Abstract
The gradual penetration of grid-forming (GFM) converters into new power systems with renewable energy sources may result in the emergence of small-signal instability issues. These issues can be elucidated using sequence impedance models, which offer a more tangible and meaningful interpretation than dq-domain impedance models and state-space models. However, existing research has primarily focused on the impact of power loops and inner control loops in GFM converters, which has not yet elucidated the precise physical interpretation of inner voltage and current loops of GFM converters in circuits. This paper derives series-parallel sequence impedance models of multi-loop GFM converters, demonstrating that the voltage loop can be regarded as a parallel impedance and the current loop as a series impedance. Consequently, the corresponding small-signal stability characteristics can be identified through Bode diagrams of sequence impedances or by examining the physical meanings of impedances in series and in parallel. The results indicate that the GFM converter with a single power loop is a candidate suitable for application in new power systems, given its reduced number of control parameters and enhanced low-frequency performance, particularly in weak grids. The results of PLECS simulations and corresponding prototype experiments verify the accuracy of the analytical analysis under diverse grid conditions.
THE increasing penetration of renewable energy sources into power grids and the high demand for switching between grid-connected and off-grid states have led to a widespread use of power electronic converters. However, this widespread use may result in harmonic oscillation and other interaction stability problems [
In practical application, it should be noted that GFM converters are produced by different companies and therefore have relatively different multiple control loops. The power outer-loops of GFM converters, which include droop control, power synchronization control (PSC), virtual synchronous generator (VSG) control, and matching control (MC), exhibit similar impedance behaviors [
The stability mechanism of multi-loop GFM converters, including small-signal stability under diverse grid conditions, can be analyzed by state-space modeling or impedance modeling [
The sequence impedance model, particularly the positive-sequence impedance model, tends to be a method more suitable for analyzing the small-signal stability of GFM converters. This model divides the grid-connected converter system into two impedances: one representing the grid and the other representing the converter [
Previous sequence impedance models [
A unified sequence impedance model is proposed to analyze the influence of different control parameters of GFM converters [
This paper proposes a series-parallel sequence impedance model of multi-loop GFM converters to separate different control loops in GFM converters like puzzles to elucidate their physical meanings in circuits. Furthermore, this model can also be employed for rapid modeling when modifying control loops, enabling swift verification of model accuracy. Additionally, it can be utilized to elucidate the distinctive positive- and negative-sequence impedance characteristics and associated small-signal instability issues of the voltage and current inner-loops and the power outer-loop. This allows for the selection of an optimal control strategy.
Accordingly, this paper is structured as follows. Section II illustrates the process of sequence impedance modeling for GFM converters. Section III proposes the model to decompose sequence impedances into series parts and parallel parts to reveal their physical meanings in circuits. Section IV analyzes the impedance characteristics of different control loops and control parameters in GFM converters, and recommends GFM converters with the single power loop. Section V provides applications of the proposed model in addressing small-signal stability issues. Section VI employs simulations and prototype experiments to validate the findings. Section VII offers a summary of the principal contributions.
As shown in

Fig. 1 Circuit model of GFM converter.
The fundamental voltage and current signals in phase A with harmonics at a certain frequency caused by disturbance or injection, i.e., and , respectively, can be represented as two-pole frequency signals and can be expressed in the time domain as:
(1) |
(2) |
where , , and are the amplitudes of the fundamental, positive-sequence, and negative-sequence voltages, respectively; , , and are the amplitudes of the fundamental, positive-sequence, and negative-sequence currents, respectively; is the frequency of the fundamental voltage; and are the frequencies of the positive- and negative-sequence harmonic voltages, respectively, and ; and are the phase angle differences between the fundamental voltage and the positive- and negative-sequence harmonic voltages, respectively; and are the phase angle differences between the fundamental voltage and the positive- and negative-sequence harmonic currents, respectively; and is the phase angle difference between the fundamental voltage and current.
In a three-phase symmetrical system, the positive- and negative-sequence components can be decoupled [

