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Impedance Decomposition for Design-Oriented Analysis of Grid-Forming Voltage-Source Converters

Grid-forming voltage-source converters (VSCs) tend to suffer from sideband oscillations in stiff grids. Conventional state-space eigenvalue analysis or impedance-based stability analysis based on generalized Nyquist stability criterion can only analyze how specific control parameters affect the overall system stability, which ends up with less insight into a controller designoriented analysis. To tackle this challenge, this article starts with the impedance modeling and stability analysis of grid-forming VSCs in stiff grids. Then, an impedance decomposition approach to characterizing the impacts of multiple control loops is proposed, which benefits in a controller design-oriented analysis for stability enhancement. Finally, simulation and experimental results are provided to validate the approach.


I. INTRODUCTION
Voltage-source converters (VSCs) become predominant in modern power systems, as the penetration of renewable energy sources increases [1]. Those VSCs are, in general, operating in the grid-following mode [2], where the VSCs feed the power to the grid via the vector current control which is synchronized with the voltage at the point of connection (PoC). However, such control schemes can lead to sideband oscillations [3] when VSCs are operating in weak grids with low short-circuit capability [3]- [5]. This phenomenon can also be termed as sub-synchronous oscillations (SSOs) if the two oscillation frequencies are within twice of the fundamental frequency [6]. The instability of grid-following VSCs in weak grids have been extensively studied [4], [7]- [10]. It is found that the phase-locked loop (PLL) used for the grid synchronization adds a negative resistance within its control bandwidth, which tends to destabilize the system in weak grids [9].
To enhance the stability of VSCs in weak grids, the gridforming control techniques are emerging in recent years as competitive alternatives [11]- [14]. Differing from the gridfollowing control, the grid-forming control regulates the VSC output voltage and achieves synchronization with the grid based on the injected power [15], such that the PLL can be removed. The recent works have proved that the grid-forming converters achieve better stability performance than the gridfollowing converters in weak grids [16]- [19]. However, the grid-forming converters may also suffer from instability with stiff grids [20]. To address the stability issue, small-signal modeling and analysis approaches have been performed on grid-forming VSCs. The existing methods for stability analysis can be classified into two categories: i) the state-space eigenvalue analysis and ii) the impedance-based analysis.
The state-space eigenvalue analysis, also named as the modal analysis, analyzes the eigenvalues of system state matrix based on the time-domain state-space model. Then, the participation factor [21] and parameter sensitivity analysis can be utilized to evaluate how state variables and control parameters affect the overall system stability [22]. It has been shown that the stability of grid-forming converters is impacted by many factors, including the droop control coefficient [23], the voltage control [21], [24], and the operating point of the system [25]. Although accurate, the state-space model characterizes the overall system dynamics, thus the analysis result only applies for a specific grid condition. The modeling and analysis need to be redone if the grid condition is changed. Therefore, the adverse effects of VSC control loops on the VSC-grid interaction explained from the VSC terminal dynamics are still unknown.
To tackle this issue, the impedance-based analysis has been adopted in the analysis of the VSC-grid interaction. In this approach, the whole system is partitioned at the PoC of VSC, where the VSC impedance and grid impedance are modeled separately to characterize the voltage-current relationships seen from each side [26]. Then, the system stability can be analyzed in two ways: a) the passivity-based analysis of the VSC impedance [27], b) the Nyquist stability criterion (NSC) applied to the minor loop gain formed by the VSC impedance and grid impedance.
The passivity-based analysis provides a sufficient condition for system stability, i.e., the VSC-grid interaction is stable if each subsystem is stable and passive [27]. The non-passive (or non-dissipative) behavior induced by VSC controls can destabilize VSC-grid interactions. The passivity-based analysis reported in [28] has identified the voltage feedforward loop in grid-forming VSCs can add negative damping in the sub-synchronous frequency range, which tends to destabilizes the system with stiff grids. However, this work overlooks the dynamic impacts of the outer-loop power control. Thus, the analysis is inadequate to reveal the instability mechanism of grid-forming VSCs.
The NSC-based analysis can accurately predict the system stability based on the VSC and grid impedances. Some works have applied this approach to address the stability of gridforming VSCs by using the dq-frame [29]- [31] or αβ-frame [32] impedance model. Such a model characterizes the full control dynamics into the VSC output impedance, yet its multi-input multi-output (MIMO) nature requires applying the generalized NSC for the stability analysis [33] or a simplified single-input single-output (SISO) impedance-based stability analysis considering the grid interaction [32]. The former one applies the analysis by calculating the eigenvalue transfer functions of the minor-loop-gain matrix, and the latter one relies on deriving an equivalent SISO model, both of which lose the physical meaning of the VSC impedance itself.
Consequently, they cannot explicitly reveal how VSC control loops interact with each other, and why their interactions affect the stability. This drawback makes it difficult to develop a holistic design approach for the multi-loop control system of VSCs.
To address the challenges with the aforementioned analysis methods, this paper explores further on the stability of gridforming VSCs in stiff grids based on the impedance modeling. The presented work is discerned from the prior art in two major aspects: a) The instability mechanism of grid-forming VSCs in stiff grids is elucidated based on the impedance singular value and passivity analysis. b) An impedance decomposition approach is proposed to characterize the VSC control-loop impacts, which contributes to a design-oriented analysis for the gridforming VSCs. The rest of the paper is organized as follows. Section II derives the αβ-frame impedance model of the droop-controlled grid-forming converter. Section III elucidates the instability mechanism of grid-forming VSCs in stiff grids based on the impedance singular value and passivity analysis. To address the stability issue, Section IV proposes an impedance decomposition method to represent the dynamic effects of multiple control loops, based on which a design-oriented analysis is conducted. Simulation and experimental validations are provided in Section V. Section VI finally concludes the paper.

