The increasing integration of renewable energy sources into the power grid has placed significant importance on the interface equipment that manages this connection. Among these, the solar inverter plays a crucial role in converting and conditioning the DC power from photovoltaic arrays for injection into the AC grid. Traditional control paradigms for grid-connected solar inverters are predominantly designed and optimized for operation under ideal, balanced three-phase grid conditions. However, the practical grid environment is often subject to disturbances. Asymmetric faults, which can arise from line-to-ground faults, single-phase switching events, or the connection of large unbalanced loads, lead to the presence of negative-sequence voltage components. This voltage unbalance poses a substantial challenge to the stable and high-performance operation of solar inverters, potentially causing excessive current stress, power oscillations, and even triggering protective shutdowns. Consequently, developing and analyzing robust control strategies for solar inverters under such asymmetric grid fault conditions is not merely an academic exercise but a critical necessity for ensuring grid reliability and the continued contribution of solar power generation.
This article presents a comprehensive design, analysis, and comparative study of control strategies for solar inverters during asymmetric grid faults. The core of the investigation lies in comparing two fundamental approaches: one formulated in the Synchronous Reference Frame (SRF) and the other in the Stationary Reference Frame. The control architecture for a grid-connected solar inverter can be systematically decomposed into three primary, interrelated functions: Grid Synchronization, Power Control, and Current Regulation. We will delve into each of these functions, explaining their operational principles and implementation nuances within both coordinate systems. The ultimate goal is to evaluate their performance, complexity, and suitability for fault ride-through operation in solar inverter applications.
1. Grid Synchronization Under Unbalanced Voltages
Accurate and fast detection of the grid voltage’s fundamental positive-sequence phase angle, frequency, and magnitude is paramount for the controller of a solar inverter. Under balanced conditions, a standard Phase-Locked Loop (PLL) in the synchronous reference frame suffices. However, during asymmetrical faults, the grid voltage contains both positive- and negative-sequence components at the fundamental frequency. A conventional SRF-PLL will interpret the oscillating negative-sequence component as a disturbance, leading to significant double-frequency oscillations in the estimated phase and frequency. This erroneous information severely degrades the performance of the subsequent control loops in the solar inverter.
1.1. Synchronous Frame Approach: Dual Decoupled SRF-PLL (DDSRF-PLL)
The DDSRF-PLL is designed to extract the positive-sequence component accurately in the presence of unbalance. Its core idea is to perform synchronous transformations on both the positive- and negative-sequence hypothetical frames simultaneously and use a decoupling network to cancel out the cross-coupling terms. The three-phase grid voltages ($v_{abc}$) are first transformed into the $\alpha\beta$ stationary frame using the Clarke transformation, $T_{abc/\alpha\beta}$.
$$T_{abc/\alpha\beta} = \frac{2}{3}\begin{bmatrix} 1 & -\frac{1}{2} & -\frac{1}{2} \\ 0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2} \end{bmatrix}$$
$$ \begin{bmatrix} v_{\alpha} \\ v_{\beta} \end{bmatrix} = T_{abc/\alpha\beta} \cdot \begin{bmatrix} v_a \\ v_b \\ v_c \end{bmatrix} $$
These $\alpha\beta$ components are then transformed into two rotating frames: one rotating at the estimated positive-sequence angular speed $\hat{\omega}t$ (the $dq^+$ frame) and another rotating at $-\hat{\omega}t$ (the $dq^-$ frame, effectively a positive-sequence frame for the negative-sequence component).
$$ \begin{bmatrix} v_d^+ \\ v_q^+ \end{bmatrix} = \begin{bmatrix} \cos\hat{\theta} & \sin\hat{\theta} \\ -\sin\hat{\theta} & \cos\hat{\theta} \end{bmatrix} \begin{bmatrix} v_{\alpha} \\ v_{\beta} \end{bmatrix} $$
$$ \begin{bmatrix} v_d^- \\ v_q^- \end{bmatrix} = \begin{bmatrix} \cos\hat{\theta} & -\sin\hat{\theta} \\ \sin\hat{\theta} & \cos\hat{\theta} \end{bmatrix} \begin{bmatrix} v_{\alpha} \\ v_{\beta} \end{bmatrix} $$
Under perfect decoupling and synchronization, the positive-sequence component appears as DC quantities in the $dq^+$ frame, while the negative-sequence component appears as DC in the $dq^-$ frame. However, cross-coupling exists. The DDSRF-PLL employs a decoupling network to calculate and subtract the cross-coupling terms ($v_{q,dc}^-$ and $v_{d,dc}^-$) from the measured $v_q^+$ and $v_d^+$ signals. The decoupled $v_q^+$ is then fed into a standard PI-based PLL to regulate it to zero, thereby locking onto the positive-sequence phase angle $\hat{\theta}$. The low-pass filters (LPFs) extract the DC components required for the decoupling network. While effective, this method involves multiple transformations, filters, and a PI regulator, which can introduce complexity and dynamic lag in the response of the solar inverter’s synchronization unit.
