As the global energy landscape shifts towards sustainable sources, solar photovoltaic (PV) generation has emerged as a cornerstone of modern power systems. The integration of large-scale PV plants, however, presents significant challenges to grid stability, particularly during fault conditions. Solar inverters, as the critical interface between PV arrays and the grid, must ensure reliable operation even under severe voltage disturbances. In this context, zero-voltage ride-through (ZVRT) capability is paramount, requiring solar inverters to remain connected and provide reactive support when grid voltages plummet to zero. Traditional control methods, such as proportional-integral (PI) control, often struggle with dynamic response and harmonic suppression during asymmetric faults. This article delves into an advanced control strategy based on model predictive modulation function (MPMF) for solar inverters, designed to enhance ZVRT performance. We explore the mathematical foundations, control design, and simulation validation, emphasizing the pivotal role of solar inverters in maintaining grid integrity.
The proliferation of solar inverters in power networks has heightened the need for robust fault ride-through capabilities. Grid codes worldwide, including China’s GB/T 19964-2012, mandate that PV stations must withstand voltage dips to zero for up to 0.15 seconds while injecting dynamic reactive current. This ZVRT requirement is more stringent than low-voltage ride-through (LVRT), as it demands uninterrupted operation during complete voltage collapse. Solar inverters must rapidly adjust their output currents to limit overcurrents and supply reactive power, aiding voltage recovery. Asymmetrical faults, such as single-phase short circuits, introduce negative-sequence components that complicate control due to harmonic generation and unbalanced currents. Existing approaches for solar inverters, including dual-sequence PI control and finite control set model predictive control (FCS-MPC), face limitations in response speed, computational burden, or harmonic distortion. This motivates the development of the MPMF-based ZVRT method, which synergizes predictive control with fixed-frequency modulation to optimize solar inverter performance.
To understand the control design, we first establish the mathematical model of a three-phase grid-connected solar inverter under asymmetric faults. The main circuit comprises a DC-link capacitor, filtering inductors, and resistive losses, interfacing with the grid through a voltage-source inverter. The dynamics in the three-phase stationary abc frame are given by:
$$ L \frac{d\mathbf{I}_{abc}}{dt} = \mathbf{U}_{abc} – \mathbf{E}_{abc} – R \mathbf{I}_{abc} $$
where \( \mathbf{U}_{abc} \), \( \mathbf{I}_{abc} \), and \( \mathbf{E}_{abc} \) represent the inverter output voltage, output current, and grid voltage vectors, respectively; \( L \) and \( R \) denote the filter inductance and resistance. For control simplification, we transform this into the stationary \( \alpha\beta \) frame using Clarke transformation:
$$ \mathbf{T}_c = \frac{2}{3} \begin{bmatrix} 1 & -\frac{1}{2} & -\frac{1}{2} \\ 0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2} \end{bmatrix} $$
yielding:
$$ L \frac{d\mathbf{I}_{\alpha\beta}}{dt} = \mathbf{U}_{\alpha\beta} – \mathbf{E}_{\alpha\beta} – R \mathbf{I}_{\alpha\beta} $$
Under ideal balanced conditions, further transformation to the synchronous rotating dq frame simplifies control by converting AC quantities to DC. However, during asymmetrical faults, grid voltages contain both positive- and negative-sequence components. For solar inverters, which typically use three-wire connections, zero-sequence currents are absent. The grid voltage can be decomposed as:
$$ \mathbf{E}_{abc} = \mathbf{E}_{abc}^+ + \mathbf{E}_{abc}^- $$
where superscripts + and − denote positive- and negative-sequence components. In the dq frame, these introduce double-frequency oscillations, complicating control. To address this, we employ sequence separation techniques. First, in the \( \alpha\beta \) frame, the positive- and negative-sequence voltages are extracted using:
$$ \mathbf{E}_{\alpha\beta}^+ = \mathbf{T}_{2s/2s+} \mathbf{E}_{\alpha\beta}, \quad \mathbf{E}_{\alpha\beta}^- = \mathbf{T}_{2s/2s-} \mathbf{E}_{\alpha\beta} $$
with transformation matrices:
$$ \mathbf{T}_{2s/2s+} = \frac{1}{2} \begin{bmatrix} 1 & -q \\ q & 1 \end{bmatrix}, \quad \mathbf{T}_{2s/2s-} = \frac{1}{2} \begin{bmatrix} 1 & q \\ -q & 1 \end{bmatrix} $$
where \( q = e^{-j\pi/2} \) is a 90° phase-shift operator. Subsequent transformation to positive and negative rotating dq frames yields DC quantities, enabling independent control. The inverter model in these frames is expressed as:
$$ \begin{aligned}
u_d^+ &= L \frac{di_d^+}{dt} + R i_d^+ – \omega L i_q^+ + e_d^+ \\
u_q^+ &= L \frac{di_q^+}{dt} + R i_q^+ + \omega L i_d^+ + e_q^+ \\
u_d^- &= L \frac{di_d^-}{dt} + R i_d^- + \omega L i_q^- + e_d^- \\
u_q^- &= L \frac{di_q^-}{dt} + R i_q^- – \omega L i_d^- + e_q^-
\end{aligned} $$
This model highlights the coupling between axes, which is mitigated in the proposed MPMF approach by operating in the \( \alpha\beta \) frame for prediction. The control of solar inverters hinges on regulating these currents to achieve ZVRT mandates.
