In the context of rapidly increasing energy demands and the depletion of fossil fuels, the integration of renewable energy sources such as wind, solar, and hydropower into the power grid has become essential. However, the intermittent nature of these sources requires energy storage systems to buffer the generated power. When multiple microgrids containing high proportions of wind, solar, and storage systems connect to the main grid, the grid-connected inverters that interface these microgrids can experience undesirable oscillations, particularly subsynchronous oscillations, which threaten the stability and security of the power system. This paper presents a comprehensive study on modeling and suppression of oscillation characteristics of grid-connected inverters under such conditions. I focus on building a small-signal equivalent circuit model to capture the oscillation behavior, analyze the mechanism of subsynchronous resonance, and propose a combined dq-decoupling and PI controller method to effectively suppress the oscillations. Extensive simulations validate the reliability and effectiveness of the proposed approach.
This work explores various types of solar inverter configurations used in high-penetration renewable systems. Different types of solar inverter, such as string inverters and central inverters, exhibit distinct impedance characteristics that influence oscillation dynamics. The study particularly emphasizes the behavior of multi-parallel inverters when multiple types of solar inverter are interconnected, revealing how their output impedances interact with the grid impedance to trigger subsynchronous oscillations. By understanding these interactions across different types of solar inverter, we can design more robust oscillation suppression strategies.
To begin, I establish the structure of a grid-connected multi-inverter system with high-proportion wind-solar-storage microgrids. The system comprises multiple inverters, each directly connected to photovoltaic panels or energy storage devices, and collectively interfacing with the main grid. The complexity of the circuit arises from the parallel operation of many inverters, which can lead to resonant interactions. I develop a small-signal equivalent circuit model that represents each inverter’s output impedance and transfer functions, enabling the analysis of oscillation characteristics under small perturbations.
The model is derived from the single-inverter configuration. Let \(I_{\text{ref}}(s)\) denote the reference current of an inverter, and define the following transfer functions: \(H_{\text{OI}}(s)\) as the parallel output impedance, \(Z_{\text{OI}}(s)\) as the electromotive force, and \(H_{\text{OE}}(s)\) as the combined output impedance from feedforward control. The harmonic voltage source is \(U_{\text{SW}}(s)\), and \(H_{\text{OSW}}(s)\) is the feedback channel output impedance. The output current of a single inverter is given by
$$
I_g(s) = B_i(s) I_{\text{ref}}(s) + B_{iu}(s) U_{\text{SW}}(s) + B_{ie}(s) E(s)
$$
where
$$
B_i(s) = \frac{H_{\text{OSW}}(s) H_{\text{OI}}(s)}{H_{\text{OSW}}(s) H_{\text{OI}}(s) + H_{\text{OE}}(s) H_{\text{OI}}(s) + H_{\text{OE}}(s) H_{\text{OSW}}(s)}
$$
$$
B_{iu}(s) = \frac{H_{\text{OI}}(s)}{H_{\text{OSW}}(s) H_{\text{OI}}(s) + H_{\text{OI}}(s) H_{\text{OE}}(s) + H_{\text{OI}}(s) H_{\text{OSW}}(s) H_{\text{OE}}(s)}
$$
$$
B_{ie}(s) = \frac{H_{\text{OSW}}(s) + H_{\text{OI}}(s)}{H_{\text{OSW}}(s) H_{\text{OI}}(s) + H_{\text{OE}}(s) H_{\text{OI}}(s) + H_{\text{OE}}(s) H_{\text{OSW}}(s)}
$$
From these, the individual impedances are extracted as
$$
H_{\text{OSW}}(s) = \frac{B_i(s)}{B_{iu}(s)} = PI(s) G_{\text{delay}}(s)
$$
$$
H_{\text{OI}}(s) = \frac{-B_i(s)}{B_{iu}(s) + B_{ie}(s)} = \frac{PI(s) G_{\text{delay}}(s)}{s^2 L_1 C_1} + \frac{PI(s) G_{\text{delay}}(s)}{s C_1 PI(s) G_{\text{delay}}(s)} – \frac{PI(s) G_{\text{delay}}(s)}{G_{\text{delay}}(s)}
$$
$$
H_{\text{OE}}(s) = \frac{B_i(s)}{B_{ie}(s)} – \frac{1}{B_{ie}(s)}
$$
where \(PI(s)\) is the transfer function of the current-loop regulator, \(G_{\text{delay}}(s)\) is the system delay, \(L_1\) is the inverter-side inductance, and \(C_1\) is the capacitance. This small-signal model provides the equivalent impedance necessary for oscillation analysis.
