In this work, I focus on the control strategy of grid-connected solar inverters under asymmetrical grid faults. The operation of solar inverters becomes challenging when the grid voltage contains negative-sequence components, because conventional vector control based on positive-sequence synchronous frames cannot effectively regulate the negative-sequence currents. As a result, the output power of solar inverters suffers from double-frequency oscillations, which may cause severe DC-link voltage ripples and threaten the stability of the system. To overcome these issues, I propose a model predictive current control (MPCC) method for solar inverters that operates directly in the stationary reference frame, avoiding the need for phase-sequence decomposition of grid currents. I first derive the mathematical model of a grid-connected solar inverter under unbalanced grid conditions, analyze the power characteristics, and then develop a control strategy that can achieve different objectives such as suppressing active-power oscillations or eliminating negative-sequence currents. Simulation results obtained on the PSCAD/EMTDC platform demonstrate that the proposed method provides excellent dynamic and steady-state performance, effectively improves the current quality of solar inverters during asymmetrical faults, and is easy to implement in digital signal processors.
1. Dynamic Modeling of Solar Inverters Under Asymmetrical Grid Faults
The main circuit of a photovoltaic (PV) generation system using solar inverters is depicted in Figure 1. The system consists of a DC-link capacitor, a three-phase voltage-source inverter, filter inductors, and equivalent resistances. The output voltage of the solar inverter is denoted by \(U_{tabc}\), the output current by \(I_{abc}\), and the grid voltage by \(U_{gabc}\). The DC-link voltage is \(U_{dc}\), while \(P_{dci}\) and \(P_{dco}\) represent the input power from the PV array and the output power from the DC-link, respectively.
According to Kirchhoff’s voltage law, the mathematical model of the solar inverter in the three-phase stationary coordinate system can be expressed as:
\[
U_{tabc} = U_{gabc} + L \frac{dI_{abc}}{dt} + R I_{abc}
\tag{1}
\]
Applying the Clarke transformation to convert Eq. (1) into the two-phase stationary \(\alpha\beta\) frame yields:
\[
U_{t\alpha\beta} = U_{g\alpha\beta} + L \frac{dI_{\alpha\beta}}{dt} + R I_{\alpha\beta}
\tag{2}
\]
During asymmetrical grid faults, any electrical quantity \(F\) (voltage or current) in the stationary frame consists of a positive-sequence component rotating counterclockwise at the synchronous angular speed \(\omega_0\) and a negative-sequence component rotating clockwise at the same angular speed:
\[
F_{\alpha\beta} = F_{\alpha\beta}^p + F_{\alpha\beta}^n = F_{dq}^p e^{j\theta} + F_{dq}^n e^{-j\theta}
\tag{3}
\]
where superscripts \(p\) and \(n\) denote positive and negative sequences, respectively, and \(\theta = \omega_0 t\). Substituting Eq. (3) into Eq. (2), the dynamic model of the solar inverter under unbalanced grid voltage is obtained as:
\[
\begin{aligned}
U_{tdq}^p &= U_{gdq}^p + L \frac{dI_{dq}^p}{dt} + j\omega_0 L I_{dq}^p \\
U_{tdq}^n &= U_{gdq}^n + L \frac{dI_{dq}^n}{dt} – j\omega_0 L I_{dq}^n
\end{aligned}
\tag{4}
\]
The complex power delivered by the solar inverter is given by:
\[
S = 1.5 \, U_{g\alpha\beta} \, I_{\alpha\beta}^*
\tag{5}
\]
By substituting the positive- and negative-sequence components of \(U_{g\alpha\beta}\) and \(I_{\alpha\beta}\) into Eq. (5), the active and reactive power can be written as:
\[
\begin{aligned}
P &= P_0 + P_{c2} \cos(2\omega t) + P_{s2} \sin(2\omega t) \\
Q &= Q_0 + Q_{c2} \cos(2\omega t) + Q_{s2} \sin(2\omega t)
\end{aligned}
\tag{6}
\]
The detailed relationships between the power components and the current sequence components under grid-voltage orientation (with \(U_{gq}^p = 0\)) are summarized in the matrix form:
\[
\begin{bmatrix}
P_0 \\[2pt] P_{c2} \\[2pt] P_{s2} \\[2pt] Q_0 \\[2pt] Q_{c2} \\[2pt] Q_{s2}
\end{bmatrix}
= \frac{3}{2}
\begin{bmatrix}
U_{gd}^p & U_{gq}^p & U_{gd}^n & U_{gq}^n \\
U_{gd}^n & U_{gq}^n & U_{gd}^p & U_{gq}^p \\
U_{gq}^n & -U_{gd}^n & -U_{gq}^p & U_{gd}^p \\
U_{gq}^p & -U_{gd}^p & U_{gq}^n & -U_{gd}^n \\
U_{gq}^n & -U_{gd}^n & U_{gq}^p & -U_{gd}^p \\
-U_{gd}^n & -U_{gq}^n & U_{gd}^p & U_{gq}^p
\end{bmatrix}
\begin{bmatrix}
I_d^p \\[2pt] I_q^p \\[2pt] I_d^n \\[2pt] I_q^n
\end{bmatrix}
\tag{7}
\]
From Eq. (7) it is clear that both active and reactive power contain double-frequency components. Since it is impossible to eliminate these oscillations in both \(P\) and \(Q\) simultaneously, I choose two practical control objectives: (i) suppression of active-power double-frequency oscillations, and (ii) suppression of negative-sequence currents. The corresponding reference current values are derived below.
