In modern power systems, the integration of renewable energy sources, particularly through solar inverters, has become increasingly critical. Solar inverters are essential components that convert DC power from photovoltaic arrays into AC power suitable for grid connection. However, the operation of solar inverters can be significantly challenged during grid faults, especially unsymmetrical faults such as single-phase or two-phase faults. These faults introduce negative-sequence components in the grid voltage, leading to unbalanced currents and power fluctuations that can degrade system performance, cause DC-link voltage ripples, and even damage equipment. Traditional control strategies for solar inverters, such as vector control based on proportional-integral (PI) regulators in positive-sequence synchronous reference frames, are designed under the assumption of balanced three-phase conditions. When unsymmetrical faults occur, these methods fail to effectively manage negative-sequence currents, resulting in double-frequency oscillations in active and reactive power. This not only stresses the DC-link capacitor but also compromises grid power quality. Therefore, developing advanced control techniques for solar inverters under unsymmetrical grid conditions is paramount for enhancing grid stability and ensuring reliable operation.
In this paper, we address the control of solar inverters during grid unsymmetrical faults by proposing a model predictive current control (MPCC) strategy. Our approach eliminates the need for sequential decomposition of grid currents, simplifying the control structure while improving dynamic and steady-state performance. We begin by establishing a mathematical model of grid-connected solar inverters under unsymmetrical fault conditions, analyzing the power characteristics to derive control objectives. Subsequently, we detail the MPCC algorithm, which leverages a predictive model and a cost function to select optimal switching states, enabling rapid tracking of reference currents. Simulation studies validate the effectiveness of our method in suppressing active power fluctuations and negative-sequence currents. The results demonstrate that solar inverters equipped with MPCC can maintain high performance even during severe grid disturbances, highlighting the engineering value of this approach for real-world applications.
The core of our study revolves around enhancing the robustness of solar inverters in adverse grid conditions. Solar inverters are typically interfaced with the grid through an L-filter or LCL-filter, and their control must adapt to voltage imbalances. Unsymmetrical faults, such as those caused by line-to-ground faults, lead to asymmetric grid voltages, which in turn induce negative-sequence currents. These currents can cause overheating, torque pulsations in rotating machinery, and resonance issues in filter networks. For solar inverters, the primary concerns include DC-link voltage instability due to power oscillations and increased harmonic distortion in output currents. Previous research has explored various control strategies, such as dual-current control in positive and negative synchronous reference frames, proportional-resonant controllers, and deadbeat control. However, these methods often involve complex sequence separation, require multiple PI regulators, or are sensitive to parameter variations. Our proposed MPCC strategy overcomes these limitations by operating in the stationary αβ-reference frame, thereby avoiding the delays associated with sequence decomposition. This not only simplifies the control algorithm but also enhances the transient response of solar inverters, making them more resilient to grid faults.
To lay the foundation for our control design, we first derive the mathematical model of a grid-connected solar inverter under unsymmetrical grid conditions. Consider the main circuit of a photovoltaic generation system, as shown in the figure above, which includes a DC-link capacitor, a three-phase voltage-source inverter, an L-filter, and the grid. The inverter output voltages \(U_{tabc}\) are connected to the grid voltages \(U_{gabc}\) through the filter inductance \(L\) and equivalent resistance \(R\). Applying Kirchhoff’s voltage law in the three-phase stationary abc-coordinate system yields:
