I begin from the operating reality that most solar inverters are designed, tuned, and validated under the assumption that the three-phase grid voltage is balanced. When I connect solar inverters to a distribution or transmission point where the phase voltages are not symmetrical, the behavior of those solar inverters changes in ways that are not merely cosmetic. The DC-link voltage of solar inverters can acquire a second harmonic ripple. The output currents of solar inverters can become three-phase asymmetrical. The total harmonic distortion of the current injected by solar inverters can rise sharply. In severe cases, the thermal and control stress placed on solar inverters can lead to device failure. I therefore treat unbalanced grid voltage not as a rare disturbance but as a design condition that solar inverters must survive and, ideally, ride through with good power quality.
My objective in this work is to develop a control strategy for solar inverters that keeps the grid-side current balanced and sinusoidal when the grid voltage contains both positive-sequence and negative-sequence components. I want the method to be fast enough for practical solar inverters, simple enough for embedded implementation, and accurate enough to satisfy grid-code expectations. I first address the synchronization problem because every current-controlled solar inverters depends on a reliable phase angle. I design a positive-sequence and negative-sequence separation phase-locked loop based on a second-order generalized integrator. I then use that separation to implement a negative-sequence voltage feedforward path in the current control of solar inverters. I also add a second-order notch filter after the DC voltage outer-loop controller so that the second harmonic ripple on the DC link does not propagate into the current references of solar inverters. I verify the sequence separation experimentally and the complete control strategy by electromagnetic transient simulation.
For clarity, I define the main quantities and symbols that I use repeatedly when I analyze solar inverters under unbalanced conditions.
| Symbol | Meaning in my analysis of solar inverters |
|---|---|
| \(v_{abc}\) | Three-phase grid voltage vector seen by solar inverters |
| \(v_{abc}^{+}\) | Positive-sequence component of grid voltage |
| \(v_{abc}^{-}\) | Negative-sequence component of grid voltage |
| \(v_{abc}^{0}\) | Zero-sequence component of grid voltage |
| \(v_{\alpha\beta}\) | Stationary-frame voltage vector |
| \(v_{\alpha\beta}^{+}\) | Positive-sequence stationary-frame voltage vector |
| \(v_{\alpha\beta}^{-}\) | Negative-sequence stationary-frame voltage vector |
| \(e_d^P,e_q^P\) | Positive-sequence grid voltage in synchronous coordinates |
| \(e_d^N,e_q^N\) | Negative-sequence grid voltage in synchronous coordinates |
| \(I_d^P,I_q^P\) | Positive-sequence current injected by solar inverters |
| \(I_d^N,I_q^N\) | Negative-sequence current injected by solar inverters |
| \(p_0,q_0\) | Average active and reactive power of solar inverters |
| \(p_{2c},p_{2s}\) | Second harmonic active-power components |
| \(q_{2c},q_{2s}\) | Second harmonic reactive-power components |
| \(V_{dc}\) | DC-link voltage of solar inverters |
| \(V_{dc}^{*}\) | DC-link voltage reference for solar inverters |
| \(\omega_g\) | Grid angular frequency |
| \(\omega’\) | Center frequency of the second-order generalized integrator |
| \(k\) | Damping gain of the second-order generalized integrator |
| \(q\) | Ninety-degree phase-shift operator |
| \(T^{+},T^{-}\) | Symmetrical-component transformation matrices |
| \(G_i(s)\) | Current-loop controller of solar inverters |
| \(G_v(s)\) | Voltage-loop controller of solar inverters |
| \(G_{\text{notch}}(s)\) | Second-order notch filter used in solar inverters |
I start with the mathematical description of an unbalanced three-phase voltage. By the method of symmetrical components, the voltage vector can be decomposed into positive-sequence, negative-sequence, and zero-sequence parts. For solar inverters connected to a three-wire system, I do not consider zero-sequence voltage in the control because there is no zero-sequence current path. This is a standard and useful simplification for three-phase solar inverters. I write the decomposition as
$$
v_{abc}=v_{abc}^{+}+v_{abc}^{-}+v_{abc}^{0}.
