Low-Frequency Oscillation Analysis of Grid-Forming Energy Storage Converters Based on a Multi-Damping Path Model

1  Introduction

The increased use of renewable energy has resulted in energy storage becoming an important method for mitigating renewable energy fluctuations and enhancing grid flexibility [1,2]. Currently, most existing energy storage systems are grid-following types that use phase-locked loops to track the voltage phase at the point of common coupling to achieve synchronization. However, phase-locked loops do not perform well in weak grids and cannot accurately track the voltage phase at the point of common coupling (PCC), which means that grid-following energy storage systems are unable to operate normally [3]. However, the continuous integration of renewable energy can lead to reduced grid strength, which means that grid-following energy storage will face adaptation difficulties in the future.

Grid-forming energy storage does not use phase-locked loops. Instead, it employs a power synchronization control strategy similar to generators and achieves synchronization through its own power deviation. In contrast to grid-following energy storage, grid-forming energy storage can autonomously establish grid voltage, has voltage source characteristics, provides inertia support and damping to the system, and can better adapt to the “double-high” characteristics of substantial renewable energy and power electronics penetration [4,5].

Previous research on grid-forming energy storage primarily focused on utilizing the inherent energy storage functions associated with peak shaving and frequency regulation. However, the grid-forming energy storage characteristics strongly depend on electronic power converters and their control strategies. In recent years, wide-band oscillations caused by interactions between electronic power converter control loops and the grid have frequently occurred and these have seriously affected the stable operation of power systems [6–8]. Therefore, the small-signal stability issues associated with grid-forming energy storage cannot be ignored. For example, Reference [9] establishes a unified modular small-signal model for grid-forming inverters and verifies its effectiveness through eigenvalue analysis and time-domain simulations. However, it fails to elucidate the dynamic mechanisms behind key eigenvalue trajectories and their impacts on system performance. Reference [10] constructed a state-space model for grid-forming energy storage for an offshore wind farm and used a sensitivity analysis to obtain the impacts of key parameters and grid strength on system dynamic behavior. Reference [11] establishes small-signal models for parallel grid-forming converters ranging from single-unit to multi-unit systems and provides mathematical stability analysis. However, it does not delve into clarifying the key physical mechanisms that influence system stability. These studies all used the system state-space model and sensitivity and eigenvalue analysis methods to analyze the small-signal stability of the system. This approach is theoretically rigorous and accurate, but computationally intensive, complex to solve, and difficult to scale up. Reference [12] clarifies the equivalent impedance of voltage and current loops by establishing a sequence impedance model for grid-forming converters and verifies the advantages of the single-power-loop structure. However, its explanation of the dynamic interaction mechanisms between control loops remains insufficiently in-depth. Reference [13] proposes a modular three-port admittance modeling framework that effectively addresses the challenge of impedance modeling in hybrid AC/DC systems containing grid-following and grid-forming converters, demonstrating good universality. However, its analysis of the internal interactions within the system and the intrinsic stability mechanisms remains insufficient. Similarly, Reference [14] established impedance matrix models for grid-forming converters in a synchronous rotating coordinate system and a coordinate system defined by the dynamic angular frequency of the power system. These studies concluded that the two impedance modeling methods were equivalent, but the latter better explained system behavior during disturbances. Furthermore, each study analyzed the small-signal stability of the system by establishing frequency-domain impedance models. Overall, the proposed models have strong scalability but increase the complexity of the stability analysis. However, the physical meaning of coupled impedance remains unclear.

Compared to eigenvalue and impedance analysis methods, the damping torque analysis method has clear physical meaning and can effectively separate and quantify the coupling effects of multiple factors. Reference [15] constructed a multi-loop damping path model of a virtual synchronous generator control system based on the embedded torque method. The results showed that there were interactive influences between the d-axis and q-axis components of the voltage control loop and current control loop of the virtual synchronous generator. The malignant competition and negative incentive of the damping torque between these interactive loops meant that the oscillation modes were unstable. However, the entire damping loop was overly complex and difficult to analyze. Reference [16] established a damping path model for a wind farm and grid-forming energy storage and concluded that grid-forming energy storage can provide certain levels of damping that can suppress wide-band oscillations caused by wind farms connected to weak grids. Reference [17], establishing a single-input single-output dynamic interaction model, reveals the excellent dynamic characteristics of grid-forming energy storage in weak grids and its impact on grid-following photovoltaics. However, due to the inherent limitations of the single-input single-output model, it is difficult to fully uncover the complex dynamic coupling relationships between the multiple control loops within these systems. In summary, the damping torque analysis method has clear physical meaning and can effectively achieve the separation and quantitative assessment of multi-factor coupling effects. However, previous studies treated the research system as a whole when conducting damping torque analyses and analyzed the damping characteristics of the entire system without examining internal interactions.

Based on this analysis of previous studies, the novel contributions made by this study are as follows:

(1)   Constructed a multi-timescale model of a grid-forming energy storage grid-connected system. Derived a full-order mathematical model for the grid-forming energy storage system and used a participation factor analysis to establish a reduced-order model and a multi-damping torque model, which provided a theoretical foundation that explained the internal dynamic interaction mechanisms associated with the system.

(2)   Quantitative evaluation of the system damping characteristics from a damping perspective. A constructed multi-damping torque model was used to systematically and quantitatively evaluate the impact of grid strength, operating conditions, and control parameters on the low-frequency oscillation damping characteristics of the grid-forming energy storage system from a damping perspective.

(3)   Comprehensive validation through theoretical analysis and time-domain simulation. The theoretical analysis results were fully verified using time-domain simulation methods. The results confirmed the accuracy and effectiveness of the proposed model and analysis approach. This showed that the conclusions were reliable and had strong engineering applicability. They also provide theoretical guidance for tuning the control parameters in grid-forming energy storage systems.