Fig. 2 Diagram of active power signals in frequency domain. (a) Positive-sequence domain. (b) Negative-sequence domain.
The voltage and current signals in the frequency domain can be expressed as (3) and (4), respectively.
(3) |
(4) |
where f is the power frequency; and the symbol corresponds to positive and negative frequencies.
In general, GFM converters are controlled by instantaneous power signals in the time domain, which can be obtained in the frequency domain by the frequency-domain convolution theorem. Therefore, the active power P can be obtained through the inverse Fourier transform of the product of (3) and (4) after the Fourier transform as:
(5) |
In a three-phase symmetrical system, the phase power can be considered in relation to the power frequency. When the power frequency is or (ignoring the negative frequency), the phase power is in the positive sequence, and the sum of three-phase power is 0; when the power frequency is 0 or , the phase power is in the zero sequence, and the sum of three-phase power is three times the power per phase, as given in (6). This component of active power is transferred to the DC side in accordance with the law of energy conservation.
(6) |
where is the transferred active power of three-phase symmetrical system.
Similarly, the frequency caused by a negative-sequence disturbance can be transferred to the DC side and should be considered.
The control structure of GFM converter with single power loop is shown in

Fig. 3 Control structure of GFM converter with single power loop.
The amplitude and the phase angle of the input signal for the sinusoidal pulse width modulation (SPWM), i.e., and , respectively, are controlled by the actual and target values of the instantaneous active and reactive power on the output side. The three-phase modulated signals , , and are expressed as:
(7) |
In accordance with Kirchhoff’s current law (KCL) and Kirchhoff’s voltage law (KVL), the relationship between the terminal current and voltage of phase A in the frequency domain can be derived as:
(8) |
(9) |
where is the filtering inductance of GFM converter; is the filtering capacitance of GFM converter; is the equivalent resistance of ; and .
In accordance with (9), the sequence impedance can be determined by examining the influence of harmonics on and through the implementation of control strategies.
can be expressed as (10) according to the P-f loop shown in
(10) |
where and are the virtual inertia and damping set by GFM converter, respectively; is the setting value for the output active power of GFM converter; and is the rated angular velocity.
The only variable influenced by small-signal disturbances in (10) is , which requires further research. According to (6), can be separated into two distinct components as , where is the constant fundamental active power, and is the harmonic active power. Similarly, can be divided as , where is the original value of phase angle, and is the phase angle deviation caused by the harmonic active power, which can be calculated as:
(11) |
In the frequency domain, can be expressed as:
(12) |
Substituting into , we have:
(13) |
where , and is the power angle of GFM converter.
Due to the power angle characteristic of generator, we have:
(14) |
(15) |
Since (13) represents in the time domain, according to (12) and the frequency-domain convolution theorem, in the frequency domain can be derived as:
(16) |
Substituting (16) into (9) yields the positive-sequence impedance:
(17) |
(18) |
(19) |
It is important to note that if the GFM converter is an ideal voltage source, should be equal to , which is analogous to (17) particularly when and are relatively small.
As shown in

