In modern power systems, the integration of renewable energy sources, particularly photovoltaic (PV) generation, has become increasingly prevalent. Solar inverters, as critical interfaces between PV arrays and the grid, play a vital role in ensuring stable and efficient power conversion. However, when asymmetrical faults occur in the grid—such as single-phase or two-phase faults—the voltage and current waveforms become unbalanced, introducing negative-sequence components. This imbalance can lead to significant challenges for traditional control strategies designed for symmetric conditions. Specifically, power oscillations at twice the grid frequency may arise, causing detrimental effects on DC-link voltage stability, capacitor lifespan, and overall system reliability. Therefore, developing advanced control methods for solar inverters under asymmetrical grid conditions is essential to maintain power quality and grid support capabilities.
Traditional vector control schemes for solar inverters, often based on proportional-integral (PI) regulators in synchronous reference frames, are effective under symmetric grid voltages. However, under asymmetrical faults, these controllers fail to account for negative-sequence components, resulting in uncontrolled power fluctuations and current distortion. Various approaches have been explored to address this issue. For instance, dual-current control methods involve separate regulation of positive- and negative-sequence currents in rotating reference frames, but they require complex sequence decomposition and multiple PI controllers, introducing delays and design complexity. Other techniques, such as proportional-resonant controllers, offer improved dynamics without sequence separation but are sensitive to grid frequency variations and component tolerances. In recent years, predictive control techniques, including deadbeat control and model predictive control (MPC), have gained attention due to their fast dynamic response and simplicity in digital implementation. This article focuses on a model predictive current control strategy for solar inverters under asymmetrical grid faults, aiming to suppress active power oscillations and negative-sequence currents without the need for explicit sequence decomposition.
To analyze the behavior of solar inverters under asymmetrical grid conditions, we first establish a mathematical model. Consider a three-phase grid-connected solar inverter system with an L filter, where the inverter output voltage, grid voltage, and output current are denoted as \(U_{tabc}\), \(U_{gabc}\), and \(I_{abc}\), respectively. The DC-link capacitor is represented by \(C\), with \(U_{dc}\) as the DC voltage. The system dynamics in the three-phase stationary frame can be described by Kirchhoff’s voltage law:
$$U_{tabc} = U_{gabc} + L \frac{dI_{abc}}{dt} + R I_{abc}$$
where \(L\) is the filter inductance and \(R\) is the equivalent resistance of the line and inductor. Transforming this equation into the two-phase stationary \(\alpha\beta\) frame using Clarke transformation yields:
$$U_{t\alpha\beta} = U_{g\alpha\beta} + L \frac{dI_{\alpha\beta}}{dt} + R I_{\alpha\beta}$$
Under asymmetrical grid faults, the grid voltage and current contain both positive-sequence and negative-sequence components. In the \(\alpha\beta\) frame, any electrical quantity \(F\) (voltage or current) can be expressed as:
$$F_{\alpha\beta} = F^p_{\alpha\beta} + F^n_{\alpha\beta} = F^p_{dq} e^{j\theta} + F^n_{dq} e^{-j\theta}$$
where the superscripts \(p\) and \(n\) denote positive- and negative-sequence components, respectively, \(F^p_{dq}\) and \(F^n_{dq}\) are the components in the rotating reference frames, and \(\theta = \omega_0 t\) with \(\omega_0\) being the grid angular frequency. Applying this to grid voltage, inverter voltage, and current, we have:
$$U_{g\alpha\beta} = U^p_{gdq} e^{j\theta} + U^n_{gdq} e^{-j\theta}$$
$$U_{t\alpha\beta} = U^p_{tdq} e^{j\theta} + U^n_{tdq} e^{-j\theta}$$
$$I_{\alpha\beta} = I^p_{dq} e^{j\theta} + I^n_{dq} e^{-j\theta}$$
Substituting these into the \(\alpha\beta\) model and separating the positive- and negative-sequence terms, the dynamic equations for solar inverters under asymmetrical faults become:
$$U^p_{tdq} = U^p_{gdq} + L \frac{dI^p_{dq}}{dt} + j\omega_0 L I^p_{dq}$$
$$U^n_{tdq} = U^n_{gdq} + L \frac{dI^n_{dq}}{dt} – j\omega_0 L I^n_{dq}$$
These equations highlight the coupling between sequences and the influence of grid imbalances on solar inverters control.
