In the pursuit of global sustainability, the “Dual Carbon” strategic objectives have emerged as a pivotal pathway, driving a rapid transformation in power systems worldwide. This shift is characterized by an ever-increasing penetration of renewable energy sources, primarily solar and wind, whose installed capacity continues to rise annually. These distributed energy resources are predominantly integrated into the distribution network via grid connected inverters. The control strategy of these grid connected inverters is fundamental, directly influencing the security, stability, and reliability of the entire power system. Consequently, developing robust and intelligent control methodologies for these inverters is of paramount importance.
Within the realm of grid-forming control for inverters, several established techniques exist, such as droop control and virtual synchronous generator (VSG) control. However, Virtual Oscillator Control (VOC) has recently garnered significant attention as a promising alternative. VOC offers several inherent advantages, including innate droop characteristics in steady-state, fast dynamic response, and strong disturbance rejection capabilities. Among the various VOC implementations, the Andronov-Hopf Oscillator (AHO)-based control is particularly notable for its structural simplicity and its ability to explicitly schedule output power while preserving the dynamic performance of traditional VOC. This makes it a highly suitable candidate for controlling modern grid connected inverters.
However, the practical deployment of grid connected inverters is often challenged by non-ideal grid conditions. One prevalent and critical scenario is an unbalanced grid voltage, which can arise from asymmetrical faults, uneven line impedances, or single-phase loads. When a grid connected inverter employing conventional VOC is subjected to such unbalanced voltages, it typically leads to severe operational issues. The most prominent problems are three-phase unbalanced output currents and significant double-frequency (2ω) oscillations in the output active and reactive power. These phenomena not only degrade power quality but also impose stress on the inverter hardware and can destabilize the grid. Therefore, enhancing the VOC strategy to ensure reliable performance under unbalanced grid conditions is an essential research endeavor. This article addresses this challenge by proposing a modified AHO-based control strategy. We will delve into the fundamental principles, detail the proposed improvements involving sequence separation and enhanced current control, and validate the strategy’s effectiveness through comprehensive simulation studies.
Fundamental Principles of Andronov-Hopf Oscillator (AHO) Control
The core of the AHO control strategy is a virtual oscillator circuit that mimics the behavior of a non-linear resonant circuit. This circuit generates stable sinusoidal voltage references for the pulse-width modulation (PWM) stage of the grid connected inverter. The basic structure comprises a resonant tank (with virtual inductance \(L_{voc}\) and capacitance \(C_{voc}\)), non-linear state-dependent sources, and a feedback network. The system’s state is defined by the capacitor voltage and a scaled inductor current.
We define the state vector \(\mathbf{x}\) as:
$$
\mathbf{x} = [x_1, x_2]^T = [v_C, \varepsilon i_L]^T
$$
where \(\varepsilon = \sqrt{L_{voc}/C_{voc}}\). The dynamics of the virtual oscillator are governed by the following non-linear differential equations, derived from Kirchhoff’s laws and the definitions of the controlled sources \(v_m\) and \(i_m\):
$$
\begin{aligned}
C_{voc} \frac{d v_C}{dt} &= – i_L + \frac{\xi}{\varepsilon \omega_n}(2X_n^2 – \|\mathbf{x}\|^2) v_C – \Delta i_1 \\
L_{voc} \frac{d i_L}{dt} &= v_C + \frac{\xi}{\omega_n}(2X_n^2 – \|\mathbf{x}\|^2) \varepsilon i_L – \varepsilon \Delta i_2
\end{aligned}
$$
Here, \(\xi\) is a constant governing the convergence speed, \(X_n\) is the root-mean-square (RMS) of the oscillation amplitude, \(\|\mathbf{x}\|\) is the Euclidean norm of the state vector, and \(\omega_n = 1/\sqrt{L_{voc}C_{voc}}\) is the nominal system frequency. The terms \(\Delta i_1\) and \(\Delta i_2\) constitute the feedback input from the inverter’s output. They are calculated by comparing the measured output current in the stationary \(\alpha\beta\) frame (\(\mathbf{i}_{\alpha\beta}\)) with a reference current (\(\mathbf{i}_{\alpha\beta}^*\)):
