In this work, we present a comprehensive investigation into the zero voltage ride through (ZVRT) capability of photovoltaic grid-connected solar inverters under asymmetrical grid faults. The increasing penetration of photovoltaic power into modern power systems imposes stringent requirements on grid-connected converters, particularly during severe voltage sags. Traditional control strategies, such as linear proportional-integral (PI) regulators, often exhibit limited dynamic performance and poor harmonic rejection under unbalanced conditions. To address these challenges, we propose a novel control algorithm based on model predictive modulation function (MPMF). This approach leverages the discrete-time model of the solar inverter in the stationary αβ reference frame, derives an optimal modulation function by minimizing a cost function, and integrates it with a fixed-switching-frequency sinusoidal pulse width modulator (SPWM). The proposed method not only enhances the transient response of reactive current injection but also significantly reduces the total harmonic distortion (THD) of the grid-connected current. In this article, we detail the mathematical formulation, control architecture, and simulation verification of the MPMF-based ZVRT strategy, comparing its performance with conventional PI control and finite-control-set model predictive control (FCS-MPC). Our results show that the proposed technique outperforms existing approaches in terms of speed, accuracy, and computational simplicity. Throughout this paper, we emphasize the role of solar inverters in maintaining grid stability during fault events, and we demonstrate how the proposed method contributes to fulfilling the latest grid codes.
1. Introduction
The global energy landscape is undergoing a substantial transformation, with renewable energy sources, particularly solar photovoltaic (PV) systems, emerging as pivotal players in the quest for sustainable power generation. The installed capacity of photovoltaic power has grown exponentially over the past decade, driven by declining costs and supportive government policies. However, this rapid integration of solar inverters into the electric power network introduces significant operational challenges. Among these, the ability of grid-connected solar inverters to remain connected during grid disturbances is of paramount importance. Traditionally, utilities required inverters to disconnect under severe voltage depressions to avoid islanding and protect equipment. Yet, as the penetration level rises, sudden disconnection of large-scale PV plants can lead to frequency and voltage instability, potentially causing cascading outages. Consequently, modern grid codes mandate low voltage ride through (LVRT) and, increasingly, zero voltage ride through (ZVRT) capabilities. For instance, the national standard GB/T 19964-2012 stipulates that photovoltaic stations must remain connected for at least 0.15 seconds when the grid voltage drops to zero, injecting reactive current to support voltage recovery.
Distinguishing between symmetrical and asymmetrical faults is crucial. Symmetrical three-phase faults are rare but severe, while asymmetrical faults, such as single-phase-to-ground or phase-to-phase faults, are more common. Under asymmetrical conditions, negative-sequence components appear in voltages and currents, introducing oscillations and causing conventional dq-control methods to fail. Therefore, advanced control strategies are required for solar inverters to handle unbalanced faults while maintaining sinusoidal, balanced currents and providing dynamic reactive power support.
Several control paradigms have been proposed for LVRT/ZVRT of photovoltaic inverters. One widely adopted technique is the positive-negative sequence dual current control performed in two synchronous reference frames rotating in opposite directions. This method enables independent regulation of positive- and negative-sequence currents, which is necessary for eliminating double-frequency ripples in active power or for achieving balanced grid currents. Implementation typically employs PI controllers in both frames. While the steady-state performance is satisfactory, the dynamic response is limited by the bandwidth of the PI regulators, and parameter tuning is non-trivial, especially under grid-impedance variations. Furthermore, PI controllers are sensitive to grid frequency fluctuations and require accurate sequence separation and phase-locked loops, which may introduce delays.
An alternative is the proportional-resonant (PR) controller operating in the stationary αβ frame. PR controllers can track sinusoidal references without sequence decomposition, thereby simplifying the control architecture. However, their performance depends heavily on the accuracy of the resonant frequency, and they do not inherently provide satisfactory harmonics rejection when the grid frequency deviates.
With the advent of high-speed digital signal processors, model predictive control (MPC) has attracted considerable interest in power electronics. Finite-control-set MPC (FCS-MPC) exploits the discrete switching states of the voltage source inverter. At each sampling instant, the algorithm evaluates all possible voltage vectors using a cost function, then applies the vector that minimizes the predicted current error. FCS-MPC offers excellent dynamic performance and flexibility in incorporating constraints. Nevertheless, applying only one voltage vector per sampling period results in a relatively high current THD, since the average voltage approximation is coarse. Moreover, the computational burden grows rapidly with the number of switching states, demanding high sampling rates and powerful controllers.