The power output of solar inverters is critical for grid stability. The complex power \(S\) in the \(\alpha\beta\) frame is given by:
$$S = 1.5 U_{g\alpha\beta} I^*_{\alpha\beta}$$
where \(I^*_{\alpha\beta}\) is the complex conjugate of the current. Substituting the sequence expressions, the active power \(P\) and reactive power \(Q\) can be derived as:
$$P = P_0 + P_{c2} \cos(2\omega_0 t) + P_{s2} \sin(2\omega_0 t)$$
$$Q = Q_0 + Q_{c2} \cos(2\omega_0 t) + Q_{s2} \sin(2\omega_0 t)$$
Here, \(P_0\) and \(Q_0\) are the average power terms, while \(P_{c2}\), \(P_{s2}\), \(Q_{c2}\), and \(Q_{s2}\) are coefficients of the double-frequency oscillations caused by asymmetrical faults. These coefficients depend on the positive- and negative-sequence voltages and currents:
$$
\begin{bmatrix}
P_0 \\
P_{c2} \\
P_{s2} \\
Q_0 \\
Q_{c2} \\
Q_{s2}
\end{bmatrix}
= \frac{3}{2}
\begin{bmatrix}
U^p_{gd} & U^p_{gq} & U^n_{gd} & U^n_{gq} \\
U^n_{gd} & U^n_{gq} & U^p_{gd} & U^p_{gq} \\
U^n_{gq} & -U^n_{gd} & -U^p_{gq} & U^p_{gd} \\
U^p_{gq} & -U^p_{gd} & U^n_{gq} & -U^n_{gd} \\
U^n_{gq} & -U^n_{gd} & U^p_{gq} & -U^p_{gd} \\
-U^n_{gd} & -U^n_{gq} & U^p_{gd} & U^p_{gq}
\end{bmatrix}
\begin{bmatrix}
I^p_d \\
I^p_q \\
I^n_d \\
I^n_q
\end{bmatrix}
$$
This formulation shows that power oscillations are inherent under asymmetrical conditions. For solar inverters, controlling these oscillations is crucial to prevent DC-link voltage ripple and ensure maximum power point tracking (MPPT) efficiency. Two primary control objectives are considered: suppressing active power fluctuations and eliminating negative-sequence currents. Each objective leads to different reference current settings for the solar inverters.
For active power fluctuation suppression, we set \(P_{c2} = P_{s2} = 0\). Assuming grid voltage orientation with \(U^p_{gq} = 0\), the reference currents in the rotating frames are:
$$I^p_{dref} = \frac{P_{ref}}{1.5 U^p_{gd} (1 – k_{dd}^2 – k_{qd}^2)}$$
$$I^p_{qref} = -\frac{Q_{ref}}{1.5 U^p_{gd} (1 – k_{dd}^2 – k_{qd}^2)}$$
$$I^n_{dref} = -k_{dd} I^p_d – k_{qd} I^p_q$$
$$I^n_{qref} = k_{dd} I^p_q – k_{qd} I^p_d$$
where \(k_{dd} = U^n_{gd} / U^p_{gd}\) and \(k_{qd} = U^n_{gq} / U^p_{gd}\). Here, \(P_{ref}\) is determined by the MPPT algorithm, typically as:
$$P_{ref} = (U_{MPPT} – U_{dc}) \left( K_p + \frac{K_I}{s} \right) U_{MPPT}$$
with \(K_p\) and \(K_I\) as PI gains, and \(U_{MPPT}\) the voltage at the maximum power point. For unity power factor operation, \(Q_{ref} = 0\).
For negative-sequence current suppression, we set \(I^n_{dref} = I^n_{qref} = 0\), yielding simpler references:
$$I^p_{dref} = \frac{P_{ref}}{1.5 U^p_{gd}}$$
$$I^p_{qref} = -\frac{Q_{ref}}{1.5 U^p_{gd}}$$
$$I^n_{dref} = 0$$
$$I^n_{qref} = 0$$
These reference currents guide the control strategy for solar inverters under asymmetrical faults.
The proposed control strategy employs model predictive current control (MPCC) to regulate solar inverters output. Unlike traditional methods, MPCC operates directly in the \(\alpha\beta\) stationary frame, avoiding the need for sequence decomposition and reducing computational delay. The core idea is to use a discrete-time model of the solar inverters to predict future currents for all possible switching states, then select the state that minimizes a cost function related to current tracking error.