$$
\begin{bmatrix} \Delta i_1 \\ \Delta i_2 \end{bmatrix} = k_i \mathbf{R}(\theta) (\mathbf{i}_{\alpha\beta} – \mathbf{i}_{\alpha\beta}^*)
$$
where \(k_i\) is a current feedback gain, and \(\mathbf{R}(\theta)\) is a rotation matrix, typically with \(\theta = \pi/2\):
$$
\mathbf{R}(\theta) = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}
$$
The reference current \(\mathbf{i}_{\alpha\beta}^*\) is generated from the active and reactive power setpoints (\(P^*, Q^*\)) and the measured output voltage \(\mathbf{v}_{\alpha\beta}\):
$$
\begin{bmatrix} i_{\alpha}^* \\ i_{\beta}^* \end{bmatrix} = \frac{2}{3\|\mathbf{v}_{\alpha\beta}\|^2} \begin{bmatrix} v_{\alpha} & v_{\beta} \\ v_{\beta} & -v_{\alpha} \end{bmatrix} \begin{bmatrix} P^* \\ Q^* \end{bmatrix}
$$
Finally, the oscillator’s state is scaled by a voltage gain \(k_v\) to produce the voltage reference for the inverter in the \(\alpha\beta\) frame:
$$
\mathbf{v}_{\alpha\beta}^{ref} = [v_{\alpha}^{ref}, v_{\beta}^{ref}]^T = k_v [v_C, \varepsilon i_L]^T
$$
By analyzing the steady-state solution of these dynamics, we can derive the inherent droop characteristics of the AHO-controlled grid connected inverter:
$$
\begin{aligned}
V &\approx \frac{V_n}{\sqrt{2}} \left(1 + \sqrt{1 – \frac{2k_i k_v^3}{3C_{voc}\xi V_n^4}(Q – Q^*)} \right)^{1/2} \\
\omega &\approx \omega_n – \frac{k_v k_i}{3C_{voc} V^2} (P – P^*)
\end{aligned}
$$
where \(V_n = k_v X_n\). These equations reveal that the frequency exhibits a droop relationship with active power, while the voltage magnitude is related to reactive power, enabling power sharing and grid support functions.
Operational Challenges for Grid Connected Inverters in Unbalanced Grids
When the point of common coupling (PCC) experiences unbalanced three-phase voltages, the conventional control strategy for a grid connected inverter faces significant performance degradation. The primary issues stem from the presence of negative-sequence components in the grid voltage. Under such conditions, the instantaneous complex power \(S\) injected by the inverter can be expressed in terms of positive- and negative-sequence voltage and current vectors in the rotating \(dq\) frame:
$$
S = \mathbf{U}_{\alpha\beta} \hat{\mathbf{I}}_{\alpha\beta} = (\mathbf{U}_{dq}^+ e^{j\omega_g t} + \mathbf{U}_{dq}^- e^{-j\omega_g t})(\hat{\mathbf{I}}_{dq}^+ e^{-j\omega_g t} + \hat{\mathbf{I}}_{dq}^- e^{j\omega_g t})
$$
Expanding this expression, the instantaneous active power \(P\) and reactive power \(Q\) are obtained:
$$
\begin{aligned}
P(t) &= P_0 + P_{c2} \cos(2\omega_g t) + P_{s2} \sin(2\omega_g t) \\
Q(t) &= Q_0 + Q_{c2} \cos(2\omega_g t) + Q_{s2} \sin(2\omega_g t)
\end{aligned}
$$
Here, \(P_0\) and \(Q_0\) are the average (dc) components of power, while the terms with coefficients \(P_{c2}\), \(P_{s2}\), \(Q_{c2}\), and \(Q_{s2}\) represent the double-frequency oscillatory components. These oscillatory components are directly linked to the negative-sequence currents and voltages. Their amplitudes are given by:
$$
\begin{bmatrix} P_0 \\ Q_0 \\ P_{c2} \\ P_{s2} \\ Q_{c2} \\ Q_{s2} \end{bmatrix} = \frac{3}{2}
\begin{bmatrix}
v_{gd}^+ & v_{gq}^+ & v_{gd}^- & v_{gq}^- \\
v_{gq}^+ & -v_{gd}^+ & v_{gq}^- & -v_{gd}^- \\
v_{gd}^- & v_{gq}^- & v_{gd}^+ & v_{gq}^+ \\
v_{gq}^- & -v_{gd}^- & -v_{gq}^+ & v_{gd}^+ \\
v_{gq}^- & -v_{gd}^- & v_{gq}^+ & -v_{gd}^+ \\
-v_{gd}^- & -v_{gq}^- & v_{gd}^+ & v_{gq}^+
\end{bmatrix}
\begin{bmatrix} i_d^+ \\ i_q^+ \\ i_d^- \\ i_q^- \end{bmatrix}
$$
For a grid connected inverter using standard VOC, the controller’s feedback is based on the total three-phase quantities. Without specific mitigation, the inverter will naturally inject currents that follow the unbalanced grid voltage, leading to severe three-phase current imbalance. Furthermore, as evident from the power equations, the presence of negative-sequence current components (\(i_d^-, i_q^-\)) inevitably generates large double-frequency power ripples. These ripples can cause overheating, mechanical stress, and violate grid codes. Therefore, an advanced control strategy must simultaneously address two key objectives: 1) ensuring balanced three-phase output currents to protect the inverter and local loads, and 2) suppressing the double-frequency power oscillations to maintain stable power delivery to the grid.