Next, I examine the subsynchronous resonance suppression mechanism. When the line impedance is predominantly inductive, the active and reactive loops of the inverter can be considered decoupled. By linearizing the active power around the equilibrium point \((O_n^*, \delta_n)\), the small-signal active power is
$$
Y_{P\delta}(s) = \frac{\Delta P}{\Delta \delta} = \frac{X_g O_n^* O_g}{L_c^2 s^2} + \frac{X_g O_n^* O_g}{2 R_g L_c s} + \frac{X_g O_n^* O_g}{R_g^2 + X_g^2}
$$
where \(O_g\) is the line current, \(R_g\) is the equivalent resistance, \(L_c\) is the perturbation capacitance, \(X_g = \omega L_g\) is the pure reactance, and \(O_n^*\) is the amplitude. The open-loop transfer function of active power is
$$
Y_P(s) = Y_{P\delta}(s) \cdot \frac{1}{s (\lambda_\omega s + D_\omega)}
$$
with \(\lambda_\omega\) and \(D_\omega\) being droop coefficients. The resonant frequency and peak are
$$
\omega_r = \frac{\omega_0 L_g}{L_g + L_0}, \quad \lambda_r = \frac{R_g^2 + X_g^2}{2 X_g R_g}
$$
To mitigate oscillations, I introduce virtual impedance into the control loop. Let \(\zeta = L_g / (L_0 + L_g)\) be the resonant coefficient, and define total reactance \(X_t = X_v + X_g\) and total resistance \(R_t = R_v + R_g\) where \(X_v = \omega_r L_v\) and \(R_v\) are virtual parameters. The modified open-loop power becomes
$$
Y_{p0}(s) = \frac{3 X_t O_n^* O_g}{L_c^2 s^2 + 2 R_t L_c s + R_t^2 + X_t^2}
$$
The new resonant frequency and peak are
$$
\omega_r = \omega_0 \sqrt{\frac{L_v + L_g}{L_0 + L_g}}, \quad \gamma_r = \frac{R_t^2 + X_t^2}{2 X_t R_t}
$$
When \(\zeta = 1\), the oscillation is synchronous; if \(\zeta > 1\), it is supersynchronous. However, the interaction of inverter output impedance with line impedance often leads to subsynchronous oscillations. The key is to increase the first-order coefficient of the oscillation loop, thereby raising the damping ratio and reducing both the resonant frequency and peak.
Based on this mechanism, I implement a dq-decoupling suppression method combined with PI control. In the three-phase ABC frame, the inverter voltage equations are
$$
U_{sA} = L_s \frac{dI_A}{dt} + U_{IA}, \quad U_{sB} = L_s \frac{dI_B}{dt} + U_{IB}, \quad U_{sC} = L_s \frac{dI_C}{dt} + U_{IC}
$$
After dq transformation,
$$
U_{Id} = U_{sd} – L_s \frac{dI_d}{dt} + \omega_0 L_s I_q, \quad U_{Iq} = U_{sq} – L_s \frac{dI_q}{dt} + \omega_0 L_s I_d
$$
The coupling between d and q axes is evident. To decouple, I introduce adjustable parameters \(\Delta U_d\) and \(\Delta U_q\):
$$
U_{Id} = U_{sd} + \omega_0 L_s I_q – \Delta U_d, \quad U_{Iq} = U_{sq} + \omega_0 L_s I_d – \Delta U_q
$$
Then, PI controllers are added to track the steady-state errors:
$$
\Delta U_d = k_{p1} \Delta I_d + k_{I1} \int \Delta I_d dt, \quad \Delta U_q = k_{p2} \Delta I_q + k_{I2} \int \Delta I_q dt
$$
with
$$
\Delta I_d = I_d + I_{d\_ref}, \quad \Delta I_q = I_q – I_{q\_ref}
$$
Finally, the suppressed output voltage dynamics become
$$
0 = L_s \frac{dI_d}{dt} – k_{p1} \Delta I_d – k_{I1} \int \Delta I_d dt, \quad 0 = L_s \frac{dI_q}{dt} – k_{p2} \Delta I_q – k_{I2} \int \Delta I_q dt
$$
This ensures complete decoupling and effective oscillation suppression.