To overcome the limitations of FCS-MPC, we propose a model predictive modulation function (MPMF) technique. Unlike FCS-MPC, which selects a discrete voltage vector, MPMF calculates a continuous modulation function (i.e., duty-cycle or reference voltage) that is synthesized by a conventional SPWM modulator. The modulation function is obtained by solving an optimization problem that minimizes the current error between the predicted output current and the reference trajectory. This approach combines the advantages of model predictive control with the fixed switching frequency and low harmonic distortion of carrier-based PWM. Since the control variable is continuous, the inverter can emulate any voltage vector within the linear modulation range, leading to better steady-state performance. Additionally, because the optimization is solved analytically via differentiation of a quadratic cost function, the computational load is substantially lower than the enumerative search in FCS-MPC.
In this paper, we focus on the application of MPMF to the ZVRT control of solar inverters during asymmetrical grid faults. We derive the discrete-time predictive model of the inverter in the αβ frame and formulate a cost function that penalizes the squared error of the predicted output current relative to the reference. By taking the derivative and setting it to zero, we obtain a closed-form expression for the optimal modulation function. To mitigate control delay, a two-step-ahead prediction is implemented. We then integrate this controller into a complete ZVRT scheme that generates positive- and negative-sequence current references according to grid code requirements. The performance of the proposed method is validated through extensive simulations under both asymmetrical and symmetrical voltage dips. Compared with PI-ZVRT and FCS-MPC-ZVRT, the MPMF method exhibits faster reactive current response, lower THD, and reduced computational complexity. The remainder of this paper is organized as follows. Section 2 details the mathematical model of the solar inverter under unbalanced grid conditions. Section 3 introduces the proposed MPMF-based zero voltage ride through strategy. Section 4 presents simulation results and comprehensive comparisons. Section 5 draws conclusions.
2. Modeling of Solar Inverters Under Grid Faults
Figure 1 illustrates the main circuit topology of a three-phase photovoltaic grid-connected inverter that we consider in this study. The inverter consists of six switches, a dc-link capacitor \(C\), and an LCL filter (with a small filter capacitance that can be neglected in the low-frequency model). The grid is represented by an equivalent Thévenin source with line impedance \(R\) and series inductance \(L\). The dc-side is fed by a photovoltaic array, but for the control design, we treat the dc-link voltage \(U_{dc}\) as constant, assuming the front-end converter or battery maintains it. The inverter output voltage \(U_{abc}\), current \(I_{abc}\), and grid voltage \(E_{abc}\) are three-phase vectors.
According to Kirchhoff’s voltage law, the dynamics of the output current in the abc frame can be written as:
$$
L \frac{dI_{abc}}{dt} = U_{abc} – E_{abc} – R I_{abc} \tag{1}
$$
where the vectors are defined as \(I_{abc} = [i_a, i_b, i_c]^T\), \(U_{abc} = [u_a, u_b, u_c]^T\), and \(E_{abc} = [e_a, e_b, e_c]^T\). The phase voltages are assumed balanced in an ideal case, but they contain positive- and negative-sequence components during unbalanced faults. To facilitate control, we first transform (1) to the stationary αβ reference frame using the Clarke transformation matrix \(T_c\):
$$
T_c = \frac{2}{3}
\begin{bmatrix}
1 & -\frac{1}{2} & -\frac{1}{2} \\
0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2}
\end{bmatrix} \tag{2}
$$
Applying (2) to (1), we obtain:
$$
L \frac{dI_{\alpha\beta}}{dt} = U_{\alpha\beta} – E_{\alpha\beta} – R I_{\alpha\beta} \tag{3}
$$
where the subscript \(\alpha\beta\) denotes the two-phase components. For balanced conditions, \(E_{\alpha\beta}\) is a vector rotating at the grid angular speed \(\omega\). In a synchronously rotating dq reference frame defined by the transformation \(T_p\):
$$
T_p = \begin{bmatrix} \cos\omega t & \sin\omega t \\ -\sin\omega t & \cos\omega t \end{bmatrix} \tag{4}
$$
Equation (3) becomes:
$$
\begin{aligned}
u_d &= e_d + L \frac{di_d}{dt} + R i_d – \omega L i_q, \\
u_q &= e_q + L \frac{di_q}{dt} + R i_q + \omega L i_d.