Proposed Improved AHO Control Strategy for Grid Connected Inverters
To tackle the aforementioned challenges, we propose a comprehensive modification to the standard AHO control structure. The enhanced strategy incorporates three key modules: a fast and accurate sequence separation block, an improved dual-sequence current control loop with reference calculation, and a pre-synchronization unit for seamless grid connection. The overall block diagram of the proposed control system for the grid connected inverter is conceptualized as follows.
1. Positive and Negative Sequence Separation using Reduced-Order Resonant (ROR) Regulators
Accurate and rapid extraction of positive- and negative-sequence components is the cornerstone of unbalanced condition management. While methods like the Second-Order Generalized Integrator (SOGI) are common, we employ Reduced-Order Resonant (ROR) regulators for their computational efficiency and independent control over sequences. The ROR transfer functions for extracting positive-sequence (at frequency \(\omega\)) and negative-sequence (at frequency \(-\omega\)) components are respectively:
$$
G_{R+}(s) = \frac{k}{s – j\omega}, \quad G_{R-}(s) = \frac{k}{s + j\omega}
$$
where \(k\) is a gain parameter. The magnitude-frequency response of these regulators shows infinite gain at their respective resonant frequencies (\(\omega\) and \(-\omega\)) and high attenuation elsewhere, enabling clean separation. The implementation structure involves transforming the measured three-phase voltages \(v_{abc}\) and currents \(i_{abc}\) into the \(\alpha\beta\) stationary frame. These \(\alpha\beta\) components are then fed into parallel ROR-based observer loops. For instance, for current separation, the system solves:
$$
\begin{aligned}
\hat{i}_{\alpha}^+ &= G_{R+}(s) (i_{\alpha} – \hat{i}_{\alpha}^+) \\
\hat{i}_{\beta}^+ &= G_{R+}(s) (i_{\beta} – \hat{i}_{\beta}^+)
\end{aligned}
$$
and similarly for negative-sequence components using \(G_{R-}(s)\). This yields the decoupled positive-sequence vector \(\mathbf{i}_{\alpha\beta}^+\) and negative-sequence vector \(\mathbf{i}_{\alpha\beta}^-\). The same process is applied to the PCC voltage \(\mathbf{v}_{g,\alpha\beta}\) to obtain \(\mathbf{v}_{g,\alpha\beta}^+\) and \(\mathbf{v}_{g,\alpha\beta}^-\). This separated information is crucial for the subsequent control loops.
2. Current Reference Calculation and Enhanced Dual-Sequence Current Control
With the sequence components available, we design an independent current control structure. The primary control objective under voltage imbalance is often to inject balanced three-phase currents. This is achieved by setting the negative-sequence current reference to zero: \(\mathbf{i}_{dq}^{-*} = [0, 0]^T\). For the positive-sequence current, we derive a reference that ensures proper power transfer and grid support. Considering the inverter output filter and grid impedance, the relationship between the inverter’s internal reference voltage (from the AHO, \(\mathbf{v}_{dq}^{ref*}\)), the grid positive-sequence voltage (\(\mathbf{v}_{g,dq}^+\)), and the positive-sequence output current (\(\mathbf{i}_{dq}^+\)) in the synchronous \(dq\) frame is:
$$
\mathbf{v}_{dq}^{ref*} = \mathbf{v}_{g,dq}^+ + (R + j\omega L) \mathbf{i}_{dq}^+
$$
where \(R\) and \(L\) represent the equivalent total resistance and inductance between the inverter and the grid. From this, we can solve for the desired positive-sequence current references that would result if the inverter tracked its voltage reference perfectly:
$$
\begin{bmatrix} i_d^{+*} \\ i_q^{+*} \end{bmatrix} = \frac{1}{R^2 + (\omega L)^2} \begin{bmatrix} R & \omega L \\ -\omega L & R \end{bmatrix} \begin{bmatrix} v_d^{ref*} – v_{gd}^+ \\ v_q^{ref*} – v_{gq}^+ \end{bmatrix}
$$
These references, \(i_d^{+*}\) and \(i_q^{+*}\), along with \(i_d^{-*}=0\) and \(i_q^{-*}=0\), are then used in the current control loop. The proposed current controller operates on both sequences independently. It typically consists of proportional-resonant (PR) controllers tuned at the fundamental frequency for the positive-sequence loop and at the negative fundamental frequency for the negative-sequence loop to ensure zero steady-state error. The output of these controllers provides the compensating voltage signals \(\Delta \mathbf{v}_{dq}^+\) and \(\Delta \mathbf{v}_{dq}^-\), which are added to the original AHO-generated voltage reference \(\mathbf{v}_{dq}^{ref*}\) after appropriate inverse transformations. This forms the final voltage command for the PWM modulator. This architecture allows the grid connected inverter to simultaneously regulate positive-sequence power and suppress negative-sequence currents.
