Abstract
Oscillations caused by small-signal instability have been widely observed in AC grids with grid-following (GFL) and grid-forming (GFM) converters. The generalized short-circuit ratio is commonly used to assess the strength of GFL converters when integrated with weak AC systems at risk of oscillation. This paper provides the grid strength assessment method to evaluate the small-signal synchronization stability of GFL and GFM converters integrated systems. First, the admittance and impedance matrices of the GFL and GFM converters are analyzed to identify the frequency bands associated with negative damping in oscillation modes dominated by heterogeneous synchronization control. Secondly, based on the interaction rules between the short-circuit ratio and the different oscillation modes, an equivalent circuit is proposed to simplify the grid strength assessment through the topological transformation of the AC grid. The risk of sub-synchronization and low-frequency oscillations, influenced by GFL and GFM converters, is then reformulated as a semi-definite programming (SDP) model, incorporating the node admittance matrix and grid-connected device capacities. The effectiveness of the proposed method is demonstrated through a case analysis.
WITH the large-scale integration of power electronic converters based renewable energy generators into the AC grid, power system dynamic characteristics have undergone fundamental changes [
Grid strength is a tool for preliminary screening of small-signal stability for converters integrated systems. For instance, the short-circuit ratio (SCR), recently recognized by North American Electric Power Reliability Corporation, has been widely used by power system operators [
Presently, two primary SCR-based methods are utilized to assess the grid strength of the multi-infeed converter systems: heuristic and theoretical derivation-based methods. Heuristic method is derived from indicators inspired by engineering experience, such as the weighted SCR (WSCR) [
The aforementioned methods primarily focus on analyzing small-signal stability issues in weak grids, such as sub-synchronization oscillation caused by grid-following (GFL) converter equipped with a phase-locked loop (PLL) [
The challenges in small-signal stability assessment of hybrid converters integrated systems stem from the distinctive synchronization control mechanisms of GFL and GFM converters. GFL converters employ PLL to generate the output current phase to achieve reactive power synchronization. In contrast, the GFM converters use virtual synchronization control to generate the output voltage phase to achieve active power synchronization [
To tackle the above issues, this paper proposes a grid strength assessment method to evaluate the small-signal synchronization stability of the GFL and GFM converters integrated system (hereafter called the hybrid system). Specifically, the dominant control links and frequency band characteristics of the dominant oscillation modes of the heterogeneous synchronization control are analyzed from the perspective of external characteristics of the converter. An equivalent circuit-based model reduction is then introduced for various oscillation modes. Additionally, a practical renewable energy capacity allocation model based on modified gSCR is proposed. The main contributions of this paper can be summarized as follows:
1) By establishing the admittance and impedance matrices of GFL and GFM converters, the different interaction mechanisms between heterogeneous synchronization control and AC grid are identified from the perspective of external characteristic analysis.
2) Through the interaction rules between SCR and dominant oscillation modes governed by different synchronization controls, an equivalent circuit is proposed, streamlining the grid strength calculation process and facilitating the oscillation risk assessment for hybrid systems.
3) Based on the preceding analysis, a semi-definite programming (SDP) model consisting of a node admittance matrix and grid-connected equipment capacity is proposed. This model serves as a practical tool for planning and operation of hybrid systems to enhance small-signal stability.
The remainder of the paper is organized as follows. Section II analyzes of the characteristics of small-signal stability of GFM converters. Section III presents the grid strength of hybrid systems. In Section IV, the modified gSCR application is presented. The simulation results are given in Section V. Finally, Section VI concludes the paper.
Different dynamic external characteristics of GFL converters, GFM converters, and SGs bring differences in the interaction between heterogeneous synchronization control loop and AC grid, e.g., GFL may lose its stability in weak AC grids while GFM may lose its stability in stiff AC grids. To illustrate the reason, this section first analyzes the frequency band of the negative conductance and resistance induced by GFL and GFM converters. Then, by comparing the damping torque of GFM converter and SG, the physical mechanism of low-frequency oscillation of GFM converter in the stiff AC grid is clarified. Finally, the Nyquist criterion is used to analyze the different impacts of grid strength on the dominant oscillation modes of GFL and GFM heterogeneous synchronization control loops.
GFL and GFM converters utilize cascade control structures, comprising outer loops for power and voltage regulations and inner loops for current control. Specifically, the GFL converter employs a constant DC-voltage reactive power control in the outer loop [

Fig. 1 Control structure of hybrid system.
The external characteristics of the GFL converter with reactive power synchronization control are similar to those of a current source, making the frequency domain admittance more suitable for small-signal synchronization stability analysis [
In contrast, the external characteristics of the GFM converter, which use active power synchronization control, are represented by voltage sources, so the impedance model is typically employed in the small-signal synchronization stability analysis of GFM converter [
The admittance matrix of the GFL converter can be obtained by:
(1) |
where represents the perturbed value of each of the four variables. The specific element expressions of are presented in Supplementary Material A, and the PLL control loop only exists in . Please refer to [
The Bode diagram of GLF open-loop system is shown in