$$
The positive-sequence component can be extracted by the transformation matrix \(T^{+}\), and the negative-sequence component by \(T^{-}\). I express these transformations as
$$
v_{abc}^{+}=T^{+}v_{abc}
=\frac{1}{3}
\begin{bmatrix}
1 & a & a^{2}\\
a^{2} & 1 & a\\
a & a^{2} & 1
\end{bmatrix}
\begin{bmatrix}
v_a\\
v_b\\
v_c
\end{bmatrix},
$$
$$
v_{abc}^{-}=T^{-}v_{abc}
=\frac{1}{3}
\begin{bmatrix}
1 & a^{2} & a\\
a & 1 & a^{2}\\
a^{2} & a & 1
\end{bmatrix}
\begin{bmatrix}
v_a\\
v_b\\
v_c
\end{bmatrix},
$$
where \(a=e^{j2\pi/3}\). I then transform these sequence quantities into the stationary \(\alpha\beta\) frame. A compact way to express the positive-sequence and negative-sequence separation is through the ninety-degree phase-shift operator \(q=e^{-j\pi/2}\). I write
$$
v_{\alpha\beta}^{+}
=
\frac{1}{2}
\begin{bmatrix}
1 & -q\\
q & 1
\end{bmatrix}
v_{\alpha\beta},
$$
$$
v_{\alpha\beta}^{-}
=
\frac{1}{2}
\begin{bmatrix}
1 & q\\
-q & 1
\end{bmatrix}
v_{\alpha\beta}.
$$
These two equations are central to my design of sequence separation for solar inverters. They tell me that if I can create an orthogonal copy of the measured voltage, I can separate the positive and negative sequences without a full Fourier analysis and without a long delay. The orthogonal copy must be accurate over the frequency range that solar inverters actually encounter. It must also be fast, because solar inverters must respond to grid faults, voltage sags, and phase jumps without losing current control.
I use a second-order generalized integrator, which I abbreviate as SOGI, to generate the orthogonal signal. The SOGI is attractive for solar inverters because it is a resonator-based structure with a simple implementation and a clear frequency response. Its center frequency is set to the nominal grid frequency. Its damping gain controls the tradeoff between selectivity and dynamic response. I define the SOGI input as a sinusoidal signal \(v\), the in-phase output as \(v’\), and the quadrature output as \(qv’\). The transfer functions from \(v\) to \(v’\) and from \(v\) to \(qv’\) are
$$
D(s)=\frac{v'(s)}{v(s)}
=
\frac{k\omega’ s}{s^{2}+k\omega’ s+\omega’^{2}},
$$
$$
Q(s)=\frac{qv'(s)}{v(s)}
=
\frac{k\omega’^{2}}{s^{2}+k\omega’ s+\omega’^{2}}.
$$
When the input frequency \(\omega\) is equal to the SOGI center frequency \(\omega’\), the magnitude and phase behavior of the two outputs have the properties I need for solar inverters. I summarize those properties in the following table.
| Condition | In-phase output \(v’\) | Quadrature output \(qv’\) |
|---|---|---|
| \(\omega=\omega’\) | Same amplitude as input | Same amplitude as input |
| \(\omega=\omega’\) | Same phase as input | Phase lag of \(90^\circ\) |
| \(\omega\neq\omega’\) | Amplitude and phase are frequency dependent | Amplitude and phase are frequency dependent |
| Purpose in solar inverters | Track the measured voltage | Provide orthogonal signal for sequence separation |
The magnitude and phase expressions that I use to assess the SOGI are
$$
|D(s)|=
\frac{k\omega’\omega}
{\sqrt{(k\omega’\omega)^{2}+(\omega^{2}-\omega’^{2})^{2}}},
$$
$$
\angle D(s)=\arctan\left(\frac{\omega’^{2}-\omega^{2}}{k\omega’\omega}\right),
$$
$$
|Q(s)|=\frac{\omega’}{\omega}|D(s)|,
$$
$$
\angle Q(s)=\angle D(s)-\frac{\pi}{2}.
$$
These expressions confirm that the SOGI gives me a quadrature signal with the correct phase relationship when the center frequency matches the grid frequency. For solar inverters, this means that I can build a positive-sequence and negative-sequence separation module with two SOGI blocks, one for the \(\alpha\) axis and one for the \(\beta\) axis. The resulting sequence separation module is simple, and it avoids the heavy computational load of some adaptive observer methods. It also avoids the phase-feedback delay that can cause large overshoot and long recovery times in some double synchronous reference frame phase-locked loops for solar inverters.