3. Pre-synchronization Module for Smooth Grid Connection
To avoid large inrush currents and voltage disturbances during the closing of the static switch that connects the inverter to the grid, a pre-synchronization phase is essential. Before grid connection, the AHO’s feedback input is switched from the normal current-based feedback to a voltage-based error signal. In this mode, the input \(\Delta \mathbf{i}\) becomes:
$$
\begin{bmatrix} \Delta i_1 \\ \Delta i_2 \end{bmatrix} = \gamma \begin{bmatrix} v_{\alpha}^{ref} – v_{g\alpha} \\ v_{\beta}^{ref} – v_{g\beta} \end{bmatrix}
$$
where \(\gamma\) is a control gain dictating the synchronization speed, and \(v_{g\alpha}, v_{g\beta}\) are the measured grid voltages. This feedback forces the AHO’s output voltage \(\mathbf{v}_{\alpha\beta}^{ref}\) to align in phase, frequency, and amplitude with the grid voltage \(\mathbf{v}_{g,\alpha\beta}\). Once the voltage difference falls within a predefined tolerance (e.g., phase error < 2°, magnitude error < 5%), the switch is commanded to close, and the feedback is seamlessly switched back to the normal current-regulation mode. This process ensures a transient-free connection for the grid connected inverter, enhancing its reliability and lifespan.
Simulation Model, Parameters, and Performance Analysis
To validate the efficacy of the proposed improved AHO control strategy for a grid connected inverter, a detailed simulation model was constructed in the Matlab/Simulink environment. The system comprises a DC source, a three-phase two-level voltage source inverter (VSI), an LCL output filter, a coupling inductor representing grid impedance, and the unbalanced grid voltage source. The control algorithms, including the standard AHO, the proposed modified AHO with sequence separation, and the pre-synchronization logic, were implemented in discrete-time.
The key parameters for the AHO controller and the power circuit are summarized in the tables below. These parameters are designed to meet typical specifications for a medium-power grid connected inverter.
| Parameter | Symbol | Value |
|---|---|---|
| Nominal Oscillation Amplitude | \(X_n\) | 1 V |
| Convergence Speed Constant | \(\xi\) | 15 |
| Nominal Frequency | \(\omega_n\) | 100π rad/s |
| Voltage Scaling Gain | \(k_v\) | 220 |
| Current Feedback Gain | \(k_i\) | 0.067 |
| Virtual Capacitance | \(C_{voc}\) | 0.27 F |
| Virtual Inductance | \(L_{voc}\) | 37.82 µH |
| Pre-synchronization Gain | \(\gamma\) | 1 |
| Active Power Setpoint | \(P^*\) | 10 kW |
| Reactive Power Setpoint | \(Q^*\) | 0 kVar |
| Parameter | Symbol | Value |
|---|---|---|
| DC Link Voltage | \(V_{dc}\) | 700 V |
| Grid Nominal Voltage (Phase RMS) | \(U_g\) | 220 V |
| Filter Inductor (Inverter-side) | \(L_f\) | 3 mH |
| Filter Capacitor | \(C_f\) | 50 µF |
| Grid-side Coupling Inductor | \(L_g\) | 5 mH |
| Filter Inductor ESR | \(R_f\) | 0.1 Ω |
| Grid Coupling Resistance | \(R_g\) | 0.3 Ω |
| Switching Frequency | \(f_{sw}\) | 10 kHz |
The simulation scenario was designed as follows: The grid connected inverter starts in islanded mode, supplying local loads. At t = 0.1 s, the pre-synchronization process is initiated. Once synchronization is achieved (around t = 0.11 s), the main contactor closes, connecting the inverter to a balanced grid. The system operates normally until t = 1.5 s, when a single-phase voltage dip is simulated to create an unbalanced grid condition: the Phase-A voltage amplitude drops to 240 V (peak ≈ 339 V), while Phases B and C remain at their nominal 311 V (peak). The simulation runs until t = 3.0 s.