$$U_{tabc} = U_{gabc} + L \frac{dI_{abc}}{dt} + R I_{abc}$$
Transforming this equation into the stationary αβ-coordinate system using Clarke transformation simplifies the analysis by reducing the three-phase system to two orthogonal components. The transformation is given by:
$$T = \sqrt{\frac{2}{3}} \begin{bmatrix} 1 & -\frac{1}{2} & -\frac{1}{2} \\ 0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2} \end{bmatrix}$$
Thus, the model in αβ-coordinates becomes:
$$U_{t\alpha\beta} = U_{g\alpha\beta} + L \frac{dI_{\alpha\beta}}{dt} + R I_{\alpha\beta}$$
Under unsymmetrical grid faults, the grid voltages and currents contain both positive-sequence and negative-sequence components. In the stationary frame, these can be expressed as:
$$U_{g\alpha\beta} = U_{g\alpha\beta}^p + U_{g\alpha\beta}^n = U_{gdq}^p e^{j\theta} + U_{gdq}^n e^{-j\theta}$$
$$I_{\alpha\beta} = I_{\alpha\beta}^p + I_{\alpha\beta}^n = I_{dq}^p e^{j\theta} + I_{dq}^n e^{-j\theta}$$
where \(\theta = \omega_0 t\), \(\omega_0\) is the grid angular frequency, and superscripts \(p\) and \(n\) denote positive and negative sequences, respectively. The terms \(U_{gdq}^p\), \(U_{gdq}^n\), \(I_{dq}^p\), and \(I_{dq}^n\) are vectors in the rotating dq-coordinates. Substituting these into the αβ-model and separating the positive and negative sequences leads to the decoupled equations in the rotating frames:
$$U_{tdq}^p = U_{gdq}^p + L \frac{dI_{dq}^p}{dt} + j\omega_0 L I_{dq}^p$$
$$U_{tdq}^n = U_{gdq}^n + L \frac{dI_{dq}^n}{dt} – j\omega_0 L I_{dq}^n$$
This representation highlights the cross-coupling terms due to the grid frequency, which must be accounted for in control design. However, our MPCC strategy operates directly in the αβ-frame, bypassing the need for this decoupling. The dynamics of solar inverters are thus captured by the discrete-time version of the αβ-model, which serves as the predictive model for current control.
Next, we analyze the power characteristics of solar inverters under unsymmetrical faults to define appropriate control objectives. The instantaneous complex power output of the inverter is given by:
$$S = \frac{3}{2} U_{g\alpha\beta} I_{\alpha\beta}^*$$
where \(I_{\alpha\beta}^*\) is the complex conjugate of the current. Expanding this using the sequence components yields expressions for active power \(P\) and reactive power \(Q\):
$$P = P_0 + P_{c2} \cos(2\omega_0 t) + P_{s2} \sin(2\omega_0 t)$$
$$Q = Q_0 + Q_{c2} \cos(2\omega_0 t) + Q_{s2} \sin(2\omega_0 t)$$
The coefficients \(P_0\), \(P_{c2}\), \(P_{s2}\), \(Q_0\), \(Q_{c2}\), and \(Q_{s2}\) depend on the positive and negative sequence voltages and currents. Specifically, they can be derived as:
$$\begin{bmatrix} P_0 \\ P_{c2} \\ P_{s2} \\ Q_0 \\ Q_{c2} \\ Q_{s2} \end{bmatrix} = \frac{3}{2} \begin{bmatrix} U_{gd}^p & U_{gq}^p & U_{gd}^n & U_{gq}^n \\ U_{gd}^n & U_{gq}^n & U_{gd}^p & U_{gq}^p \\ U_{gq}^n & -U_{gd}^n & -U_{gq}^p & U_{gd}^p \\ U_{gq}^p & -U_{gd}^p & U_{gq}^n & -U_{gd}^n \\ U_{gq}^n & -U_{gd}^n & U_{gq}^p & -U_{gd}^p \\ -U_{gd}^n & -U_{gq}^n & U_{gd}^p & U_{gq}^p \end{bmatrix} \begin{bmatrix} I_d^p \\ I_q^p \\ I_d^n \\ I_q^n \end{bmatrix}$$
The presence of double-frequency components (\(2\omega_0\)) in active and reactive power is undesirable as it causes DC-link voltage ripple and mechanical stress. For solar inverters, mitigating these oscillations is crucial for maintaining maximum power point tracking (MPPT) efficiency and prolonging capacitor lifespan. However, it is generally impossible to eliminate both active and reactive power ripples simultaneously due to the degrees of freedom in current control. Therefore, we focus on two primary control objectives for solar inverters: suppressing active power fluctuations or eliminating negative-sequence currents. Each objective leads to different reference current settings, as summarized in Table 1.