2  Mathematical Model of a Grid-Forming Energy Storage Grid-Connected System

2.1 Grid-Forming Energy Storage Grid-Connected System

The isolation effect of the DC/AC converter and the large support capacitor on the DC side meant that this study equivalently modeled the DC/DC module as an ideal DC voltage source and did not consider the dynamic process on the DC side when establishing a small-signal model of a grid-forming energy storage grid-connected system. The model for the grid-forming energy storage grid-connected system is shown in Fig. 1.

Figure 1: Grid-forming energy storage grid-connected system model

Where udc represents the DC voltage; eabc represents the three-phase output voltage of the grid-forming energy storage system; iabc represents the three-phase output current of the grid-forming energy storage system; uabc represents the three-phase voltage at the PCC of the grid-forming energy storage system; Vabc represents the three-phase grid voltage; and Lf and Lg represent filter inductance and line inductance, respectively.

2.2 Mathematical Model of the Grid-Forming Energy Storage Grid-Connected System

The mathematical model for the system shown in Fig. 1 contains five parts: grid-forming energy storage filter inductance, active power synchronization, a reactive power-voltage control loop, voltage/current vector control, and an AC grid. The dynamic equation for filter inductance is shown in Eq. (1):

{Lfdiddt=ed−ud+ω0LfiqLfdiqdt=eq−uq−ω0Lfid,(1)

where Lf represents the filter inductance; id and iq represent the d-axis and q-axis components of the output current of the grid-forming converter in the main circuit coordinate system, respectively; ed and eq represent the d-axis and q-axis components of the output voltage of the grid-forming energy storage system in the main circuit coordinate system, respectively; and ud and uq represent the d-axis and q-axis components of the voltage at the PCC of the grid-forming energy storage system in the main circuit coordinate system, respectively.

Active power synchronization is used to achieve self-synchronization in grid-forming energy storage systems. Its output phase is used for the park and inverse transformations of three-phase signals to realize vector control of grid-forming energy storage. Its dynamic equation is shown in Eq. (2):

{Jdωdt=Pref−Po−Dp(ω−ω0)dθdt=ω,(2)

where J represents the virtual moment of inertia; ω represents the virtual angular acceleration; Pref represents the reference power of the grid-forming converter; Dp represents the virtual damping coefficient; ω0 represents the virtual reference angular acceleration; θ represents the virtual phase angle; and Po represents the output active power of the grid-forming energy storage system, with its expression given as follows:

Po=1.5udid+1.5uqiq.(3)

The reactive power-voltage control loop is used to generate the d-axis voltage reference value at the PCC to achieve voltage vector orientation. Its dynamic equation is shown in Eq. (4):

UM=((Uref−Ut)Kv−Qo)Kq+U0,(4)

where Kv represents the voltage droop coefficient; Kq represents the reactive power-voltage regulation coefficient; U0 represents the reference voltage; UM represents the d-axis voltage reference value at the PCC generated by the reactive power-voltage control loop; Uref represents the reference value of the phase voltage amplitude; Ut represents the phase voltage amplitude; and Qo represents the output reactive power of the grid-forming energy storage system, which is calculated as follows:

{Ut=ud2+uq2Qo=1.5uqid−1.5udiq.(5)

The grid-forming energy storage system has a voltage d-axis orientation. Based on the output voltage from the reactive power-voltage control loop, the active current reference value icdref and the reactive current reference value icqref in the control coordinate system are dynamically generated through the voltage outer loop control. The mathematical model is as follows:

{icdref=(UM−ucd)(kp1+ki1s)icqref=(uqref−ucq)(kp2+ki2s),(6)

where ucd and ucq represent the d-axis and q-axis components of the voltage at the PCC of the grid-forming energy storage system in the control coordinate system, respectively; kp1 and ki1 represent the proportional and integral coefficients of the active voltage outer loop, respectively; and kp2 and ki2 represent the proportional and integral coefficients of the reactive voltage outer loop, respectively. The q-axis voltage reference value uqref at the PCC was set to 0.

The current inner loop control, based on the voltage constraint equation, regulates the output voltage of the grid-forming energy storage system to ensure the active/reactive currents track their reference values. The mathematical model for the d-axis and q-axis components (ecd and ecq) of the grid-forming energy storage system output voltage in the control coordinate system is as follows:

{ecd=(icdref−icd)(kp3+ki3s)−ωLficq+ucdecq=(icqref−icq)(kp4+ki4s)+ωLficd+ucq,(7)

where icd and icq represent the d-axis and q-axis components of the output current of the grid-forming energy storage system in the control coordinate system, respectively; kp3 and ki3 represent the proportional and integral coefficients of the active current inner loop, respectively; and kp4 and ki4 represent the proportional and integral coefficients of the reactive current inner loop, respectively.

The mapping relationship between the voltage and current d/q-axis components in the control coordinate system and the voltage and current d/q-axis components in the main circuit coordinate system is as follows:

{ucd=ud+uq0θucq=uq−ud0θ,{icd=id+iq0θicq=iq−id0θ,(8)

where ud0 and uq0 represent the d-axis and q-axis components of the initial voltage value at the PCC of the grid-forming energy storage system, respectively, and id0 and iq0 represent the d-axis and q-axis components of the initial output current value for the grid-forming energy storage system, respectively.

The dynamic model of the AC grid line inductance is represented by the following equation:

{Lgdiddt=ud−Vd+ω0LgiqLgdiqdt=uq−Vq−ω0Lgid,(9)

where Vd and Vq represent the d-axis and q-axis components of the grid voltage in the main circuit coordinate system, respectively.

Eqs. (1)–(9) constitute the mathematical model for the grid-forming energy storage grid-connected system. The linearized state-space model at the equilibrium point is as follows:

ΔX∙=AΔX+BΔU,(10)

where ∆X = [∆id, ∆iq, ∆x1, ∆x2, ∆x3, ∆x4, ∆ω, ∆θ] represents the small-signal form of the state variables. The state variables ∆id and ∆iq denote the small-signal disturbance components of the active current and reactive current, respectively; ∆x1–∆x4 are defined as intermediate state variables in the control system, corresponding to the dynamic processes of the active power-voltage outer loop, reactive power-voltage outer loop, active current inner loop, and reactive current inner loop; ∆ω represents the small-signal disturbance component of the virtual angular velocity, and ∆θ represents the small-signal disturbance component of the virtual phase angle. A and B denote the state matrix and input matrix of the system, respectively, where the positive real part eigenvalues of the state matrix A reflect the oscillatory instability modes of the system, and the risk of instability increases as the real part of the eigenvalues shifts toward the positive direction of the real axis. The detailed expression of the state matrix A is provided in Appendix A.

