The integration of renewable energy sources into the power grid is pivotal for addressing the dual challenges of fossil fuel depletion and environmental degradation. The grid tied inverter serves as the critical interface between distributed generation systems, such as photovoltaic arrays, and the utility grid. In single-phase systems, the performance and stability of the grid tied inverter are directly governed by the accuracy and speed with which it can synchronize its output with the grid voltage’s amplitude, frequency, and phase. Precise detection of these grid parameters is therefore non-negotiable for safe, stable, and efficient operation, ensuring the injected current is perfectly in phase with the grid voltage.
Phase-Locked Loop (PLL) techniques are the cornerstone for grid synchronization. Among them, the Second-Order Generalized Integrator based Frequency-Locked Loop (SOGI-FLL) has gained significant traction for single-phase grid tied inverter applications due to its inherent frequency adaptive capability. This feature allows it to track grid frequency variations, a common occurrence in weak grids or under high penetration of renewables. However, the conventional SOGI-FLL structure exhibits a critical vulnerability: its performance severely degrades in the presence of non-ideal grid conditions, specifically DC offsets and harmonic distortions. These disturbances are prevalent in real-world grids due to transformer saturation, nonlinear loads, and switching events.
When the grid voltage contains a DC component, the conventional SOGI’s orthogonal output signals experience a significant offset. This offset propagates through the FLL, causing oscillations in the estimated frequency and phase. Similarly, harmonic components distort the input signal, leading to inaccuracies in the generated synchronization signals. For a grid tied inverter, these inaccuracies translate into poor power quality, increased current harmonics, and potential instability, compromising the entire energy conversion system’s reliability.
This article presents a comprehensive analysis of an enhanced voltage control method for single-phase grid tied inverter systems based on a significantly improved SOGI-FLL structure. The proposed method directly addresses the limitations of the conventional approach by integrating two dedicated suppression mechanisms: a DC control loop and a cascaded resonant filter stage. The core contribution lies in the synergistic operation of these two elements, enabling robust and accurate grid synchronization under a wide range of adverse grid conditions, including frequency steps, DC offsets, and harmonic pollution.
Fundamental Principles and Limitations of the Conventional SOGI-FLL
The conventional SOGI-FLL is a synergistic structure comprising a Second-Order Generalized Integrator (SOGI) block for orthogonal signal generation and a Frequency-Locked Loop (FLL) for adaptive frequency tracking. Its standard structure is depicted in the block diagram below and forms the baseline for our analysis.
The SOGI block takes the single-phase grid voltage $v_g$ as input and generates two orthogonal outputs: $v_\alpha$ (in-phase) and $v_\beta$ (quadrature-phase). The transfer functions from input $v_g$ to these outputs are given by:
$$ G_\alpha(s) = \frac{v_\alpha(s)}{v_g(s)} = \frac{k \omega’ s}{s^2 + k \omega’ s + \omega’^2} $$
$$ G_\beta(s) = \frac{v_\beta(s)}{v_g(s)} = \frac{k \omega’^2}{s^2 + k \omega’ s + \omega’^2} $$
Here, $k$ is the damping factor (typically set to $\sqrt{2}$ for optimal bandwidth), and $\omega’$ is the resonant frequency of the SOGI, which is adjusted by the FLL. When $\omega’$ is exactly equal to the grid frequency $\omega_g$, $v_\alpha$ matches $v_g$ in both amplitude and phase, while $v_\beta$ has the same amplitude but a -90° phase shift, forming a perfect orthogonal pair. The error signal $\epsilon_v$ is defined as the difference between the input and the in-phase output.
The FLL’s role is to drive the frequency error to zero. It operates based on the product of the quadrature signal $v_\beta$ and the error signal $\epsilon_v$. The frequency error estimate $\epsilon_f$ is derived as:
$$ \epsilon_f = \epsilon_v \cdot v_\beta $$
A stability analysis using the Barkhausen criterion or describing functions shows that when $\omega’ > \omega_g$, $\epsilon_f > 0$, and when $\omega’ < \omega_g$, $\epsilon_f < 0$. This error is fed through an integral gain $-k_1$ to adjust $\omega’$ until $\epsilon_f = 0$, achieving frequency lock: $\omega’ = \omega_g$.
While elegant, this structure has fundamental flaws in non-ideal grids. Analyzing the transfer function $G_\beta(s)$ at DC ($s = j0$) reveals the root of the DC offset problem:
$$ G_\beta(j0) = \frac{v_\beta(j0)}{v_g(j0)} = k $$
This equation indicates that any DC component present in $v_g$ is amplified by a factor $k$ and appears directly in the $v_\beta$ output. This corrupted $v_\beta$ signal is then used in the FLL’s frequency error calculation ($\epsilon_f = \epsilon_v v_\beta$), introducing a steady-state error that causes the estimated frequency $\omega’$ to oscillate around the true value. The system cannot distinguish between the error caused by frequency mismatch and the error induced by the DC offset.