The performance of the proposed control strategy is compared against the conventional AHO control. The key metrics analyzed are: 1) Three-phase output current waveform and balance, 2) Total Harmonic Distortion (THD) of the output current, and 3) The magnitude of double-frequency oscillations in active and reactive power.
Simulation results clearly demonstrate the superiority of the proposed method for the grid connected inverter. Under the unbalanced grid condition (after t=1.5 s), the conventional AHO control leads to highly unbalanced currents, with phase magnitudes varying significantly (e.g., from 13 A to 33 A). This results in a current THD exceeding acceptable limits for grid codes. Furthermore, the output power exhibits large double-frequency ripples, with peak-to-peak oscillations of approximately 1.9 kW in active power and 2.0 kVar in reactive power.
In stark contrast, the grid connected inverter employing the proposed improved AHO strategy maintains balanced three-phase currents even under the unbalanced voltage condition. The phase currents remain nearly equal in magnitude, with a maximum deviation of only about 1 A. The current THD is significantly reduced to 1.66%, which is well within the typical limit of 5% prescribed by standards like IEEE 1547. The power oscillation suppression is also remarkable. The peak-to-peak ripple in active power is reduced to about 1.0 kW (a 47.4% reduction compared to the conventional method), and in reactive power to about 1.3 kVar (a 35% reduction). The pre-synchronization module functioned perfectly, enabling a smooth connection at t=0.11 s without any noticeable voltage or current transients.
To quantify the performance improvement, we summarize the key comparative results in the table below.
| Performance Metric | Conventional AHO Control | Proposed Improved AHO Control | Improvement |
|---|---|---|---|
| Current Balance (Max-Min phase magnitude) | ~20 A (Unbalanced) | ~1 A (Balanced) | Significantly Balanced |
| Output Current THD | >5% (Estimated) | 1.66% | >60% reduction |
| Active Power Ripple Amplitude (approx. peak-to-peak) | 1.9 kW | 1.0 kW | Reduced by 47.4% |
| Reactive Power Ripple Amplitude (approx. peak-to-peak) | 2.0 kVar | 1.3 kVar | Reduced by 35% |
| Grid Connection Transient | Not Specifically Addressed | Smooth, no inrush current | Enabled via pre-sync |
The mathematical basis for the power ripple reduction can be revisited using the earlier power component equations. By forcing the negative-sequence current references to zero (\(i_d^{-*}=0, i_q^{-*}=0\)), the terms \(P_{c2}, P_{s2}, Q_{c2}, Q_{s2}\) that depend on \(i_d^-\) and \(i_q^-\) are minimized. However, they are not driven to zero completely because the coefficients also involve negative-sequence grid voltages (\(v_{gd}^-, v_{gq}^-\)), which are non-zero due to the imbalance. The residual oscillation is due to the interaction between the positive-sequence current and the negative-sequence voltage. Further suppression could be achieved by setting different current reference objectives (e.g., constant instantaneous power), but at the cost of unbalanced currents. The chosen objective of balanced currents represents a practical and widely adopted compromise for the health of the grid connected inverter itself and any local three-phase loads.
Conclusion
This article has presented a comprehensive analysis and solution for a critical operational challenge faced by modern grid connected inverters: maintaining high performance under unbalanced grid voltage conditions. We focused on the promising Virtual Oscillator Control, specifically the AHO variant, and identified its limitations in such scenarios, namely unbalanced output currents and significant double-frequency power oscillations. To overcome these limitations, we proposed a modified AHO control strategy. The core enhancements include the integration of fast and accurate Reduced-Order Resonant (ROR) regulators for real-time separation of positive- and negative-sequence voltage and current components, the derivation of a current reference calculation block to enforce balanced current injection, and the design of an independent dual-sequence current control loop. Additionally, a pre-synchronization module was incorporated to ensure transient-free grid connection, a vital feature for practical deployment of any grid connected inverter.
Through detailed simulation studies in Matlab/Simulink, the proposed strategy was rigorously validated. The results conclusively demonstrate that the improved control enables the grid connected inverter to produce balanced three-phase currents with low harmonic distortion (1.66% THD) even when the grid voltages are severely unbalanced. Simultaneously, it substantially mitigates the double-frequency ripples in both active and reactive power, reducing their oscillation amplitudes by 47.4% and 35%, respectively, compared to the conventional AHO approach. The pre-synchronization mechanism effectively eliminates inrush currents during grid connection. Therefore, the proposed modified Virtual Oscillator Control strategy offers a robust, high-performance solution for grid connected inverters operating in realistic, non-ideal grid environments, contributing to the stable and efficient integration of renewable energy sources and supporting the broader objectives of sustainable power system development.