The core of the MPMF-ZVRT method lies in predicting optimal modulation functions to minimize current tracking errors. Unlike FCS-MPC, which evaluates discrete voltage vectors, MPMF derives continuous modulation signals, reducing harmonic distortion and computational load for solar inverters. We define the modulation function vector \( \mathbf{M}_{\alpha\beta} = [m_\alpha, m_\beta]^T \), related to the inverter output voltage by \( \mathbf{U}_{\alpha\beta} = \mathbf{M}_{\alpha\beta} U_{dc} \), where \( U_{dc} \) is the DC-link voltage. Using forward Euler discretization with sampling time \( T_s \), the current prediction model in the \( \alpha\beta \) frame is:
$$ \begin{aligned}
i_\alpha(k+1) &= \frac{T_s}{L} \left( m_\alpha(k) U_{dc} – e_\alpha(k) \right) + \left(1 – \frac{R T_s}{L}\right) i_\alpha(k) \\
i_\beta(k+1) &= \frac{T_s}{L} \left( m_\beta(k) U_{dc} – e_\beta(k) \right) + \left(1 – \frac{R T_s}{L}\right) i_\beta(k)
\end{aligned} $$
To compensate for computational delays, a two-step prediction is employed, where the modulation function at time \( k+1 \) minimizes the current error at \( k+2 \). The predicted currents at \( k+2 \) are:
$$ \begin{aligned}
i_\alpha(k+2) &= \frac{T_s}{L} \left( m_\alpha(k+1) U_{dc} – e_\alpha(k+1) \right) + \left(1 – \frac{R T_s}{L}\right) i_\alpha(k+1) \\
i_\beta(k+2) &= \frac{T_s}{L} \left( m_\beta(k+1) U_{dc} – e_\beta(k+1) \right) + \left(1 – \frac{R T_s}{L}\right) i_\beta(k+1)
\end{aligned} $$
Grid voltages at \( k+1 \) are estimated using:
$$ \begin{bmatrix} e_\alpha(k+1) \\ e_\beta(k+1) \end{bmatrix} = \begin{bmatrix} \cos(\omega T_s) & -\sin(\omega T_s) \\ \sin(\omega T_s) & \cos(\omega T_s) \end{bmatrix} \begin{bmatrix} e_\alpha(k) \\ e_\beta(k) \end{bmatrix} $$
The evaluation function for minimizing current errors is chosen as the sum of squared errors:
$$ J = \left[ i_\alpha(k+2) – i_{\alpha\text{ref}}(k+2) \right]^2 + \left[ i_\beta(k+2) – i_{\beta\text{ref}}(k+2) \right]^2 $$
where \( i_{\alpha\text{ref}} \) and \( i_{\beta\text{ref}} \) are reference currents in the \( \alpha\beta \) frame, derived from sequence-based references. According to grid codes, during voltage dips, solar inverters must inject reactive current while limiting active current to prevent overcurrent. The reference currents in positive- and negative-sequence dq frames are set as:
$$ \begin{aligned}
i_{q\text{ref}}^+ &= 1.5 \times (0.9 – U_T) I_N \\
i_{d\text{ref}}^+ &= \sqrt{(1.05 \times I_N)^2 – (i_{q\text{ref}}^+)^2} \\
i_{d\text{ref}}^- &= 0, \quad i_{q\text{ref}}^- = 0
\end{aligned} $$
with \( U_T \) being the per-unit grid voltage and \( I_N \) the rated current. These are transformed to the \( \alpha\beta \) frame via inverse Park and Clarke transformations. To find the optimal modulation functions, we differentiate \( J \) with respect to \( m_\alpha(k+1) \) and \( m_\beta(k+1) \) and set derivatives to zero, yielding:
$$ \begin{aligned}
m_\alpha(k+1) &= \frac{L}{U_{dc} T_s} i_{\alpha\text{ref}}(k+2) – \frac{L}{U_{dc} T_s} \left\{ \frac{T_s}{L} \left[ m_\alpha(k) U_{dc} – e_\alpha(k) \right] + \left(1 – \frac{R T_s}{L}\right) i_\alpha(k) \right\} + \frac{1}{U_{dc}} e_\alpha(k+1) \\
m_\beta(k+1) &= \frac{L}{U_{dc} T_s} i_{\beta\text{ref}}(k+2) – \frac{L}{U_{dc} T_s} \left\{ \frac{T_s}{L} \left[ m_\beta(k) U_{dc} – e_\beta(k) \right] + \left(1 – \frac{R T_s}{L}\right) i_\beta(k) \right\} + \frac{1}{U_{dc}} e_\beta(k+1)
\end{aligned} $$
These modulation functions are then fed to a fixed-frequency sinusoidal pulse-width modulation (SPWM) block to generate gate signals for the solar inverter. This approach ensures rapid tracking of reference currents with low harmonic distortion, enhancing the ZVRT capability of solar inverters.