II. MODELING OF GRID-FORMING VSCS
A three-phase grid-forming VSC is shown in Fig. 1 with a single-line representation, which is connected to a grid through the grid impedance L g . The droop control with lowpass filters (LPFs) [34], i.e., one of the typical grid-forming control schemes, is utilized for the power synchronization, and the capacitor-voltage and inductor-current dual-loop control [35] is employed for the voltage regulation. The impedance model of the VSC is derived in the αβ frame, and the derivation starts from the inner voltage loop to the outer power loop. All the circuit and control parameters are listed in Tables  I-II.

A. Voltage Control Modeling
First, considering the inner-loop voltage control only, the three-phase VSC system can be characterized with a SISO model in the αβ frame, since the plant and control dynamics in the α axis and the β axis are identical with no cross-couplings [36]. Fig. 2 shows the derived small-signal model of the dualloop voltage-controlled VSC [37]. Z ol , G uv , G ii , and G ui are the open-loop transfer functions of the plant, which are

Dual-loop voltage control
Power control where Z L1 and Y Cf denote the impedance and admittance of the filter inductor and filter capacitor, respectively. The current control adopts a proportional (P) controller and the voltage control adopts a proportional + resonant (PR) controller [38]. They are implemented in the αβ frame, whose transfer functions are denoted by Gi and Gv, respectively. Gd represents the impact of the time delay [39]. The transfer functions of the controllers and the time delay are given as