1.1 Reference Currents for Active-Power Oscillation Suppression
Setting \(P_{c2} = P_{s2} = 0\) and orienting the positive-sequence voltage on the \(d\)-axis (\(U_{gq}^p = 0\)), the current references become:
\[
\begin{aligned}
I_{d,ref}^p &= \frac{P_{ref}}{1.5 U_{gd}^p (1 – k_{dd}^2 – k_{qd}^2)} \\[4pt]
I_{q,ref}^p &= -\frac{Q_{ref}}{1.5 U_{gd}^p (1 – k_{dd}^2 – k_{qd}^2)} \\[4pt]
I_{d,ref}^n &= -k_{dd} I_d^p – k_{qd} I_q^p \\[4pt]
I_{q,ref}^n &= k_{dd} I_q^p – k_{qd} I_d^p
\end{aligned}
\tag{8}
\]
where \(k_{dd} = U_{gd}^n / U_{gd}^p\) and \(k_{qd} = U_{gq}^n / U_{gd}^p\). The active power reference \(P_{ref}\) is obtained from the maximum power point tracking (MPPT) algorithm:
\[
P_{ref} = \left( U_{MPPT} – U_{dc} \right) \left( K_p + \frac{K_I}{s} \right) U_{MPPT}
\tag{9}
\]
1.2 Reference Currents for Negative-Sequence Suppression
Setting \(I_{d,ref}^n = I_{q,ref}^n = 0\), the references simplify to:
\[
\begin{aligned}
I_{d,ref}^p &= \frac{P_{ref}}{1.5 U_{gd}^p} \\[4pt]
I_{q,ref}^p &= -\frac{Q_{ref}}{1.5 U_{gd}^p} \\[4pt]
I_{d,ref}^n &= 0 \\[4pt]
I_{q,ref}^n &= 0
\end{aligned}
\tag{10}
\]
For unity power factor operation, I set \(Q_{ref} = 0\).
2. Model Predictive Current Control Strategy for Solar Inverters
The goal of the current controller is to make the output current of the solar inverter accurately track the reference currents even during asymmetrical grid faults. Since the grid current in the \(\alpha\beta\) frame contains both positive- and negative-sequence components, conventional controllers that require sequence separation introduce delays and errors. The MPCC approach directly predicts the future current values for all possible switching states of the solar inverter and selects the optimal state by minimizing a cost function.
I use the single-pole binary logic switching function \(g_i\) (\(i = a, b, c\)) to describe the switching state of each bridge leg:
\[
g_i =
\begin{cases}
1 & \text{upper switch ON} \\
0 & \text{lower switch ON}
\end{cases}
\tag{11}
\]
The output voltage of the solar inverter can be expressed as:
\[
u_t = \left( g_a + a g_b + a^2 g_c \right) U_{dc}
\tag{12}
\]
where \(a = e^{j \frac{2\pi}{3}}\). Transforming to the \(\alpha\beta\) frame, I get:
\[
\begin{aligned}
U_{t\alpha} &= \sqrt{\frac{2}{3}} \left( g_a – \frac{1}{2}g_b – \frac{1}{2}g_c \right) U_{dc} \\
U_{t\beta} &= \frac{\sqrt{2}}{2} \left( g_b – g_c \right) U_{dc}
\end{aligned}
\tag{13}
\]
The eight possible switching combinations of \(g_a, g_b, g_c\) produce eight 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 |
Discretizing Eq. (2) with sampling time \(T_s\), I obtain the prediction model for the output current at the next sampling instant \(k+1\):
\[
\begin{aligned}
I_{\alpha}(k+1) &= \left( 1 – \frac{T_s R}{L} \right) I_{\alpha}(k) + \frac{T_s}{L} \left( U_{t\alpha}(k) – U_{g\alpha}(k) \right) \\
I_{\beta}(k+1) &= \left( 1 – \frac{T_s R}{L} \right) I_{\beta}(k) + \frac{T_s}{L} \left( U_{t\beta}(k) – U_{g\beta}(k) \right)
\end{aligned}
\tag{14}
\]
To select the optimal switching state, I define the cost function:
\[
c = \left| I_{\alpha ref} – I_{\alpha}(k+1) \right| + \left| I_{\beta ref} – I_{\beta}(k+1) \right|
\tag{15}
\]
For each sampling period, I evaluate the eight voltage vectors from Table 1 using Eq. (14) and Eq. (15). The switching state corresponding to the minimum cost function is applied in the next control cycle.
