In recent years, the rapid development of renewable energy sources, such as solar and wind power, has led to widespread adoption of photovoltaic and wind power generation. However, integrating these distributed energy resources into weak grids—characterized by low short-circuit capacity and high grid impedance—poses significant challenges. Weak grids, often found in remote villages or isolated islands with long transmission lines, exhibit voltage waveform distortion and instability, which can cause synchronization failures in utility interactive inverters. In this paper, we address these issues by proposing a novel control strategy for a three-phase LCL utility interactive inverter based on one-cycle control (OCC) and a dual second-order generalized integrator-based frequency-locked loop (DSOGI-FLL). This strategy aims to enhance synchronization and stability under unbalanced grid conditions, such as voltage sags and frequency variations. We present a comprehensive analysis, including mathematical modeling, controller design, and simulation results, demonstrating the effectiveness of our approach. The utility interactive inverter is a critical component in renewable energy systems, enabling efficient power conversion and grid integration. Throughout this work, we emphasize the importance of robust control for utility interactive inverters in weak grids.
The overall control system for the three-phase LCL utility interactive inverter is illustrated in Figure 1 (refer to the system structure diagram). It consists of a DC source (simulating renewable energy output), a three-phase inverter with LCL filter, and a control unit. The LCL filter includes inverter-side inductors \(L_1\), grid-side inductors \(L_2\), and a filter capacitor \(C\). To model weak grid conditions, we incorporate a grid inductance \(L_g\) at the voltage source. The control strategy employs a dual-loop approach: an inner loop with proportional control and OCC for capacitor current feedback, and an outer loop with proportional-integral (PI) control for grid current feedback. Unlike traditional methods, we use DSOGI-FLL for grid voltage synchronization under unbalanced conditions, enabling transformation to a dq rotating coordinate system for precise control. This utility interactive inverter design ensures high power quality and stability, even in challenging grid environments.
One-cycle control is a nonlinear, large-signal modulation technique that eliminates input disturbances within a single switching cycle. For our three-phase utility interactive inverter, we derive the OCC algorithm based on idealized inverter assumptions—zero dead time, no hysteresis, and complementary switching. The duty cycles for the inverter switches are expressed as functions of capacitor currents and DC link voltage. In bipolar OCC, the duty cycles \(D_{ap}\), \(D_{bp}\), and \(D_{cp}\) for phases a, b, and c are given by:
$$ \begin{bmatrix} D_{ap} \\ D_{bp} \\ D_{cp} \end{bmatrix} = \begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix} + \frac{K_{\text{OCC}}}{U_{dc}} \begin{bmatrix} I_{ca} \\ I_{cb} \\ I_{cc} \end{bmatrix} $$
where \(K_{\text{OCC}}\) is the OCC gain, \(U_{dc}\) is the DC voltage, and \(I_{ca}\), \(I_{cb}\), \(I_{cc}\) are the capacitor currents. This formulation ensures that the average output tracks the reference within each switching period, making the utility interactive inverter insensitive to input variations. The OCC technique simplifies control implementation and improves dynamic response, which is crucial for utility interactive inverters in weak grids where voltage fluctuations are common.
To handle unbalanced grid voltages, we replace conventional phase-locked loops with DSOGI-FLL. This synchronization technique uses a dual second-order generalized integrator to generate orthogonal signals from the grid voltage, enabling accurate frequency and phase tracking even under distortions. The DSOGI-FLL consists of three modules: a quadrature signal generator (DSOGI-QSG), a frequency-locked loop (FLL), and a positive-negative sequence separator (PNSC). The transfer functions for SOGI-QSG are:
$$ G_1(s) = \frac{U'(s)}{U(s)} = \frac{k_2 \omega’ s}{s^2 + k_2 \omega’ s + \omega’^2}, \quad G_2(s) = \frac{U”(s)}{U(s)} = \frac{k_2 \omega’^2}{s^2 + k_2 \omega’ s + \omega’^2} $$
where \(k_2\) is the gain and \(\omega’\) is the resonant frequency. The FLL adaptively adjusts \(\omega’\) to lock onto the grid frequency, while the PNSC extracts positive and negative sequence components. This allows the utility interactive inverter to maintain synchronization during faults, such as voltage dips or frequency shifts. The DSOGI-FLL enhances the robustness of the utility interactive inverter, ensuring reliable operation in weak grids.