| Control Objective | Positive-Sequence Currents | Negative-Sequence Currents | Conditions |
|---|---|---|---|
| Suppress Active Power Fluctuations | \(I_{d}^{p,ref} = \frac{P_{ref}}{1.5 U_{gd}^p (1 – k_{dd}^2 – k_{qd}^2)}\) | \(I_{d}^{n,ref} = -k_{dd} I_d^p – k_{qd} I_q^p\) | \(U_{gq}^p = 0\), \(k_{dd} = U_{gd}^n / U_{gd}^p\), \(k_{qd} = U_{gq}^n / U_{gd}^p\) |
| \(I_{q}^{p,ref} = -\frac{Q_{ref}}{1.5 U_{gd}^p (1 – k_{dd}^2 – k_{qd}^2)}\) | \(I_{q}^{n,ref} = k_{dd} I_q^p – k_{qd} I_d^p\) | ||
| Eliminate Negative-Sequence Currents | \(I_{d}^{p,ref} = \frac{P_{ref}}{1.5 U_{gd}^p}\) | \(I_{d}^{n,ref} = 0\) | \(U_{gq}^p = 0\), \(Q_{ref} = 0\) for unity power factor |
| \(I_{q}^{p,ref} = -\frac{Q_{ref}}{1.5 U_{gd}^p}\) | \(I_{q}^{n,ref} = 0\) |
In these equations, \(P_{ref}\) is the reference active power, typically set by the MPPT algorithm for solar inverters, and \(Q_{ref}\) is the reference reactive power, often zero for unity power factor operation. The terms \(k_{dd}\) and \(k_{qd}\) represent the ratios of negative-sequence to positive-sequence voltages, which quantify the degree of grid unbalance. By setting the reference currents accordingly, solar inverters can achieve either balanced power output or balanced currents, depending on system requirements.
Now, we present our model predictive current control strategy for solar inverters. MPCC is a finite-control-set method that uses a discrete model of the system to predict future behavior over a horizon and selects the control action that minimizes a cost function. For solar inverters, this involves predicting the αβ-currents for all possible switching states of the inverter and choosing the state that minimizes the error between predicted and reference currents. The key steps are as follows:
First, we discretize the continuous-time model of the solar inverter. Using forward Euler approximation with sampling time \(T_s\), the discrete model in αβ-coordinates is:
$$I_\alpha(k+1) = \left(1 – \frac{T_s R}{L}\right) I_\alpha(k) + \frac{T_s}{L} \left( U_{t\alpha}(k) – U_{g\alpha}(k) \right)$$
$$I_\beta(k+1) = \left(1 – \frac{T_s R}{L}\right) I_\beta(k) + \frac{T_s}{L} \left( U_{t\beta}(k) – U_{g\beta}(k) \right)$$
This serves as the predictive model, where \(k\) denotes the current sampling instant. The inverter output voltages \(U_{t\alpha}\) and \(U_{t\beta}\) depend on the switching states of the three-phase bridge. Using single-polarity binary switching functions \(g_a, g_b, g_c \in \{0,1\}\) for each leg (1 for upper switch ON, 0 for lower switch ON), the αβ-voltages are derived from the DC-link voltage \(U_{dc}\):
$$U_{t\alpha} = \sqrt{\frac{2}{3}} \left( g_a – \frac{1}{2}g_b – \frac{1}{2}g_c \right) U_{dc}$$
$$U_{t\beta} = \frac{\sqrt{2}}{2} (g_b – g_c) U_{dc}$$
There are \(2^3 = 8\) possible switching combinations, resulting in eight distinct voltage vectors, as listed in Table 2. Note that two combinations (000 and 111) produce zero voltage, corresponding to the null vectors.