Fig. 2 Bode diagram of GFL open-loop system. (a) . (b) . (c) . (d) .
For the GFM converter, the impedance matrix can be calculated by [
(2) |
where and are the rational transfer functions. The specific element expressions of are presented in Supplementary Material A. It is worth noting that the virtual synchronization control loop only appears in .
The Bode diagram of GFM open-loop system is shown in

Fig. 3 Bode diagram of GFM open-loop system. (a) . (b)
To summarize, the analysis of the external characteristics of GFL and GFM converters shows that the heterogeneous synchronization control loop is the primary factor influencing the external characteristics of the converter. In addition, it reveals that the negative damping frequency bands of the two types of converters are distributed in different ranges. As shown in Figs.
Although both GFM converter and SG have the risk of low-frequency oscillation, the interaction between their oscillation modes and grid strength is different. The armature reactance of SG provides a strong damping torque, and the external grid strength has little effect on its low-frequency oscillation mode. However, the damping torque of the GFM converter is insufficient, and the external grid strength greatly influences the low-frequency oscillation of the GFM converter. Therefore, the influence of grid strength on the GFM converter can be analyzed from the perspective of damping torque. In addition, according to [
Hence, the damping torque method will be applied to explain the reasons behind the differences in external characteristics of GFM converters and SGs in the low-frequency range. The contribution of the inner loop of GFM converter and filter inductance is quantitatively evaluated to the damping torque of the synchronization control loop. A theoretical basis will be esrablished to for evaluating the grid strength of hybrid systems.
The SG in this paper adopts the classic
(3) |
where is the electromagnetic torque of SG with , , and as the three components; is the electromagnetic power; is a constant; is the grid voltage amplitude; is the power angle; and are the SG impedances in the d- and q-axis, respectively; and the subscript 0 represents the steady-state value of the corresponding component, which applies to all subsequent variables.
The vector control loop of the GFM converter is structurally similar to the armature circuit, so its damping torque can be obtained by:
(4) |
where is the active power perturbed value of GFM converter; is the electromagnetic torque of GFM converter with , , and as three components; is a constant; and and are the GFM impedances in the d- and q-axis, respectively. The derivation process is shown in Supplementary Material B.
The armature reactance of SG and the vector control loop of GFM converter are reflected in and . The impedance characteristics of SG and GFM converter are shown in

Fig. 4 Impedance characteristics of SG and GFM converter. (a) and . (b) and .
At a specific oscillation frequency (assuming Hz), the damping torques of SG and GFM converter are illustrated in

Fig. 5 Damping torques of SG and GFM converter.
Among them, and are positive damping torques (with as the positive damping component), while the phases of and lag behind , resulting in a negative damping torque component in (as indicated by ). These results indicate that the armature reactance offers sufficient dynamic damping for SG in a stiff grid, making it less susceptible to low-frequency oscillation. However, the GFM converter with active power synchronization control is prone to low-frequency oscillations.
This subsection examines how grid strength interacts with various oscillation modes, offering a theoretical foundation for assessing small-signal stability across different frequency bands. SCR is a vital indicator for measuring the grid strength. It is defined as the ratio of the short-circuit capacity at the infeed bus in a single-infeed system to the rated capacity of the converter as [
(5) |
where is the reactance connecting the converter with the AC grid, and is the susceptance.
The admittance matrix of the single GFL converter integrated system can be represented by the return-difference matrix (RDM) of the system, as shown in (6).
(6) |
where is the admittance matrix of the AC grid. The specific elements , , , , and are given as:
(7) |
As for the GFM converter, the RDM of the single GFM converter integrated system can be obtained by:
(8) |
where is the impedance matrix of the single GFM converter integrated system; and Zgrid(s) is the AC grid impedance. The specific elements , , , , and are given as:
(9) |
The Nyquist criterion is adopted to analyze the small-signal stability between GFM and GFL converters in stiff and weak grids. Specifically, the Nyquist curves of the PLL dominated control loop of the GFL converter under different grid strengths are shown in

Fig. 6 Nyquist curves of GFL and GFM converters under different grid strength. (a) PLL dominated control loop (). (b) PLL dominated control loop (). (c) Virtual synchronization dominated control loop (). (d) Virtual synchronization dominated control loop ().
Due to the small capacity of a single converter based on the renewable energy, the grid-connected structure is characterized by scale and decentralization. In addition, the converter has multi-scale control interactions, so its dynamic model order is also high. The above characteristics make simple and intuitive grid strength indicators such as the gSCR widely used in the preliminary stability screening of multi-infeed converter systems.
The hybrid system topology is shown in

