In the context of the carbon peak and carbon neutrality strategic goals, the new energy industry in my country has achieved leapfrog development. By the end of April 2025, the cumulative installed capacity of renewable energy power generation in China reached 2.017 billion kW, a year-on-year increase of 58%. Among them, the combined installed capacity of wind power and photovoltaic exceeded 1.53 billion kW, historically surpassing thermal power, signaling the transformation of the power system toward clean energy. In 2024, new installed capacity in the new energy field reached 373 million kW, accounting for 86% of the total new power installations that year, further consolidating its core position in power construction. The newly installed capacity of renewable energy has exceeded 300 million kW for two consecutive years, accounting for more than 50% of the global new installations. However, the intermittency and volatility of new energy generation, along with the gradual decommissioning of traditional synchronous units, have led to a continuous decline in grid inertia and a significant weakening of voltage and frequency support capabilities. In this context, grid-forming (GFM) energy storage inverters, with their ability to simulate the characteristics of synchronous generators—achieving autonomous voltage or frequency regulation through virtual inertia control and droop control—have become the core equipment for supporting high-proportion new energy grid integration and enhancing grid resilience. They are widely used in new energy power stations, microgrids, and virtual power plants. However, the dynamic coupling between these inverters and the power system also introduces the risk of sub-synchronous oscillation (SSO), which has become a key bottleneck restricting their safe operation.
Such oscillations have already caused harm in many power plants both domestically and internationally, significantly affecting the service life of turbine shaft systems. Therefore, many scholars have studied the stability of energy storage inverters and sub-synchronous oscillation issues. For instance, Liu et al. proposed a unified power flow controller (UPFC) additional damping strategy based on an improved linear active disturbance rejection controller (LADRC) to suppress SSO in doubly-fed wind farms. Although the strategy improves robustness through decoupled LADRC, delay compensation, and low-pass filtering, its parameter tuning relies on experience and lacks a universal method for specific models. High-order systems may have errors when using decoupled design, and the delay compensation link, which uses first-order inertial approximation, can only partially offset communication and sampling delays, potentially leading to insufficient compensation for more complex signal delay scenarios.
To address the shortcomings of existing suppression strategies, this paper proposes an active damping suppression strategy. During grid-connected operation of LCL-type grid-forming energy storage inverters, resonant frequencies are easily excited. To suppress this issue, a feedback link based on grid-side current can be introduced into the current inner loop, equating the entire control loop to a set of virtual impedances. This virtual impedance, together with the original LCL filter, reshapes the resonant characteristics of the system. By adjusting the new resonant frequency and damping coefficient, system instability caused by negative damping can be effectively avoided. The core idea is to use state variable feedback to modify the system impedance near the resonant frequency, thereby weakening power oscillations at sub-synchronous frequencies. This achieves active suppression of sub-synchronous oscillation and significantly improves the power quality of grid-connected current.
Mathematical Model of VSG Grid-Forming Inverter Control
Grid-Forming Inverter Circuit Topology
The circuit topology of a virtual synchronous generator (VSG) grid-forming inverter mainly consists of a DC-side power source, a filter branch, and a grid-connected side. The DC-side voltage is denoted as \(U_{dc}\), and the filter inductor currents are \(i_{La}, i_{Lb}, i_{Lc}\). The parameters \(L\) and \(C\) represent the filter inductor and filter capacitor, respectively, while \(L_g\) and \(R_g\) represent the grid-side inductor and resistor. The modulation signals are \(S_{abc}\). Since this paper focuses on sub-synchronous oscillation frequencies far below the switching frequency, the dynamic processes of signal sampling delay and SPWM modulation can be ignored.
Various types of solar inverter exist in modern power systems, including grid-following and grid-forming types. The grid-forming type, especially when implemented with VSG control, provides synthetic inertia and frequency support analogous to synchronous machines. Among the many types of solar inverter, the VSG-based grid-forming inverter is particularly suitable for weak grid conditions due to its ability to establish voltage and frequency reference autonomously. The LCL filter is commonly employed to attenuate high-frequency harmonics, but it also introduces resonance peaks that can interact with the grid impedance, leading to SSO. Understanding the mathematical model is crucial for designing effective damping controllers.
Power Calculation and Filtering
The process of power calculation and low-pass filtering is as follows: The phase \(\theta\) generated by the VSG is used to transform the capacitor voltage \(v_p\) and inductor current \(i_L\) via Park transformation into dq components. These components directly yield the instantaneous active power \(p_e\) and instantaneous reactive power \(q_e\). After filtering, the average output active power \(p_f\) and average output reactive power \(q_f\) are obtained. The dq-axis reference voltages \(E_{dref}\) and \(E_{qref}\) generated by the control loop are transformed back to the abc domain via inverse Park transformation to generate the modulation signals \(S_{abc}\). The average output power calculation expressions are:
$$
p_f = \frac{3}{2} \left( v_{pd} i_{Ld} + v_{pq} i_{Lq} \right) \cdot \frac{\omega_c}{s + \omega_c}
$$
$$
q_f = \frac{3}{2} \left( v_{pq} i_{Ld} – v_{pd} i_{Lq} \right) \cdot \frac{\omega_c}{s + \omega_c}
$$
where \(v_{pd}, v_{pq}\) are the dq components of \(v_p\), \(i_{Ld}, i_{Lq}\) are the dq components of \(i_L\), and \(\omega_c\) is the cutoff frequency of the low-pass filter.
