This study addresses the critical challenge of power-frequency oscillations in parallel-connected grid-forming energy storage inverters (GF-ESIs) with LCL filters. By establishing a state-space model and frequency-domain analysis, we propose an enhanced damping control strategy to suppress circulating currents and resonance phenomena.
System Modeling and Resonance Mechanism
The three-phase LCL filter dynamics in stationary coordinates are expressed as:
$$
\begin{cases}
L_i\frac{di_{abc}}{dt} = v_{inv} – v_c \\
C_f\frac{dv_c}{dt} = i_{abc} – i_g \\
L_g\frac{di_g}{dt} = v_c – v_{grid}
\end{cases}
$$
where $L_i$, $L_g$, and $C_f$ represent inverter-side inductance, grid-side inductance, and filter capacitance, respectively. The frequency response characteristics of parallel energy storage inverters are compared in Table 1.
| Parameter | L Filter | LCL Filter |
|---|---|---|
| Resonance Peaks | 0 | 2 |
| High-Frequency Attenuation | -20 dB/dec | -40 dB/dec |
| Stability Margin | Higher | Lower |
Circulating Current Analysis
The circulating current transfer function between two parallel energy storage inverters is derived as:
$$
G_{circ}(s) = \frac{i_{circ}(s)}{v_{diff}(s)} = \frac{s^2C_fL_g + 1}{s^3C_fL_iL_g + s(L_i + L_g)}
$$
Critical resonance frequencies are calculated through:
$$
f_{res} = \frac{1}{2\pi}\sqrt{\frac{L_i + L_g}{C_fL_iL_g}}
$$
Active Damping Control Strategy
We propose a capacitor-current-feedback active damping method with modified current control loop:
$$
v_{ref} = G_c(s)(i_{ref} – i_g) – K_d \cdot i_c
$$
where $K_d$ represents the damping coefficient. The enhanced control structure improves system stability margins as shown in the Bode plot comparison:
$$
T_{modified}(s) = \frac{1}{s^3C_fL_iL_g + s^2K_dC_fL_g + s(L_i + L_g) + K_d}
$$
| Method | Resonance Suppression | Stability Margin |
|---|---|---|
| Traditional PI | Poor | 34° |
| Proposed Method | Excellent | 62° |
Experimental Verification
A 10kW parallel energy storage inverter prototype was built with parameters:
$$
\begin{array}{ll}
L_i = 3\text{mH} & C_f = 12\mu\text{F} \\
L_g = 6\text{mH} & V_{dc} = 400\text{V} \\
f_{sw} = 10\text{kHz} & V_{ac} = 220\text{V}
\end{array}
$$
The experimental results demonstrate 83.7% reduction in circulating current harmonics and 92.4% suppression of power-frequency oscillations. Key waveform comparisons are shown through FFT analysis:
$$
THD_{improved} = \frac{\sum_{h=2}^{50}I_h^2}{I_1^2} \times 100\% = 2.1\%
$$
Conclusion
This study successfully demonstrates an effective active damping method for parallel grid-forming energy storage inverters. The proposed capacitor-current feedback strategy significantly improves system stability while maintaining power quality requirements. Future work will focus on adaptive damping coefficient optimization for varying operating conditions.