I now connect the sequence separation to the control of solar inverters. Under unbalanced grid voltage, the instantaneous active and reactive power exchanged by solar inverters contains not only average terms but also second harmonic terms. If I choose to suppress the negative-sequence current, I set the negative-sequence current references of solar inverters to zero. In that case, the power expressions become functions of the positive-sequence current and both positive-sequence and negative-sequence voltages. I write the average active power and average reactive power as
$$
p_0=\frac{3}{2}\left(e_d^P I_d^P+e_q^P I_q^P\right),
$$
$$
q_0=\frac{3}{2}\left(e_q^P I_d^P-e_d^P I_q^P\right).
$$
The second harmonic active-power components are
$$
p_{2c}=\frac{3}{2}\left(e_d^N I_d^P+e_q^N I_q^P\right),
$$
$$
p_{2s}=\frac{3}{2}\left(-e_d^N I_q^P+e_q^N I_d^P\right).
$$
The second harmonic reactive-power components are
$$
q_{2c}=\frac{3}{2}\left(e_q^N I_d^P-e_d^N I_q^P\right),
$$
$$
q_{2s}=\frac{3}{2}\left(-e_q^N I_q^P-e_d^N I_d^P\right).
$$
I collect these power components in a table because they explain why solar inverters suffer when the grid is unbalanced. The average power determines the main energy transfer. The second harmonic terms determine the ripple that appears on the DC link and, through the voltage loop, on the current references of solar inverters.
| Power term | Expression | Effect on solar inverters |
|---|---|---|
| \(p_0\) | \(\frac{3}{2}(e_d^P I_d^P+e_q^P I_q^P)\) | Average active power delivered by solar inverters |
| \(q_0\) | \(\frac{3}{2}(e_q^P I_d^P-e_d^P I_q^P)\) | Average reactive power exchanged by solar inverters |
| \(p_{2c}\) | \(\frac{3}{2}(e_d^N I_d^P+e_q^N I_q^P)\) | Cosine second harmonic active-power ripple |
| \(p_{2s}\) | \(\frac{3}{2}(-e_d^N I_q^P+e_q^N I_d^P)\) | Sine second harmonic active-power ripple |
| \(q_{2c}\) | \(\frac{3}{2}(e_q^N I_d^P-e_d^N I_q^P)\) | Cosine second harmonic reactive-power ripple |
| \(q_{2s}\) | \(\frac{3}{2}(-e_q^N I_q^P-e_d^N I_d^P)\) | Sine second harmonic reactive-power ripple |
Because the second harmonic power terms are generally not zero when the grid is unbalanced, the DC-link voltage of solar inverters contains a second harmonic component. If I allow that ripple to pass through the voltage outer loop, the current reference of solar inverters will also contain harmonic components. The resulting grid current can contain third, fifth, seventh, and higher order harmonics. This is why I place a notch filter after the voltage controller. The notch filter is tuned to the second harmonic frequency, which is twice the grid frequency. I write the notch filter transfer function as
$$
G_{\text{notch}}(s)=
\frac{s^{2}+\omega_0^{2}}{s^{2}+2\zeta\omega_0 s+\omega_0^{2}},
$$
where \(\omega_0=2\omega_g\) and \(\zeta\) is the damping ratio. I select \(\zeta\) so that the filter strongly attenuates the second harmonic while preserving an acceptable voltage-loop bandwidth. The voltage controller itself can have a relatively low cutoff frequency. Together, the low-bandwidth voltage controller and the second-order notch filter reduce the propagation of DC-link ripple into the current commands of solar inverters.
My unbalanced control system uses negative-sequence voltage feedforward. The idea is straightforward. The current controller of solar inverters already uses positive-sequence voltage feedforward to improve disturbance rejection. When the grid is unbalanced, I add a negative-sequence feedforward term based on the separated negative-sequence voltage. I define the negative-sequence current references as
$$
I_d^{N*}=0,
\qquad
I_q^{N*}=0.
$$
Thus, the control objective for solar inverters is to suppress negative-sequence current. The positive-sequence current references come from the outer loops. I express the active current reference as
$$
I_d^{P*}=G_v(s)\left(V_{dc}^{*}-V_{dc}\right),
$$
while the reactive current reference \(I_q^{P*}\) is set according to the desired reactive power or power factor. The current controller for solar inverters can then be written in synchronous coordinates as
$$
v_d^{P*}=e_d^P+\omega L I_q^P+G_i(s)\left(I_d^{P*}-I_d^P\right),
$$
$$
v_q^{P*}=e_q^P-\omega L I_d^P+G_i(s)\left(I_q^{P*}-I_q^P\right).
$$
To include the negative-sequence feedforward, I add the measured or estimated negative-sequence voltage components to the voltage command. I write
$$
v_d^{*}=v_d^{P*}+e_d^{N},
$$
$$
v_q^{*}=v_q^{P*}+e_q^{N}.
$$
This feedforward path allows solar inverters to cancel the negative-sequence disturbance more quickly than a feedback-only controller. It also keeps the structure simple. I do not need four current PI controllers and a separate voltage PI controller arranged in a dual synchronous frame. I do not need a positive-sequence and negative-sequence current separation module in the current feedback path. I only need a clean sequence separation of the grid voltage. That is the main reason the method is attractive for practical solar inverters: it reduces the number of tuning parameters while still providing a clear physical response to unbalanced voltage.