\end{aligned} \tag{5}
$$
Here, \(e_d\) and \(e_q\) are the direct- and quadrature-axis components of the grid voltage. Under balanced conditions, \(e_d = E_m\) and \(e_q = 0\). The instantaneous active and reactive power output from the solar inverter is given by:
$$
P = \frac{3}{2} E_m I_m \cos\varphi = \frac{3}{2} e_d i_d, \quad
Q = -\frac{3}{2} E_m I_m \sin\varphi = -\frac{3}{2} e_d i_q \tag{6}
$$
Thus, control of \(i_d\) and \(i_q\) effectively controls active and reactive power. However, when an asymmetrical fault occurs, \(e_d\) and \(e_q\) contain double-frequency oscillations due to negative-sequence components. The grid voltage vector can be decomposed into positive- and negative-sequence components:
$$
E_{abc} = E_{abc}^+ + E_{abc}^-
\tag{7}
$$
In the stationary frame, these components can be extracted using the following filters:
$$
\begin{aligned}
E_{\alpha\beta}^+ &= T_{2s/2s}^+ E_{\alpha\beta}, \quad T_{2s/2s}^+ = \frac{1}{2} \begin{bmatrix} 1 & -q \\ q & 1 \end{bmatrix}, \\
E_{\alpha\beta}^- &= T_{2s/2s}^- E_{\alpha\beta}, \quad T_{2s/2s}^- = \frac{1}{2} \begin{bmatrix} 1 & q \\ -q & 1 \end{bmatrix}
\end{aligned} \tag{8}
$$
where \(q = e^{-j\pi/2}\) represents a 90° phase-shift operator. Subsequently, each sequence is transformed into its own synchronous frame by using the corresponding Park matrices:
$$
\begin{aligned}
E_{dq}^+ &= T_{\alpha\beta/dq}^+ E_{\alpha\beta}^+, \quad T_{\alpha\beta/dq}^+ = \begin{bmatrix} \cos\omega t & \sin\omega t \\ -\sin\omega t & \cos\omega t \end{bmatrix}, \\
E_{dq}^- &= T_{\alpha\beta/dq}^- E_{\alpha\beta}^-, \quad T_{\alpha\beta/dq}^- = \begin{bmatrix} \cos\omega t & -\sin\omega t \\ \sin\omega t & \cos\omega t \end{bmatrix}.
\end{aligned} \tag{9}
$$
This transformation yields constant values in steady state, allowing the design of PI controllers in both frames. The resulting voltage components are:
$$
\begin{bmatrix} e_d^+ \\ e_q^+ \\ e_d^- \\ e_q^- \end{bmatrix} =
\begin{bmatrix} U^+ \\ 0 \\ U^- \cos\phi \\ -U^- \sin\phi \end{bmatrix}
\tag{10}
$$
where \(U^+\) and \(U^-\) are the amplitudes of the positive- and negative-sequence voltages, and \(\phi\) is the initial phase angle of the negative-sequence voltage. Similarly, the inverter current can be decomposed into positive- and negative-sequence components in their respective frames. The complete model of the solar inverter in the dual synchronous frames is:
$$
\begin{aligned}
u_d^+ &= L \frac{di_d^+}{dt} + R i_d^+ – \omega L i_q^+ + e_d^+, \\
u_q^+ &= L \frac{di_q^+}{dt} + R i_q^+ + \omega L i_d^+ + e_q^+, \\
u_d^- &= L \frac{di_d^-}{dt} + R i_d^- – \omega L i_q^- + e_d^-, \\
u_q^- &= L \frac{di_q^-}{dt} + R i_q^- – \omega L i_d^- + e_q^-.
\end{aligned} \tag{11}
$$
Equation (11) reveals the cross-coupling between the d and q axes, which complicates the design of a direct model predictive controller. For the MPMF approach, we prefer the stationary αβ frame because it avoids rotation and decouples the control variables. The predictive model is built directly from (3), which is a first-order system with a simple discretization. The main challenge is to generate appropriate current references in the αβ frame from the required positive- and negative-sequence active and reactive current setpoints. We address this in the next section.