Fig. 4 Control structure of GFM converter with distinct inner-loops. (a) Single voltage loop. (b) Single current loop. (c) Voltage and current dual-loop.
Given that the topology of GFM converter remains unchanged, the relationship between the terminal current and voltage of phase A is analogous to (9), which can be expressed as:
(20) |
The modulated signal differs from (7) because multi-loop GFM converters employ a different SPWM. The inverse Park’s transformation of (7) can be expressed as:
(21) |
where and are the d- and q-axis voltages of modulated signals, respectively; and can be neglected in the three-phase three-wire system. can be derived as:
(22) |
Substituting into (22), can be expressed as:
(23) |
(24) |
(25) |
where and are the original values for the d- and q-axis voltages of modulated signals, respectively.
and are determined by the inner-loops of GFM converters. As illustrated in
(26) |
(27) |
(28) |
where is the reference voltage of modulated signals; is the reference value of the loop impedance; and are the measured values for the d- and q-axis voltages of modulated signals, respectively; and are the measured values for the d- and q-axis currents of modulated signals, respectively; and and are the proportional-integral (PI) transfer functions of voltage and current loops, respectively, which can be expressed as:
(29) |
(30) |
where and are the proportional and integral parameters of voltage loop, respectively; and and are the proportional and integral parameters of current loop, respectively.
Without small-signal interferences, the actual values for the d- and q-axis voltages and the currents of modulated signals can be expressed as:
(31) |
(32) |
(33) |
(34) |
Considering the impact of small-signal interference on the Park’s transformation through θ, the measured values of d- and q-axis voltages of modulated signals can be derived as:
(35) |
(36) |
Therefore, the measured values for the d- and q-axis voltages and currents of modulated signals considering the small-signal interference can be expressed as:
(37) |
(38) |
Substituting (31) to (35) and (12) into (37) and (38), we have:
(39) |
(40) |
(41) |
(42) |
Substituting (39)-(42) into (26)-(28), and according to (20) and (22), the positive-sequence impedances of GFM converter with single voltage loop, single current loop, and voltage and current dual-loop, i.e., , , and , respectively, can be derived as (43)-(45).
(43) |
(44) |
(45) |
where is the modulation ratio of GFM converter; is the PI transfer function of voltage and current dual-loop; and is the DC-side voltage.
The all-in-one impedance model, as shown in (43)-(45), appears to be overly intricate for the analysis of small-signal characteristics. Accordingly, this paper proposes a series-parallel sequence impedance model of multi-loop GFM converters to reveal the interrelationship between characteristics from a physical standpoint. An examination of (43)-(45) reveals that the numerators and denominators exhibit analogous components. The numerators can be partitioned into two parts based on the plus sign, whereas the denominators cannot. The denominators can be transformed into the numerators through the reciprocal change, and then can be split into two parts. Therefore, (46) and (47) can be derived from (43) and (45), respectively.
(46) |
(47) |
To simplify the form, (48)-(50) can be derived from (44), (46), and (47), respectively.
(48) |
(49) |
(50) |
where
(51) |
(52) |
(53) |
(54) |
(55) |
(56) |
(57) |
(58) |
where ; and .
In a physical sense, given that (51)-(58) all represent impedance, the splitting terms of the numerators can be regarded as impedances in series, and the splitting terms of the denominators can be regarded as admittance in series, i.e., impedances in parallel.
As shown in (48)-(50), the positive-sequence impedance of the GFM converter with a single voltage loop can be conceptualized as two impedances in parallel. Similarly, the positive-sequence impedances of GFM converters with a single current loop can be conceptualized as two impedances in series. Furthermore, the positive-sequence impedances of GFM converters with voltage and current dual-loop can be seen as two impedances in parallel, where each impedance can be viewed as two impedances in series.
Additionally, (51), (53), and (55) exhibit analogous forms to (17), i.e., the impedance of GFM converter with the single power loop. They share: ① a common filter component and power loop component, i.e., and , respectively; ② the constant component, i.e., for the single power loop, for the single voltage loop, for the voltage and current dual-loop, and for the single current loop; and ③ the PI controller component, i.e., for the single voltage loop, for the single current loop, and for the voltage and current dual-loop. Similarly, a comparison of (52) and (57), or (54) and (56), reveals that they exhibit similar forms, which can be classified as series impedances and parallel impedances, respectively. Therefore, it can be concluded that, in GFM converters, the voltage loop acts as a parallel impedance, while the current loop acts as a series impedance.
Additionally, two more points of interest regarding the positive-sequence impedances of GFM converters warrant consideration. One noteworthy point is that the positive-sequence impedances maintain coupling terms. The coupling term in the current loop results in the appearance of additional coupling terms such as in both the numerator and the denominator due to the frequency-domain convolution calculation. This leads to the positive-sequence impedance being nearly infinite around the fundamental frequency. Another point is that appears in each numerator of the series and parallel components, indicating that the voltage and current loops exhibit similar capacitive resistance characteristics. However, these intriguing observations do not fully capture the true nature of GFM converters. To gain a more comprehensive understanding and to further apply the proposed model in small-signal analysis, it is essential to analyze the proposed model through corresponding schematic diagrams and Bode diagrams.
Given that each circuit element is composed of resistance, capacitance, and inductance, which can be represented by no more than three dimensions, the proposed model of GFM converter with voltage and current dual-loop, which comprises four elements, can be considered to have less than 12 dimensions. This is a considerably more compact representation than those presented in [
The system parameters of GFM converter with different control loops are shown in
Parameter | Value | Parameter | Value |
---|---|---|---|
(rad·s) | 100π | (N·s/m) | 20 |
(kW) | 10 |
(kg· | 0.05 |
(kVA) | 0 | 0.2 | |
(V) | 311.13 | 20 | |
(V) | 690 | 1 | |
(A) | 10.71 | 5 | |
(mH) | 3.2 | (Ω) | 1.5 |
(μF) | 20 | (Ω) | 0.1 |
(mH) | 3.2 |

Fig. 5 Bode diagram for positive-sequence impedance of GFM converter with different control loops. (a) Single power loop. (b) Single voltage loop. (c) Single current loop. (d) Voltage and current dual-loop.