B. Power Control Modeling
To model the control dynamic of the power loop, the αβframe complex vectors are defined with v=v α +jv β and v * =v α −jv β [10], [40]. Then, the instantaneous active power and reactive power can be calculated with the complex vectors as     The LPFs and the droop controllers are applied to P and Q to obtain θ and V 0 , respectively. The detailed control diagrams are given in Fig. 3, based on which the transfer functions for the P and Q loops can be represented by Then, the αβ-frame voltage reference is generated as where GP(p) and GQ(p) denote the time-domain operators of the transfer functions GP(s) and GQ(s) with p=d/dt. θ1 denotes the time-variant phase of the VSC output voltage at the fundamental frequency. It is seen from (10) and (11) that P and Q are calculated by the nonlinear equations that consist of the multiplications of voltages and currents, and thus the linearization should be also done when substituting (10) and (11) where Vdq and I2dq are defined as the steady-state values of the VSC output voltage and current in the synchronous reference frame of θ 1 . It is noted that the dynamic of v ref is related not only to v and i 2 , but also to 1 2 j e   v and 1 2 2 j e   i . To derive a unified form for the ac voltage and current variables, the complex conjugate and the angle rotation of 1 2 j e  are applied successively to (16), which yields Combining (16) and (17), the dynamic relationship from the VSC output voltage and current to the voltage reference can be derived as (18), which is given at the bottom of this page. Then the two complex transfer matrices, i.e. Z ref (s) and G ref (s), can be defined accordingly with p replaced by s.
It is seen from (18) that, in the αβ frame, all the ac variables should be denoted by two complex vectors. Consequently, the dynamic of the power synchronization loop is characterized by a MIMO model. The existence of off-diagonal elements in Zref and Gref implies the frequency-coupling effects introduced by the power-loop control.

C. VSC Impedance Model
To integrate the inner voltage control loop into the power control loop, the SISO models of (8) and (9) are extended into the MIMO forms in the complex space, which leads to Since the inner-loop voltage control is linear time-invariant, there are no frequency coupling terms in the off-diagonal elements in (19) and (20). Then, the full model of the VSC can be depicted by Fig. 4, where Z o and G vv represent the dynamics of the inner-loop voltage control, Zref and Gref represent the dynamics of the outer-loop power control. The output impedance of the VSC is then derived as ZVSC is a MIMO model represented in the αβ frame. The four elements of (21) are defined as Z VSC11 , Z VSC12 , Z VSC21 , and Z VSC22 .

D. Impedance Model Validation
One of the advantages of the impedance-based analysis over the state-space analysis is that the VSC can be treated as a "black-box" system, and its impedance can be obtained from frequency scanning. Perturbing the VSC with frequencyscanned small signals around its operating point/ trajectory, the frequency response of the linearized small-signal impedance model of the VSC can be calculated, by applying the discrete Fourier transformation (DFT) analysis of the measured voltage and current waveforms at the PoC [41].
Using the impedance measurement method proposed in [40], the derived αβ-frame impedance model is validated via switching models in MATLAB/Simulink. Since the frequency coupling behavior is introduced by the power control dynamics, which is strongest around the fundamental frequency, the frequency is scanned within (0 Hz, 200 Hz). The measured impedance is plotted by asterisks shown in Fig.  5, which shows in good agreement with the analytical model. It verifies the accuracy of the VSC modeling.

III. INSTABILITY MECHANISM IN STIFF GRIDS
This section formulates the problem by discussing the instability mechanism of the grid-forming VSC in stiff grids.

A. Combined Passivity and Small-Gain Stability Criterion
The combined small-gain and passivity stability criterion [42] is introduced first: Criterion 1: For an interconnected system formed by the return ratio of L(s)=Q(s)P(s), if P(s) is strictly passive, and Q(s) is stable and passive in specific frequency bands, the closed-loop system is stable if the non-passive frequency bands of Q(s), where σ means the maximum singular value. It is noted that Criterion 1 is merely a sufficient condition for stability, which may not be readily met in practical stable systems. However, it is inspired that the stability analysis with the generalized NSC can be simply focused in the non-passive frequency bands of Q(s), which is carried out in Part B. Furthermore, the small-gain condition based on the singular value analysis can be used to clarify the instability mechanism of grid-forming VSCs in stiff grids, which is conducted in Part C.