I compare the conventional control strategy and my proposed strategy in the following table. I refer to the conventional strategy as Type I and my negative-sequence voltage feedforward strategy as Type II.
| Feature | Type I: conventional control of solar inverters | Type II: proposed negative-sequence feedforward for solar inverters |
|---|---|---|
| Grid-voltage assumption | Balanced three-phase voltage | Unbalanced three-phase voltage with positive and negative sequences |
| Sequence separation | Usually absent or only in the phase-locked loop | SOGI-based positive and negative sequence separation |
| Negative-sequence current | Not directly controlled | Suppressed to zero |
| Current-loop structure | Single synchronous frame PI controller | Single synchronous frame PI controller plus negative-sequence voltage feedforward |
| DC-link ripple treatment | Limited attenuation | Second-order notch filter after voltage controller |
| Tuning complexity | Low for balanced grid, poor for unbalanced grid | Moderate and systematic for unbalanced grid |
| Current quality | Unbalanced current and high THD under voltage sag | Balanced current and low THD under voltage sag |
I also examined alternative methods that are often applied to solar inverters under unbalanced voltage. Some methods use a decoupled double synchronous reference frame phase-locked loop. Those methods can achieve high steady-state accuracy, but they depend on phase feedback and can exhibit large overshoot and long recovery when the grid phase jumps. Some methods use an adaptive observer. Those methods can be accurate, but the computational burden is high for many solar inverters. Some methods use positive-sequence and negative-sequence dual current controllers. Those methods can achieve good control, but the number of current PI controllers and the coordination of their gains make tuning difficult for solar inverters. Some methods use proportional resonant controllers in the stationary frame. Those controllers avoid synchronous-frame decoupling, but their frequency adaptability is limited when the grid frequency deviates. Some methods control only positive-sequence current. That approach can reduce DC-link voltage ripple to some extent, but it cannot fully achieve constant-power grid connection, and it often requires a complicated high-pass filter for voltage compensation. I therefore prefer the SOGI-based sequence separation and negative-sequence voltage feedforward because it gives a good balance between performance and implementation complexity for solar inverters.
I validated the positive-sequence and negative-sequence separation experimentally. I implemented the SOGI-based separation module on a digital signal controller platform. The input signal contained a positive-sequence component with an amplitude of \(100\text{ V}\) and a negative-sequence component with an amplitude of \(50\text{ V}\). I aligned the synchronous d-axis with the grid voltage phase in order to lock the phase angle. The sequence separation module produced the positive-sequence amplitude, the negative-sequence amplitude, and the d-axis voltage in a form that could be used by the controller of solar inverters. I summarize the experimental conditions and observations in the following table.
| Item | Value or observation |
|---|---|
| Input positive-sequence amplitude | \(100\text{ V}\) |
| Input negative-sequence amplitude | \(50\text{ V}\) |
| Sequence separation method | SOGI-based positive and negative sequence extraction |
| Synchronous alignment | d-axis aligned with grid voltage phase |
| Dynamic response | Fast settling and low overshoot in the extracted sequence quantities |
| Steady-state accuracy | Accurate positive-sequence and negative-sequence amplitudes |
| Relevance to solar inverters | Provides reliable phase and amplitude information for current control |
The experimental result supports my claim that the SOGI-based method is suitable for solar inverters. It gives me a positive-sequence angle and a negative-sequence voltage estimate without a complicated adaptive law. It also avoids the long transient that can occur in some phase-locked loops when the grid phase changes. For solar inverters, fast and accurate sequence information is important because the current controller must decide almost immediately how to distribute current among phases during a voltage sag. If the sequence separation is slow, the current response of solar inverters will be late and distorted. If the sequence separation is inaccurate, the negative-sequence feedforward will inject an incorrect voltage component and may worsen the current balance. My experimental result indicates that the SOGI-based module provides the required speed and accuracy.
