The integration of large-scale photovoltaic (PV) systems into modern power grids introduces challenges related to stability and power quality. Among these, forced low-frequency oscillations (FLOs) caused by maximum power point tracking (MPPT) control dynamics have emerged as a critical concern. This study investigates the mechanism of FLOs in single-stage PV systems utilizing virtual synchronous generator (VSG) control, emphasizing the role of MPPT-induced interharmonics and control parameter interactions.
MPPT Dynamics and Interharmonic Generation
MPPT algorithms, particularly perturbation-based methods like perturb-and-observe (P&O), inherently generate periodic voltage references that introduce interharmonics in the DC-link voltage. For a sampling frequency \( f_{\text{MPPT}} \), the dominant interharmonic frequencies in the DC voltage reference \( V_{\text{dc,ref}} \) are expressed as:
$$ f_n = \frac{(2k – 1)f_{\text{MPPT}}}{4}, \quad k = 1, 2, 3, \ldots $$
These interharmonics propagate through the VSG control loops, creating oscillatory components in the active power reference \( P_{\text{ref}} \). The relationship between \( V_{\text{dc,ref}} \) and \( P_{\text{ref}} \) is governed by the DC voltage controller dynamics:
$$ K_v(s) = \frac{1}{s} \cdot \frac{\beta_1(s + \beta_2/\beta_3)^2}{(s + \beta_2)^2} $$
Typical MPPT parameters and their spectral impacts are summarized in Table 1.
| Parameter | Value | Impact |
|---|---|---|
| Sampling Period \( T_{\text{MPPT}} \) | 0.5 s | Primary interharmonic at 0.5 Hz |
| Voltage Step \( \Delta U \) | 18 V | Amplitude modulation index: 2.67% |
| DC Controller \( \beta_1, \beta_2, \beta_3 \) | 1295.5, 298, 55.5 | Phase margin: 45° at 10 Hz |
Closed-Loop Dynamics of VSG-Controlled Systems
The power transfer characteristics of VSG-controlled inverters are modeled through coupled active/reactive power dynamics:
$$ \begin{bmatrix} \Delta P \\ \Delta Q \end{bmatrix} = \begin{bmatrix} H_{P\delta} & H_{PE} \\ H_{Q\delta} & H_{QE} \end{bmatrix} \begin{bmatrix} \Delta\delta \\ \Delta E \end{bmatrix} $$
where \( H_{P\delta} \), \( H_{PE} \), \( H_{Q\delta} \), and \( H_{QE} \) represent sensitivity coefficients. The closed-loop transfer functions for active and reactive power are derived as:
$$ \rho_{11}(s) = \frac{H_{P\delta}H_{\delta P}(1 + H_{QE}H_{EQ})}{1 + H_{P\delta}H_{\delta P} + H_{QE}H_{EQ} + H_{PE}H_{EQ}H_{Q\delta}H_{\delta P}} $$
$$ \rho_{21}(s) = \frac{H_{Q\delta}H_{\delta P}}{1 + H_{P\delta}H_{\delta P} + H_{QE}H_{EQ} + H_{PE}H_{EQ}H_{Q\delta}H_{\delta P}} $$
Resonance Amplification Mechanism
The VSG’s virtual inertia \( J \) and damping coefficient \( k_p \) significantly influence FLO characteristics:
$$ \text{Resonant Frequency: } f_r = \frac{1}{2\pi}\sqrt{\frac{K}{J\omega_{\text{ref}}}} $$
$$ \text{Peak Amplification: } M_{\text{peak}} = \frac{1}{2\xi} \quad \text{where } \xi = \frac{k_p}{2\sqrt{J\omega_{\text{ref}}K} $$
Key parametric relationships are illustrated in Table 2.
| Parameter | Variation | \( f_r \) Trend | \( M_{\text{peak}} \) Trend |
|---|---|---|---|
| Virtual Inertia \( J \) | ↑ | ↓ | ↑ |
| Damping Coefficient \( k_p \) | ↑ | – | ↓ |
| Line Resistance Ratio \( R/X \) | ↑ | – | ↑ |
Validation Through Frequency-Domain Analysis
The equivalence between closed-loop transfer functions and traditional second-order models is demonstrated through Bode analysis. For a system with \( J = 4 \, \text{kg·m}^2 \) and \( k_p = 4000 \, \text{W/Hz} \):
$$ \text{Natural Frequency: } f_n = 1.0 \, \text{Hz} $$
$$ \text{Damping Ratio: } \xi = 0.238 $$
Simulation results confirm the amplification factors predicted by the transfer function model (Table 3).
| \( P_{\text{ref}} \) Perturbation | Frequency (Hz) | Active Power Gain | Reactive Power Coupling |
|---|---|---|---|
| 200 W Step | 0.5 | 1.1× | 0.55× |
| 200 W Step | 1.0 | 2.1× | 1.0× |
| 200 W Step | 1.5 | 1.5× | 0.75× |
Mitigation Strategies and Design Implications
To suppress FLOs in MPPT-controlled PV systems:
1. Optimize \( f_{\text{MPPT}} \) to avoid alignment with \( f_r \)
2. Implement adaptive virtual inertia: \( J_{\text{adapt}} = k \cdot |dP/dt|^{-1} \)
3. Enhance power decoupling through modified VSG control:
$$ \Delta Q_{\text{comp}} = k_{\text{decoup}} \cdot \frac{dP}{dt} $$
The analysis establishes that MPPT-induced interharmonics interact with VSG control dynamics to create frequency-specific amplification windows. Proper coordination between MPPT sampling rates and VSG parameters is essential for maintaining grid stability in high-penetration PV systems.