B. Stability Analysis based on Generalized NSC and Passivity
For the VSC-grid interconnected system, the system return ratio is defined as where Z g representing the grid impedance can be similarly derived in a MIMO form according to (20). In such a system, Zg is strictly passive, while ZVSC may be non-passive in specific frequency bands. The passivity of ZVSC can be checked by the positive realness of the passivity index [43], which is the minimum eigenvalue of the Hermitian matrix of ZVSC, i.e., Based on the generalized NSC [44], the system is stable if and only if the two eigenvalue transfer functions of L satisfy the NSC. Then, the VSC-grid system stability can be checked by combining the Bode diagrams of the eigenvalue transfer functions of L and the passivity index of Z VSC .
A critically unstable case (Case A) is studied first. The VSC parameters listed in Table II with the voltage control bandwidth of 57 Hz are utilized, and the grid impedance with the short-circuit ratio (SCR) of 10 is considered, i.e., L g = 6 mH. The steady-state operating points of the droop-controlled VSC are given as follows.  SCR = 10: Fig. 6 plots the Bode diagrams of the eigenvalue transfer functions of L and the passivity index of ZVSC. From Fig. 6(a), there are three non-passive frequency bands, around −50 Hz, 50 Hz and 150 Hz, respectively. It can be seen that in the two non-passive frequency bands around −50 Hz and 50 Hz, λ1 and λ 2 do not cross over ±180°, thus the frequency responses are not important to stability. Therefore, the stability analysis can be focused in the non-passive frequency band around the fundamental frequency, which is shown in the zoomed in plot in Fig. 6(b).
The NSC can be then applied to Fig. 6(b). From the Bode diagram of λ1, there is a right-half-plane (RHP) pole identified at 50 Hz, since the phase steps up at the resonant peak [45]. This RHP pole can be also observed from Fig. 5 in each impedance element, which is induced by the (I+G vv G ref ) −1 in (21). It is noted that the existence of RHP poles does not violate the fact that the grid-forming VSC should be opencircuit stable, since in the open-circuit case, I 2dq = 0, such that the (I+G vv G ref ) −1 = I. It is thus implied that the existence of RHP poles in the grid-forming VSC impedance is dependent on the VSC operating points. From the phase response of λ 1 , in the non-passive frequency band, there are two clockwise crossings (CCs) and one anti-clockwise crossing (ACC) over 180°, as denoted by the red dots. The ACC compensates the impact of the RHP pole, thus does not affect the stability. For the two CCs, it can be found that the corresponding magnitude responses are both critically larger than 0 dB, which indicates a critically unstable system.
It is worth noting that the non-passive frequency band for grid-forming VSCs is not wide around the fundamental frequency, which can limit the variation range of the potential oscillation modes. Consequently, the sideband oscillations induced by the grid-forming VSC-grid interaction are SSOs [6].

C. Stability Affected by Short Circuit Ratio
To illustrate the impact of SCR on the stability, Case B considering a grid impedance of L g = 8 mH (SCR = 7) is studied. The steady-state operating points of the VSC under this grid condition are given as follows.  SCR = 7: I d = 10.5 A, I q = 0 A, V d = 190.54 V, V q = 0 V.
The stability analysis based on the generalized NSC and the passivity analysis is provided in Fig. 7, where only zoomed in plot around the fundamental frequency is shown for brevity. It can be seen that the phase crossover frequencies at the CCs are almost unchanged compared with Fig. 6(b), yet the corresponding magnitude response of λ1 is less than 0 dB. It is indicated that when the grid is weaker (SCR reduced), the system stability is improved.
However, the analysis based on the eigenvalue transfer functions of L cannot elucidate much insight into the interaction of the VSC and grid, which makes it difficult to understand the instability mechanism. Therefore, the singular value analysis is further conducted, which is inspired by Criterion 1. Based on the definition of L, it can be derived that  is satisfied at the phase crossover frequencies in the nonpassive frequency bands of ZVSC. Fig. 8 shows the singular value analysis of ZVSC and Zg compared with the passivity index of Z VSC . Since under Cases A-B with SCR changing, the operating points of the VSC are almost unchanged, indicating that the profile of ZVSC is almost unchanged. And the phase response of Z g is mainly inductive around the fundamental frequency, thus the phase crossover frequencies within the non-passive frequency band of ZVSC are almost unchanged, which are simply marked by the red dashed lines at 46.6 Hz and 53.4 Hz. The singular value is an indicator of the matrix magnitude. It can be seen that the VSC impedance features high magnitude around the fundamental frequency, which tends to intersect with the grid impedance and violate the small-gain condition in Criterion 1. As the SCR increases, i.e., the grid is stiffer, the frequency band for is widen, which tends to reduce the system gain margin at the phase crossover frequencies. This explains why the grid-forming VSC may suffer from instability in stiff grids. However, the singular value analysis still fails in analyzing the impacts of VSC control loops, since the VSC impedance is calculated as a whole. This challenge will be addressed in next section.

