In modern power systems, the increasing integration of renewable energy sources, such as wind, solar, and storage-based multi-microgrids, has introduced significant challenges to grid stability. Among these, oscillations in grid-tied inverters are a critical issue that can compromise power quality and system reliability. This article explores the modeling and suppression of oscillation characteristics in grid-tied inverters when high proportions of wind-solar-storage multi-microgrids are connected. I will delve into the small-signal equivalent circuit modeling of grid-tied inverters, analyze the oscillation suppression mechanisms, and propose effective mitigation strategies using dq decoupling and PI control methods. The goal is to enhance the oscillation suppression effect of grid-tied inverters, ensuring stable operation in complex grid environments.
The structure of grid-tied inverters with high penetration of wind-solar-storage multi-microgrid access typically involves multiple inverters connected in parallel to the grid. Each grid-tied inverter interfaces with renewable sources, such as photovoltaic panels or wind turbines, and energy storage systems, converting DC power to AC for grid integration. However, when numerous microgrids are接入, the interaction between inverter outputs and grid impedance can lead to subsynchronous oscillations, which are detrimental to system performance. To address this, I first construct a small-signal equivalent circuit model for the grid-tied inverter. This model captures the dynamic behavior of the grid-tied inverter under oscillation conditions, allowing for impedance analysis and control design.
The small-signal equivalent circuit model of a grid-tied inverter is derived from its structural components, including inductors, capacitors, and control loops. For a single grid-tied inverter, the output current in the frequency domain can be expressed as:
$$I_g(s) = B_i(s) I_{ref}(s) + B_{iu}(s) U_{SW}(s) + B_{ie}(s) E(s)$$
where \( B_i(s) \), \( B_{iu}(s) \), and \( B_{ie}(s) \) are transfer functions representing the input current to output current, harmonic voltage source to output current, and disturbance to output current, respectively. These functions are defined as:
$$B_i(s) = \frac{H_{OSW}(s) H_{OI}(s)}{H_{OSW}(s) H_{OI}(s) + H_{OE}(s) H_{OI}(s) + H_{OE}(s) H_{OSW}(s)}$$
$$B_{iu}(s) = \frac{H_{OI}(s)}{H_{OSW}(s) H_{OI}(s)} + \frac{H_{OI}(s)}{H_{OE}(s) H_{OI}(s)} + \frac{H_{OI}(s)}{H_{OE}(s) H_{OSW}(s)}$$
$$B_{ie}(s) = \frac{H_{OSW}(s) + H_{OI}(s)}{H_{OSW}(s) H_{OI}(s)} + \frac{H_{OSW}(s) + H_{OI}(s)}{H_{OE}(s) H_{OI}(s)} + \frac{H_{OSW}(s) + H_{OI}(s)}{H_{OE}(s) H_{OSW}(s)}$$
Here, \( H_{OSW}(s) \), \( H_{OI}(s) \), and \( H_{OE}(s) \) denote the output impedances related to parallel output, electromotive force, and feedforward control, respectively. The output impedance of the grid-tied inverter is crucial for oscillation analysis. For instance, the output impedance \( H_{OSW}(s) \) can be derived as:
$$H_{OSW}(s) = \frac{B_i(s)}{B_{iu}(s)} = \frac{1}{PI(s) G_{delay}(s)}$$
where \( PI(s) \) is the transfer function of the current loop regulator, and \( G_{delay}(s) \) represents the system delay. Similarly, \( H_{OI}(s) \) and \( H_{OE}(s) \) are expressed as:
$$H_{OI}(s) = \frac{[-B_i(s)]}{[B_{iu}(s) + B_{ie}(s)]} = \frac{PI(s) G_{delay}(s)}{s^2 L_1 C_1} + \frac{PI(s) G_{delay}(s)}{s C_1 PI(s) G_{delay}(s)} – \frac{PI(s) G_{delay}(s)}{G_{delay}(s)}$$
$$H_{OE}(s) = B_i(s) / B_{ie}(s) – 1 / B_{ie}(s)$$
These equations form the basis for analyzing the oscillation characteristics of the grid-tied inverter. The model highlights how parameters like inductance \( L_1 \) and capacitance \( C_1 \) influence the inverter’s behavior, especially under high penetration scenarios where multiple grid-tied inverters interact.