3  Grid-Forming Energy Storage Multi-Damping Path Model

3.1 Multi-Timescale Decomposition Analysis of the Model

To simplify the subsequent analysis, the linearized state-space model of the grid-forming energy storage grid-connected system obtained in Section 2.2 was reduced in order. The eigenvalues of the system when the grid strength is 7.2 and Po = 0.5 MW are listed in Table 1.

The system contains one subsynchronous oscillation mode λ5/6 (12.2 Hz) and two low-frequency oscillation modes λ3/4 (3.4 Hz) and λ7/8 (0.2 Hz). The eigenvalue for λ3/4 dominates the low-frequency oscillation mode of the system.

The participation factors associated with the eigenvalue for λ3/4 are shown in Fig. 2. The dominant low-frequency oscillation eigenvalue for λ3/4 is primarily influenced by the active voltage outer loop and reactive voltage outer loop control stages. The power synchronization stage has a minimal effect and the current inner loop control has almost no influence. Furthermore, when the bandwidth of the current inner loop is significantly higher than that of the voltage outer loop, the dynamic behavior of the current inner loop can be largely ignored, which means that the current rapidly tracks its reference value and achieves quasi-steady-state control.

Figure 2: Participating factors associated with the dominant eigenvalue for λ3/4

Based on the above analysis, the updated reduced-order state-space model that ignores the dynamics of the current inner loop is as follows:

ΔX1∙=A1ΔX1+B1ΔU1,(11)

where ∆X1 = [∆x1, ∆x2, ∆ω, ∆θ] and A1 and B1 represent the state and input matrices of the system, respectively. The detailed expression of the state matrix A1 is provided in Appendix B. The eigenvalues of the reduced-order model under a grid strength of 7.2 and Po = 0.5 MW are shown in Table 2. The eigenvalue for λ3/4 dominates the low-frequency oscillation mode of the system with an oscillation frequency of 3.58 Hz. The deviation from the oscillation frequency produced by the detailed model (3.4 Hz) is 0.18 Hz, which falls within an acceptable range and confirms the accuracy and effectiveness of the reduced-order model.

3.2 Multi-Damping Path Model

A single-input single-output transfer function block diagram of the grid-forming energy storage system was established based on the reduced-order model of the system, as shown in Fig. 3:where G1 = kp1 + ki1/s represents the active voltage outer loop control link and G2 = kp2 + ki2/s represents the reactive voltage outer loop control link.

Figure 3: Block diagram of the single input and single output transfer functions for grid-forming energy storage

The output active power can be expressed using system-related state variables. The detailed derivation process is provided in Appendix C. The expression for output active power is as follows:

ΔPo=DpΔω+(K1+K2s)Δθ+K3sΔud+K4sΔuq+K5sΔiq,(12)

where K1–K5 represent the power amplification coefficients, and their specific expressions are detailed in Eq. (A21) of Appendix C.

The expressions for ∆ud, ∆uq, ∆iq, and ∆θ are given by the following equations:

{Δud=Gud(s)ΔθΔuq=Guq(s)ΔθΔiq=Giq(s)ΔθΔθ=1sΔω,(13)

where Gud(s) represents the transfer function from ∆θ to ∆ud; Guq(s) represents the transfer function from ∆θ to ∆uq; and Giq(s) represents the transfer function from ∆θ to ∆iq. The specific expressions for Gud(s), Guq(s), and Giq(s) are detailed in Eqs. (A15) and (A17) of Appendix C.

Fig. 4 shows a single-input single-output model of the grid-forming energy storage system that is based on the above analyses. This model incorporates five power feedback loops.

Figure 4: Single input single output model of grid-forming energy storage

Reference [18] first applied the damping torque analysis method to analyze the impact of excitation control systems on low-frequency oscillations in synchronous machines. The torque generated by excitation changes can be divided into two components: the synchronizing torque ∆MS∆δ, which is proportional to ∆δ, and the damping torque ∆MD∆δ, which is proportional to the rotational speed s∆δ. A block diagram of the second-order equivalent model for the synchronous machine is shown in Fig. 5.

Figure 5: Block diagram of a second-order equivalent model for a synchronous machine

Fig. 5 shows that the virtual rotor dynamics of the grid-forming energy storage grid-connected system. ∆Pin − ∆Po = Js∆ω has a structure that is similar to the rotor dynamics of a synchronous machine: ∆Mm − ∆Me = TJs∆ω. Based on this similarity, the damping torque method was adopted to analyze the low-frequency oscillation characteristics of the grid-forming energy storage grid-connected system. The relationship between the virtual angular acceleration ∆ω and the output power ∆Po is as follows:

ΔPoi=Φi(s)Δω,(14)

where the expressions for Φi(s) (I = 1, 2, 3, 4, 5) are as follows:

{Φ1(s)=DpΦ2(s)=1s(K1+K2s)Φ3(s)=1s(Gud(s)K3s)Φ4(s)=1s(Guq(s)K4s)Φ5(s)=1s(Giq(s)K5s).(15)

When a low-frequency oscillation disturbance occurs, all the state variables associated with the system oscillate at the same frequency ωs. Therefore, by substituting s = jωs into the power feedback transfer function, the power feedback transfer function can be expressed in the following form:

ΔPoiΔω=Re[Φi(jωs)]+j Im[Φi(jωs)].(16)

According to Eq. (16) and s = jωs, Eq. (14) can be transformed into the following form:

ΔPoi=Re[Φi(jωs)]Δω+−Im[Φi(jωs)]ωssΔω.(17)

Therefore, the power feedback loop of the system can be decomposed into the following form:

ΔPoi=(Di(ωs)+Ki(ωs)s)Δω.(18)

A multi-damping path model for the grid-forming energy storage grid-connected system can then be obtained, as shown in Fig. 6.