IV. IMPEDANCE DECOMPOSITION FOR DESIGN-ORIENTED ANALYSIS
This section further investigates the control-loop impacts on the stability of grid-forming VSCs, by proposing an impedance decomposition approach, which enables a designoriented analysis.

A. Control-Loop-Based Impedance Decomposition
The closed-loop VSC impedance model in Fig. 4 consists of two feedforward loops and one feedback loop. For the feedforward loops, one is characterized by Z o representing the voltage control dynamics, and the other is characterized by ZrefGvv mainly impacted by the power control dynamics around the fundamental frequency. According to the expression of Z ref in (18), the contributions of the active power 46 . (26) Then, Fig. 4 is reformulated as Fig. 9(a), where three feedforward loops are used to represent the three control loop dynamics, respectively. For abbreviation, VC represents the inner-loop voltage control, APC represents the active power control, and RPC represents the reactive power control. Consequently, the VSC impedance shown in (21) can be decomposed into several impedances in series, i.e., where each impedance represents the dynamic of each feedforward loop. Then, the equivalent circuit of the gridforming VSC by decomposing different control loops is depicted as Fig. 9(b). It should be noted that the above control impedance decomposition is merely based on the feedforward loop separation. These control loops are still coupled with each other through the feedback loop, which results in a common factor in each decomposed impedance, i.e., (I+GvvGref) −1 .

B. Control-Loop Impact Analysis
The singular value analysis in Section III-C has indicated that the impedance magnitude is critical to the stability. However, the grid-forming VSC impedance features high magnitude around the fundamental frequency, which disagrees with the common conception that a voltage-controlled VSC should have a low output impedance within its control bandwidth. The reason can be explained by analyzing the control-loop impacts based on the decomposed impedances.
According to (27), the decomposed impedances differ from each other in the feedforward loops. Therefore, the different impacts of control loops on the VSC impedance shaping can be discussed by comparing the magnitude responses of Z o , Z refP G vv and Z refQ G vv . Before comparing them together, the voltage control impact on Z o and the power control impact on Z refP and Z refQ are discussed first.

1) Voltage Control Impact
Zo represents the voltage control impact, which is a diagonal matrix. The second diagonal element is merely frequency shifted from the first diagonal element, thus the voltage control impact on the output impedance shaping can be clarified by the SISO model given in (9).
Since the VC output impedance is influenced by the voltage control design, the VC open-loop gain calculated by (28) and the VC output impedance are both plotted in Fig. 10, considering the two sets of VC parameters listed in Table II. The VC control bandwidth (BW) is defined based on magnitude crossover frequencies.
In Fig. 10, for the VC loop gain, it can be seen that due to the impact of the LC resonance, the bandwidth cannot be designed sufficiently high [46]. Tv1 achieves a gain margin (GM) of 6 dB, with only a limited bandwidth of 57 Hz. T v2 increases the bandwidth to 167 Hz, yet sacrifices the GM to 4.8 dB.
For the VC output impedance, its magnitude is low merely in a narrow frequency band around f 1 = 50 Hz. The width of this frequency band is in positive correlation to the VC bandwidth. In the very low frequency range approaching to 0 Hz, the magnitude response of Z o approaches to which behaves as a virtual resistance emulated by the innerloop voltage and current controllers. Following the dual-loop voltage control design guideline [46], K pv is usually designed much small to reduce the impact of the LC resonance on the VC loop gain design, such that KpiKpv << 1 is satisfied. Thus,  Fig. 10 Bode diagrams of VC loop gains and VC output impedances the low-frequency (much lower than f1) impedance is mainly shaped by the current controller gain, which is usually designed as large as possible, since the current controller gain is aimed to provide active damping for the LC resonance [38]. In the high-frequency range (much higher than f 1 ), it can be found that the impedance is mainly shaped by the LC resonance including the time-delay effects [37]. Consequently, the limited VC bandwidth results in a non-negligible Zo around f 1 , which can interact with the power control impedances and further affect the VSC output impedance.