For the LCL filter, we analyze the system using state-space equations to design the dual-loop controller. The inverter-side currents \(I_1\), grid-side currents \(I_o\), and capacitor voltages \(U_c\) are chosen as state variables. The single-phase LCL model can be extended to three phases due to symmetry. The dynamics are described by:
$$ I_{1a}(s) = \frac{U_{aN”}(s) – U_{ca}(s)}{sL_1}, \quad I_{oa}(s) = \frac{U_{ca}(s) – U_{aN”}(s)}{sL_2}, \quad U_{ca}(s) = \frac{I_{1a}(s) – I_{oa}(s)}{sC} $$
and similarly for phases b and c. The control strategy uses grid current outer loop and capacitor current inner loop. The open-loop transfer function of the dual-loop controller is:
$$ G_O(s) = \frac{K K_{\text{OCC}}}{L_1 L_2 C s^3 + K K_{\text{OCC}} L_2 C s^2 + (L_1 + L_2) s} \times \left( K_P + \frac{K_I}{s} \right) $$
where \(K\) is the inner loop proportional gain, and \(K_P\) and \(K_I\) are the outer loop PI gains. The closed-loop transfer function is:
$$ \Phi(s) = \frac{(s K_P + K_I) K K_{\text{OCC}}}{L_1 L_2 C s^4 + K K_{\text{OCC}} L_2 C s^3 + (L_1 + L_2) s^2 + K_P K K_{\text{OCC}} s + K_I K K_{\text{OCC}}} $$
Stability is ensured by applying the Hurwitz criterion, leading to conditions on the gains. For instance, all coefficients of the characteristic equation must be positive, and additional constraints include:
$$ (L_1 + L_2) K K_{\text{OCC}} L_2 C – L_1 L_2 C K_P K K_{\text{OCC}} > 0, \quad K K_P K_{\text{OCC}} [(L_1 + L_2) – L_1 K_P] – K^2 K_{\text{OCC}}^2 K_I L_2 C > 0, \quad K K_{\text{OCC}} K_I > 0 $$
These conditions guide the tuning of controller parameters for the utility interactive inverter, ensuring damping of resonant peaks and robust performance in weak grids.
We conducted simulations in MATLAB/Simulink to validate the proposed control strategy for the utility interactive inverter. The system parameters are summarized in Table 1, which includes values for the LCL filter, controller gains, and grid conditions. These parameters are typical for a medium-power utility interactive inverter in renewable energy applications.
| Parameter | Symbol | Value |
|---|---|---|
| DC Link Voltage | \(U_{dc}\) | 700 V |
| Inverter-Side Inductance | \(L_1\) | 2 mH |
| Grid-Side Inductance | \(L_2\) | 1 mH |
| Filter Capacitance | \(C\) | 10 μF |
| Grid Inductance (Weak Grid) | \(L_g\) | 5 mH |
| OCC Gain | \(K_{\text{OCC}}\) | 0.1 |
| Inner Loop Gain | \(K\) | 10 |
| Outer Loop Proportional Gain | \(K_P\) | 0.5 |
| Outer Loop Integral Gain | \(K_I\) | 100 |
| DSOGI-FLL Gain | \(k_2\) | 1.414 |
| FLL Gain | \(\Gamma\) | 100 |
| Grid Voltage (Nominal) | \(U_s\) | 311 V (peak) |
| Grid Frequency | \(f\) | 50 Hz |
The simulation scenarios include balanced grid voltage, 50% sag in phase a voltage, and a 2 Hz frequency increase. These test the utility interactive inverter’s response to weak grid disturbances. The performance metrics are total harmonic distortion (THD) of grid current, power factor, and dynamic response time. Table 2 summarizes the simulation results under different conditions, highlighting the effectiveness of our control strategy for the utility interactive inverter.
| Scenario | THD Before Event (%) | THD After Event (%) | Power Factor | Response Time (ms) |
|---|---|---|---|---|
| Balanced Voltage | 2.49 | N/A | >0.99 | <5 |
| 50% Sag in Phase a | 3.96 | 3.88 | >0.98 | <10 |
| Frequency Increase by 2 Hz | 3.09 | 2.47 | >0.99 | <8 |
Under balanced grid voltage, the utility interactive inverter achieves synchronization with minimal THD (2.49%), meeting IEEE standards for utility interactive inverters. The grid currents closely track the voltages, as shown in the waveform plots. For the voltage sag scenario, the DSOGI-FLL quickly adapts to the unbalanced condition, maintaining synchronization and stable current injection. The THD remains below 4%, demonstrating the robustness of the utility interactive inverter. During the frequency transition, the DSOGI-FLL locks onto the new frequency within milliseconds, and the utility interactive inverter adjusts its output accordingly, with THD improving to 2.47%. These results confirm that the proposed control strategy enhances the utility interactive inverter’s performance in weak grids.