| \(g_a\) | \(g_b\) | \(g_c\) | \(U_{t\alpha}\) | \(U_{t\beta}\) |
|---|---|---|---|---|
| 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 |
For each switching state, we use the predictive model to compute the predicted currents \(I_\alpha(k+1)\) and \(I_\beta(k+1)\). The reference currents \(I_{\alpha,ref}\) and \(I_{\beta,ref}\) are obtained by transforming the dq-reference currents from Table 1 to the αβ-frame. Specifically, after determining \(I_d^{p,ref}, I_q^{p,ref}, I_d^{n,ref}, I_q^{n,ref}\) based on the chosen control objective, we apply inverse Park and Clarke transformations:
$$I_{\alpha,ref} = I_{\alpha}^{p,ref} + I_{\alpha}^{n,ref}$$
$$I_{\beta,ref} = I_{\beta}^{p,ref} + I_{\beta}^{n,ref}$$
To separate the positive and negative sequence components of grid voltages for reference current calculation, we employ a second-order generalized integrator (SOGI) based method. This technique extracts \(U_{g\alpha}^p, U_{g\beta}^p, U_{g\alpha}^n, U_{g\beta}^n\) without introducing significant delay, enabling accurate computation of the reference currents for solar inverters.
The cost function \(c\) is defined as the absolute error between predicted and reference currents:
$$c = |I_{\alpha,ref} – I_\alpha(k+1)| + |I_{\beta,ref} – I_\beta(k+1)|$$
We evaluate this cost for all eight voltage vectors and select the switching state that minimizes \(c\). This state is applied during the next sampling period. The block diagram of the MPCC strategy for solar inverters is illustrated conceptually, integrating the predictive model, cost function, and switching selection. This approach ensures that the output currents of solar inverters closely track their references, even during unsymmetrical faults, without requiring sequence decomposition. The algorithm is computationally efficient and well-suited for digital implementation in modern solar inverter controllers.
To validate our MPCC strategy for solar inverters, we conducted extensive simulations using a detailed model of a 0.5 MW photovoltaic system. The parameters are summarized in Table 3. The system includes a DC-link capacitor, an L-filter, and a grid connection at 0.27 kV. The simulations were performed under various unsymmetrical fault scenarios, such as single-phase faults, to assess the performance of solar inverters in suppressing power fluctuations and negative-sequence currents.
| Parameter | Value |
|---|---|
| Rated Power | 0.5 MW |
| DC-Link Capacitance | 20,000 µF |
| Filter Inductance \(L\) | 0.18 mH |
| Equivalent Resistance \(R\) | 0.0005 Ω |
| Grid Voltage (AC side) | 0.27 kV |
| Sampling Period \(T_s\) | 20 µs |
| Grid Frequency \(\omega_0\) | 50 Hz (314 rad/s) |
First, we tested the dynamic and steady-state performance of solar inverters under balanced 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, shown in Table 4, indicate that the αβ-currents tracked their references rapidly, with settling times under 5 ms. The three-phase currents remained symmetric, and active and reactive power were decoupled, confirming the effectiveness of MPCC for solar inverters in normal operation.
| Metric | Value |
|---|---|
| Current Tracking Error (RMS) | < 0.5% |
| Settling Time for Power Step | 4 ms |
| Total Harmonic Distortion (THD) of Currents | 0.61% |
| Power Decoupling | Achieved |
Next, we introduced an unsymmetrical fault: a single-phase (A-phase) short-circuit fault at 1.0 s, cleared at 1.4 s. The grid voltage asymmetry increased from 0.3% to 12% during the fault. We compared our MPCC strategy with a traditional symmetric control strategy that uses PI regulators in the positive-sequence frame without sequence separation. The key performance indicators are summarized in Table 5 for the two control objectives: suppressing active power fluctuations and eliminating negative-sequence currents.
| Control Strategy | Active Power Ripple (2ω₀ amplitude) | DC-Link Voltage Ripple (2ω₀ amplitude) | Current THD | Negative-Sequence Current Reduction |
|---|---|---|---|---|
| Symmetric Control (PI) | 0.004 MW | 0.006 V | 11.4% (avg) | None |
| MPCC (Suppress Power Fluctuations) | 0.001 MW (75% reduction) | 0.0046 V (23% reduction) | 1.57% (avg) | Partial |
| MPCC (Eliminate Negative-Sequence Currents) | 0.0035 MW | 0.0055 V | 1.65% (avg) | 98.3% reduction |
The results demonstrate that solar inverters using MPCC significantly outperform traditional methods. When targeting active power fluctuation suppression, the double-frequency component in active power was reduced by 75%, leading to a smoother DC-link voltage and enhanced MPPT stability. The current THD remained below 2%, indicating high power quality. For the objective of eliminating negative-sequence currents, the asymmetry in output currents was reduced to 1.7%, effectively ensuring balanced three-phase currents despite the unbalanced grid voltages. This is crucial for protecting connected equipment and complying with grid codes. Furthermore, the MPCC strategy showed robustness to parameter variations. As indicated in Table 6, variations in filter inductance \(L\) had minimal impact on current THD, affirming the reliability of solar inverters equipped with MPCC in practical scenarios where parameters may drift.