3. Proposed MPMF-Based Zero Voltage Ride Through Method
3.1 Principle of Model Predictive Modulation Function
Before diving into the derivation, we explain the concept of the modulation function. In a voltage-source inverter, the output voltage can be expressed as the product of the dc-link voltage \(U_{dc}\) and a modulation function \(M\). For a three-phase two-level inverter, the phase voltages are related to the switching states via:
$$
u_a = \frac{U_{dc}}{2} (2S_a – 1), \quad u_b = \frac{U_{dc}}{2} (2S_b – 1), \quad u_c = \frac{U_{dc}}{2} (2S_c – 1)
\tag{12}
$$
where \(S_x \in \{0,1\}\) for \(x = a,b,c\). By using space-vector modulation, the average voltage produced over a switching period is a linear combination of the eight basic voltage vectors. If we denote the duty-cycle or modulation function by \(M\), then the average output voltage vector \(U_{\alpha\beta}\) is \(U_{dc} M_{\alpha\beta}\). Instead of selecting a single discrete vector (as in FCS-MPC), MPMF computes a continuous modulation function \(M_{\alpha\beta}(k+1)\) that minimizes the predicted current error at the next sampling instant. This modulation function is then input to a standard SPWM comparator, which generates the switching signals. Figure 2 illustrates the relationship between the modulation function and the switching states over one sampling period. Since multiple switching states are used within one period, the average vector is accurately synthesized, leading to lower current ripple and THD. Moreover, the analog computation avoids enumerating the candidate vectors, greatly reducing the computational cost.
3.2 Predictive Model and Cost Function
We begin with the αβ-frame model given by (3). To implement digital control, we discretize this equation using the first-order forward Euler method with a sampling period \(T_s\). Let \(k\) denote the current sampling instant. Then the predicted current at \(k+1\) is:
$$
\begin{aligned}
i_{\alpha}(k+1) &= \frac{T_s}{L} \big( u_{\alpha}(k) – e_{\alpha}(k) \big) + \left(1 – \frac{R T_s}{L}\right) i_{\alpha}(k), \\
i_{\beta}(k+1) &= \frac{T_s}{L} \big( u_{\beta}(k) – e_{\beta}(k) \big) + \left(1 – \frac{R T_s}{L}\right) i_{\beta}(k).
\end{aligned} \tag{13}
$$
Replacing \(u_{\alpha}(k)\) with \(U_{dc} m_{\alpha}(k)\), and similarly for the β component, yields:
$$
\begin{aligned}
i_{\alpha}(k+1) &= \frac{T_s U_{dc}}{L} m_{\alpha}(k) – \frac{T_s}{L} e_{\alpha}(k) + \left(1 – \frac{R T_s}{L}\right) i_{\alpha}(k), \\
i_{\beta}(k+1) &= \frac{T_s U_{dc}}{L} m_{\beta}(k) – \frac{T_s}{L} e_{\beta}(k) + \left(1 – \frac{R T_s}{L}\right) i_{\beta}(k).
\end{aligned} \tag{14}
$$
The control objective is to make the current vector \(I_{\alpha\beta}(k+1)\) track a reference \(I_{\alpha\beta}^{ref}(k+1)\) as closely as possible. Therefore, we define a cost function as the sum of squared errors:
$$
J = \big( i_{\alpha}(k+1) – i_{\alpha}^{ref}(k+1) \big)^2 + \big( i_{\beta}(k+1) – i_{\beta}^{ref}(k+1) \big)^2.
\tag{15}
$$
Because \(J\) is a quadratic and convex function of \(m_{\alpha}(k)\) and \(m_{\beta}(k)\), the optimal modulation function can be found analytically by setting the partial derivatives to zero:
$$
\frac{\partial J}{\partial m_{\alpha}(k)} = 0, \quad \frac{\partial J}{\partial m_{\beta}(k)} = 0.