1.2. Stationary Frame Approach: Dual Second-Order Generalized Integrator Frequency-Locked Loop (DSOGI-FLL)
The DSOGI-FLL operates entirely in the stationary $\alpha\beta$ frame, offering a structurally simpler alternative for the solar inverter. Instead of tracking the phase angle, it directly estimates the grid frequency. The key element is the Second-Order Generalized Integrator (SOGI), which acts as an adaptive band-pass filter tuned to the estimated frequency $\hat{\omega}$. For a given input $v$, it generates two outputs: a filtered in-phase signal $v’$ and a quadrature signal $qv’$, shifted by 90°. Its transfer functions are:
$$ D(s) = \frac{v’}{v}(s) = \frac{k\hat{\omega}s}{s^2 + k\hat{\omega}s + \hat{\omega}^2} $$
$$ Q(s) = \frac{qv’}{v}(s) = \frac{k\hat{\omega}^2}{s^2 + k\hat{\omega}s + \hat{\omega}^2} $$
where $k$ determines the bandwidth. Two SOGIs are used to process the $v_\alpha$ and $v_\beta$ components, generating $v_\alpha’$, $qv_\alpha’$, $v_\beta’$, and $qv_\beta’$. A Frequency-Locked Loop (FLL) is then employed to adapt $\hat{\omega}$. The FLL algorithm is based on the error orthogonality principle in the stationary frame. A common update law is:
$$ \dot{\hat{\omega}} = -\Gamma \cdot \left( \frac{v_\alpha – v_\alpha’}{v_\alpha’^2 + (qv_\alpha’)^2} \cdot qv_\alpha’ + \frac{v_\beta – v_\beta’}{v_\beta’^2 + (qv_\beta’)^2} \cdot qv_\beta’ \right) $$
where $\Gamma$ is the adaptive gain. The positive-sequence components ($v_{\alpha\beta}^+$) can be easily reconstructed from these signals: $v_\alpha^+ = (v_\alpha’ – qv_\beta’)/2$ and $v_\beta^+ = (v_\beta’ + qv_\alpha’)/2$. The phase angle $\hat{\theta}$ for transformation, if needed, can be obtained via $\hat{\theta} = \arctan(v_\beta^+ / v_\alpha^+)$. The absence of a PI regulator for phase tracking and the direct frequency estimation often result in a faster and more straightforward dynamic response for the solar inverter’s synchronization system.
| Feature | DDSRF-PLL (Synchronous Frame) | DSOGI-FLL (Stationary Frame) |
|---|---|---|
| Primary Control Object | Phase Angle ($\theta$) | Frequency ($\omega$) |
| Core Structure | Dual rotating transformations, Decoupling network, PI regulator, LPFs | Two SOGI adaptive filters, Frequency update law |
| Complexity | Higher (multiple coordinate transforms, decoupling) | Lower (operates in natural frame) |
| Dynamic Response | Can be slower due to PI regulator and filter delays | Typically faster, direct frequency adaptation |
| Output Information | $\hat{\theta}$, $\hat{\omega}$, positive-/negative-sequence $dq$ components | $\hat{\omega}$, $\hat{\theta}$, positive-/negative-sequence $\alpha\beta$ components |
2. Power Control and Current Reference Generation
During asymmetric faults, the instantaneous power delivered by the solar inverter becomes oscillatory at twice the fundamental frequency due to the interaction between voltage and current sequences. The control objective for the solar inverter must be clearly defined. A common and flexible strategy is the Positive-Negative Sequence Control (PNSC). It allows the designer to set specific goals, such as eliminating active power oscillations, reactive power oscillations, or achieving balanced sinusoidal currents.
The three-phase instantaneous power theory is used. The instantaneous active ($p$) and reactive ($q$) powers can be calculated from the $\alpha\beta$ components of voltage and current. Under unbalanced conditions, these powers contain constant and oscillatory terms related to the positive ($+$) and negative ($-$) sequences.