I then built a simulation model to verify the complete control strategy for solar inverters. I used an electromagnetic transient simulation environment and modeled a three-level solar inverters system. I selected parameters that are representative of a low-voltage or medium-power solar inverters application. I list the main simulation parameters in the following table.
| Parameter | Value |
|---|---|
| Maximum grid voltage \(E_{\max}\) | \(220\text{ V}\) |
| DC-link stabilizing capacitor \(C\) | \(600\,\mu\text{F}\) |
| Rated system power \(P_N\) | \(10\text{ kW}\) |
| Grid-side filter inductor \(L\) | \(0.45\text{ mH}\) |
| Grid frequency | Nominal frequency with small excursion considered |
| Inverter topology | Three-level solar inverters |
| Unbalanced condition | Phase A voltage drops to \(50\%\) of its original value at \(0.3\text{ s}\) |
| Control comparison | Type I conventional control and Type II proposed control |
I first simulated the conventional Type I strategy. Under the Type I strategy, the solar inverters do not explicitly suppress negative-sequence current. When the A-phase voltage drops, the output current becomes three-phase asymmetrical. The current total harmonic distortion rises significantly. I evaluated the total harmonic distortion using
$$
\text{THD}=
\sqrt{
\frac{\sum_{h=2}^{H} I_h^{2}}{I_1^{2}}
}
\times 100\%,
$$
where \(I_h\) is the harmonic current of order \(h\) and \(I_1\) is the fundamental current. I also evaluated the current unbalance using the ratio of negative-sequence current to positive-sequence current,
$$
\text{Current unbalance}
=
\frac{I^{-}}{I^{+}}
\times 100\%.
$$
For the Type I strategy, the current unbalance and THD became unacceptable for grid-connected solar inverters. The DC-link voltage of the solar inverters contained a visible second harmonic ripple. The current waveform contained low-order harmonics, especially third, fifth, and seventh harmonics. This result confirmed that conventional control of solar inverters is not sufficient when the grid voltage is unbalanced.
I then simulated the Type II strategy. In this strategy, the SOGI-based sequence separation provides the positive-sequence phase and the negative-sequence voltage. The negative-sequence current references of the solar inverters are set to zero. The negative-sequence voltage feedforward cancels the negative-sequence disturbance in the current loop. The second-order notch filter attenuates the second harmonic in the DC voltage loop. I summarize the comparison between the two strategies in the following table.
| Performance indicator | Type I conventional solar inverters | Type II proposed solar inverters |
|---|---|---|
| Three-phase current symmetry after voltage sag | Poor, visibly asymmetrical | Good, nearly symmetrical |
| Current THD after voltage sag | High and above typical grid-code limits | About \(2.3\%\) |
| DC-link second harmonic ripple | Large and coupled into current reference | Reduced by the notch filter |
| Negative-sequence current | Not controlled | Suppressed toward zero |
| Transient response to voltage sag | Current distortion and unbalance | Fast recovery and balanced current |
| Implementation complexity | Low but inadequate | Moderate and practical |
| Suitability for solar inverters | Limited under unbalanced voltage | Strong under unbalanced voltage |
The simulation results for the Type II strategy show that the grid current remains balanced after the A-phase voltage drop. The current THD remains around \(2.3\%\) in steady state. This value is much lower than the Type I result and is acceptable for many grid-connected solar inverters. The notch filter and the voltage controller work together to prevent the second harmonic DC-link ripple from entering the current references of the solar inverters. The negative-sequence voltage feedforward allows the current loop to reject the negative-sequence disturbance without requiring a separate negative-sequence current controller. The overall structure is simpler than a full dual synchronous reference frame controller and more robust than a simple positive-sequence-only controller.
I also examined the harmonic content of the current in more detail. Under Type I control, the unbalanced voltage produces current harmonics that are not present under balanced conditions. The second harmonic power ripple on the DC link of the solar inverters modulates the current reference. This modulation appears in the AC current as third harmonic, fifth harmonic, seventh harmonic, and higher order components. Under Type II control, the notch filter reduces the second harmonic in the voltage loop, and the negative-sequence feedforward reduces the negative-sequence voltage disturbance. As a result, the harmonic currents are much smaller. I show a representative comparison in the following table.
| Harmonic order | Type I solar inverters | Type II solar inverters |
|---|---|---|
| Fundamental | Present, but phases unbalanced | Balanced three-phase fundamental |
| Third harmonic | Elevated | Strongly reduced |
| Fifth harmonic | Elevated | Reduced |
| Seventh harmonic | Elevated | Reduced |
| Total harmonic distortion | High | Approximately \(2.3\%\) |
The dynamic response of the proposed method is important for solar inverters. When the grid voltage changes suddenly, the phase-locked loop of the solar inverters must provide a stable angle. The SOGI-based sequence separation avoids the slow phase feedback that can cause large overshoot. The negative-sequence voltage feedforward acts immediately on the voltage command, so the current controller of the solar inverters does not have to wait for the feedback loop to accumulate error. The notch filter is placed in the voltage outer loop, which is slower than the current loop, so it does not slow down the current response. This separation of time scales is useful for solar inverters because the current loop must be fast for protection and power quality, while the voltage loop can be slower and more selective.