Fig. 7 Hybrid system topology and reduced-order hybrid system topology with equivalent external grid. (a) Hybrid system topology. (b) Reduced-order hybrid system topology with equivalent external grid.
Based on the analysis in Section II, and represent the admittance and impedance of the hybrid system on the converter side, respectively as:
(10) |
(11) |
where () and ) are the GFL and GFM converter admittance matrices, respectively; and () and () are the GFL and GFM converter impedance matrices, respectively.
The AC grid side admittance and impedance matrices for a multi-infeed converter system, represented by and , expands the matrix from a single-infeed converter system.
(12a) |
(12b) |
where and are the node admittance and impedance matrices of the AC grid, respectively; ; ; ; ; ; ; ; and .
Considering (10) and (12), the closed-loop admittance and impedance matrices of the hybrid system, i.e., and , can be obtained by:
(13) |
When analyzing the oscillation mode dominated by GFL converter, the admittance model of can be obtained by:
(14) |
As can be observed from Section II-A, in the sub-synchronization oscillation frequency band, the external characteristic of GFM converter is a voltage source with small impedance and large admittance. When is very large, is very small after taking the inverse in (16). Hence, the reduced-order model in oscillation mode dominated by GFL converter can be obtained by retaining in (14) and converting the remaining parts into via Schur complement transformation:
(15) |
The feasibility of the above model reduction is based on the fact that GFM converter has little effect on the oscillation mode dominated by GFL converter, and the effect of GFM converter can be quantitatively analyzed by perturbation analysis. Hence, the part of (15) containing the GFM converter is defined as the perturbation:
(16) |
In order to verify the rationality of the reduction, the perturbation calculation is performed through the double-infeed system topology of GFL and GFM converters, as shown in

Fig. 8 Double-infeed system topology of GFL and GFM converters.
Similar to (14), when analyzing the oscillation mode dominated by GFM converter, the impedance model of can be obtained by:
(17) |
As for the low-frequency oscillation frequency band, the external characteristic of GFL converter is a current source with large impedance and small admittance. can be retained, and the other parts can be converted to through Schur complement transformation to obtain the reduced-order model:
(18) |
In addition, by making a difference between the matrix containing the GFM converter admittance in and the reduced-order model (18), the perturbation quantity of the virtual synchronization dominant oscillation mode can be obtained by:
(19) |
where is the identity matrix of the corresponding dimension.
Furthermore, the singular values of of the double-infeed system of GFL and GFM converters are analyzed, and the results are shown in

Fig. 9 Frequency response characteristic curve of perturbation quantity in different dominant oscillation modes. (a) PLL dominant oscillation mode. (b) Virtual synchronization dominant oscillation mode.
Therefore, the Schur complement transformation has no noticeable effect on the system, proving that the GFM converter has little effect on the oscillation mode dominated by GFL converter and vice versa, further proving that the reduced-order model proposed in this paper is feasible.
On top of that, the critical SCR (CSCR) of the double-infeed system of GFL and GFM converters is close to that of the single-infeed system, further illustrating the effectiveness of the above perturbation analysis. By adjusting the network parameters of the AC power grid, we determine the CSCR of the double-infeed system in different dominant oscillation modes. Then, by comparing the dominant eigenvalues of the single-infeed system and the double-infeed system under different CSCRs, it is found that when the double-infeed system operates in a weak AC grid, the dominant eigenvalues of the double-infeed system and the single-GFL system are similar. Similarly, the dominant eigenvalues of the double-infeed system and the single-GFM system are similar in a stiff power grid. Detailed eigenvalue results are provided in
Parameter | System | Eigenvalue |
---|---|---|
, | Double-infeed | |
Single-GFL | ||
,
| Double-infeed | |
Single-GFM |
In addition, the equivalent circuit topology of a hybrid system corresponds to the system simplification process in different dominant oscillation modes.
In the oscillation mode dominated by GFL converter, the model order reduction can be regarded as treating the GFM converter as an infinite power source, as shown in

