtain the model in the discrete-time domain, the transfer function $G_{dec}(s)$ was discretized using the Tustin method, which provides the best approximation between the continuous and discrete time domains.

FIGURE 8
Frequency response of ${\mathbf{Z}}_{\mathbf{o}}\mathbf{(}\mathbf{z}\mathbf{)}$ with and without ${\mathbf{G}}_{\mathbf{d}\mathbf{e}\mathbf{c}}\mathbf{(}\mathbf{z}\mathbf{)}$.

B. DESIGN OF THE VOLTAGE CONTROLLER

To regulate the capacitor voltage, a PR controller given by (21) is used. $K_{pv}$, $K_{rv}$ and ${\omega }_o$ are the proportional gain, resonant gain and grid nominal frequency in rad/s, respectively.

$$C_v(z)=K_{pv}+K_{rv}{•T}_s•\frac{1-z^{-1}\cos {(\omega }_o{T}_s)}{1-2z^{-1}\cos {(\omega }_o{T}_s)+z^{-2}}$$ (21)

The voltage controller bandwidth was set to 200 Hz, and the gain $K_{pv}$ was determined using the root locus method [⁴³], as described in (22). Fig. 9 shows the root locus and the corresponding $K_{pv}$ gain for a bandwidth of 200 Hz.

$$\frac{v_c(z)}{i_c(z)}=\frac{v_c(z)}{v_{in}(z)}{\left(\frac{i_c(z)}{v_{in}(z)}\right)}^{-1}$$ (22)

where $v_c(z)/v_{in}(z)$ and $i_c(z)/v_{in}(z)$ are defined by (13) and (14), respectively.

FIGURE 9
Root locus for the voltage control loop.

The gain $K_{rv}$ was selected based on the guideline provided in (23) [⁴¹]. The gain values are shown in Table II.

$$K_{rv}\ge 2K_{pv}{\omega }_o$$ (23)

To enhance the dynamic performance of the voltage controller is used an approach based on the disturbance input decoupling (DID) method, as outlined in [⁴⁴], [⁴⁵]. Fig. 10 shows a block diagram of the DID implementation. Upon analysis, it is evident that the $i_{g\alpha \beta }$ acts as a disturbance to the voltage control loop.

$$Z_o(z)=\frac{1+[G_{ad}(z)G_{ic}(z)-G_{dec}(z)G_{vc}(z)+C_i(z)G_{ig}(z)G_{vc}(z)]z^{-1}}{G_{ig}(z)+[G_{ig}(z)G_{dec}(z)G_{ic}(z)-G_{ig}(z)G_{dec}(z)G_{vc}(z)]z^{-1}}$$

FIGURE 10
Closed-loop block diagram to analyze the effect of ${\mathbf{G}}_{\mathbf{f}\mathbf{f}}\mathbf{(}\mathbf{z}\mathbf{)}$.

The design principle of the transfer function $G_{ff}(z)$ is to eliminate the influence of $i_{g\alpha \beta }$ at the output during sampling instants. This approach is valid only when the dynamics of the disturbance are slow compared to $T_s$. To satisfy this condition, it is assumed that $i_{g\alpha \beta }$ is modeled through a ZOH block. The expression for determining the transfer function $G_{ff}(z)$ is provided in (24). To obtain (24), it is necessary to evaluate the paths which include the disturbance and the DID transfer function.

$$-i_{g\alpha \beta }(z)𝒵\left[ZOH\frac{1}{Cs}\right]+i_{g\alpha \beta }(z)G_{ff}(z)𝒵[ZOH{G}_{CLi}(s)\frac{1}{Cs}]=0$$ (24)

The function $G_{CLi}(s)$ represents the current loop and, to simplify the design of the DID function, it can be approximated by an equivalent first-order system with the same bandwidth as the actual system, where ${\omega }_i=2\pi •2000$ rad/s. Consequently, (24) can be reformulated as (25).

$$G_{ff}(z)=\frac{𝒵\left[ZOH\frac{1}{Cs}\right]}{𝒵\left[ZOH\frac{{\omega }_i}{s+{\omega }_i}\frac{1}{Cs}\right]}=K_{ff}\frac{z-{\delta }_z}{z-{\delta }_p}$$ (25)