I next consider the design of the voltage controller and the notch filter for solar inverters. The voltage outer loop regulates the DC-link voltage. Its bandwidth must be low enough to avoid interaction with the current loop and to avoid amplifying the second harmonic. The notch filter must be centered exactly at twice the grid frequency. If the grid frequency varies, the notch filter should ideally adapt its center frequency. For solar inverters connected to a stiff grid, a fixed notch at the nominal second harmonic is often sufficient. For solar inverters connected to a weak or islanded grid, an adaptive notch may be beneficial. I express the second harmonic frequency as
$$
\omega_0 = 2\omega_g.
$$
The quality factor of the notch filter determines how narrow the attenuation band is. A very narrow notch attenuates the second harmonic strongly but may be sensitive to frequency drift. A wider notch is more robust but may reduce the voltage-loop phase margin. I therefore choose a moderate damping ratio. In my design, I coordinate the voltage controller cutoff frequency, the notch damping, and the current-loop bandwidth so that the solar inverters maintain stability under both balanced and unbalanced voltage conditions. I summarize the tuning logic in the following table.
| Controller or filter | Design objective for solar inverters | Practical consideration |
|---|---|---|
| Current controller \(G_i(s)\) | Fast current tracking and disturbance rejection | Bandwidth limited by switching frequency and filter resonance |
| Voltage controller \(G_v(s)\) | Regulate DC-link voltage without amplifying ripple | Low bandwidth to decouple from current loop |
| Notch filter \(G_{\text{notch}}(s)\) | Attenuate second harmonic from DC-link voltage | Center at \(2\omega_g\), moderate damping |
| SOGI sequence separation | Fast positive and negative sequence extraction | Center at nominal grid frequency, choose \(k\) for speed and selectivity |
| Negative-sequence feedforward | Cancel negative-sequence voltage disturbance | Requires accurate negative-sequence voltage estimate |
I also compare my method with a positive-sequence-only control strategy. In a positive-sequence-only strategy, the solar inverters regulate only the positive-sequence current and do not attempt to control the negative-sequence current. This can reduce the DC-link voltage ripple to some extent because the negative-sequence current is not deliberately injected. However, it cannot fully achieve constant-power grid connection. The negative-sequence voltage still interacts with the positive-sequence current and produces second harmonic power. To compensate, a more complex high-pass filter or voltage compensation network is needed. This reduces the practical appeal for solar inverters. In contrast, my method directly suppresses the negative-sequence current and uses feedforward to handle the negative-sequence voltage. The control target is clear, and the implementation is simpler.
I further analyze the power quality requirements that solar inverters must meet. Grid codes typically limit the current total harmonic distortion and the current unbalance. When the grid voltage is unbalanced, solar inverters are often required to remain connected and to provide balanced current as far as possible. If solar inverters simply follow the conventional balanced-grid control, the current unbalance may exceed limits. If solar inverters shut down too quickly, they reduce the available renewable energy. Therefore, a control strategy that allows solar inverters to ride through unbalanced voltage while maintaining current quality is valuable. My proposed method provides that capability. It keeps the current balanced, reduces the THD, and avoids the need for a complicated multi-loop structure.
I also consider the effect of the negative-sequence voltage feedforward on the modulation index of solar inverters. When the negative-sequence voltage is added to the voltage command, the peak modulation index may increase. The DC-link voltage must be high enough to synthesize the required voltage. In my simulation, the DC-link voltage is regulated by the outer loop, and the feedforward term is limited so that the modulator does not saturate. For solar inverters with limited DC-link voltage, the negative-sequence feedforward can be scaled or prioritized. The control designer can also coordinate the feedforward with current limiting. This is important because solar inverters must protect themselves during severe voltage sags. I summarize the practical limits in the following table.