To validate the proposed method, I conducted simulations using StarSim Real-Time software, emulating a high-proportion wind-solar-storage microgrid system. The grid-connected inverters comprised multiple types of solar inverter operating in parallel. The reliability of the model was first assessed by comparing the maximum gain points at different operating frequencies. The results are shown in Table 1.
| Frequency (Hz) | Measured Gain (dB) | Model Gain (dB) |
|---|---|---|
| 1.5 | 16 | 16 |
| 5.0 | 8 | 8 |
| 10.0 | 1 | 1 |
The model accurately predicted the gain, demonstrating high reliability. Next, I observed the current frequency response before and after suppression. The raw oscillation exhibited a peak near 104 Hz with an amplitude of -2.5 dB, while the suppressed response remained smooth and without fluctuations.
The output current waveforms are compared in Figure 1. Before suppression, the current fluctuated between 15 A and 75 A. After applying the proposed method, the current stabilized within 18 A to 62 A with no visible oscillations. This confirms the method’s effectiveness in damping the subsynchronous oscillations.
Table 2 summarizes the peak voltage of phases A and B before and after suppression over a 20-second period. The suppression reduced the peak voltages and stabilized them around 3.11 kV (phase A) and 3.08 kV (phase B), significantly lower than the rising trend observed in the unsuppressed case.
| Time (s) | Phase A before (kV) | Phase A after (kV) | Phase B before (kV) | Phase B after (kV) |
|---|---|---|---|---|
| 2 | 3.23 | 3.19 | 3.11 | 3.08 |
| 4 | 3.55 | 3.22 | 3.11 | 3.08 |
| 6 | 3.67 | 3.44 | 3.11 | 3.08 |
| 8 | 3.82 | 3.57 | 3.11 | 3.08 |
| 10 | 3.91 | 3.11 | 3.61 | 3.08 |
| 12 | 4.05 | 3.66 | 3.11 | 3.08 |
| 14 | 4.12 | 3.91 | 3.11 | 3.08 |
| 16 | 4.58 | 4.12 | 3.11 | 3.08 |
| 18 | 5.09 | 4.36 | 3.11 | 3.08 |
| 20 | 6.22 | 4.59 | 3.11 | 3.08 |
Furthermore, the fluctuation voltage (measured as the deviation from the nominal value) was reduced to zero after suppression, as illustrated in the simulation. Before suppression, the fluctuation voltage varied between -18 V and +16 V in a periodic pattern; after suppression, it remained at 0 V, indicating perfect elimination of voltage oscillations.
In summary, I have developed a comprehensive modeling and suppression strategy for grid-connected inverters under high-proportion wind-solar-storage multi-microgrid access. The small-signal equivalent circuit model accurately captures the oscillation characteristics, and the combined dq-decoupling with PI control effectively damps subsynchronous oscillations. The simulation results demonstrate that the method reliably suppresses current and voltage fluctuations, reduces peak voltages, and eliminates fluctuation voltage. This work provides an effective solution for enhancing the stability of modern power systems with high renewable penetration, and it is applicable across various types of solar inverter configurations commonly used in microgrids.