Fig. 6 RT-Box experiment platform.
By comparing the distinction between the amplitude and phase of impedances at low and high frequencies (where the dividing line is twice the fundamental frequency [
From the standpoint of quantitative analysis, the stability margins with different control loops are shown in Supplementary Material A Table SAI and Fig. SA1.
In general, the grid impedance connected with the GFM converter exhibits inductive characteristics, which indicates that the sum of capacitive output impedances of GFM converter and the inductive grid impedance may be close to zero, thereby causing instability. Therefore, the current loop is of significance as a series component in modifying low-frequency impedance characteristics.
The above low-frequency positive-sequence impedance characteristics can also be derived from the proposed model. As previously stated, the proposed model of GFM converter includes a low-frequency inductive impedance due to the inductive component in (17), (51), (53), and (55). In the proposed model of GFM converter with either a single voltage loop or a single current loop, the components expressed as (52) and (54) are included, which have a similar capacitive impedance and are used to reduce the inductance component and increase the capacitive component. This is achieved by the capacitive component in the numerator and the inductive component or the resistor-inductance component in the denominator. Nevertheless, (52) behaves as a parallel impedance while (54) behaves as a series one. In general, the parallel impedance caused by the voltage loop will reduce the amplitude, whereas the series impedance caused by the current loop will increase the amplitude. Since the inductive impedance resulting from the power loop has a relatively small amplitude at low frequencies, the decreasing effect of the voltage loop on the impedance amplitude of the power loop is considerably less than the increasing effect of the current loop. Therefore, the voltage loop appears to be inconsequential as a series component in altering low-frequency impedance characteristics.
At high frequencies, however, the GFM converters with different control loops exhibit similar capacitive impedance characteristics with relatively high amplitude, as shown in
(59) |
(60) |

Fig. 7 Bode diagram of positive-sequence impedances of GFM converter and grid with 180° phase shift.
where Psc is the short-circuit capacity of grid; is the effective value of ; is the equivalent impedance of grid; and PN is the rated active transmission power of GFM converter.
This indicates that the GFM converter with distinct voltage or current loops exhibits similar small-signal instability issues at high frequencies. Therefore, the impact of different control loops should be concentrated more at low frequencies. Previous results derived from the proposed model and corresponding Bode diagrams indicate that the single power loop is the preferred option due to fewer parameters involved, which makes control easier and provides prior small-signal stability at low frequency. Furthermore, the single power loop can control the inner electric potential by its amplitude E and phase instead of ud and uq, which can provide more stable voltage support during instability.
In general, the negative-sequence impedances exhibit fewer characteristics [
The influences of control parameters J, DP, kvp, and kvi on positive-sequence impedance of GFM converters are shown in

Fig. 8 Influences of control parameters on positive-sequence impedance of GFM converters. (a) . (b) . (c) . (d) .
A sensitivity analysis can offer valuable insights into the robustness of the proposed model under different parameters. Therefore, the parameter-based sensitivity analysis method introduced in [
(61) |
where is the sensitivity of each parameter; is the phase margin of GFM converter; represents various control parameters such as ; is the original value of each control parameter; and is the parameter perturbation.
The parameter-based sensitivity analysis results can be obtained in Table II. It can be observed that the small-signal instability is predominantly associated with control parameters J and DP. However, a comparison between
Control parameter | Spara |
---|---|
-5.7 | |
0.3 | |
0.1 | |
-0.1 |
As shown in Table III, the phase and amplitude characteristics of different control loops and parameters in the proposed model are summarized.
Impedance model | Control loop and parameter | Characteristic | |
---|---|---|---|
Low-frequency | High-frequency | ||
Positive-sequence | Power loop | Inductive, small-amplitude | *Capacitive, *high-amplitude |
Voltage loop (parallel) | Capacitive, high-amplitude | Capacitive, high-amplitude | |
Current loop (series) | *Capacitive, *high-amplitude | Capacitive, small-amplitude | |
J (series) | Capacitive, small-amplitude | Inductive, small-amplitude | |
DP (series) | Inductive, small-amplitude | Capacitive, small-amplitude | |
kvp (parallel) | Inductive, high-amplitude | Capacitive, high-amplitude | |
kvi (parallel) | Capacitive, high-amplitude | Inductive, high-amplitude | |
Negative-sequence | Power loop | Inductive, small-amplitude | *Capacitive, *high-amplitude |
Voltage loop (parallel) | Capacitive, high-amplitude | Capacitive, high-amplitude | |
Current loop (series) | Capacitive, small-amplitude | Capacitive, small-amplitude |
Note: * means the corresponding loop plays a leading role in the proposed model.
The impedance analysis in Section III indicates that the proposed model can be effectively employed to analyze and modify the small-signal stability of GFM converters with high speed and efficacy. However, the aforementioned improvement is limited in scope. It is more beneficial to equip people with the knowledge to conduct further research than to conduct the results for them. Therefore, it is necessary to restate how to use the proposed model to improve the small-signal stability characteristics.
As shown in