| Practical issue | Effect on solar inverters | Mitigation in my strategy |
|---|---|---|
| Modulation saturation | Voltage command may exceed available DC-link voltage | Limit feedforward and current references |
| Current limiting | Solar inverters must avoid overcurrent during sag | Use current limiter in positive-sequence reference |
| Frequency variation | SOGI and notch center frequency may be off | Adapt center frequency if required |
| Measurement noise | Sequence separation may become noisy | Filter SOGI outputs and use reasonable damping |
| DC-link ripple | Second harmonic may enter current reference | Notch filter after voltage controller |
| Grid impedance | Feedforward may interact with filter resonance | Coordinate current-loop gain and feedforward gain |
My experimental and simulation results lead me to several conclusions about solar inverters under unbalanced grid voltage. First, the SOGI-based sequence separation is a practical way to obtain positive-sequence and negative-sequence voltage information for solar inverters. It is simple, fast, and accurate when the center frequency is set correctly. Second, negative-sequence current suppression is a clear control objective for solar inverters. Setting the negative-sequence current references to zero prevents the solar inverters from injecting unbalanced current into the grid. Third, negative-sequence voltage feedforward improves the disturbance rejection of the current loop without adding a large number of controllers. Fourth, a second-order notch filter after the voltage controller is effective in reducing the second harmonic ripple that would otherwise modulate the current references of solar inverters. Fifth, the proposed method keeps the current THD around \(2.3\%\) after an A-phase voltage drop to \(50\%\), which is far better than the conventional strategy.
I want to emphasize that the proposed method is not merely a simulation concept. The sequence separation was implemented and tested on a digital controller. The control structure uses standard blocks that are familiar to engineers working with solar inverters: a phase-locked loop, a voltage outer loop, a current inner loop, and a pulse-width modulator. The new elements are the SOGI-based sequence separation, the negative-sequence voltage feedforward, and the notch filter. These elements are compatible with existing solar inverters control architectures. They can be added with modest software changes and without a complete redesign of the power stage.
I also note that the proposed strategy is particularly suitable for solar inverters that must operate under weak-grid or unbalanced-grid conditions. In such conditions, the negative-sequence voltage can be significant, and the current unbalance can be large if the solar inverters do not actively suppress it. By using negative-sequence voltage feedforward, the solar inverters respond quickly to the negative-sequence disturbance. By using the notch filter, the solar inverters prevent the DC-link ripple from degrading the current waveform. The result is a more stable and more grid-friendly solar inverters system.
For completeness, I express the overall control law of the solar inverters in a compact form. The positive-sequence current references are generated by the DC voltage loop and the reactive power command. The negative-sequence current references are zero. The current controller produces positive-sequence voltage commands with feedforward of the positive-sequence grid voltage and cross-coupling compensation. The negative-sequence voltage is added as a feedforward term. The final voltage command is
$$
\begin{bmatrix}
v_d^{*}\\
v_q^{*}
\end{bmatrix}
=
\begin{bmatrix}
e_d^P+\omega L I_q^P+G_i(s)(I_d^{P*}-I_d^P)+e_d^N\\
e_q^P-\omega L I_d^P+G_i(s)(I_q^{P*}-I_q^P)+e_q^N
\end{bmatrix}.
$$
This equation summarizes the core of my control strategy for solar inverters. It shows that the negative-sequence voltage is compensated directly, while the positive-sequence current is regulated by the PI controller. The DC voltage loop provides the active current reference through
$$
I_d^{P*}=G_v(s)G_{\text{notch}}(s)\left(V_{dc}^{*}-V_{dc}\right).
$$
The reactive current reference is set independently. If the objective is unity power factor, then \(I_q^{P*}=0\). If the objective is reactive power support, then \(I_q^{P*}\) is determined by the reactive power command. The negative-sequence current references remain zero:
$$
I_d^{N*}=0,
\qquad
I_q^{N*}=0.
$$
This control law is simple enough to implement in real time. It does not require a positive-sequence and negative-sequence current decomposition in the feedback path. It only requires a reliable voltage sequence separation. The SOGI provides that separation. This is the key simplification that makes the method practical for solar inverters.
I also consider the case where the grid frequency deviates from its nominal value. The SOGI center frequency can be adjusted by a frequency-locked loop or by the phase-locked loop output. If the frequency deviation is small, the SOGI still provides acceptable quadrature signals, although the amplitude and phase errors increase. For solar inverters connected to a stable grid, the frequency deviation is usually small. For solar inverters connected to a weak grid, the frequency may vary more widely, and an adaptive SOGI center frequency is beneficial. Similarly, the notch filter can be made adaptive so that its center frequency tracks twice the grid frequency. I summarize the frequency adaptation options in the following table.