Figure 6: Grid-forming energy storage multi-damping path model

Di(ωs) is the equivalent damping coefficient for each branch and Ki(ωs) is the equivalent synchronizing coefficient for each branch. The expressions for Di(ωs) and Ki(ωs) are given by the following equations:

{D1(jωs)=DpD2(jωs)=Re(1jωs(K1+K2jωs))D3(jωs)=Re(1jωs(Gud(jωs)K3jωs))D4(jωs)=Re(1jωs(Guq(jωs)K4jωs))D5(jωs)=Re(1jωs(Giq(jωs)K5jωs)),{K1(jωs)=0K2(jωs)=−Im(1jωs(K1+K2jωs))ωsK3(jωs)=−Im(1jωs(Gud(jωs)K3jωs))ωsK4(jωs)=−Im(1jωs(Guq(jωs)K4jωs))ωsK5(jωs)=−Im(1jωs(Giq(jωs)K5jωs))ωs.(19)

3.3 Stability Criterion

Based on the grid-forming energy storage multi-damping-path model, this section derives the system stability indicators. To simplify the analysis, the multi-damping-path model of the grid-forming energy storage is equivalently represented as a second-order model using the superposition principle of transfer functions, as shown in Fig. 7.

Figure 7: Block diagram of the second order equivalent model for grid-forming energy storage

D(ωs) and K(ωs) are the damping coefficient and synchronizing coefficient, respectively. Their specific expressions are given by the following equations:

{D(ωs)=D1(ωs)+D2(ωs)+D3(ωs)+D4(ωs)+D5(ωs)K(ωs)=K1(ωs)+K2(ωs)+K3(ωs)+K4(ωs)+K5(ωs).(20)

Fig. 7 shows that the closed-loop transfer function of the system is as follows:

ΔθΔPin=1Js2+D(ωs)s+K(ωs),(21)

and the characteristic roots of Eq. (21) are

s1,2=−D(ωs)2J±12(D(ωs)J)2−4K(ωs)J.(22)

Based on the analysis of the characteristic roots (Eq. (22)), the stability criteria can be summarized as follows:

(1) When K(ωs) < 0, the equation has a positive real root and the system experiences a runaway loss of synchronization.

(2) When K(ωs) > 0 and D(ωs) < 0, the equation has a pair of conjugate complex roots with a positive real part and the system experiences oscillatory instability.

(3) When K(ωs) > 0 and D(ωs) > 0, the system remains stable.

4  Grid-Forming Energy Storage Low-Frequency Oscillation Analysis

In this section, a quantitative evaluation is conducted to assess the impact of control parameters, operating conditions, and grid strength on the low-frequency oscillation characteristics of grid-forming energy storage systems. The influence patterns of key control parameters on system low-frequency oscillations are systematically elucidated. Based on the grid-forming energy storage grid-connected system model shown in Fig. 1, an electromagnetic transient simulation model is built in the PSCAD/EMTDC platform to verify the correctness of the aforementioned damping model analysis conclusions. The key parameters of the model are listed in Table 3.

4.1 Analysis of the Damping Characteristics under Different Grid Strength Conditions

The quantitative analysis results for the five Damping Paths constructed for the grid-forming energy storage grid-connected system when the Short-Circuit Ratio (SCR) ranged from 7.2 to 8.4 are shown in Fig. 8 and Table 4. Damping Paths 3 and 5 dominated the total damping of the system and primarily provided negative damping. As the grid strength increased, the negative damping intensified, which degraded system stability. Furthermore, Damping Paths 3 and 5 were strongly correlated with the active voltage ∆ud, and changes in ∆ud strongly influenced the active power output of the system. When the grid strength was less than 7.9, Damping Paths 3 and 5 introduced negative damping, but the inherent virtual damping (Damping Path 1) compensated for the negative damping introduced as the grid strength increased. When the grid strength exceeded 7.82, the inherent virtual damping could not counteract the negative damping contributed by Damping Paths 3 and 5, which led to negative total damping and system instability.

Figure 8: Characteristic damping curves under different power grid intensities

The response curves for the active power Po and reactive power Qo of the system under the different grid strengths at t = 15 s when the grid strength was 5.1 to 7.6 (blue curve) and 8.0 (red curve) are shown in Fig. 9. When the grid strength was 8.0, the Po and Qo of the system showed low-frequency oscillatory instability with an oscillation frequency of 3.4 Hz. However, when the grid strength was 7.6, Po and Qo showed damped oscillations and the system was stable. The simulation results were consistent with the theoretical analysis and showed that grid-forming energy storage is prone to low-frequency oscillations under strong grid conditions.

Figure 9: Po and Qo response curves under different power grid intensities

4.2 Analysis of the Damping Characteristics under Different Operating Conditions

The quantitative analysis results for the five damping paths constructed for the grid-forming energy storage grid-connected system when the operating condition ranged from 0.1 to 1.3 MW are shown in Fig. 10 and Table 5. Damping Paths 3 and 5 dominated the total damping of the system and primarily provided negative damping. As the operating power decreased, the negative damping increased, which degraded system stability. Damping Paths 3 and 5 were highly correlated with the active voltage (∆ud) and changes in ∆ud strongly influenced the active power output of the system. When the operating power exceeded 0.51 MW, Damping Paths 3 and 5 provided negative damping, but the inherent virtual damping (Damping Path 1) compensated for the negative damping introduced as the operating power decreased. However, when the operating power fell below 0.51 MW, Damping Path 1 could not counteract the negative damping due to Damping Paths 3 and 5, resulting in negative total damping and system instability.