To understand the oscillation suppression mechanism, I analyze the subsynchronous resonance in grid-tied inverters. When the line impedance is predominantly inductive, the active and reactive power loops of the grid-tied inverter become independent. Considering a balance point \( (O^*_n, \delta_n) \) for the n-th grid-tied inverter, the active power perturbation can be linearized as:
$$Y_{P\delta}(s) = \frac{\Delta P}{\Delta \delta} = \frac{X_g O^*_n O_g}{L_c^2 s^2} + \frac{X_g O^*_n O_g}{2 R_g L_c s} + \frac{X_g O^*_n O_g}{R_g^2 + X_g^2}$$
where \( \Delta P \) is the small-signal increment of active power, \( \Delta \delta \) is the phase perturbation, \( O_g \) is the line current, \( R_g \) is the equivalent resistance, \( L_c \) is the disturbance capacitance, and \( X_g = \omega L_g \) is the inductive reactance. The electromagnetic active open-loop transfer function of the grid-tied inverter is then:
$$Y_P(s) = Y_{P\delta}(s) \cdot \frac{1}{s(\lambda_\omega s + D_\omega)}$$
where \( \lambda_\omega \) and \( D_\omega \) are droop control coefficients. This function indicates that the grid-tied inverter exhibits phase lag during oscillations, leading to resonant frequencies and peaks. The resonant frequency \( \omega_r \) and peak \( \lambda_r \) are given by:
$$\omega_r = \omega_0 \cdot \frac{L_g}{L_g + L_0}$$
$$\lambda_r = \frac{(R_g^2 + X_g^2)}{2 X_g R_g}$$
Here, \( \omega_0 \) is the initial resonant frequency, and \( L_0 \) is the equivalent inductance of the grid-tied inverter. When virtual impedance is introduced to account for multi-microgrid access, the total impedance changes, affecting oscillation dynamics. The updated active power function becomes:
$$Y_{p0}(s) = \frac{3 X_t O^*_n O_g}{L_c^2 s^2 + 2 R_t L_c s + R_t^2 + X_t^2}$$
where \( X_t = X_v + X_g \) is the total inductive reactance, \( R_t = R_v + R_g \) is the total resistance, with \( R_v \) and \( X_v = \omega_r L_v \) as virtual resistance and reactance, respectively. The resonant frequency and peak then modify to:
$$\omega_r = \omega_0 \cdot \sqrt{\frac{L_v + L_g}{L_0 + L_g}}$$
$$\gamma_r = \frac{(R_t^2 + X_t^2)}{2 X_t R_t}$$
This analysis shows that increasing the damping ratio in the grid-tied inverter control loop can suppress subsynchronous oscillations by reducing resonant peaks and frequencies.
For practical oscillation suppression in grid-tied inverters, I employ a dq decoupling method combined with PI control. In the ABC coordinate system, the grid and inverter voltages are related by:
$$U_{sA} = L_s \frac{dI_A}{dt} + U_{IA}$$
$$U_{sB} = L_s \frac{dI_B}{dt} + U_{IB}$$
$$U_{sC} = L_s \frac{dI_C}{dt} + U_{IC}$$
After dq transformation, the fluctuating voltages are:
$$U_{Id} = U_{sd} – L_s \frac{dI_d}{dt} + \omega_0 L_s I_q$$
$$U_{Iq} = U_{sq} – L_s \frac{dI_q}{dt} + \omega_0 L_s I_d$$
where \( I_d \) and \( I_q \) are d-axis and q-axis currents, and \( U_{sd} \) and \( U_{sq} \) are dq-axis voltages. To decouple and suppress oscillations, I introduce variable parameters \( \Delta U_d \) and \( \Delta U_q \), leading to:
$$U_{Id} = U_{sd} + \omega_0 L_s I_q – \Delta U_d$$
$$U_{Iq} = U_{sq} + \omega_0 L_s I_d – \Delta U_q$$
Further, using PI controllers to track steady-state errors, the parameters are defined as:
$$\Delta U_d = k_{p1} \Delta I_d + k_{I1} \int \Delta I_d \, dt$$
$$\Delta U_q = k_{p2} \Delta I_q + k_{I2} \int \Delta I_q \, dt$$
with \( \Delta I_d = I_d + I_{d\_ref} \) and \( \Delta I_q = I_q – I_{q\_ref} \), where \( k_{p1}, k_{p2}, k_{I1}, k_{I2} \) are PI constants. Substituting these into the voltage equations yields the suppressed output fluctuations:
$$0 = L_s \frac{dI_d}{dt} – k_{p1} \Delta I_d – k_{I1} \int \Delta I_d \, dt$$
$$0 = L_s \frac{dI_q}{dt} – k_{p2} \Delta I_q – k_{I2} \int \Delta I_q \, dt$$
This approach effectively stabilizes the grid-tied inverter by mitigating current and voltage oscillations.