2) Power Control Impact
The power control impact is characterized by the impedance matrix Zref, which can be decomposed into ZrefP and ZrefQ to quantify the interaction of APC and RPC, respectively. Considering V dq =V 0 in steady state, it can be seen that the interaction between APC and RPC can be analyzed by comparing the frequency responses of V 0 G P (s−jω 1 ) and G Q (s−jω 1 ) according to (25), (26). Assuming the same LPF cut-off frequencies in APC and RPC, the ratio of their magnitude responses can be derived as where rP and rQ are the allowed maximum percentages of the frequency deviation and voltage deviation. Consequently, the magnitude crossover frequency of (30) can be found as f = f1±rP•f0/rQ. It is implied that, when |f-f1| < rP•f0/rQ, the APC impact is stronger, and a large gain is introduced as f approaches to f 1 due to the integrator used in APC. When |ff 1 | > r P •f 0 /r Q , the RPC impact is stronger.

3) Control-Loop Impacts on VSC Impedance
To further compare the control-loop impacts on VSC impedance, the Bode diagrams of Zo, ZrefPGvv and ZrefQGvv are plotted together in Fig. 11. According to the magnitude analysis, it can be seen that when f goes farther from the fundamental frequency, the VC's impact is much stronger than other loops, indicating VC is dominant. When f is approaching to the fundamental frequency, the APC's impact is mostly   Fig. 11, it is learned that the high magnitude of the grid-forming VSC impedance around the fundamental frequency is mainly resulted from the integrator adopted in the APC. The magnitude analysis also provides an overall understanding on how the control loops affect the VSC impedance in a wide frequency range, which can be helpful to understand their impacts on stability: the magnitude analysis can be compared with the system oscillation modes to analyze the sensitivity of different control loops to stability. For instance, according to the analysis in Section III, the system oscillation modes, which should be close to the phase crossover frequencies in critically unstable conditions, are found around 46.6 Hz and 53.4 Hz. The two frequencies locate within the red dashed ellipses in Fig. 11, thus the stability studied in this case is sensitive to both VC and APC parameters. Since the RPC's impact is weaker, the stability should be less sensitive to the RPC parameters.
It is noted that the above results are merely carried out based on the droop-controlled VSC. If the grid-forming VSC adopts different control schemes or different types of controllers, e.g., if an integrator is adopted in the RPC, then the RPC's impact around the fundament frequency is not negligible compared with the APC. Although the results may vary from cases to cases, such comparative analysis approach can still be applied to gain more insights into the control-loop impacts for a design-oriented study.