Figure 10: Characteristic damping curves under different working conditions

The response curves for the active power increment (∆Po) and reactive power increment (∆Qo) of the system under different operating conditions at t = 15 s and when the grid strength ranged from 5.1 to 7.8 are shown in Fig. 11. The blue curve represents Po = 1.3 MW and the red curve represents Po = 0.1 MW. When Po = 0.1 MW, the active power Po and reactive power Qo of the system showed low-frequency oscillatory instability with an oscillation frequency of 3.3 Hz. However, when Po = 1.3 MW, Po and Qo showed damped oscillations and the system was stable. The simulation results were consistent with the theoretical analysis and showed that grid-forming energy storage is prone to low-frequency oscillations under low operating conditions.

Figure 11: ∆Po and ∆Qo response curves under different working conditions

4.3 Analysis of the Damping Characteristics under Different Reactive Power-Voltage Regulation Coefficients (Kq)

The quantitative analysis results for the five damping paths when the reactive power-voltage regulation coefficient Kq ranged from 0.07 to 0.13, are shown in Fig. 12 and Table 6. Damping Paths 3 and 5 dominated the total damping of the system and primarily provided negative damping. As Kq decreased, the negative damping intensified, which degraded system stability. Damping Paths 3 and 5 were highly correlated with the active voltage ∆ud, and changes in ∆ud strongly influenced the active power output of the system. When Kq exceeded 0.099, Damping Paths 3 and 5 led to negative damping; however the inherent virtual damping (Damping Path 1) compensated for the negative damping introduced as Kq decreased. When Kq fell below 0.099, Damping Path 1 could not counteract the negative damping contributed by Damping Paths 3 and 5, which led to negative total damping and system instability.

Figure 12: Damping characteristic curves at different Kq levels

The response curves for Po and Qo under different reactive power-voltage regulation coefficients (Kq) at t = 15 s when the grid strength increased from 5.1 to 7.8 are shown in Fig. 13. The blue curve represents Kq = 0.13, and the red curve represents Kq = 0.07. When Kq = 0.07, Po and Qo showed low-frequency oscillatory instability with an oscillation frequency of 3.1 Hz, but when Kq = 0.13, Po and Qo showed that the oscillations had been damped and the system was stable. The simulation results were consistent with the theoretical analysis and showed that appropriate increases in Kq within a certain range can enhance the stability of grid-forming energy storage systems.

Figure 13: Response curves for Po and Qo at different Kq levels

4.4 Analysis of the Damping Characteristics under Different Active Voltage Outer Loop Proportional Coefficients (kp1)

The quantitative analysis results for the five damping paths when the active voltage outer loop proportional coefficient (kp1) ranged from 0.07 to 0.13 are shown in Fig. 14 and Table 7. Damping Paths 3 and 5 dominated the total damping of the system and primarily provided negative damping. As kp1 decreased, the negative damping increased, which degraded system stability. Damping Paths 3 and 5 were highly correlated with ∆ud, and changes in ∆ud strongly influenced the active power output. When kp1 exceeded 0.48, Damping Paths 3 and 5 led to negative damping, but Damping Path 1 compensated for the negative damping that was introduced as kp1 decreased. However, when kp1 fell below 0.48, Damping Path 1 could not counteract the negative damping contributed by Damping Paths 3 and 5, resulting in negative total damping and system instability.

Figure 14: Characteristic damping curves at different kp1 levels

The response curves for Po and Qo under different active voltage outer loop proportional coefficients (kp1) at t = 15 s when the grid strength increased from 5.1 to 7.8 are shown in Fig. 15. The blue curve represents kp1 = 0.8 and the red curve represents kp1 = 0.2. When kp1 = 0.2, Po and Qo showed low-frequency oscillatory instability with an oscillation frequency of 3.4 Hz, but when kp1 = 0.8, Po and Qo showed that the oscillations had been damped and the system was stable. The simulation results were consistent with the theoretical analysis and showed that appropriate increases in kp1 within a certain range can enhance the stability of grid-forming energy storage systems.

Figure 15: Response curves for Po and Qo at different kp1 levels

4.5 Analysis of the Damping Characteristics under Different Reactive Voltage Outer Loop Proportional Coefficients (kp2)

The quantitative analysis results for the five damping paths when the reactive voltage outer loop proportional coefficient (kp2) ranged from 0.07 to 0.13 are shown in Fig. 16 and Table 8. Damping Paths 3 and 5 dominated the total damping of the system and primarily provided negative damping. As kp2 decreased, the negative damping intensified, which degraded system stability. Damping Paths 3 and 5 were highly correlated with ∆ud and changes in ∆ud strongly influenced the active power output. When kp2 exceeded 0.49, Damping Paths 3 and 5 led to negative damping, but Damping Path 1 compensated for the negative damping introduced as kp2 decreased. However, when kp2 fell below 0.49, Damping Path 1 could not counteract the negative damping contributed by Damping Paths 3 and 5, resulting in negative total damping and system instability.

Figure 16: Characteristic damping curves at different kp2 levels

The response curves for Po and Qo under different reactive voltage outer loop proportional coefficients (kp2) at t = 15 s when the grid strength increased from 5.1 to 7.8 are shown in Fig. 17. The blue curve represents kp2 = 0.8, and the red curve represents kp2 = 0.2. When kp2 = 0.2, Po and Qo showed low-frequency oscillatory instability with an oscillation frequency of 3.4 Hz. However, when kp2 = 0.8, Po and Qo showed that the oscillations had been damped and the system was stable. The simulation results were consistent with the theoretical analysis and showed that appropriate increases in the kp2 values within a certain range can enhance the stability of grid-forming energy storage systems.

Figure 17: Po and Qo response curves at different kp2 levels

4.6 Multi-Parameter Coupling Effects on Damping Characteristics

Under a grid strength of 8.2 and an operating condition of 0.5 MW, this paper quantitatively analyzes the influence of the coupling characteristics between the active power-voltage outer-loop proportional coefficient kp1 and the reactive power-voltage outer-loop proportional coefficient kp2 on the system damping, based on the theoretically derived Eq. (19) and the damping stability criterion. The results are shown in Fig. 18. Fig. 18 illustrates the distribution characteristics of the system damping as kp1 and kp2 vary. The study reveals that kp1 and kp2 have a comparable impact on the system damping: when either kp1 or kp2 increases, the system damping is significantly enhanced; conversely, a decrease in either parameter leads to a corresponding reduction in damping. In particular, when kp1 or kp2 is set too small, the system damping decreases sharply and exhibits negative damping, thereby inducing oscillatory instability. This finding provides an important theoretical basis for the optimal design of controller parameters in grid-forming energy storage systems, indicating that appropriately increasing both the active power-voltage outer-loop proportional coefficient kp1 and the reactive power-voltage outer-loop proportional coefficient kp2 contributes to enhanced system damping and improved stability margin.