$$ p = \frac{3}{2}(v_\alpha i_\alpha + v_\beta i_\beta) = P_0 + \tilde{P}_{2\omega} $$
$$ q = \frac{3}{2}(v_\alpha i_\beta – v_\beta i_\alpha) = Q_0 + \tilde{Q}_{2\omega} $$
In the PNSC strategy implemented in the stationary frame, the reference currents are generated directly based on the desired power setpoints and the decomposed sequence voltages. For instance, if the goal is to inject constant active power ($P^*$) and zero reactive power ($Q^*=0$) while accepting unbalanced currents, the positive- and negative-sequence reference currents in the $\alpha\beta$ frame are calculated as:
$$ \begin{bmatrix} i_{\alpha}^* \\ i_{\beta}^* \end{bmatrix} = \frac{2}{3} \cdot \frac{P^*}{(v_{\alpha}^+)^2+(v_{\beta}^+)^2} \begin{bmatrix} v_{\alpha}^+ \\ v_{\beta}^+ \end{bmatrix} + \frac{2}{3} \cdot \frac{P^*}{(v_{\alpha}^-)^2+(v_{\beta}^-)^2} \begin{bmatrix} v_{\alpha}^- \\ v_{\beta}^- \end{bmatrix} $$
Here, $P^*$ is typically provided by the DC-link voltage regulator of the solar inverter to maintain power balance. Alternative formulations allow for the elimination of power oscillations by setting appropriate negative-sequence current references. The key advantage of generating references in the stationary frame is that they are naturally sinusoidal signals, ready to be tracked by a suitable current regulator without further transformation.
3. Current Regulation in Different Frames
The final and most critical inner loop of the solar inverter controller is the current regulation loop. Its task is to force the actual grid currents to follow the references generated by the power control block with minimal error and high bandwidth.
3.1. Synchronous Frame Current Control using PI Regulators
In the synchronous frame approach, the sinusoidal current references ($i_{\alpha\beta}^*$) and measured currents ($i_{\alpha\beta}$) are transformed into the positive-sequence $dq$ frame using the angle $\hat{\theta}$ from the DDSRF-PLL. In this frame, the fundamental positive-sequence components become DC quantities. Standard PI controllers can then achieve zero steady-state error for DC signals. The control law in the $dq$ frame for a solar inverter is given by:
$$ v_{d}^{**} = \left( K_p + \frac{K_i}{s} \right)(i_d^* – i_d) – \omega L i_q + v_d $$
$$ v_{q}^{**} = \left( K_p + \frac{K_i}{s} \right)(i_q^* – i_q) + \omega L i_d + v_q $$
where $K_p$ and $K_i$ are the PI gains, $L$ is the filter inductance, and $\omega L i_q$, $\omega L i_d$ are the cross-coupling terms, and $v_d$, $v_q$ are the grid voltage feedforward terms. The outputs $v_{dq}^{**}$ are then transformed back to the stationary frame to generate the Pulse Width Modulation (PWM) signals. To control negative-sequence currents (if required), an identical dual $dq$ control structure operating at $-\hat{\omega}$ is needed. This leads to a complex system with multiple coordinate transformations, decoupling, and feedforward requirements.
3.2. Stationary Frame Current Control using PR Regulators
The stationary frame approach offers a more natural and simpler solution for the current control loop in a solar inverter. Since the current references are sinusoidal signals at the fundamental grid frequency, a controller with infinite gain at that specific frequency is ideal. A Proportional-Resonant (PR) controller fulfills this role. Its transfer function in the $\alpha\beta$ frame is:
$$ G_{PR}(s) = K_p + \frac{2K_i\omega_c s}{s^2 + 2\omega_c s + \omega_0^2} $$
where $K_p$ is the proportional gain, $K_i$ is the resonant gain, $\omega_0$ is the resonant frequency (set to the grid fundamental frequency, e.g., $2\pi\cdot50$ rad/s), and $\omega_c$ is a small cutoff frequency to provide a finite bandwidth around resonance. A separate PR controller is used for each $\alpha$ and $\beta$ axis. The control law is straightforward:
$$ v_{\alpha}^{**} = G_{PR}(s)(i_\alpha^* – i_\alpha) + v_{\alpha} $$
$$ v_{\beta}^{**} = G_{PR}(s)(i_\beta^* – i_\beta) + v_{\beta} $$
Notably, there is no need for coordinate transformations within the current loop. The cross-coupling inherent in the $dq$ frame control is absent here because the $\alpha$ and $\beta$ axes are orthogonal and stationary. The grid voltage feedforward ($v_{\alpha}$, $v_{\beta}$) is still beneficial for disturbance rejection. This structure is inherently simpler, easier to implement, and provides excellent tracking of sinusoidal references for the solar inverter.
| Aspect | Synchronous Frame PI Control | Stationary Frame PR Control |
|---|---|---|
| Control Variable | DC quantities (in $dq$ frame) | AC quantities (in $\alpha\beta$ frame) |
| Core Controller | PI Regulator ($K_p + K_i/s$) | PR Regulator ($K_p + \frac{2K_i\omega_c s}{s^2+2\omega_c s+\omega_0^2}$) |
| Transformations Needed | Mandatory (abc->$\alpha\beta$->dq and inverse) | Only abc->$\alpha\beta$ (for measurement) |
| Cross-Coupling Compensation | Required ($\pm\omega L i_{q/d}$ terms) | Not Required |
| Structure for Dual-Sequence Control | Dual $dq$ controllers (one at $\omega$, one at $-\omega$) | Single PR controller per axis handles both sequences inherently* |
| Algorithmic Complexity | High | Low |
*The PR controller’s resonant peak at $\omega_0$ provides high gain for both positive- and negative-sequence components, which are at the same frequency in the stationary frame.