First, we model the switching behavior of the solar inverters. Let \(g_i\) (for \(i = a, b, c\)) represent the switching state of each leg, with \(g_i = 1\) for the upper switch on and \(g_i = 0\) for the lower switch on. The inverter output voltage in the three-phase frame is:
$$u_t = (g_a + a g_b + a^2 g_c) U_{dc}$$
where \(a = e^{j\frac{2\pi}{3}}\). Transforming to the \(\alpha\beta\) frame:
$$U_{t\alpha} = \sqrt{\frac{2}{3}} \left( g_a – \frac{1}{2} g_b – \frac{1}{2} g_c \right) U_{dc}$$
$$U_{t\beta} = \frac{\sqrt{2}}{2} (g_b – g_c) U_{dc}$$
The eight possible switching combinations yield distinct voltage vectors, as summarized in Table 1.
| \(g_a\) | \(g_b\) | \(g_c\) | \(U_{t\alpha}\) | \(U_{t\beta}\) |
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0.8165 \(U_{dc}\) | 0 |
| 0 | 1 | 0 | 0.40825 \(U_{dc}\) | 0.7071 \(U_{dc}\) |
| 0 | 1 | 1 | -0.40825 \(U_{dc}\) | 0.7071 \(U_{dc}\) |
| 1 | 0 | 0 | -0.8165 \(U_{dc}\) | 0 |
| 1 | 0 | 1 | -0.40825 \(U_{dc}\) | -0.7071 \(U_{dc}\) |
| 1 | 1 | 0 | 0.40825 \(U_{dc}\) | 0.7071 \(U_{dc}\) |
| 1 | 1 | 1 | 0 | 0 |
Next, we discretize the dynamic equation for solar inverters. Using forward Euler approximation with sampling time \(T_s\):
$$L \frac{dI_{\alpha\beta}}{dt} \approx L \frac{I_{\alpha\beta}(k+1) – I_{\alpha\beta}(k)}{T_s}$$
Substituting into the \(\alpha\beta\) model and rearranging, the predictive model becomes:
$$I_{\alpha}(k+1) = \left(1 – \frac{T_s R}{L}\right) I_{\alpha}(k) + \frac{T_s}{L} (U_{t\alpha}(k) – U_{g\alpha}(k))$$
$$I_{\beta}(k+1) = \left(1 – \frac{T_s R}{L}\right) I_{\beta}(k) + \frac{T_s}{L} (U_{t\beta}(k) – U_{g\beta}(k))$$
This model predicts the currents at the next sampling instant based on the present states and applied voltage. For each of the eight voltage vectors from Table 1, we compute predicted currents and evaluate a cost function \(c\) defined as:
$$c = |I_{\alpha ref} – I_{\alpha}| + |I_{\beta ref} – I_{\beta}|$$
The voltage vector that minimizes \(c\) is selected and applied during the next sampling period. The reference currents \(I_{\alpha ref}\) and \(I_{\beta ref}\) are obtained by transforming the sequence references from earlier. Specifically, after calculating \(I^p_{dref}\), \(I^p_{qref}\), \(I^n_{dref}\), and \(I^n_{qref}\) based on the control objective, we convert them to the \(\alpha\beta\) frame:
$$I^p_{\alpha ref} = I^p_{dref} \cos(\theta) – I^p_{qref} \sin(\theta)$$
$$I^p_{\beta ref} = I^p_{dref} \sin(\theta) + I^p_{qref} \cos(\theta)$$
$$I^n_{\alpha ref} = I^n_{dref} \cos(-\theta) – I^n_{qref} \sin(-\theta)$$
$$I^n_{\beta ref} = I^n_{dref} \sin(-\theta) + I^n_{qref} \cos(-\theta)$$
Then, the total references are:
$$I_{\alpha ref} = I^p_{\alpha ref} + I^n_{\alpha ref}$$
$$I_{\beta ref} = I^p_{\beta ref} + I^n_{\beta ref}$$
To obtain the sequence components of grid voltage for reference calculation, we use a second-order generalized integrator (SOGI) based sequence separation method. This approach extracts \(U^p_{g\alpha}\), \(U^p_{g\beta}\), \(U^n_{g\alpha}\), and \(U^n_{g\beta}\) without significant delay, enabling real-time control for solar inverters.
The overall control structure for solar inverters integrates MPCC with the sequence separation and reference generation blocks. This design ensures that solar inverters can rapidly track current references under both symmetrical and asymmetrical grid conditions, enhancing the robustness of PV systems.
To validate the proposed method, we conducted simulation studies using a detailed model of a 0.5 MW PV system. The parameters are as follows: rated power 0.5 MW, DC-link capacitor 20,000 μF, filter inductance 0.18 mH, equivalent resistance 0.0005 Ω, grid voltage 0.27 kV (line-to-line), and sampling period 20 μs. The solar inverters control was implemented in a PSCAD/EMTDC environment, focusing on both symmetrical and asymmetrical fault scenarios.
First, we evaluated the dynamic performance of solar inverters under symmetrical grid conditions. The reference active power was stepped from 0.2 MW to 0.3 MW at 0.3 s, and the reference reactive power from 0 Mvar to 0.15 Mvar at 0.5 s. The results, as shown in Figure 1 (simulated waveforms), demonstrate that the output currents quickly tracked the references, with smooth transitions and minimal coupling between active and reactive power. This confirms the excellent dynamic and steady-state characteristics of MPCC for solar inverters.