Figure 18: Effect of kp1 and kp2 variations on system damping

Under a grid strength of 8.2 and an operating condition of 0.5 MW, this paper quantitatively evaluates the influence of coupling between the proportional coefficient kp1 and integral coefficient ki1 in the active power-voltage outer loop on system damping characteristics, based on the theoretically derived Eq. (19) and the damping stability criterion. The results are shown in Fig. 19, which illustrates the distribution characteristics of system damping with variations in kp1 and ki1. The analysis demonstrates that system damping exhibits a significant increasing trend with the rise of either kp1 or ki1, whereas damping decreases correspondingly when either parameter is reduced. Particularly, when kp1 or ki1 is set too small, system damping drops markedly and may even develop negative damping, consequently inducing oscillatory instability. This regularity provides important theoretical guidance for optimizing controller parameters in grid-forming energy storage systems, indicating that appropriate increases in both kp1 and ki1 contribute to enhanced system damping and improved stability margins.

Figure 19: Effect of kp1 and ki1 variations on system damping

Under a grid strength of 8.2 and an operating condition of 0.5 MW, this paper quantitatively investigates the influence of coupling between the reactive power-voltage regulation coefficient Kq and the voltage droop coefficient Kv on system damping characteristics, based on the theoretically derived Eq. (19) and the damping stability criterion. The results are shown in Fig. 20, which illustrates the distribution characteristics of system damping with variations in Kq and Kv. The findings demonstrate that system damping strengthens with an increase in either Kq or Kv, and weakens with their decrease. Specifically, the voltage droop coefficient Kv exhibits a relatively minor influence on damping, whereas the reactive power-voltage regulation coefficient Kq plays a dominant role in determining system damping. When Kq is sufficiently large, the system consistently maintains positive damping to ensure stable operation. However, when Kq falls below a critical threshold, system damping deteriorates significantly and may even develop negative damping, thereby inducing oscillatory instability. This regularity provides important theoretical guidance for optimizing controller parameters in grid-forming energy storage systems, indicating that prioritizing appropriate Kq values contributes to enhanced system damping and improved stability margins.

Figure 20: Effect of Kq and Kv variations on system damping

4.7 Advantages over the Existing Solutions

To accurately address low-frequency oscillations, a critical challenge constraining the stable grid integration of grid-forming energy storage systems, and to systematically summarize the characteristics and limitations of existing research methods, Table 9 comprehensively analyzes mainstream research literature and systematically summarizes the advantages and disadvantages of typical methods. Eigenvalue analysis and impedance analysis are two typical methods for studying low-frequency oscillations in grid-forming energy storage systems. The eigenvalue analysis method is based on a detailed state-space model of the system and evaluates system stability by solving the model’s eigenvalues. However, the stability information provided by this method is mostly qualitative, making it difficult to precisely quantify the degree of instability of oscillation modes or to reveal the underlying instability mechanisms. The impedance analysis method divides the system into equipment subsystems and grid subsystems, establishes equivalent impedance models for both, and applies the Nyquist criterion for stability discrimination. Although this method has clear physical significance in frequency domain analysis, it struggles to accurately reflect the impact of internal dynamic coupling on stability and cannot intuitively explain the instability mechanism. To overcome the limitations of the above methods, this paper proposes a multi-damping-path model for grid-forming energy storage integration based on the damping torque analysis method. This model simplifies the eighth-order state-space model of the system into a second-order equivalent model, significantly reducing the analytical complexity of low-frequency oscillation problems. Based on the analytical expression of the damping coefficient established in Eq. (19), quantitative evaluation of the influence of control parameters, operating conditions, and grid strength on low-frequency oscillation damping can be achieved.

The constructed model reasonably explains the phenomenon that the system is prone to low-frequency oscillations when grid strength increases but remains stable when grid strength is weak. Research shows that under fixed control parameters, when the grid strength gradually increases to a critical value (7.82 under the conditions set in this paper), the system will enter an unstable state and exhibit low-frequency oscillations. On this basis, this paper further analyzes the influence characteristics of various parameters in the grid-forming energy storage system on stability. The results indicate that, within a certain range, appropriately increasing the active power-voltage outer-loop proportional coefficient kp1, the active power-voltage outer-loop integral coefficient ki1, the reactive power-voltage outer-loop proportional coefficient kp2, and the reactive power-voltage regulation coefficient Kq of the grid-forming energy storage helps enhance system damping and improve the stable operation margin. This study provides a theoretical basis and methodological support for the adaptive adjustment of control parameters in grid-forming energy storage integration systems under different operating conditions and grid strengths.

4.8 Engineering Challenges and Technical Prospects

With the continuous increase in renewable energy penetration, grid-forming energy storage has been identified as a key device for constructing new-type power systems due to its dual potential in enhancing grid flexibility and stability. It not only effectively addresses the integration challenges of renewable energy sources such as wind and solar, meeting multi-timescale power balance requirements, but also provides critically needed rotational inertia and damping support to suppress wide-band oscillations caused by high proportions of power electronic devices.

However, as described in this paper, grid-forming energy storage faces small-signal stability issues under strong grid conditions, particularly low-frequency oscillation risks, which represent a core challenge for its large-scale application. The root cause of this challenge lies in the complex dynamic interaction mechanisms among its internal multiple control loops, making traditional stability analysis methods difficult to accurately quantify the contribution of each control parameter to system damping, resulting in a lack of clear theoretical guidance for parameter setting. The multi-damping torque model proposed in this paper aims to address this challenge, providing basis for optimizing control parameters of grid-forming energy storage from the perspective of damping path decoupling.