The reference currents \(I_{\alpha ref}\) and \(I_{\beta ref}\) are obtained after coordinate transformation of the sequence references from Eq. (8) or Eq. (10):
\[
\begin{aligned}
I_{\alpha ref} &= I_{\alpha ref}^p + I_{\alpha ref}^n \\
I_{\beta ref} &= I_{\beta ref}^p + I_{\beta ref}^n
\end{aligned}
\tag{16}
\]
To extract the positive- and negative-sequence voltage components required for the reference computation, I use the second-order generalized integrator (SOGI) method. This eliminates the need for complex low-pass or notch filters and avoids delays introduced by conventional sequence separation. The overall control block diagram is illustrated in Figure 2.
3. Simulation Verification
To validate the effectiveness of the proposed MPCC strategy for solar inverters, I built a 0.5 MW photovoltaic generation system model in PSCAD/EMTDC. The simulation parameters are listed in Table 2.
| Rated power | 0.5 MW |
| DC-link capacitance | 20,000 \(\mu\)F |
| Filter inductance | 0.18 mH |
| Equivalent resistance | 0.0005 \(\Omega\) |
| AC-side voltage (line-to-line rms) | 0.27 kV |
| Sampling period | 20 \(\mu\)s |
First, I investigated the sensitivity of the MPCC method to inductance parameter mismatch. The total harmonic distortion (THD) of the phase-A current was evaluated in two cases: (1) the controller inductance matches the actual inductance, and (2) the controller inductance remains unchanged while the main-circuit inductance varies. The results are shown in Table 3.
| \(L_f\) (mH) | THD (%) | |
|---|---|---|
| Case 1 (matched) | Case 2 (mismatched) | |
| 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 |
It can be observed that the THD remains below 2.7% even with a mismatched inductance, and the difference between the two cases is less than 0.6%. This confirms that the MPCC method has strong robustness against parameter variations, which is critical for practical solar inverters operating in changing environmental conditions.
Next, I tested the dynamic response of the MPCC under balanced grid conditions. The active power reference was stepped from 0.2 MW to 0.3 MW at \(t = 0.3\) s, while the reactive power reference was stepped from 0 Mvar to 0.15 Mvar at \(t = 0.5\) s. The simulation results showed that the \(\alpha\beta\) current components track their references rapidly and that the active and reactive power are fully decoupled. This confirms the excellent dynamic and steady-state characteristics of the proposed controller for solar inverters.
An asymmetrical fault (phase-A-to-ground) was then applied at \(t = 1.0\) s and cleared at \(t = 1.4\) s. The grid voltage unbalance factor increased from 0.3% to 12% during the fault. I compared three control strategies:
- Symmetric control: conventional PI control based on the positive-sequence synchronous frame, without any sequence separation or fault-specific correction.
- Proposed asymmetric control (Objective 1): MPCC with reference currents computed to suppress active-power double-frequency oscillations.
- Proposed asymmetric control (Objective 2): MPCC with reference currents computed to suppress negative-sequence currents.
Figure 3 shows the comparison of active power output. Under the symmetric control strategy, the active power oscillates with a double-frequency component whose amplitude reaches approximately 0.004 MW after stabilization. With the proposed objective-1 control, the oscillation amplitude is reduced to 0.001 MW, a reduction of 75%.
The DC-link voltage behavior is compared in Table 4. According to the power balance equation \(C U_{dc} \frac{dU_{dc}}{dt} = P_{dci} – P_{dco}\), a reduction in active-power ripple directly reduces the DC-link voltage ripple. The measured second-harmonic ripple in the DC-link voltage was 0.006 V for symmetric control and 0.0046 V for the proposed asymmetric control, which agrees well with the theoretical prediction.
| Control strategy | Active-power 2nd-order ripple amplitude (MW) | DC-link voltage 2nd-order ripple (V) | Current THD (phase A/B/C, %) | Current unbalance factor (%) |
|---|---|---|---|---|
| Symmetric control | 0.004 | 0.006 | 11.8 / 11.4 / 11.2 | 5 |
| Proposed objective 1 | 0.001 | 0.0046 | 1.56 / 1.57 / 1.60 | 10 |
| Proposed objective 2 | — | — | <2.0 | 1.7 |
In addition to suppressing power oscillations, the proposed MPCC significantly improves the output current quality. With symmetric control, the phase-current THD values are about 11.8%, 11.4%, and 11.2%, respectively. With the proposed objective-1 control, these THD values drop to 1.56%, 1.57%, and 1.60%. Figure 4 shows the injected grid current waveforms for the symmetric and proposed methods.