\tag{16}
$$
Solving (16) gives the modulation function that minimizes the current error at the next step. However, due to computation and sampling delays, it is desirable to predict the current two steps ahead. We use the same Euler approximation with a one-step delay compensation. The predicted current at time \(k+2\) is:
$$
\begin{aligned}
i_{\alpha}(k+2) &= \frac{T_s}{L} \big( u_{\alpha}(k+1) – e_{\alpha}(k+1) \big) + \left(1 – \frac{R T_s}{L}\right) i_{\alpha}(k+1), \\
i_{\beta}(k+2) &= \frac{T_s}{L} \big( u_{\beta}(k+1) – e_{\beta}(k+1) \big) + \left(1 – \frac{R T_s}{L}\right) i_{\beta}(k+1).
\end{aligned} \tag{17}
$$
The grid voltage at the future instant \(k+1\) can be estimated from a known rotation:
$$
\begin{bmatrix} e_{\alpha}(k+1) \\ e_{\beta}(k+1) \end{bmatrix} =
\begin{bmatrix} \cos\omega T_s & -\sin\omega T_s \\ \sin\omega T_s & \cos\omega T_s \end{bmatrix}
\begin{bmatrix} e_{\alpha}(k) \\ e_{\beta}(k) \end{bmatrix}.
\tag{18}
$$
We then formulate a two-step-ahead cost function:
$$
J_1 = \big( i_{\alpha}(k+2) – i_{\alpha}^{ref}(k+2) \big)^2 + \big( i_{\beta}(k+2) – i_{\beta}^{ref}(k+2) \big)^2.
\tag{19}
$$
Substituting (17) into (19), and then differentiating with respect to \(m_{\alpha}(k+1)\) and \(m_{\beta}(k+1)\), we obtain the explicit formulas for the optimal modulation functions:
$$
\begin{aligned}
m_{\alpha}(k+1) &= \frac{L}{U_{dc} T_s} i_{\alpha}^{*}(k+2) – \frac{L}{U_{dc} T_s} \left[ \frac{T_s}{L} \left( u_{\alpha}(k) – e_{\alpha}(k) \right) + \left(1 – \frac{R T_s}{L}\right) i_{\alpha}(k) \right] + \frac{1}{U_{dc}} e_{\alpha}(k+1), \\
m_{\beta}(k+1) &= \frac{L}{U_{dc} T_s} i_{\beta}^{*}(k+2) – \frac{L}{U_{dc} T_s} \left[ \frac{T_s}{L} \left( u_{\beta}(k) – e_{\beta}(k) \right) + \left(1 – \frac{R T_s}{L}\right) i_{\beta}(k) \right] + \frac{1}{U_{dc}} e_{\beta}(k+1).
\end{aligned} \tag{20}
$$
Here, \(i_{\alpha}^{*}(k+2)\) and \(i_{\beta}^{*}(k+2)\) are the commanded current references at time \(k+2\). In practice, we need to compute these references from the required positive- and negative-sequence currents and transform them back to the αβ frame. The expression (20) can be rearranged into a more compact form:
$$
\begin{aligned}
m_{\alpha}(k+1) &= \frac{L}{U_{dc} T_s} \left[ i_{\alpha}^{*}(k+2) – i_{\alpha}(k) \right] + \frac{e_{\alpha}(k) + e_{\alpha}(k+1)}{U_{dc}} + \frac{R}{U_{dc}} i_{\alpha}(k), \\
m_{\beta}(k+1) &= \frac{L}{U_{dc} T_s} \left[ i_{\beta}^{*}(k+2) – i_{\beta}(k) \right] + \frac{e_{\beta}(k) + e_{\beta}(k+1)}{U_{dc}} + \frac{R}{U_{dc}} i_{\beta}(k).
\end{aligned} \tag{21}
$$
Note that the term involving \(u_{\alpha}(k)\) cancels because \(u_{\alpha}(k)\) is the applied voltage from the previous step; this formula does not require an actual measurement of \(u_{\alpha}(k)\) since it is replaced by the known modulation function from the previous interval. The derivation assumes that the inductor is linear and that the grid voltage changes slowly, which is reasonable for a sampling frequency of 10 kHz.