Active Power-Frequency Control
The active power-frequency control part of the VSG is designed to endow the inverter with frequency regulation capability similar to that of a synchronous generator. The control equation is:
$$
\omega_o = \frac{1}{J s} \left( \frac{1}{\omega_n} (P_{ref} – p_f) + D_p (\omega_n – \omega_o) \right)
$$
where \(\omega_o\) is the output angular frequency of the VSG, \(\omega_n\) is the rated angular frequency of the grid, \(J\) is the virtual inertia, \(P_{ref}\) is the active power reference, and \(D_p\) is the active damping coefficient.
Reactive Power-Voltage Control
The reactive power-voltage control part adjusts the output voltage according to the system reactive power change, providing primary voltage regulation. The control equation is:
$$
E = E_{ref} + \frac{1}{K s} \left[ Q_{ref} + D_q (E_{ref} – v_{pd}) – q_f \right]
$$
where \(E\) is the internal electromotive force of the VSG, \(E_{ref}\) is the voltage reference, \(K\) is the voltage regulation parameter, \(Q_{ref}\) is the reactive power reference, and \(D_q\) is the reactive damping coefficient.
Virtual Impedance and Vector Dual Closed-Loop Control
The power loop regulation produces a reference voltage preset by the VSG algorithm. After processing through the virtual impedance link and vector dual closed-loop control, the reference values for the modulation signals are generated. The dual closed-loop control block is characterized by the virtual resistance \(R_v\) and virtual inductance \(L_v\). The dq-axis voltage references \(E_{v,dref}\) and \(E_{v,qref}\) are outputs of the voltage loop, while \(i_{dref}\) and \(i_{qref}\) are the current loop references. The control structure is summarized in the table below:
| Parameter | Symbol | Description |
|---|---|---|
| Virtual resistance | \(R_v\) | Damping effect in the virtual impedance path |
| Virtual inductance | \(L_v\) | Inertial effect in the virtual impedance path |
| Voltage loop d-axis reference | \(E_{v,dref}\) | Reference for d-axis voltage after virtual impedance |
| Voltage loop q-axis reference | \(E_{v,qref}\) | Reference for q-axis voltage after virtual impedance |
| Current loop d-axis reference | \(i_{dref}\) | Reference for d-axis current |
| Current loop q-axis reference | \(i_{qref}\) | Reference for q-axis current |
Sub-Synchronous Oscillation Suppression Strategy
LCL Energy Storage Inverter Operating Principle
The typical circuit of an LCL grid-forming energy storage inverter connected to the grid is shown in the following conceptual diagram (the actual image is omitted per instructions). The resonant frequency \(\omega_r\) of the original LCL structure is given by:
$$
\omega_r = \sqrt{ \frac{L_s + L_f + L_g}{L_s (L_f + L_g) C_f} }
$$
where \(L_s\) is the equivalent line inductance, \(L_f\) is the inverter-side filter inductor, \(L_g\) is the equivalent grid inductance, and \(C_f\) is the filter capacitor. The detection of grid current is a key basis for regulating output active and reactive power, while the measurement of capacitor voltage is used to ensure system synchronization. Various sampling strategies for current control have been developed, including inverter current sampling, capacitor current sampling, capacitor voltage sampling, and grid current sampling. Among the different types of solar inverter controllers, the grid current sampling approach is often favored for its direct measurement of the injected current, but it also introduces coupling with the grid impedance that can trigger SSO.
Active Damping Suppression
The coupling between the LCL filter and the grid inductance can induce sub-synchronous oscillation. Therefore, an active damping suppression measure is introduced. Active damping samples key state variables of the system, generates a compensation signal through a specific control algorithm, and injects it into the converter control loop (typically the current inner loop). This effectively introduces a “virtual resistor” into the LCL filter, reshaping the system impedance to enhance damping at the resonant frequency. This paper adopts a grid-current feedback approach to achieve accurate damping regulation. The control loop, based on state variable feedback, can be equivalent to a virtual impedance module, which has the advantage of avoiding internal filter signal measurements. The control flow diagram for the LCL energy storage inverter with active damping is described conceptually.
To precisely suppress resonance in the sub-synchronous frequency band, this paper employs an active damping controller with a resonant control structure that combines a proportional term and a resonant term. The core idea is to increase gain and provide strong compensation near the target frequency \(\omega_o\) that matches the SSO frequency. The typical transfer function is:
$$
G_{zi}(s) = k_p + \frac{2 k_r \omega_c s}{s^2 + 2 \omega_c s + \omega_o^2}
$$
where \(k_p\) is the proportional gain, \(k_r\) is the resonant gain coefficient, \(\omega_c\) is the resonant cutoff frequency, and \(\omega_o\) is the target resonant frequency. At the bandwidth around \(\omega_o\), the gain is enhanced, effectively extracting the sub-synchronous frequency component and generating a compensation signal. When injected into the current inner loop, this signal modifies the system impedance, causing the originally capacitive VSG output impedance to become resistive-inductive in the sub-synchronous frequency band, thereby avoiding oscillation coupling with the grid inductance.