From the perspective of engineering application and standardization, grid-forming technology is gradually moving toward standardization. To ensure the safe and stable operation of power systems, various countries have successively issued relevant technical standards. As shown in Table 10, Chinese standards explicitly require that grid-forming equipment must provide effective damping control when the system experiences low-frequency oscillations in the range of 0.2–2.5 Hz with amplitudes greater than 0.003 Hz. This regulation not only imposes mandatory requirements on equipment performance but also demonstrates the practical significance of this paper’s research content, namely the accurate analysis of damping characteristics.

The future development of grid-forming energy storage technology depends on the deep integration of “mechanism research” and “engineering application”. On one hand, it is necessary to further develop refined analysis methods similar to the multi-damping torque model to reveal oscillation mechanisms under more complex operating conditions. This paper currently focuses only on low-frequency oscillation analysis of single grid-forming energy storage unit grid-connected systems, while small-signal stability analysis for multiple grid-forming energy storage unit grid-connected systems in practical engineering still requires further research. On the other hand, due to limited conditions, this paper suggests that follow-up research should further promote the transformation of these theoretical achievements into parameter setting guidelines and standard testing procedures that are easy to engineering apply, ultimately forming a complete technical chain from modeling, analysis to control, and verification, providing solid support for constructing new-type power systems with high renewable energy penetration.

5  Conclusion

This study addressed the issue of low-frequency oscillations in grid-forming energy storage systems under strong grid conditions. A multi-damping torque model for grid-forming energy storage was constructed, a quantitative analysis of low-frequency oscillation stability was performed at various grid strengths and under different operating conditions and control parameters, and the mechanism associated with the low-frequency oscillations during grid-forming energy storage under strong grid conditions was revealed. The main conclusions are as follows:

(1)   A multi-damping torque model for grid-forming energy storage systems was constructed and was used to identify the mechanism associated with low-frequency oscillations in such systems under strong grid conditions. As grid strength increased, the negative damping characteristics significantly intensified. However, the system was unstable and induced low-frequency oscillations when the total system damping became negative.

(2)   The impact of operating conditions on the damping characteristics of grid-forming energy storage was quantitatively evaluated. The results showed that grid-forming energy storage was prone to low-frequency oscillation instability under low operating conditions. As the system operating power decreased, the inherent virtual damping of the system often became insufficient to counteract the negative damping components. This resulted in negative total damping, caused system instability, and led to low-frequency oscillations.

(3)   The influence of different control parameters on the low-frequency oscillation stability of grid-forming energy storage was quantitatively analyzed. Appropriate increases in the reactive power-voltage regulation coefficient (Kq), the active voltage outer loop proportional coefficient (kp1), the active power-voltage outer-loop integral coefficient (ki1), and the reactive voltage outer loop proportional coefficient (kp2) effectively enhanced the low-frequency oscillation stability of grid-forming energy storage. The results from this study provide theoretical guidance for the tuning of control parameters in grid-forming energy storage systems.

Acknowledgement: We thank International Science Editing (http://www.internationalscienceediting.com, accessed on 01 January 2025) for editing this manuscript.

Funding Statement: This research was funded by Supported by the Science and Technology Project of State Grid Corporation of China (2025220202000070).

Author Contributions: The authors confirm contribution to the paper as follows: Conceptualization—ideas, formulation or evolution of overarching research goals and aims, Qiang Liu and Yongqiang Zhou; Writing—original draft—preparation, creation and/or presentation of the published work, specifically writing the initial draft (including substantive translation), Chaoyang Lu; Writing—review & editing—preparation, creation and/or presentation of the published work by those from the original research group, specifically critical review, commentary or revision—including pre- or post-publication stages, Zhen Yan; Project administration—management and coordination responsibility for the research activity planning and execution, Gangui Yan and Yupeng Wang; Software and validation—oversight and leadership responsibility for the research activity planning and execution, including mentorship external to the core team and verification, whether as a part of the activity or separate, of the overall replication/reproducibility of results/experiments and other research outputs, Cheng Yang. All authors reviewed the results and approved the final version of the manuscript.

Availability of Data and Materials: Not applicable.

Ethics Approval: Not applicable.

Conflicts of Interest: The authors declare no conflicts of interest to report regarding the present study.

Appendix A

The state-space matrix A is shown below:

A=[d1d2d3d4d5d6d7d8d9d100d110d12d13d14c1c2c3c4c5c6c7c8−b9−b100−b110−b12−b13ud0−b14kp1c1−1kp1c2kp1c3+ki1kp1c4kp1c5kp1c6kp1c7kp1c8−kp2b9−kp2b10−10−kp2b11+ki20−kp2b12−kp2b13kp2(ud0−b14)+id0−1.5b1id0+1.5ud0J−1.5b2id0J−1.5b3id0J−1.5b4id0J−1.5b5id0J−1.5b6id0J−1.5b7id0+DpJ−1.5b8id0J00000010]

where Lfg, a1–a14, b1–b14, c1–c8, and d1–d14 are shown below:

Lfg=LgLf+Lg,(A1)

a1=−kp3kp1KvKq−kp3kp1+1,a2=−1.5kp3kp1Kqid0,a3=−kp3,a4=1.5kp3kp1Kqud0−ω0Lf,a5=kp3ki1,a6=ki3,a7=ω0Lfid0,a8=−kp4kp2+1,a9=ω0Lf,a10=−kp4,a11=kp4ki2,a12=ki4,a13=Lfid0,a14=kp4kp2ud0+kp4id0,(A2)

b1=Lfg2a2a9+(1−Lfga8)Lfga3(1−Lfga8)(1−Lfga1),b2=Lfg2a2a10+(1−Lfga8)Lfga4(1−Lfga8)(1−Lfga1),b3=Lfga5(1−Lfga1),b4=Lfg2a2a11(1−Lfga8)(1−Lfga1),b5=Lfga6(1−Lfga1),b6=Lfg2a2a12(1−Lfga8)(1−Lfga1),b7=Lfg2a2a13(1−Lfga8)(1−Lfga1),b8=Lfg2a2a14+(1−Lfga8)Lfga7(1−Lfga8)(1−Lfga1),b9=Lfga91−Lfga8,b10=Lfga101−Lfga8,b11=Lfga111−Lfga8,b12=Lfga121−Lfga8,b13=Lfga131−Lfga8,b14=Lfga141−Lfga8,(A3)