3.3 Current Reference Generation for ZVRT
According to grid codes such as GB/T 19964-2012 and similar international standards, during a voltage fault the solar inverter must inject reactive current to support grid voltage. The dynamic reactive current reference \(i_q^*\) depends on the voltage sag magnitude \(U_T\) in per unit. We adopt the following piecewise function:
$$
\begin{cases}
i_{q}^{ref} \geq 1.5 \times (0.9 – U_T) I_N, & 0.2 \le U_T \le 0.9, \\
i_{q}^{ref} \geq 1.05 \times I_N, & U_T < 0.2, \\
i_{q}^{ref} = 0, & U_T > 0.9.
\end{cases} \tag{22}
$$
Here, \(I_N\) is the rated output current of the solar inverter. In addition, to prevent overcurrent, the active current reference is computed as:
$$
i_d^{ref} = \sqrt{ (1.05 I_N)^2 – (i_q^{ref})^2 }
\tag{23}
$$
For unbalanced faults, it is common to force the negative-sequence current to zero so that the generated current remains balanced and sinusoidal. This also mitigates active power oscillations and dc-link ripples. Therefore, we set:
$$
i_d^{-, ref} = 0, \quad i_q^{-, ref} = 0.
\tag{24}
$$
The positive-sequence active and reactive current references are set to \(i_d^{+, ref} = i_d^{ref}\) and \(i_q^{+, ref} = i_q^{ref}\). These references are in the respective synchronous dq frames. To obtain the total αβ reference, we transform the positive- and negative-sequence references back to the αβ frame using:
$$
\begin{aligned}
i_{\alpha}^{ref} &= i_d^{+, ref} \cos\omega t – i_q^{+, ref} \sin\omega t + i_d^{-, ref} \cos\omega t + i_q^{-, ref} \sin\omega t, \\
i_{\beta}^{ref} &= i_d^{+, ref} \sin\omega t + i_q^{+, ref} \cos\omega t – i_d^{-, ref} \sin\omega t + i_q^{-, ref} \cos\omega t.
\end{aligned} \tag{25}
$$
However, since the negative-sequence references are zero, the total αβ reference simplifies to the positive-sequence component:
$$
i_{\alpha}^{ref} = i_d^{ref} \cos\omega t – i_q^{ref} \sin\omega t, \quad
i_{\beta}^{ref} = i_d^{ref} \sin\omega t + i_q^{ref} \cos\omega t.
\tag{26}
$$
During a fault, if the voltage sag is zero (\(U_T=0\)), we set \(i_q^{ref} = 1.05 I_N\) and \(i_d^{ref} = 0\), resulting in a purely reactive current output. This condition satisfies the ZVRT requirement. The references at time \(k+2\) are generated by shifting the phase angle by \(2\omega T_s\), which is straightforward.
3.4 Control Architecture
The overall block diagram of the proposed MPMF-ZVRT controller is shown conceptually in the following description. The three-phase grid voltages and currents are measured and transformed to the αβ frame. The positive- and negative-sequence components of the grid voltage are extracted using a software phase-locked loop and a sequence decomposition algorithm (not detailed here). The voltage sag depth \(U_T\) is calculated from the positive-sequence amplitude. According to (22)-(26), the current references are generated. The MPMF controller then computes the optimal modulation functions \(m_{\alpha}(k+1)\) and \(m_{\beta}(k+1)\) using (21), and feeds them to an SPWM modulator that operates at a fixed switching frequency (e.g., 2.5 kHz). The resulting gate signals drive the solar inverter.
We emphasize that the proposed method does not require any switching sequence enumeration, in contrast to FCS-MPC. The optimization is deterministic and operates in closed form, which makes it suitable for hardware implementation with modest computational resources. The complete algorithm is summarized in Table 1, where the main steps are listed.
| Step | Operation |
|---|---|
| 1 | Sampling of \(e_a, e_b, e_c\) and \(i_a, i_b, i_c\) at instant \(k\). |
| 2 | Transform to \(\alpha\beta\) using Clarke transformation. |
| 3 | Detect sequence components and compute voltage sag \(U_T\). |
| 4 | Set references \(i_d^{ref}, i_q^{ref}\) according to grid code. |
| 5 | Compute \(\alpha\beta\) current references via (26). |
| 6 | Predict \(e_{\alpha\beta}(k+1)\) using (18). |
| 7 | Calculate \(m_{\alpha}(k+1), m_{\beta}(k+1)\) using (21). |
| 8 | Apply modulation functions to SPWM modulator. |
4. Simulation Results and Discussion
We validated the proposed MPMF-ZVRT method using a detailed simulation model built in MATLAB/Simulink. The simulated system is a 250 kW photovoltaic plant connected to a 35 kV grid through a step-up transformer and an LCL filter. The main parameters used in the simulation are listed in Table 2. We intentionally modeled a relatively high dc-link voltage (480 V) and a switching frequency of 2500 Hz to match practical solar inverter designs. The sampling frequency of the controller was set to 10 kHz, ensuring adequate resolution for the predictive algorithm.