4. Integrated System Comparison and Analysis
Having dissected the individual functional blocks, we can now synthesize two complete control strategies for a solar inverter under asymmetric faults and compare their overall characteristics.
Strategy A: Synchronous Frame Based Control
This strategy for the solar inverter employs a DDSRF-PLL for synchronization, PNSC (implemented in $dq$ frame or stationary frame with subsequent transformation) for power/reference generation, and dual PI controllers in the synchronous frame for current regulation. Its main advantage is the familiarity and widespread use of PI control for DC signals. However, it suffers from significant complexity. The entire control chain relies heavily on accurate and rapid coordinate transformations using the estimated angle from the PLL. Any error or delay in the PLL (such as the double-frequency ripple during fault transients) propagates through all subsequent transformations, degrading performance. The need for dual controllers for sequence separation in both the PLL and current loop adds to the computational burden and parameter tuning difficulty.
Strategy B: Stationary Frame Based Control
This strategy for the solar inverter utilizes a DSOGI-FLL for frequency/sequence detection, PNSC executed directly in the $\alpha\beta$ frame for reference generation, and PR controllers for current regulation. Its principal strength lies in its structural simplicity and elegant decoupling. The DSOGI-FLL provides fast and direct frequency/sequence extraction without a nested PI loop for phase. The current references remain as clean sinusoidal signals in their natural frame. The PR controllers then track these sinusoids perfectly without any rotational transformations. The control algorithm is more intuitive, requires fewer mathematical operations, and is less sensitive to synchronization dynamics because the current loop does not depend on a tracked phase angle for its operation—it only needs an accurate frequency $\hat{\omega}$ to tune the resonant peak, which the FLL provides robustly.
| Evaluation Criterion | Synchronous Frame Strategy (DDSRF-PLL + PI) | Stationary Frame Strategy (DSOGI-FLL + PR) |
|---|---|---|
| Overall Complexity | High | Low |
| Implementation Effort | Significant (coordinate transforms, decoupling, dual loops) | Moderate (mainly filter and resonant controller implementation) |
| Dynamic Response | Governed by PLL’s PI bandwidth; can be slower during transients | Governed by FLL adaptive gain and PR bandwidth; typically faster |
| Robustness to PLL/FLL Errors | Highly sensitive (errors affect all transformations) | Less sensitive (current loop independent of phase) |
| Computational Load | Higher (multiple trigonometric functions, matrix multiplications) | Lower |
| Tuning Difficulty | Higher (multiple interacting PI loops, decoupling gains, filter cutoffs) | Lower (fewer interacting parameters: $k$, $\Gamma$ for FLL; $K_p$, $K_i$, $\omega_c$ for PR) |
5. Conclusion
This article has presented a detailed comparative study of control methodologies for solar inverters operating under asymmetric grid fault conditions. Two principal architectures were analyzed: one rooted in the synchronous reference frame and the other in the stationary reference frame. The analysis systematically broke down the control system into its three essential functions—grid synchronization, power control, and current regulation—and explored the implementation and implications of each approach for the solar inverter.
The synchronous frame strategy, employing DDSRF-PLL and PI current controllers, is a robust and well-established method. However, its reliance on accurate and timely coordinate transformations for all measured and reference signals introduces inherent complexity, computational overhead, and potential vulnerability to synchronization dynamics during fault transients. In contrast, the stationary frame strategy, based on DSOGI-FLL and PR current controllers, offers a more streamlined and inherently decoupled solution for the solar inverter. It operates more directly on the natural AC variables, eliminating the need for rotational transformations within the critical current control loop. This results in a simpler control structure, easier implementation, and typically a more dynamic response to grid disturbances.
While both strategies are capable of achieving the fundamental goal of controlled power injection under unbalanced voltages, the evidence from architectural analysis strongly favors the stationary frame approach for solar inverter applications where simplicity, dynamic performance, and ease of digital implementation are paramount. The DSOGI-FLL provides fast and accurate sequence separation, and the PR controller offers perfect sinusoidal tracking without the baggage of reference frame rotations, making the combined stationary frame strategy a superior choice for enhancing the fault ride-through capability and overall robustness of modern solar inverters.