To validate the MPMF-ZVRT method, simulations were conducted on a 250 kW, 35 kV PV system in MATLAB/Simulink. The solar inverter parameters are summarized in Table 1, which includes key components such as filter inductors, DC-link capacitor, and switching frequency. Both asymmetric (single-phase-to-ground) and symmetric (three-phase) faults were applied, with grid voltage dropping to zero for 0.15 seconds as per ZVRT requirements.
| Parameter | Value |
|---|---|
| Grid Rated Voltage | 35 kV |
| Grid Frequency | 50 Hz |
| AC Filter Inductance | 0.12 mH |
| AC Filter Capacitance | 60 µF |
| DC-Link Capacitance | 65 mF |
| DC-Link Voltage | 480 V |
| Switching Frequency | 2500 Hz |
| Rated Power | 250 kW |
| Rated Current | 6 A |
| Overcurrent Protection | 7.2 A |
| Overvoltage Protection | 600 V |
| Transformer Rating | 250 kVA |
| Transformer Ratio | 250/35000 |
| Sampling Frequency | 10000 Hz |
Under a single-phase fault, the solar inverter using MPMF-ZVRT maintained balanced three-phase currents with no negative-sequence components, as shown in simulation waveforms. The current amplitude was limited to 6.3 A, below the overcurrent threshold of 7.2 A, while reactive current tracked its reference within 18 ms, meeting the grid code requirement of 30 ms response. For symmetric faults, currents remained symmetrical, and reactive current response time was 15 ms. In contrast, conventional PI-based ZVRT for solar inverters exhibited sluggish response (25 ms) and higher oscillations, while FCS-MPC-ZVRT showed improved dynamics (18 ms) but higher harmonic distortion. The superior performance of MPMF-ZVRT stems from its continuous modulation optimization, which reduces current ripple and enhances stability for solar inverters.
Harmonic analysis further underscores the advantages of MPMF-ZVRT for solar inverters. Total harmonic distortion (THD) of grid currents was evaluated under steady-state post-fault conditions. As summarized in Table 2, MPMF-ZVRT achieved a THD of 1.56%, significantly lower than PI-ZVRT (9.07%) and FCS-MPC-ZVRT (4.86%). This reduction in harmonics is critical for solar inverters, as it minimizes grid pollution and complies with power quality standards. The low THD is attributed to the smooth modulation functions that approximate ideal sinusoidal references, unlike the discrete switching in FCS-MPC.
| Control Method | THD (%) | Reactive Current Response Time (ms) | Computational Load |
|---|---|---|---|
| PI-ZVRT | 9.07 | 25 | Low |
| FCS-MPC-ZVRT | 4.86 | 18 | High |
| MPMF-ZVRT | 1.56 | 15 | Moderate |
The computational efficiency of MPMF-ZVRT is another benefit for solar inverters. Unlike FCS-MPC, which requires evaluating multiple voltage vectors per control cycle, MPMF solves a simple derivative equation, reducing processor burden. This enables implementation on cost-effective digital signal processors commonly used in commercial solar inverters. Moreover, the fixed switching frequency of SPWM simplifies filter design and electromagnetic compatibility, making it practical for large-scale PV plants.
In conclusion, the MPMF-based ZVRT method offers a robust solution for enhancing the fault ride-through capability of solar inverters. By leveraging model predictive control in the modulation function domain, it achieves fast dynamic response, low harmonic distortion, and computational simplicity. Solar inverters equipped with this strategy can reliably support grid stability during severe asymmetric faults, fulfilling stringent ZVRT requirements. Future work may explore adaptive tuning of prediction horizons or integration with energy storage systems to further optimize solar inverter performance. As solar penetration grows, advanced control techniques like MPMF will be indispensable for ensuring the resilience and reliability of modern power networks.
The mathematical rigor and simulation validation presented herein underscore the transformative potential of predictive control in solar inverter applications. By continuously refining these algorithms, we can unlock greater efficiency and grid support functions from solar inverters, paving the way for a sustainable energy future. The journey towards smarter solar inverters is ongoing, with innovations in control theory playing a pivotal role in shaping grid-interactive power electronics.