| Parameter | Symbol | Value |
|---|---|---|
| Nominal grid voltage (rms line-to-line) | \(E_{LL}\) | 35 kV |
| Grid frequency | \(f\) | 50 Hz |
| Filter inductance | \(L\) | 0.12 mH |
| Filter capacitance | \(C_f\) | 60 μF |
| DC-link capacitance | \(C_{dc}\) | 65 mF |
| DC-link voltage | \(U_{dc}\) | 480 V |
| Switching frequency | \(f_{sw}\) | 2500 Hz |
| Rated active power | \(P_N\) | 250 kW |
| Rated output current (rms) | \(I_N\) | 6 A |
| Overcurrent protection threshold | \(I_{max}\) | 7.2 A |
| DC-link overvoltage protection | \(U_{dc,max}\) | 600 V |
| Transformer capacity | \(S_N\) | 250 kVA |
| Transformer ratio | — | 250 V / 35 kV |
| Sampling frequency of controller | \(f_s\) | 10 kHz |
4.1 Performance Under Asymmetrical Fault
We considered a single-phase-to-ground fault where phase a voltage drops to zero. The fault was applied at \(t=0.05\) s and cleared after 0.15 s, which satisfies the ZVRT time requirement. Figure 3 shows the grid current waveforms and the reactive current response using the proposed MPMF-ZVRT method. The top subplot displays the three-phase grid voltages, where phase a voltage collapses to zero. The middle subplot shows the inverter output currents. At the moment of the fault, the currents exhibit a slight transient oscillation, but they settle within a very short time. The steady-state amplitude is approximately 6.3 A, which is below the overcurrent limit of 7.2 A, and importantly, the currents remain balanced with no negative-sequence components. The bottom subplot illustrates the actual reactive current \(i_q\) tracking the reference of \(1.05 I_N\). The response time is about 18 ms, which is well within the 30 ms limit specified by the grid code. These results confirm that the MPMF-ZVRT method can successfully ride through a zero-voltage asymmetrical fault.
To provide a more quantitative evaluation, we compared the proposed method with two conventional methods: PI-ZVRT and FCS-MPC-ZVRT. The reactive current response under the same fault condition is plotted in Figure 4 for all three methods. Clearly, the PI controller exhibits a significant oscillatory transient and takes about 25 ms to reach the reference. FCS-MPC achieves a faster response of around 18 ms, but it suffers from higher current ripple. In contrast, the MPMF method reaches the reference in only 15 ms with minimal overshoot and very little ripple. This improvement is attributed to the continuous control variable and fixed switching frequency of the SPWM modulator, which avoids the coarse quantization of the voltage vector.
| Control Method | Reactive current recovery time (ms) | Overshoot (%) | Steady-state ripple (A) |
|---|---|---|---|
| PI-ZVRT | 25 | 12 | 0.45 |
| FCS-MPC-ZVRT | 18 | 6 | 0.30 |
| MPMF-ZVRT (proposed) | 15 | 3 | 0.12 |
4.2 Harmonic Analysis
One of the most important metrics for grid-connected solar inverters is the total harmonic distortion (THD) of the output current. Higher THD leads to increased losses, electromagnetic interference, and potential violation of grid standards. We performed a Fast Fourier Transform (FFT) analysis on the steady-state current waveforms for each control method. Figure 5 shows the harmonic spectra up to 10 kHz. The PI-ZVRT method produces a THD of 9.07%, with significant low-order harmonics concentrated below 1000 Hz. The FCS-MPC-ZVRT method reduces the THD to 4.86%, but there are still notable harmonics caused by the variable switching behavior. The proposed MPMF-ZVRT method achieves a THD of only 1.56%, which is significantly lower and is within the limit specified by IEEE 519 (typically 5% for this current level). This improvement is due to the fixed switching frequency and the averaging effect of the modulation function, which effectively reduces low-order harmonics.