C. Design-Oriented Controller Tuning
After knowing the control-loop impacts on the VSC impedance, a design-oriented controller design can be conducted for stability enhancement: the controller tuning can be prioritized on the sensitive control loops. Cases C-E are analyzed, respectively, to investigate the impacts of the VC, the APC, and the RPC on the stability. The key changing parameters compared with Case A are listed in Table III. The stability analysis with the generalized NSC is shown in Fig. 12, which can be compared with the analysis result for Case A shown in Fig. 6(b). It is seen from Fig.  12(a)-(b) that by increasing the VC bandwidth or decreasing the APC droop coefficient, the system stability can be effectively improved. This supports the previous control-loop impact analysis, that the VC and APC are both sensitive to the stability. Such control parameter tuning effects can be clarified from the small-gain perspective: since increasing the VC bandwidth and reducing the APC droop coefficient both reduce the impedance magnitudes of Zo and ZrefP significantly around the fundamental frequency, the VSC impedance magnitude can thus be reduced accordingly. The singular value analysis in Section III has indicated that the high VSC magnitude is the root cause of instability of grid-forming VSCs in stiff grids in the VSC non-passive region. Therefore, reducing the VSC impedance magnitude can improve the system stability. However, it should be noted that the smallgain principle for the parameter tuning is only a sufficient condition. It may fail in some cases if the phase response is influenced more significantly, e.g., if second-order poles or zeros exist around the oscillation modes, then an accurate stability analysis with the NSC is needed. As for the RPC loop, reducing the RPC droop coefficient even for 10 times, the system still maintains in a critically unstable condition, as shown in Fig. 12(c). It proves that the RPC is not sensitive to the stability.
A flowchart can thus be summarized in Fig. 13 for a designoriented control design based on the impedance decomposition. It is noted that the approach gains more insights into the control-loop impacts of grid-forming VSCs, which can be also applied to VSCs with other schemes of grid-forming control. The analysis can provide some guidance for the controller tuning, yet the accurate stability analysis still relies on the entire system model. Furthermore, the impedance decomposition approach is not limited to grid-forming VSCs. In duality, grid-following VSCs can be decomposed into several admittances in parallel, which has been introduced in [47] to analyze the control-loop impacts on the VSC passivity.

V. CASE VALIDATIONS
This section validates the previous stability analysis of Cases A-E by experiments or simulations. Fig. 14 shows the experimental setup. A grid simulator (Chroma 61845) is used to generate an ideal three-phase voltage source. The VSC is fed by a DC source and the control is implemented in dSPACE (DS1007). The circuit parameters and the VSC control parameters are the same as given in Tables I-II  ratios (SCRs) defined at the PoC are considered, i.e., L g = 6 mH (SCR ≈ 10) and Lg = 8 mH (SCR ≈ 7).
The experimental waveforms for Case A are provided in Fig. 15. It can be found that the system is critically unstable with sideband oscillations, which verifies the stability analysis shown in Fig. 6. Moreover, the discrete Fourier transformation (DFT) analysis identifies the oscillation frequency of 3.3 Hz in the power waveforms, which turns out to be 46.7 Hz and 53.3 Hz in the αβ frame. The sideband oscillation frequencies are also well predicted by the analysis result in Fig. 6. Therefore, the proposed impedance model of the VSC can accurately analyze the system stability.
The experimental waveforms for Case B when the grid becomes weaker are provided in Fig. 16. Compared with Fig.  15, the system can be stabilized, which verifies the analysis in Fig. 7.
The impacts of VC bandwidth, APC droop coefficient, and RPC droop coefficient are further studied by Cases C-E, and the corresponding waveforms are given in Figs. 17-19, respectively. Fig. 17 verifies that the increase of the VC loop bandwidth can enhance the stability, which agrees with the analysis in Fig. 12(a). Fig. 18 indicates that the decrease of the APC droop coefficient also stabilizes the system, which agrees with the analysis in Fig. 12(b). However, in Fig. 9, the system still maintains in a critically unstable condition, with the RPC droop coefficient decreased. This also verifies the stability analysis in Fig. 12(c). DFT

V. CONCLUSION
The complex-valued αβ-frame impedance model of the droop-controlled grid-forming VSC has been established in this paper to investigate the stability in stiff grids. Based on the impedance decomposition approach, the VC, APC and RPC's impacts have been decomposed into three impedance terms in series. Through the magnitude analysis of the decomposed impedances, it has been found that for the droopcontrolled VSC: 1) The integrator used in the APC results in high magnitude in the VSC output impedance around the fundamental frequency, which tends to result in the instability in stiff grids. 2) The VC and APC is more sensitive to the instability than the RPC. Based on the above findings, the design-oriented controller design by tuning VC and APC parameters in priority has been implemented for stability enhancement, which has been validated by the stability analysis, simulations and experiments.