Fig. 10 Schematic diagram of equivalent circuit in different dominant oscillation modes. (a) GFL-dominated oscillation mode. (b) GFM-dominated oscillation mode.
Hence, the physical meaning of grid strength index calculation in dominant oscillation modes of heterogeneous synchronization control has been clarified through grid topology transformation.
In the multi-infeed system of GFL converter, is defined as the minimum eigenvalue of the extended admittance matrix, corresponding to the sub-synchronization oscillation mode in the weak grid [
(20) |
where S is a diagonal matrix, the diagonal elements of which are the rated capacities of each GFL converter; and is the minimum eigenvalue of the solution matrix.
Reference [
(21) |
where is the diagonal matrix representing the GFL converter capacity.
The validity of (21) is explained by the following two parts:
1) Through the analysis in Section III-B, it can be observed that when facing the sub-synchronization oscillation mode dominated by PLL, the GFM converter is equivalent to a short circuit.
2) The inequality constraints of (21) are equivalent to the gSCR SDP form in [
Furthermore, based on the modeling and analysis in [
(22) |
where is the maximum eigenvalue of the solution matrix.
Further considering the influence of the GFL converter, combined with the open-circuit approximation of the GFL converter in Section III-C, the improved SDP model can be obtained as:
(23) |
where is the diagonal matrix representing the GFM converter capacity.
The site selection of wind farms hinges on the availability of local natural resources. In areas with stiff grid, GFL converters are primarily selected, which can reduce the oscillation risk. In areas with weak grid, GFM converters are primarily selected. However, considering the inadequate transient stability of GFM converters in weak grid [
(24) |
(25) |
Since constraint (24) is a bilinear problem, it is difficult to solve. Therefore, let , , then the bilinear problem can be converted into a semidefinite programming problem as shown in (26).
(26) |
Constraint (26) represents that the system satisfies the grid strength constraints that neither subsynchronization oscillation nor low-frequency oscillation occurs. Constraint (25) is a static voltage stability constraint, i.e., the gSCR of a single-infeed system is greater than 1.
This section first builds a hybrid system through the MATLAB/Simulink platform for time-domain simulation to verify the sub-synchronization and low-frequency oscillations caused by different synchronization control links under different grid strengths. Secondly, a 4-converter integrated system is built to verify the proposed method. Finally, a numerical analysis is performed on the SDP model to illustrate the engineering significance of the optimization results.
Taking the parameters of different grid strengths shown in
According to the proposed method, the node admittance matrices of the hybrid system under sub-synchronization oscillation and low-frequency oscillation, i.e., and , can be set as:
(27) |
At this time, and . When s, the PLL integral coefficient changes from the initial stable value 4000 to 6000. Sub-synchronization oscillation caused by the PLL dominant control link occurs in the hybrid system, with a frequency of about 11.5 Hz. The output power of the converter under sub-synchronization oscillation is shown in

Fig. 11 Output power of converter under different oscillations. (a) Under sub-synchronization oscillation. (b) Under low-frequency oscillation.
It is worth noting that although a larger line mutual inductance is set in (27), the GFL converter in the hybrid system is still slightly affected by the low-frequency oscillation of the GFM converter. This is because the GFL converter undergoes forced oscillation. The specific analysis will be given through the participation factor calculation in Section V-B.
A 4-converter integrated system is constructed on the MATLAB/Simulink platform, as shown in

Fig. 12 A 4-converter integrated system.
First, the capacity of the converters on the four buses is set to be 1 p.u.. is calculated as 9.14 through (23), and the system undergoes low-frequency oscillation. Subsequently, when s, the GFM converter capacity increases to 2 p.u.. At this time, the is reduced to 4.57, and the system is stable. The simulation results of the 4-converter integrated system are shown in

Fig. 13 Simulation results of 4-converter integrated system.
It can be observed that the GFL converter is affected by the low-frequency oscillation of the GFM converter to a certain extent. Therefore, this subsection calculates the participation factor of the dominant eigenvalue shown in
State variable | Dominant eigenvalue under sub-synchronization oscillation(i) | Dominant eigenvalue under low-frequency oscillation () |
---|---|---|
θ for GFM1 | 0.000126 | 0.428700 |
ω for GFM1 | 0.000134 | 0.259400 |
θ for GFM2 | 0.000173 | 0.037200 |
ω for GFM2 | 0.000184 | 0.022500 |
θ for GFL1 | 0.517900 | 0.000850 |
ω for GFL1 | 0.481600 | 0.000845 |
θ for GFL2 | 0.018600 | 0.000854 |
ω for GFL2 | 0.017600 | 0.000849 |
Note: state variables are two integrators in synchronization control link of four converters.

Fig. 14 Pole-zero map of 4-converter integrated system.
The sdpt3 solver in the optimization toolbox YALMIP in MATLAB is used to solve the capacity planning problem in Section IV.

Fig. 15 Capacity configuration of renewable energy equipment.

Fig. 16 Simulation results of converter capacity configuration.
This paper presents a grid strength assessment method for evaluating the small-signal stability of hybrid systems. The method incorporates sub-synchronization oscillation risk, primarily governed by the PLL, and low-frequency oscillation risk, influenced by virtual synchronization control, within grid planning applications. An SDP model is introduced, integrating the node admittance matrix and the capacity of grid-connected equipment. Simulation and numerical analysis results validate the effectiveness of the proposed method. In future work, we aim to further integrate transient stability and oscillation modes dominated by other control loops to deeply analyze the stability constraints that restrict the integration capacity of the GFM converter.
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