Regarding harmonics, the SOGI’s $G_\alpha(s)$ acts as a band-pass filter centered at $\omega’$. While it attenuates frequencies far from $\omega’$, its attenuation at specific harmonic frequencies (e.g., 3rd, 5th) is finite, especially if the $k$ factor is set for a wide bandwidth for fast dynamics. These harmonic components leak into $v_\alpha$ and $v_\beta$, distorting the sinusoidal waveforms and, consequently, corrupting the frequency error signal $\epsilon_f$. The FLL, in trying to minimize this corrupted error, forces $\omega’$ to a value that does not correspond to the true fundamental frequency, leading to erroneous synchronization for the grid tied inverter.
The following table summarizes the impact of disturbances on the conventional SOGI-FLL:
| Grid Disturbance | Effect on SOGI Outputs | Consequence for FLL & Synchronization |
|---|---|---|
| DC Offset | DC bias appears in $v_\beta$. | Oscillations in estimated frequency ($\omega’$). |
| Harmonics (3rd, 5th, etc.) | Distortion in $v_\alpha$ and $v_\beta$ waveforms. | Steady-state frequency error and phase jitter. |
| Frequency Step | Transient error in $v_\alpha$, $v_\beta$ amplitude/phase. | Slow or oscillatory frequency tracking. |
Proposed Enhanced SOGI-FLL Structure with Dual Suppression
To overcome the aforementioned limitations, an enhanced structure is proposed. The core idea is to pre-process the grid voltage signal to remove harmful components before they enter the core SOGI-FLL and to augment the SOGI with a dedicated DC suppression feedback loop. This results in a system with three key parts: 1) A Cascaded Resonant Filter (CRF) stage for harmonic rejection, 2) A DC Control Loop integrated into the SOGI, and 3) The standard FLL for frequency adaptation.
1. Cascaded Resonant Filter (CRF) for Harmonic Pre-Filtering
The harmonic distortion in the grid voltage is primarily composed of odd harmonics (3rd, 5th, 7th, etc.). To suppress these, a bank of resonant filters tuned to reject specific harmonic frequencies is placed in cascade at the input. Each individual filter is a notch filter with the transfer function:
$$ H_m(s) = \frac{s^2 + (m \omega_n)^2}{s^2 + 2 \omega_c s + (m \omega_n)^2} $$
where $m$ is the harmonic order (3, 5, 7…), $\omega_n$ is the nominal grid frequency (e.g., $100\pi$ rad/s), and $\omega_c$ is the cutoff frequency determining the bandwidth of the notch. The overall transfer function of the CRF stage for $M$ harmonics is:
$$ G_{CRF}(s) = \prod_{m=3,5,7,…}^{M} H_m(s) $$
The choice of $\omega_c$ involves a trade-off. A small $\omega_c$ creates a sharp, narrow notch, providing excellent harmonic rejection but potentially slowing the system’s dynamic response and making it sensitive to frequency deviations. A larger $\omega_c$ offers a wider notch and better dynamic response but reduces the attenuation at the harmonic frequency. For a grid tied inverter operating in a grid with standard frequency variations, $\omega_c$ can be designed to achieve a satisfactory compromise. The output of this stage, $v_g’$, is a cleaner version of the grid voltage with significantly reduced harmonic content, which is then fed to the next stage.
2. SOGI with Integrated DC Control Loop
The filtered voltage $v_g’$ may still contain a DC offset. To address this, the conventional SOGI structure is modified by adding a DC control branch. This branch estimates the DC component present at the input of the SOGI and subtracts it in a negative feedback arrangement. The modified structure alters the transfer functions.
The new transfer functions from the input $v_g’$ to the outputs $v_\alpha$, $v_\beta$, and the estimated DC component $v_{dc}$ are derived as follows:
$$ G’_\alpha(s) = \frac{v_\alpha(s)}{v_g'(s)} = \frac{k_2 \omega’ s^2}{s^3 + (k + k_2)\omega’ s^2 + \omega’^2 s + k_2 \omega’^3} $$
$$ G’_\beta(s) = \frac{v_\beta(s)}{v_g'(s)} = \frac{k \omega’^2 s}{s^3 + (k + k_2)\omega’ s^2 + \omega’^2 s + k_2 \omega’^3} $$
$$ G_{dc}(s) = \frac{v_{dc}(s)}{v_g'(s)} = \frac{k_2 \omega’ (s^2 + \omega’^2)}{s^3 + (k + k_2)\omega’ s^2 + \omega’^2 s + k_2 \omega’^3} $$
where $k_2$ is the gain of the DC control path. The critical improvement is evident in $G’_\beta(j0)$. Evaluating at DC:
$$ G’_\beta(j0) = 0 $$
This proves that the DC component in $v_g’$ is completely blocked from reaching the $v_\beta$ output. The signal $v_{dc}$ effectively becomes an estimate of the input’s DC offset. At the fundamental frequency ($s = j\omega’$), the transfer functions simplify such that $v_\alpha$ and $v_\beta$ remain an orthogonal pair with unity gain, meaning the DC control loop does not affect the generation of the fundamental synchronization signals. $G_{dc}(s)$ acts as a band-stop filter, passing DC and blocking the fundamental frequency, making it an ideal estimator for the DC offset in this application.