To validate the proposed method, I conducted simulation experiments using StarSim Real-Time software to emulate a grid environment with high penetration of wind-solar-storage multi-microgrids. The performance of the grid-tied inverter was evaluated in terms of gain, current frequency response, output current, voltage peaks, and fluctuation voltages. Below are tables summarizing key results.
| Frequency (Hz) | Maximum Gain (dB) – Actual | Maximum Gain (dB) – Proposed Method |
|---|---|---|
| 1.5 | 16.0 | 16.0 |
| 5.0 | 8.5 | 8.5 |
| 10.0 | 1.0 | 1.0 |
| 15.0 | -5.0 | -5.0 |
| 20.0 | -10.0 | -10.0 |
The table shows that the proposed method accurately replicates the actual gain points across frequencies, demonstrating high reliability in modeling the grid-tied inverter. The gain decreases with increasing frequency, indicating stable performance.
| Time (s) | Phase A Voltage (kV) – Before Suppression | Phase A Voltage (kV) – After Suppression | Phase B Voltage (kV) – Before Suppression | Phase B Voltage (kV) – After Suppression |
|---|---|---|---|---|
| 2 | 3.23 | 3.19 | 3.22 | 3.08 |
| 4 | 3.55 | 3.22 | 3.44 | 3.08 |
| 6 | 3.67 | 3.44 | 3.57 | 3.08 |
| 8 | 3.82 | 3.57 | 3.61 | 3.08 |
| 10 | 3.91 | 3.11 | 3.66 | 3.08 |
| 12 | 4.05 | 3.11 | 3.91 | 3.08 |
| 14 | 4.12 | 3.11 | 4.12 | 3.08 |
| 16 | 4.58 | 3.11 | 4.36 | 3.08 |
| 18 | 5.09 | 3.11 | 4.59 | 3.08 |
| 20 | 6.22 | 3.11 | 4.82 | 3.08 |
After applying the suppression method to the grid-tied inverter, the voltage peaks stabilize at 3.11 kV for Phase A and 3.08 kV for Phase B, significantly reducing fluctuations compared to the unsuppressed case. This confirms the effectiveness of the dq decoupling and PI control in mitigating oscillations.
Additionally, the output current of the grid-tied inverter was analyzed. Before suppression, the current waveform exhibited large fluctuations between 15 A and 75 A. After suppression, the current became stable, ranging only from 18 A to 62 A without波动. The current frequency response also improved; for instance, at 0 Hz, the magnitude was -32.5 dB, and it peaked at -2.5 dB around 10^4 Hz. The suppressed curve was smoother, indicating reduced oscillation. The fluctuation voltage, which varied between -18 V and 16 V before suppression, dropped to 0 V after suppression, showcasing excellent performance.
In summary, the proposed modeling and suppression method for grid-tied inverters with high penetration of wind-solar-storage multi-microgrids proves highly reliable. By leveraging small-signal equivalent circuit models and advanced control strategies, oscillations in grid-tied inverters are effectively mitigated. The grid-tied inverter’s output waveforms become stable, with minimized current and voltage fluctuations. Future work could focus on optimizing PI parameters for different grid conditions and extending the method to larger-scale systems with multiple grid-tied inverters. Overall, this approach enhances the stability and reliability of modern power networks, supporting the integration of renewable energy sources.