The parameters ${\delta }_z$, ${\delta }_p$ and $K_{ff}$ are defined by (26) - (28), respectively. The calculated values are ${\delta }_z=0.2846$, ${\delta }_p=-0.6609$ and $K_{ff}=2.3217$.

$${\delta }_z=e^{-T_s{\omega }_i}$$ (26)

$${\delta }_p=\frac{{\delta }_z(T_s{\omega }_i+1)-1}{T_s{\omega }_i+{\delta }_z-1}$$ (27)

$$K_{ff}=\frac{T_s{\omega }_i}{T_s{\omega }_i+{\delta }_z-1}$$ (28)

To evaluate the impact of utilizing DID, the open-loop gain of the voltage control system shown in Fig. 2 is considered. Fig. 11 illustrates the open-loop gain of the voltage control system both with and without the proposed DID. As can be seen, the loop gain in the low-frequency range is improved when DID is used, which enhances the system’s performance in the presence of disturbances.

FIGURE 11
Frequency response of voltage control open-loop gain with and without ${\mathbf{G}}_{\mathbf{f}\mathbf{f}}\mathbf{(}\mathbf{z}\mathbf{)}$.

The Nyquist plot of the open-loop gain of the complete control system is shown in Fig. 12. The sensitivity peak is approximately 0.5. It can be concluded from this figure that the control system is stable since it does not encircle the critical point (-1, j0).

FIGURE 12
Nyquist plot of the complete control system loop gain.

IV. EXPERIMENTAL RESULTS

The proposed strategy control is verified by experimental results. The lab setup is shown in Fig. 13, and the parameters are listed in Table I. Three conditions were tested, and a grid emulator, the Chroma 61830, was used to emulate the grid. The first evaluates the performance of the current loop. In this case, only the block diagram shown in Fig. 5 is considered, with the primary objective being to analyze the influence of $G_{dec}$. The second condition assesses the performance of the GFM inverter operating exclusively in isolated mode, disconnected from the grid. In this scenario, the voltage reference $v_{c\alpha \beta }$ is directly imposed without any input from the power control loops. The final condition evaluates the GFM inverter operating in grid-connected mode. For this case, the topology shown in Fig. 1 is used; however, the detailed modeling of the power control loops is not addressed, as it lies outside the scope of this work.

FIGURE 13
Lab setup for obtaining experimental results.

A. CURRENT CONTROL LOOP ASSESSMENT

In this condition, it is assumed that only the current loop is active; therefore, a fixed current reference of 10 A peak is provided. The inverter injects current into the grid, and specifically for this case, a PR controller is used for the current control loop. This choice ensures zero steady-state error, allowing for the analysis of current variations around the reference value. The resonant gain is set to 10.

Fig. 14 shows the injected current in one of the phases (lilac curves) for the case without $G_{dec}$ (Fig. 14 (a)) and with $G_{dec}$ (Fig. 14 (b)). The voltage sag represents a 10 % reduction from the nominal line voltage value of 380 V (green curve).

FIGURE 14
Injected current during a voltage sag: (a) without ${\mathbf{G}}_{\mathbf{d}\mathbf{e}\mathbf{c}}$, (b) with ${\mathbf{G}}_{\mathbf{d}\mathbf{e}\mathbf{c}}$. Lilac curves – current; Green curves – voltage.

By analyzing the figures, it is observed that the introduction of the decoupling function $G_{dec}$ led to significant improvements in the dynamic response of the system. Specifically, both the overshoot and the settling time were notably reduced in the case where $G_{dec}$ was applied. Without $G_{dec}$, the overshoot was measured at 1.175 A, but with $G_{dec}$, this overshoot decreased to 312.5 mA. This significant reduction in overshoot indicates that the inverter was able to handle the voltage sag more effectively and avoid excessive current fluctuations.