For the objective of suppressing negative-sequence currents, the simulation results are presented in Figure 5. In this case, the negative-sequence current is effectively eliminated, and the current unbalance factor is reduced to 1.7%, while the THD remains below 2%. This is particularly beneficial for solar inverters connected to weak grids that require balanced current injection during faults.
4. Discussion and Comparative Analysis
To provide a comprehensive view of the advantages of the proposed MPCC method for solar inverters, Table 5 summarizes the key features of different control strategies that have been applied to solar inverters under unbalanced grid conditions.
| Method | Reference frame | Sequence decomposition required? | Number of PI controllers | Harmonic rejection | Implementation complexity |
|---|---|---|---|---|---|
| Dual current control [2-3] | Positive- and negative-synchronous | Yes | 4 | Moderate | High |
| Dual current with LCL [4] | Positive- and negative-synchronous | Yes | 4 | Moderate (resonance issue) | Very high |
| Proportional-resonant [5] | Stationary \(\alpha\beta\) | No | 2 (PR) | Sensitive to grid frequency | Moderate |
| Deadbeat + SVM [7-8] | Stationary \(\alpha\beta\) | No | 1 (voltage vector) | Good | Moderate |
| Proposed MPCC | Stationary \(\alpha\beta\) | No | 0 (cost function) | Excellent | Low |
From Table 5 it is evident that the proposed MPCC method entirely avoids the need for phase-sequence decomposition of grid currents, which is a major advantage over dual-current-control methods. By employing a single cost function that combines both \(\alpha\) and \(\beta\) current errors, the controller directly handles positive- and negative-sequence components simultaneously. This not only simplifies the control architecture but also reduces the computational burden and improves the transient response.
Another important feature of MPCC for solar inverters is its flexibility. By simply changing the reference current formulation, the same predictive model and cost function can be used to achieve different control objectives. This is in contrast to linear controllers that require separate tuning and structural modifications. The simulation results demonstrated that both objectives — suppressing active-power ripple and suppressing negative-sequence current — can be seamlessly realized with the same MPCC framework.
The robustness of MPCC to parameter mismatch is also worth highlighting. Since the prediction model depends on the inductance \(L\), I verified that even when the actual inductance deviates significantly from the controller value, the THD of the output current remains low. This robustness is particularly valuable in practice because the filter inductance of solar inverters may degrade over time or vary with temperature and current level.
5. Conclusion
In this paper, I have presented a comprehensive study on the application of model predictive current control for solar inverters under asymmetrical grid faults. I first derived the mathematical model of the solar inverter in the stationary \(\alpha\beta\) frame, including both positive- and negative-sequence components. The power analysis revealed the existence of double-frequency oscillations under unbalanced voltages, and I derived explicit formulas for reference currents corresponding to two practical control objectives. The MPCC strategy uses a discrete-time prediction model to forecast the future current values for all eight switching states of the solar inverter, and selects the optimal state by minimizing a simple cost function. This method does not require any phase-sequence decomposition of the grid currents, thus avoiding the delays and errors associated with conventional techniques.
Simulations conducted on the PSCAD/EMTDC platform confirmed the excellent dynamic and steady-state performance of the proposed method. Under balanced conditions, the MPCC enabled fast tracking of power reference changes with perfect decoupling between active and reactive power. Under asymmetrical fault conditions, the proposed strategy reduced the active-power ripple by 75% compared to symmetric control, decreased the DC-link voltage ripple in accordance with theoretical predictions, and improved the grid current THD from over 11% to below 1.6%. When the objective was to suppress negative-sequence currents, the current unbalance factor was reduced to 1.7% while maintaining a THD below 2%.
Moreover, the proposed MPC-based approach is highly robust to inductance parameter variations, which was verified by simulation. The digital implementation is straightforward because the control algorithm simply evaluates eight switching states in every sampling period and applies the one that minimizes the cost function. This makes the MPCC method highly attractive for practical solar inverters, where low cost, high reliability, and excellent power quality are essential. Overall, the model predictive current control method provides a flexible and powerful solution for the control of solar inverters during unbalanced grid conditions, and it holds great engineering value for future photovoltaic power conversion systems.