Next, we examined the impact of parameter variations on solar inverters performance. Since MPCC relies on an accurate model, changes in inductance \(L\) could affect current tracking. We simulated scenarios where the actual inductance varied while the controller inductance was either updated or kept constant. Table 2 summarizes the total harmonic distortion (THD) of phase-A current under different conditions.
| Inductance \(L_f\) (mH) | THD (%) (Controller Updated) | THD (%) (Controller Fixed) |
|---|---|---|
| 0.05 | 2.10 | 2.70 |
| 0.09 | 1.25 | 1.35 |
| 0.18 | 0.61 | 0.61 |
| 0.21 | 0.44 | 0.45 |
| 0.36 | 0.54 | 0.43 |
The THD remains below 3% in all cases, indicating that MPCC for solar inverters is robust to parameter mismatches, with minimal performance degradation. This is crucial for practical applications where system parameters may drift over time.
For asymmetrical fault analysis, we simulated a single-phase-to-ground fault on phase A, occurring at 1.0 s and cleared at 1.4 s. The grid voltage asymmetry increased from 0.3% to 12% during the fault. We compared two control strategies: a symmetric control strategy (using only positive-sequence PI regulators) and the proposed asymmetric MPCC strategy with two objectives—suppressing active power oscillations and eliminating negative-sequence currents.
Under the symmetric control strategy, active power exhibited significant double-frequency oscillations, with an amplitude of approximately 0.004 MW in steady state. The DC-link voltage also showed ripple, with a peak-to-peak variation of about 0.006 V. In contrast, with MPCC targeting active power fluctuation suppression, the oscillation amplitude reduced by 75% to 0.001 MW, and the DC-link voltage ripple decreased correspondingly. The relationship between power oscillations and DC voltage can be expressed as:
$$C U_{dc} \frac{dU_{dc}}{dt} = P_{dci} – P_{dco}$$
where \(P_{dci}\) is the input power from the PV array (constant under MPPT) and \(P_{dco}\) is the output power. Reducing \(P_{dco}\) oscillations directly attenuates \(U_{dc}\) fluctuations, protecting the DC capacitor and improving MPPT efficiency for solar inverters.
Furthermore, current quality improved markedly with MPCC. With symmetric control, the current THD exceeded 11%, and the waveform was distorted. With MPCC for solar inverters, the THD dropped below 2%, and the currents became more sinusoidal, even under fault conditions. This highlights the ability of MPCC to enhance grid current quality while managing asymmetrical faults.
When targeting negative-sequence current suppression, the solar inverters injected balanced three-phase currents during the fault. The negative-sequence current magnitude was reduced to near zero, but active power oscillations were larger compared to the first objective. This trade-off is inherent in control design for solar inverters, as perfect compensation of both power and current is not simultaneously achievable under asymmetrical voltages. Nonetheless, MPCC provides flexibility to prioritize based on system requirements.
The simulation results confirm that MPCC for solar inverters effectively addresses asymmetrical grid faults. Key advantages include:
- Fast current tracking without sequence decomposition, reducing computational burden.
- Robustness to parameter variations, ensuring reliability in real-world solar inverters.
- Ability to suppress either power oscillations or negative-sequence currents, enhancing grid support functions.
- Simple digital implementation, as it requires only a discrete model and cost function evaluation.
In conclusion, the integration of model predictive current control into solar inverters offers a powerful solution for handling asymmetrical grid faults. By leveraging a predictive model and optimized switching selection, solar inverters can maintain high performance under unbalanced conditions, contributing to grid stability and power quality. Future work could explore adaptive MPCC schemes for solar inverters to handle time-varying grid impedances or extend the method to multi-level inverter topologies. Overall, this approach holds significant engineering value for advancing the control of solar inverters in modern power systems.
From a broader perspective, the deployment of such advanced control strategies in solar inverters is essential as renewable penetration increases. Asymmetrical faults, though less common than symmetrical ones, can have disproportionate impacts on inverter-based resources. Therefore, ensuring that solar inverters are equipped with robust fault ride-through capabilities is critical for grid resilience. The MPCC method, with its simplicity and effectiveness, represents a step forward in making solar inverters more adaptive and reliable. Additionally, the principles discussed here could be applied to other types of inverters, such as those used in wind energy or energy storage systems, further broadening the impact of predictive control technologies.
In summary, this article has presented a comprehensive analysis and control method for solar inverters under asymmetrical grid faults. We derived mathematical models, formulated control objectives, and implemented a model predictive current control strategy. Simulation studies validated the approach, showing improved dynamic response and power quality. As the energy transition progresses, solar inverters will continue to play a pivotal role, and advanced control methods like MPCC will be key to unlocking their full potential in diverse grid conditions.