3. Complete System and Parameter Design
The complete enhanced controller for the grid tied inverter is formed by connecting the CRF, the DC-controlled SOGI, and the FLL. The FLL operates identically as before, using the now “clean” signals $v_\alpha$ and $\epsilon_v$ to generate the frequency error $\epsilon_f = \epsilon_v v_\beta$ and adjust $\omega’$. Key parameters for the entire system must be carefully selected:
| Parameter | Symbol | Typical Value / Selection Criteria | Role |
|---|---|---|---|
| SOGI Damping Factor | $k$ | $\sqrt{2} \approx 1.414$ | Determines SOGI bandwidth and transient response. |
| FLL Integral Gain | $k_1$ | $1$ to $10$ (System dependent) | Controls the speed of frequency adaptation. |
| DC Control Gain | $k_2$ | ~$0.2$ to $1.0$ | Sets the bandwidth of the DC estimation loop. |
| CRF Cutoff Frequency | $\omega_c$ | $20$ to $30$ rad/s (e.g., $25.4$) | Balances harmonic rejection depth with response time. |
| Targeted Harmonics | $m$ | 3, 5, 7, 9 | Specifies which harmonic orders are filtered. |
The design process involves a sequential approach. First, $k$ is set for desired fundamental tracking dynamics. Then, $k_2$ is chosen to ensure stable DC rejection without affecting the fundamental. Finally, $\omega_c$ for the CRF is selected based on the required harmonic attenuation and the expected rate of grid frequency change. The enhanced structure’s key transfer functions and their ideal characteristics are summarized below:
| Function | Mathematical Expression | Ideal Property for $\omega’=\omega_g$ |
|---|---|---|
| $G_{CRF}(j m \omega_n)$ | $\prod \frac{-(m\omega_n)^2 + (m\omega_n)^2}{-(m\omega_n)^2 + 2j\omega_c m\omega_n + (m\omega_n)^2} = 0$ | Zero gain at harmonic frequencies. |
| $G’_\alpha(j \omega_g)$ | $ \approx 1 \angle 0^\circ$ | Unity gain, zero phase shift for fundamental. |
| $G’_\beta(j \omega_g)$ | $ \approx 1 \angle -90^\circ$ | Unity gain, -90° phase shift for fundamental. |
| $G’_\beta(j 0)$ | $0$ | Zero gain at DC. |
| $G_{dc}(j 0)$ | $1 \angle 0^\circ$ | Accurate DC component estimation. |
Performance Evaluation: Comparative Analysis
The superiority of the proposed enhanced SOGI-FLL is demonstrated through a comparative analysis against two benchmark methods: the conventional SOGI-PLL (which lacks frequency adaptation) and the Dual-SOGI-FLL (DSOGI-FLL), a common improvement that uses two SOGIs for better harmonic rejection but can be complex. The evaluation considers three critical non-ideal grid scenarios relevant to a grid tied inverter.
Scenario 1: Grid Frequency Step Change
This test evaluates dynamic performance. The grid frequency abruptly changes from 50 Hz to 55 Hz.
- Conventional SOGI-PLL: Struggles significantly. The output $v_\beta$ suffers a large amplitude drop (~40V), and the estimated frequency shows a steady-state error, oscillating within ±2.8% of the new frequency.
- DSOGI-FLL: Shows good stability in $v_\beta$ but poor frequency tracking. The frequency estimate has a large initial deviation (~8%) and settles to an incorrect value (55.5 Hz) after a long period (~1.5 s).
- Proposed Enhanced SOGI-FLL: Demonstrates excellent performance. The orthogonal signals $v_\alpha$ and $v_\beta$ remain stable and accurately represent the new frequency. The estimated frequency locks onto the new 55 Hz value within 0.08 seconds with a maximum deviation of less than 0.5%.
Scenario 2: Grid Voltage with DC Offset
A 30V DC component is superimposed on the grid voltage.