c1=(−KvKq−1)b1−1.5Kqb9id0,c2=(−KvKq−1)b2−1.5Kqb10id0+1.5Kqud0,c3=(−KvKq−1)b3,c4=(−KvKq−1)b4−1.5Kqb11id0,c5=(−KvKq−1)b5,c6=(−KvKq−1)b6−1.5Kqb12id0,c7=(−KvKq−1)b7−1.5Kqb13id0,c8=(−KvKq−1)b8−1.5Kqb14id0,(A4)

d1=1Lf+Lg(a1b1+a2b9+a3),d2=1Lf+Lg(a1b2+a2b10+a4)+ω0,d3=1Lf+Lg(a1b3+a5),d4=1Lf+Lg(a1b4+a2b11),d5=1Lf+Lg(a1b5+a6),d6=1Lf+Lg(a1b6+a2b12),d7=1Lf+Lg(a1b7+a2b13),d8=1Lf+Lg(a1b8+a2b14+a7),d9=1Lf+Lg(a8b9+a9)−ω0,d10=1Lf+Lg(a8b10+a10),d11=1Lf+Lg(a8b11+a11),d12=1Lf+Lg(a8b12+a12),d13=1Lf+Lg(a8b13+a13),d14=1Lf+Lg(a8b14+a14)(A5)

Appendix B

The state-space matrix A1 is shown below:

A1=[h1h20h3−f6−f50−f4+ud0h4/Jh5/J−Dp/Jh6/J0010]

where e1–e8, f1–f6, g1–g6, and h1–h6 are shown below:

e1=−ω0Lgkp2,e2=ω0Lgid0+ω0Lgkp2ud0,e3=ω0Lgki2,e4=ω0Lgkp1KvKq+ω0Lgkp1,e5=1.5ω0Lgkp1kp2Kqud0+1.5ω0LgKqkp1id0+1,e6=−1.5ω0Lgkp1Kqud0id0−1.5ω0Lgkp1Kqud0kp2ud0+ω0Lgiq0,e7=−1.5ω0Lgkp1Kqud0ki2,e8=−ω0Lgki1,(A6)

f1=e6e1−e5e2e5−e4e1,f2=e7e1−e5e3e5−e4e1,f3=e8e1e5−e4e1,f4=e6−e4e2e4e1−e5,f5=e7−e4e3e4e1−e5,f6=e8e4e1−e5,(A7)

g1=−kp1KvKqf1−kp1f1−1.5kp1kp2Kqud0f4−1.5Kqkp1id0f4+1.5kp1Kqud0id0+1.5kp1Kqud0kp2ud0−iq0,g2=−kp1KvKqf2−kp1f2−1.5kp1kp2Kqud0f5−1.5Kqkp1id0f5+1.5kp1Kqud0ki2g3=−kp1KvKqf3−kp1f3−1.5kp1kp2Kqud0f6−1.5Kqkp1id0f6+ki1,g4=−kp2f4+id0+kp2ud0,g5=−kp2f5+ki2,g6=−kp2f6,(A8)

h1=−KvKqf3−f3−1.5Kqid0f6+1.5ud0Kqg6,h2=−KvKqf2−f2−1.5Kqid0f5+1.5ud0Kqg5,h3=−KvKqf1−f1−1.5Kqid0f4+1.5ud0Kqg4,h4=−1.5f3id0−1.5ud0g3,h5=−1.5f2id0−1.5ud0g2,h6=−1.5f1id0−1.5ud0g1.(A9)

Appendix C

Detailed Derivation of Multi-Damping-Path Analysis

The output power ∆Po can be expressed as a function of the reduced-order model state variables ∆X1 = [∆x1, ∆x2, ∆ω, ∆θ] according to the transfer function block diagram of grid-forming energy storage shown in Fig. A1.

ΔPo=DpΔω+1.5id0Xg2kp2ki1+1.5ud0ki1−Xg2kp1kp2(−KvKq−1)+1.5Kqkp1Xg(kp2ud0+id0)+1Δx1+−1.52Xg2Kqkp1ki2id02−1.5id0Xgki2−1.5ud0Xgkp1ki2(−KvKq−1)+1.52kp1Kqud02ki2−Xg2kp1kp2(−KvKq−1)+1.5Kqkp1Xg(kp2ud0+id0)+1Δx2+(1.52kp1Kqud02−1.52Xg2id02Kqkp1id0−1.5id0Xg−1.5ud0Xgkp1(−KvKq−1))(id0+kp2ud0)−Xg2kp1kp2(−KvKq−1)+1.5Kqkp1Xg(kp2ud0+id0)+1Δθ.(A10)

Figure A1: Block diagram of the single input and single output transfer functions for grid-forming energy storage

The state variables ∆x1, ∆x2, and ∆ω need to be expressed in terms of the state variable ∆θ. The equations for the intermediate state variables ∆x1 and ∆x2 are shown below:

{dΔx1dt=(−KvKq−1)Δud−1.5Kqid0Δuq+1.5ud0KqΔiqdΔx2dt=−Δuq+ud0Δθ⇒{sΔx1=(−KvKq−1)Δud−1.5Kqid0Δuq+1.5ud0KqΔiqsΔx2=−Δuq+ud0Δθ.(A11)

The state variables ∆ud, ∆uq, and ∆iq need to be derived and expressed in terms of the state variable ∆θ. The expressions for ∆ud, ∆uq, ∆id, and ∆iq are shown below:

{Δud=−XgΔiqΔuq=XgΔid,{Δid=−KvKqG1Δud−1.5Kqid0G1Δuq+1.5ud0KqG1Δiq−G1ΔudΔiq=(ud0G2+id0)Δθ−ΔuqG2,(A12)