| Function | Fixed-frequency implementation | Adaptive implementation |
|---|---|---|
| SOGI center frequency | Set to nominal grid frequency | Track estimated grid frequency |
| Notch filter center frequency | Set to twice nominal grid frequency | Track twice estimated grid frequency |
| Suitability for solar inverters | Good for stiff grids with small frequency deviation | Better for weak grids with larger frequency variation |
| Computational cost | Low | Moderate |
| Implementation complexity | Simple | Requires frequency estimation |
I further discuss the role of the DC-link capacitor in solar inverters under unbalanced voltage. The second harmonic power ripple flows into the DC link and causes a voltage ripple. The capacitor size affects the amplitude of that ripple. A larger capacitor reduces the voltage ripple but increases cost and volume. The control strategy cannot eliminate the physical power ripple completely when the negative-sequence current is suppressed, because the negative-sequence voltage still interacts with the positive-sequence current. However, the control strategy can prevent the voltage ripple from propagating into the current references. The notch filter does this by attenuating the second harmonic in the voltage feedback path. The current loop then receives a cleaner active current reference. The DC-link voltage may still ripple slightly, but the AC current remains balanced and sinusoidal. This is a practical tradeoff for solar inverters.
I also compare the proposed method with a strategy that seeks constant active power. If the control objective is constant active power, the solar inverters may need to inject negative-sequence current to cancel the second harmonic power. That approach can reduce the DC-link ripple, but it causes unbalanced current. In many grid codes, unbalanced current injection is undesirable or limited. Therefore, I choose negative-sequence current suppression as the primary objective. This choice prioritizes current quality over perfect DC-link ripple elimination. The notch filter then handles the residual DC-link ripple. For solar inverters, this is often the preferred balance because grid codes usually emphasize current quality and balanced injection. I summarize the tradeoff in the following table.
| Control objective | Effect on current balance | Effect on DC-link ripple | Typical use in solar inverters |
|---|---|---|---|
| Negative-sequence current suppression | Balanced current | Ripple remains but is filtered from control path | Preferred for strict current quality |
| Constant active power | May require negative-sequence current | Lower ripple | Used when DC-link stress dominates |
| Positive-sequence-only control | Current may be unbalanced | Partial ripple reduction | Simple but limited |
| Dual current control | Can achieve either objective | Depends on references | Flexible but complex |
My simulation results confirm that the Type II strategy is effective for solar inverters. After the A-phase voltage drops to \(50\%\), the current remains balanced. The THD remains around \(2.3\%\). The DC-link voltage ripple is attenuated in the control loop. The negative-sequence current is suppressed. The overall system remains stable. These results demonstrate that the proposed control strategy is correct and practical for solar inverters under unbalanced grid voltage.
I conclude that the combination of SOGI-based sequence separation, negative-sequence voltage feedforward, and second-order notch filtering provides a strong solution for solar inverters. The method is simpler than a dual synchronous reference frame controller with four current PI regulators. It is more robust than a proportional resonant controller with poor frequency adaptability. It is more complete than a positive-sequence-only control strategy. It addresses the phase-locking problem, the current-balance problem, and the DC-link ripple problem in a coordinated way. For solar inverters operating in modern distribution networks, this coordinated approach is valuable because unbalanced voltage is common and power quality requirements are strict.
I summarize the final performance of the proposed control strategy for solar inverters in the following table.
| Performance aspect | Result for solar inverters |
|---|---|
| Positive-sequence phase tracking | Fast and accurate through SOGI-based separation |
| Negative-sequence voltage estimation | Accurate enough for feedforward compensation |
| Grid current balance | Maintained after A-phase voltage sag |
| Current THD | Approximately \(2.3\%\) |
| DC-link voltage ripple | Attenuated by second-order notch filter |
| Negative-sequence current | Suppressed toward zero |
| Controller complexity | Moderate and practical for solar inverters |
| Simulation validation | Electromagnetic transient results confirm the theory |
In my view, the most important lesson is that solar inverters should not be controlled as if the grid is always balanced. The sequence separation must be fast, the negative-sequence disturbance must be rejected, and the DC-link ripple must be prevented from entering the current reference. When these three tasks are handled together, solar inverters can maintain balanced current and low harmonic distortion even under severe voltage unbalance. The SOGI-based method gives me the sequence information I need. The negative-sequence voltage feedforward gives me the disturbance rejection I need. The notch filter gives me the ripple attenuation I need. Together, they form a coherent control strategy for solar inverters under unbalanced grid voltage.
I therefore recommend this strategy for solar inverters that must comply with modern grid codes, especially where voltage unbalance is likely. The method can be implemented with standard digital control hardware. It requires no extra current sensors beyond those already used by solar inverters. It does not require a complicated observer or a large number of PI controllers. It provides a clear path from measurement to sequence separation to current control to modulation. For these reasons, I consider the proposed control strategy a practical and effective solution for solar inverters operating under unbalanced grid voltage.