To further analyze the control strategy, we derive key formulas for the utility interactive inverter. The OCC algorithm ensures that the average inverter output voltage per switching cycle equals the reference. For a three-phase system, the modulation indices can be expressed as:
$$ m_a = \frac{2 U_{ca}}{U_{dc}}, \quad m_b = \frac{2 U_{cb}}{U_{dc}}, \quad m_c = \frac{2 U_{cc}}{U_{dc}} $$
where \(U_{ca}\), \(U_{cb}\), \(U_{cc}\) are the capacitor voltages. The DSOGI-FLL provides the synchronization angle \(\theta\) for Park transformations. The grid current references in dq frame are set based on active and reactive power commands. For unity power factor, the q-axis current reference is zero, and the d-axis reference is derived from power balance:
$$ I_d^* = \frac{2 P^*}{3 U_d}, \quad I_q^* = 0 $$
where \(P^*\) is the active power reference and \(U_d\) is the d-axis grid voltage. The PI controller in the outer loop regulates the grid currents to these references. The inner loop OCC compensates for capacitor current deviations, enhancing system damping. The overall control law for the utility interactive inverter can be summarized as:
$$ \begin{bmatrix} D_{ap} \\ D_{bp} \\ D_{cp} \end{bmatrix} = \begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix} + \frac{K_{\text{OCC}}}{U_{dc}} \left( \mathbf{T}^{-1} \left( \begin{bmatrix} K_P (I_d^* – I_d) + K_I \int (I_d^* – I_d) dt \\ K_P (I_q^* – I_q) + K_I \int (I_q^* – I_q) dt \end{bmatrix} \right) \right) $$
where \(\mathbf{T}\) is the inverse Park transformation matrix. This integrated approach ensures precise control of the utility interactive inverter under various grid conditions.
The stability analysis of the utility interactive inverter involves examining the poles of the closed-loop system. Using the parameters from Table 1, we compute the characteristic equation roots. For example, with the given gains, the poles are located in the left-half plane, indicating stability. The damping ratio \(\zeta\) and natural frequency \(\omega_n\) can be derived from the second-order dominant poles. For the inner loop, the transfer function from capacitor current to duty cycle has a bandwidth determined by \(K_{\text{OCC}}\). We can approximate:
$$ \omega_{\text{bandwidth}} \approx \frac{K K_{\text{OCC}}}{L_1} $$
This bandwidth should be higher than the grid frequency but lower than the switching frequency to avoid noise amplification. For the outer loop, the PI controller introduces a zero at \(s = -K_I/K_P\), improving transient response. The utility interactive inverter’s robustness to grid impedance variations is critical in weak grids. The sensitivity function \(S(s)\) measures how disturbances affect the output:
$$ S(s) = \frac{1}{1 + G_O(s)} $$
We aim to keep \(|S(j\omega)|\) small at low frequencies to reject grid harmonics. The proposed design achieves this through high gain at fundamental frequency, thanks to the PI controller and DSOGI-FLL synchronization.
In comparison to other control methods, such as conventional PI with PLL or repetitive control, our OCC-based strategy offers faster dynamic response and better disturbance rejection. Table 3 compares different control techniques for utility interactive inverters in weak grids, based on factors like THD, response time, and implementation complexity. Our method balances performance and simplicity, making it suitable for practical utility interactive inverter applications.
| Control Method | THD (%) | Response Time (ms) | Complexity | Robustness to Unbalance |
|---|---|---|---|---|
| Proposed OCC with DSOGI-FLL | <4 | <10 | Medium | High |
| Traditional PI with PLL | 5-10 | 20-50 | Low | Low |
| Repetitive Control | <3 | >50 | High | Medium |
| Model Predictive Control | <3 | <5 | High | High |
The utility interactive inverter’s efficiency is also a key consideration. Power losses in the inverter switches and filter components affect overall system performance. We estimate losses using conduction and switching loss models. For IGBT-based inverters, the total loss \(P_{\text{loss}}\) can be approximated as:
$$ P_{\text{loss}} = 6 \left( V_{ce} I_{\text{avg}} + R_{ce} I_{\text{rms}}^2 + f_{\text{sw}} E_{\text{sw}} \right) $$
where \(V_{ce}\) is the collector-emitter saturation voltage, \(R_{ce}\) is the on-state resistance, \(I_{\text{avg}}\) and \(I_{\text{rms}}\) are the average and RMS currents, \(f_{\text{sw}}\) is the switching frequency, and \(E_{\text{sw}}\) is the switching energy per pulse. With our control strategy, the utility interactive inverter operates at near-unity power factor, minimizing reactive power flow and reducing losses. The OCC technique also helps in optimizing switching patterns, further enhancing efficiency.
Future work could focus on adaptive tuning of controller gains for the utility interactive inverter based on real-time grid impedance estimation. Machine learning algorithms could be integrated to predict grid disturbances and adjust control parameters proactively. Additionally, hardware-in-the-loop testing would validate the strategy under realistic conditions. The utility interactive inverter plays a vital role in smart grids, and advanced control methods will continue to evolve.
In conclusion, we have presented a comprehensive control strategy for a three-phase LCL utility interactive inverter in weak grids. By combining one-cycle control with DSOGI-FLL synchronization, the utility interactive inverter achieves fast dynamic response, low THD, and robust performance under voltage and frequency variations. The mathematical analysis and simulation results confirm the effectiveness of our approach. This work contributes to the development of reliable utility interactive inverters for renewable energy integration, supporting the transition to sustainable power systems. The utility interactive inverter is not just a converter but a key enabler of grid stability, and our strategy enhances its functionality in challenging environments.