Fig. 9 Schematic diagram of positive-sequence impedances of GFM converters with different control loops. (a) Single power loop. (b) Single voltage loop. (c) Single current loop. (d) Voltage and current dual-loop.
Accordingly, the proposed model aims to streamline the analysis of the impedance characteristics of the multi-loop GFM converter under small-signal stability conditions. By using the proposed model, the characteristics of different inner-loop components can be investigated and enhanced separately, akin to modifying the characteristics of a series or parallel resistance in a circuit to alter the impedance characteristics of the entire circuit. Furthermore, the blue and red arrows in
Given the impact of interaction between the voltage and current loops on the characteristics of sequence impedance is minimal [
As shown in

Fig. 10 Low-frequency small-signal instability with different values of SCR. (a) Single voltage loop. (b) Single current loop.
As shown in

Fig. 11 Output active power of GFM converter with a single power loop with different values of SCR.
As shown in

Fig. 12 Experimental platform.
The potential sources of error in the experiments mainly come from: ① sampling error in the measurement of the voltage and current; ② misalignment of the reference frame in the process of measuring the sequence impedance; and ③ a discrepancy between the fundamental frequency of the actual system and the ideal frequency of 50 Hz in the FFT analysis [
Nevertheless, under the control of the rapid prototype controller YXSPACE-SP2000 and the corresponding converter device applied in this paper, the synchronized measurements of the three-phase voltages and currents reach a sampling frequency of 20 kHz. Consequently, given that the theoretical and experimental results are in close agreement, it can be concluded that the prototype closely matches the real-world implementation.
A 50 Hz fundamental voltage with an intermittent voltage of varying frequencies is injected by the harmonic injection power supply PRE1515M to conduct a sequence impedance frequency sweep experiment. Subsequently, the FFT is employed to analyze the corresponding voltage and current data, as shown in

Fig. 13 Voltage and current data for frequency sweep experiment.
As shown in

Fig. 14 Low-frequency small-signal instability of GFM converters. (a) Single current loop. (b) Voltage and current dual-loop.
As shown in

Fig. 15 High-frequency small-signal instability of GFM converters. (a) Single voltage loop. (b) Single current loop.
The preceding experiments further validate that the proposed model of multi-loop GFM converters is an effective means of analyzing their small-signal stability.
This paper presents a series-parallel impedance model of multi-loop GFM converters. The analysis of the small-signal stability of GFM converters through this model allows the derivation of the following results.
1) The sequence impedances of multi-loop GFM converters can be decomposed into impedances in series and in parallel with different control loops. The voltage loop can be conceptualized as impedances in parallel, whereas the current loop can be conceptualized as impedances in series.
2) GFM converters with a single power loop exhibit inductive impedance characteristics at low frequencies. Conversely, the GFM converters with the voltage loop and current loop exhibit capacitive impedance characteristics at low frequencies. Given the relatively small amplitude of the single power loop, the current loop exerts a more considerable influence as parallel impedance than the voltage loop in modifying low-frequency sequence impedances from inductive to capacitive. This may potentially give rise to small-signal instability and low-frequency oscillations, particularly in weak grids.
3) The GFM converters with voltage loop and current loop display capacitive impedance characteristics at high frequencies. The relatively high amplitude of the voltage loop and the relatively low amplitude of the current loop have a negligible impact on the power loop as either parallel or series impedance. Therefore, multi-loop GFM converters display similar high-frequency capacitive impedance characteristics, which may result in high-frequency small-signal instability in both weak and strong grids. Therefore, GFM converters with a single power loop are recommended from the perspective of small-signal stability, since they offer superior low-frequency characteristics in weak grids and simpler parameters.
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