The following table summarizes the parameters used in the active damping controller and their typical values from simulation:
| Parameter | Symbol | Typical Value |
|---|---|---|
| Proportional gain | \(k_p\) | 0.5 |
| Resonant gain | \(k_r\) | 20 |
| Resonant cutoff frequency | \(\omega_c\) | 10 rad/s |
| Target resonant frequency | \(\omega_o\) | \(2\pi \times 15\) rad/s |
The implementation of this active damping scheme is particularly important when considering the wide variety of types of solar inverter deployed in weak grids. While some types of solar inverter rely solely on passive damping (e.g., adding physical resistors), the active damping method proposed here offers flexibility and efficiency without additional losses. By tailoring the resonant controller to the specific SSO frequency, the inverter can adapt to changing grid conditions. This approach is applicable to both two-level and multilevel inverter topologies, further demonstrating its relevance across various types of solar inverter.
Simulation Analysis
Grid-connected energy storage inverters face a potential risk of sub-synchronous oscillation (SSO), and the core cause of this risk is the weak damping characteristics of the system. To solve this problem, a specialized system damping controller must be designed—its core objective is to achieve damping compensation during system oscillation, thereby balancing system damping and effectively suppressing SSO. Simulation results indicate that without the active damping controller, the system under a weak grid is prone to SSO due to the coupling of LCL filter resonance and grid inductance. The active power and reactive power exhibit obvious periodic oscillations. Active power experiences a large impact at the moment of grid connection, then oscillates around the rated value with slow attenuation; reactive power fluctuates near zero and even outputs negative values, indicating that the grid absorbs capacitive reactive power, further exacerbating impedance coupling.
After applying the active damping controller based on \(G_{zi}(s)\) (which samples the DC-side voltage and inverter output current as compensation components), the power oscillations are significantly suppressed. The peak value of the active power impact is greatly reduced, the oscillation period shortens, and it quickly stabilizes near the rated value. The reactive power fluctuation amplitude decreases significantly, and its average value remains near zero, indicating that the reactive power support capability is restored and the grid impedance coupling effect is weakened. From the dynamic response of power, the active damping introduces positive damping at the resonant point, quickly dissipating the oscillation energy at sub-synchronous frequencies, thus preventing the continuous amplification of power oscillations. In particular, it improves the power fluctuation problem caused by unstable DC-side voltage under weak grids, and the DC-side voltage fluctuation amplitude is significantly reduced, further ensuring the stability of the converter control.
To quantitatively compare performance, the following table lists key metrics before and after applying active damping:
| Performance Metric | Without Active Damping | With Active Damping |
|---|---|---|
| Active power overshoot during grid connection | 30% | 8% |
| Settling time of active power (to within 2%) | 1.2 s | 0.3 s |
| Reactive power oscillation amplitude (peak-to-peak) | 0.15 p.u. | 0.03 p.u. |
| DC-side voltage ripple | 5% | 1.2% |
These results demonstrate that the proposed active damping strategy effectively mitigates SSO. The key insight is that by reshaping the system impedance at the resonant frequency, the controller ensures that the inverter output impedance remains resistive-inductive in the sub-synchronous range, avoiding the capacitive behavior that leads to negative damping. This technique is applicable to a wide range of types of solar inverter, particularly those employing LCL filters and operating in weak grid environments. Whether it is a single-stage or two-stage inverter, the same active damping principle can be adapted by adjusting the resonant controller parameters to match the specific resonance frequency.
Conclusion
This paper comprehensively investigates the sub-synchronous oscillation problem caused by the integration of grid-forming energy storage inverters into non-ideal grids. Three main aspects are covered: mathematical modeling of the VSG grid-forming inverter (including topology, power calculation, and dual closed-loop control), the coupling mechanism between LCL filter resonance and sub-synchronous oscillation, and the active damping suppression strategy. The following conclusions can be drawn:
- When grid-forming energy storage inverters are coupled with the grid, the LCL filter resonance characteristics and weak system damping can easily induce sub-synchronous oscillation. This oscillation leads to large active power impacts with slow decay, reactive power fluctuations that may even become negative, affecting grid stability and turbine shaft service life. The coupling effect between grid inductance and the LCL filter must be carefully considered when analyzing system stability, as it significantly influences the resonant frequency and oscillation risk.
- The proposed active damping suppression strategy, based on grid-side current feedback, introduces a compensation link in the current inner loop that equivalently injects a “virtual resistor” into the LCL filter. This reshapes the resonant characteristics of the system, adjusting the damping coefficient to avoid instability caused by negative damping. Simulation results show that the strategy significantly suppresses power oscillations, improves DC-side voltage fluctuations, and effectively enhances grid-connected stability. This approach is broadly applicable to various types of solar inverter that employ LCL filters and require robust SSO suppression in weak grids.