- Conventional SOGI-PLL: The DC offset directly appears as a bias in the $v_\beta$ output. The frequency estimate is highly unstable, oscillating with a maximum error of ~9%.
- DSOGI-FLL: Effectively suppresses the DC offset in $v_\beta$ within 0.025s. However, the frequency estimate still exhibits significant oscillation (±2%).
- Proposed Enhanced SOGI-FLL: The integrated DC control loop completely eliminates the offset from $v_\beta$ within 0.025s. Consequently, the frequency estimate is nearly unaffected, showing a minor transient deviation (<2%) and recovering stability within 0.03s.
Scenario 3: Grid Voltage with Harmonic Distortion
The grid voltage is polluted with 3rd (44V), 5th (44V), 7th (22V), and 9th (22V) harmonics.
- Conventional SOGI-PLL: Performance is severely degraded. The output waveforms $v_\alpha$ and $v_\beta$ are visibly distorted, and the frequency estimate is wildly inaccurate, with oscillations up to 22% error.
- DSOGI-FLL: Shows reasonable waveform recovery but not perfect. The $v_\beta$ amplitude is increased by ~4%, and the frequency estimate has an 8% error with sustained oscillations.
- Proposed Enhanced SOGI-FLL: The cascaded resonant filter effectively strips away the harmonics. The outputs $v_\alpha$ and $v_\beta$ are clean, sinusoidal signals. The frequency locks accurately with a maximum deviation of only ~2% and a settling time of 0.025s.
The quantitative results from all scenarios are consolidated in the table below, clearly highlighting the advantages of the proposed method for a robust grid tied inverter controller.
| Evaluation Metric | Conventional SOGI-PLL | DSOGI-FLL | Proposed Enhanced SOGI-FLL |
|---|---|---|---|
| Frequency Step Response | Slow, large amplitude error in $v_\beta$, freq. error ~6%. | Stable $v_\beta$, but freq. locks to wrong value (55.5 Hz). | Fast lock (<0.08s), accurate freq. tracking (<0.5% error). |
| DC Offset Rejection | Poor. $v_\beta$ has DC bias. Freq. oscillation ~9%. | Good DC removal from $v_\beta$ (0.025s). Freq. oscillation ~3%. | Excellent DC removal (0.025s). Minor freq. transient (<2%). |
| Harmonic Immunity | Very Poor. Distorted outputs. Freq. error ~22%. | Moderate. Some waveform distortion. Freq. error ~8%. | Excellent. Clean outputs. Freq. error ~2%, fast recovery. |
| Output Signal Quality ($v_\alpha$, $v_\beta$) | Degraded under all non-ideal conditions. | Improved but not perfect under harmonics. | High-quality, sinusoidal under all tested conditions. |
| Suitability for Grid Tied Inverter | Low. Unreliable in practical, distorted grids. | Medium. Better but limited in harsh conditions. | High. Robust and adaptive for real-world grid challenges. |
Conclusion and Implementation Perspective
This article has detailed the development and analysis of an enhanced voltage synchronization method based on a modified SOGI-FLL structure, specifically designed to address the critical challenges faced by single-phase grid tied inverter systems. The proposed architecture integrates two targeted suppression mechanisms: a cascaded resonant filter for pre-emptive harmonic elimination and a novel DC control loop embedded within the SOGI for effective DC offset rejection.
The mathematical analysis of the transfer functions proves that the enhanced structure achieves the desired characteristics: $G’_\beta(j0) = 0$ for perfect DC blocking and $G_{CRF}(j m \omega_n) = 0$ for harmonic cancellation, while preserving the ideal orthogonal signal generation at the fundamental frequency. Comparative evaluations under frequency steps, DC offsets, and harmonic distortion demonstrate a decisive performance advantage over conventional SOGI-PLL and DSOGI-FLL methods. The enhanced controller provides rapid frequency locking (within 0.025s in most cases), maintains frequency deviation below 2%, and delivers clean, stable synchronization signals under diverse non-ideal grid conditions.
For practical implementation in a grid tied inverter control system, the enhanced SOGI-FLL offers a compelling solution. Its improved robustness directly translates to higher quality injected current, better compliance with grid codes (e.g., IEEE 1547), and increased system reliability. While the structure is slightly more computationally intensive than the basic SOGI-FLL, the added complexity is justified by the significant performance gain in real-world environments where grid purity cannot be assumed. The method enhances the adaptive capability of the inverter, allowing it to maintain stable operation and high power quality even during grid disturbances, which is essential for the increasing penetration of renewable energy sources. Future work may focus on optimizing the parameter auto-tuning for the CRF and DC loop under varying grid stiffness and exploring its application in more complex three-phase or unbalanced grid scenarios.