| Filter Inductance \(L\) (mH) | Current THD (%) with Matched Controller \(L\) | Current THD (%) with Fixed Controller \(L = 0.18\) mH |
|---|---|---|
| 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 |
In addition to these quantitative metrics, the transient response of solar inverters during fault inception and clearance was examined. With symmetric control, the DC-link voltage exhibited overshoot and undershoot, potentially disrupting MPPT algorithms. In contrast, MPCC ensured smooth transitions, with no significant voltage spikes, as the predictive model quickly adapted to the changing grid conditions. This highlights the superior dynamic performance of solar inverters under MPCC, which is essential for maintaining continuous operation during grid disturbances.
The mathematical formulation of the power outputs can be further refined to understand the trade-offs between control objectives. From the power equations, we derive the conditions for zero active power ripple. Setting \(P_{c2} = 0\) and \(P_{s2} = 0\) yields a set of linear equations in terms of the current sequences. Solving these under the assumption of positive-sequence voltage orientation (\(U_{gq}^p = 0\)) leads to the reference currents given earlier. Similarly, for zero negative-sequence currents, we set \(I_d^n = I_q^n = 0\). The interplay between these objectives can be visualized using the following expressions for power coefficients:
$$P_{c2} = \frac{3}{2} \left( U_{gd}^n I_d^p + U_{gq}^n I_q^p + U_{gd}^p I_d^n + U_{gq}^p I_q^n \right)$$
$$P_{s2} = \frac{3}{2} \left( U_{gq}^n I_d^p – U_{gd}^n I_q^p – U_{gq}^p I_d^n + U_{gd}^p I_q^n \right)$$
These equations underscore that independent control of positive and negative-sequence currents in solar inverters allows for flexible management of power quality. Moreover, the MPCC algorithm inherently handles these couplings without explicit decoupling, thanks to its direct minimization of current errors in the stationary frame.
For implementation in digital signal processors (DSPs), the computational load of MPCC for solar inverters is manageable. The algorithm requires evaluation of eight predictions per sampling period, each involving simple arithmetic operations. With modern DSPs, this can be executed within tens of microseconds, making it feasible for high-switching-frequency solar inverters. Additionally, the absence of PWM modulators simplifies the control loop, as the switching states are directly applied. However, attention must be paid to the sampling frequency and model accuracy to avoid instability. We recommend using a sampling frequency at least ten times the switching frequency to ensure precise prediction. The discrete model parameters can be calibrated online using adaptive techniques to account for variations in \(L\) and \(R\), further enhancing the robustness of solar inverters.
In conclusion, our study presents a comprehensive model predictive current control strategy for solar inverters operating under grid unsymmetrical faults. We have derived a detailed mathematical model, analyzed power characteristics, and formulated control objectives to either suppress active power fluctuations or eliminate negative-sequence currents. The proposed MPCC method eliminates the need for sequence separation, reducing complexity and improving dynamic response. Simulation results confirm that solar inverters with MPCC achieve excellent current tracking, low harmonic distortion, and effective mitigation of fault-induced disturbances. The method is computationally efficient, easily implementable in digital controllers, and robust to parameter variations, making it highly suitable for real-world solar inverter applications. Future work could explore the extension of MPCC to solar inverters with LCL filters, integration with energy storage systems, and experimental validation under various grid fault scenarios. Ultimately, advancing such control techniques is vital for the reliable and efficient integration of solar energy into modern power grids.