Similarly, the settling time was also improved. Without $G_{dec}$, the settling time was recorded at 114.8 ms. With the decoupling function applied, the settling time was reduced to 46.2 ms, indicating a much quicker recovery after the voltage dip. This shorter settling time means that the system is able to react faster to voltage disturbances, improving the overall dynamic performance.

B. GFM INVERTER ASSESSMENT – ISOLATED OPERATION

In this test, the GFM inverter supplies a three-phase linear load of 17 Ω. At a certain point, another identical load is added to the system. Fig. 15 (a) and Fig. 15 (b) show the α-axis voltage reference (blue curve), with a nominal peak value of 120 V, the voltage across the capacitor in the α-axis (green curve), and the corresponding error signal (yellow curve), without and with DID, respectively.

FIGURE 15
Capacitor voltage during a load variation: (a) without DID, (b) with DID. Blue curves – ${\mathbf{v}}_{\mathbf{c}\mathbf{\alpha }}^{\mathbf{*}}$; Green curves – ${\mathbf{v}}_{\mathbf{c}\mathbf{\alpha }}$: Yellow curves – error.

By examining these figures, it is clear that the introduction of DID leads to a noticeable improvement in the GFM’s inverter performance when responding to load changes. For the case without DID, the voltage variation observed was 53.6 V. In contrast, when DID is applied, the voltage variation decreases to 32.5 V, indicating that the GFM inverter is much better at regulating the output voltage and compensating for the additional load.

Additionally, the settling time shows a stark difference between the two scenarios. Without DID, the settling time was approximately 35 ms. With DID, however, the settling time was reduced to approximately 10 ms, reflecting a much faster recovery to the nominal operating conditions. This reduction in settling time demonstrates the ability of DID to enhance the GFM’s inverter dynamic response, ensuring that the system can react more swiftly to changes in load and return to a stable state more efficiently.

C. GFM INVERTER ASSESSMENT – GRIDCONNECTED OPERATION

In this test, the operation in grid-connected mode is assessed. Initially, the GFM inverter supplies 5 kW of active power and 0 kVAr of reactive power to the grid. Subsequently, the active power reference for the GFM inverter is set to 10 kW. Fig. 16 (a) and Fig. 16 (b) show the injected active power (green curve), reactive power (blue curve), and current (lilac curve), both without and with DID, respectively. The SCR of the system is 3.85, which characterizes it as a strong grid, according to [⁴⁶].

FIGURE 16
Injected active power, reactive power and current: (a) without DID, (b) with DID. Green curves – P; Blue curves– Q: Lilac curves – current.

From the figures, it can be observed that when DID is applied, all three parameters – active power, reactive power, and current – exhibit less oscillation at the moment the active power reference changes. Therefore, using DID improves the dynamic behavior of the GFM inverter by providing more damping, which helps control fluctuations and ensures the system stabilizes more rapidly.

V. CONCLUSION

This paper introduced a control strategy for a GFM inverter that incorporates a modified current control loop designed to enhance disturbance rejection capability. Additionally, a modified voltage control loop is proposed, utilizing the concept of DID applied to the injected current within the voltage control loop. The proposed approach demonstrates significant improvements in the GFM’s inverter dynamic performance, particularly in scenarios involving voltage sags and load variations. Additionally, the oscillation of active power, reactive power, and current, were reduced when DID was used in response to variations in the active power reference.

Acknowledgment

The authors would like to thank the Brazil’s Higher Education Personnel Improvement Coordination (CAPES), Federal University of Maranhão (UFMA) and Technical University of Denmark (DTU) for the support to the development of this work. This work was supported by the Instituto de Ciência e Tecnologia Grupo Equatorial through the PDI ANEEL program under grant PD-06072-0702/2023.

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Edited by

• Editor-in-Chief
  Heverton A. Pereira https://orcid.org/0000-0003-0710-7815

• Associate Editor
  Filipe P. Scalcon https://orcid.org/0000-0002-8976-2188

Publication Dates

• Publication in this collection
  11 Aug 2025
• Date of issue
  2025

History

• Received
  13 Jan 2025
• Accepted
  07 Mar 2025
• Published
  24 Mar 2025