| Control Method | THD (%) | Fundamental current (A) |
|---|---|---|
| PI-ZVRT | 9.07 | 6.263 |
| FCS-MPC-ZVRT | 4.86 | 6.293 |
| MPMF-ZVRT (proposed) | 1.56 | 6.299 |
4.3 Performance Under Symmetrical Fault
Although our focus is on asymmetrical faults, the proposed method can also handle symmetrical voltage dips with ease, since a symmetrical fault is just a special case where the negative-sequence components are zero. We simulated a three-phase voltage dip to zero for 0.15 s. The results are presented in Figure 6. The inverter output current remains symmetrical and does not exceed 6.3 A during the fault, and the reactive current reaches the reference within 15 ms. This confirms the universality of the MPMF-ZVRT approach.
4.4 Discussion on Computational Complexity
Compared with FCS-MPC, which requires evaluating eight (or more) candidate voltage vectors per sampling period, the MPMF algorithm computes the modulation function directly from closed-form equations (21). The computations involve only a few multiplications and additions per axis, plus a square root (or trigonometric calculation) for the reference generation. Therefore, the computational load is significantly reduced, making it feasible to implement on low-cost digital signal processors. The fixed switching frequency of the SPWM also simplifies the design of output filters and ensures predictable electromagnetic compatibility.
We can quantify the computational burden in terms of the number of control-loop operations. In a typical FCS-MPC implementation, each iteration requires predicting the current for each of the seven distinct voltage vectors (excluding redundant ones), calculating the cost function, comparing, and selecting the minimum. This amounts to at least seven times the prediction operations and numerous comparisons. In our MPMF approach, the prediction is implicit in the algebraic expression, and the optimal modulation function is found without branching. Table 5 summarizes a rough comparison of the operations in a single control cycle.
| Operation | FCS-MPC | MPMF |
|---|---|---|
| Predictions | 7 (or more) | 2 |
| Cost function evaluations | 7 | 0 (analytic) |
| Comparisons | 7 | 1 |
| Trigonometric functions | 1 | 1 |
| PWM modulation | No | Yes |
Our simulation results demonstrate that the MPMF method outperforms both PI and FCS-MPC in terms of dynamic response, harmonic content, and computational simplicity. The main advantages are:
- Fast dynamic response: The predictive nature of the controller provides one-step-ahead correction, yielding a reactive current response time as low as 15 ms, which satisfies the strictest grid codes.
- Low THD: The continuous modulation function synthesized by SPWM results in a nearly sinusoidal current with THD below 2%, far superior to the discrete-vector approaches.
- Robustness to unbalanced faults: By forcing the negative-sequence current to zero, the output current remains balanced and the inverter does not experience excessive overcurrent.
- Computational efficiency: The closed-form solution eliminates the need for an exhaustive search, reducing the controller latency and allowing a higher sampling frequency or lower-cost hardware.
5. Conclusion
In this paper, we have presented a comprehensive study of a model predictive modulation function control strategy for the zero voltage ride through of photovoltaic grid-connected solar inverters under asymmetrical grid faults. We established a rigorous mathematical model of the inverter in the stationary αβ frame and derived a predictive control algorithm that computes the optimal modulation function by minimizing a quadratic cost function. The proposed MPMF-ZVRT controller was integrated with a fixed-switching-frequency SPWM modulator, achieving both high dynamic performance and low harmonic distortion. Simulation results under a single-phase-to-ground fault and a three-phase fault demonstrated that the method effectively limits the peak current, maintains balanced output currents, and injects the required reactive current within 15 ms, well below the 30 ms limit defined by grid codes. Compared with traditional PI control and FCS-MPC, the MPMF method offers significant improvements in response speed, THD reduction (from 9.07% and 4.86% down to 1.56%), and computational simplicity. We believe that the proposed method constitutes a practical and effective solution for modern solar inverters that must comply with stringent ZVRT requirements while maintaining high power quality.
Future work will focus on experimental validation on a hardware-in-the-loop testbed and extending the method to handle dc-link voltage dynamics and weak-grid conditions. We also plan to investigate the robustness of the MPMF approach against parameter uncertainties and measurement noise.